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Question: What are the symptoms of alpha thalassemia major?

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Context 1: Thalassaemia minor | definition of Thalassaemia minor by Medical dictionary
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Thalassemia describes a group of inherited disorders characterized by reduced or absent amounts of hemoglobin, the oxygen-carrying protein inside the red blood cells  There are two basic groups of thalassemia disorders: alpha thalassemia and beta thalassemia  These conditions cause varying degrees of anemia, which can range from insignificant to life threatening 
All types of thalassemias are considered quantitative diseases of hemoglobin, because the quantity of hemoglobin produced is reduced or absent  Usual adult hemoglobin is made up of three components: alpha globin, beta globin, and heme  Thalassemias are classified according to the globin that is affected, hence the names alpha and beta thalassemia  Although both classes of thalassemia affect the same protein, the alpha and beta thalassemias are distinct diseases that affect the body in different ways 
Beta thalassemia may be the most best-known type of thalassemia and is also called Cooley's anemia  It is caused by a change in the gene for the beta globin component of hemoglobin  Beta thalassemia causes variable anemia that can range from moderate to severe, depending in part on the exact genetic change underlying the disease  Beta thalassemia can be classified based on clinical symptoms  Beta thalassemia major usually causes severe anemia that can occur within months after birth  If left untreated, severe anemia can result in insufficient growth and development, as well as other common physical complications that can lead to a dramatically decreased life-expectancy  Fortunately, in developed countries beta thalassemia is usually identified by screening in the newborn period, before symptoms have developed  Children who are identified early can be started on ongoing blood transfusion therapy as needed  Although transfusion therapy prevents many of the complications of severe anemia, the body is unable to eliminate the excess iron contained in the transfused blood  Over time, the excess iron deposits in tissues and organs, resulting in damage and organ failure  Another medication must be administered to help the body eliminate the excess iron and prevent iron-over-load complications  Beta thalassemia intermedia describes the disease in individuals who have moderate anemia that only requires blood transfusions intermittently, if at all 
Alpha thalassemia is the result of changes in the genes for the alpha globin component of hemoglobin  There are two main types of alpha thalassemia disease: hemoglobin H disease and alpha thalassemia major  The two diseases are quite different from beta thalassemia as well as from one another  Individuals with hemoglobin H disease can experience events of hemolytic anemia—anemia caused by the rapid breakdown of the red blood cells  These events are thought to be triggered by various environmental causes, such as infection and/or exposure to certain chemicals  Hemoglobin H disease is in most cases milder than beta thalassemia  It does not generally require transfusion therapy  Alpha thalassemia major is a very serious disease that results in severe anemia that begins even before birth  Most affected babies do not survive to be born or die shortly after birth 
The thalassemias are among the most common genetic diseases worldwide  Both alpha and beta thalassemia have been described in individuals of almost every ancestry, but the conditions are more common among certain ethnic groups  Unaffected carriers of all types of thalassemia traits do not experience health problems  In fact, the thalassemia trait is protective against malaria, a disease caused by blood-borne parasites transmitted through mosquito bites  According to a widely accepted theory, most genetic changes—mutations—that cause thalassemia occurred multiple generations ago  Coincidentally, these mutations increased the likelihood that carriers would survive malaria infection  Survivors passed the mutation onto their offspring, and the trait became established throughout areas where malaria is common  As populations migrated, so did the thalassemia traits 
Beta thalassemia trait is seen most commonly in people with the following ancestry: Mediterranean (including North African, and particularly Italian and Greek), Middle Eastern, Indian, African, Chinese, and Southeast Asian (including Vietnamese, Laotian, Thai, Singaporean, Filipino, Cambodian, Malaysian, Burmese, and Indonesian)  Alpha-thalassemia trait is seen with increased frequency in the same ethnic groups  However, there are different types of alpha thalassemia traits within these populations  The frequency of hemoglobin H disease and alpha thalassemia major depends on the type of alpha thalassemia trait  The populations in which alpha thalassemia diseases are most common include Southeast Asians and Chinese (particularly Southern Chinese) 
It is difficult to obtain accurate prevalence figures for various types of thalassemia within different populations  This difficulty arises due to testing limitations in determining exact genetic diagnoses, as well as the fact that many studies have focused on small, biased hospital populations 
Two studies reflect prevalence figures that can be helpful counseling families and determining who to screen for beta thalassemia  Between the years of 1990 and 1996, the State of California screened more than 3 1 million infants born in the state for beta thalassemia  Approximately 1 in 114,000 infants had beta thalassemia major, with prevalence rates being highest among Asian Indians (about one in 4,000), Southeast Asians (about one in 10,000), and Middle Easterners (about one in 7,000)  Another type of beta thalassemia disease, E/beta thalassemia, was represented in approximately one in 110,000 births, all of which occurred in families of Southeast Asian ancestry  Among Southeast Asians, the prevalence of E/beta thalassemia was approximately one in 2,600 births  This is in keeping with the observation that hemoglobin E trait carrier rates are relatively high within the Southeast Asian population: 16% in a study of 768 immigrants to California, and up to 25% in some specific Southeast Asian populations such as Cambodians  While these California studies address some of the limitations of earlier population studies, the pattern observed in California is expected to be different in other areas of the United States and the world  For example, Italians are underrepresented in this population when compared to the population of the East Coast of the United States 
Determining prevalence figures for alpha thalassemia is even more difficult due to increased limitations in diagnostic testing  All types of alpha thalassemia disease are most common among people of Southeast Asian and Chinese descent, for reasons that become clearer with an understanding of the underlying genetics of alpha thalassemia  One study of 500 pregnant women in Northern Thailand estimated a frequency of one in 500 pregnancies affected by alpha thalassemia major, for example  Prevalence of alpha thalassemia disease is significantly lower in the United States primarily because of immigration patterns; although at least one state, California, has observed growing hemoglobin H disease incidence rates that are high enough to justify universal newborn screening for the condition 
Humans normally make several types of the oxygen-carrying protein hemoglobin  An individual's stage in development determines whether he or she makes primarily embryonic, fetal, or adult hemoglobins  All types of hemoglobin are made of three components: heme, alpha (or alpha-like) globin, and beta (or beta-like) globin  All types of thalassemia are caused by changes in either the alpha- or beta-globin gene  These changes cause little or no globin to be produced  The thalassemias are, therefore, considered quantitative hemoglobin diseases  All types of thalassemias are recessively inherited, meaning that a genetic change must be inherited from both the mother and the father  The severity of the disease is influenced by the exact thalassemia mutations inherited, as well as other genetic and environmental factors  There are rare exceptions, notably with beta thalassemia, where globin gene mutations exhibit a dominant pattern of inheritance in which only one gene needs to be altered in order to see disease expression  Scientists continue to study the causes  For instance, a new mutation for alpha-thalassemia was discovered for the first time among Iranian patients in 2004 
BETA-THALASSEMIA  Most individuals have two normal copies of the beta globin gene, which is located on chromosome 11 and makes the beta globin component of normal adult hemoglobin, hemoglobin A  There are approximately 100 genetic mutations that have been described that cause beta thalassemia, designated as either beta0 or beta + mutations  No beta globin is produced with a beta0 mutation, and only a small fraction of the normal amount of beta globin is produced with a beta + mutation 
When an individual has one normal beta globin gene and one with a beta thalassemia mutation, he or she is said to carry the beta thalassemia trait  Beta thalassemia trait, like other hemoglobin traits, is protective against malaria infection  Trait status is generally thought not to cause health problems, although some women with beta thalassemia trait may have an increased tendency toward anemia during pregnancy 
When two members of a couple carry the beta thalassemia trait, there is a 25% chance that each of their children will inherit beta thalassemia disease by inheriting two beta thalassemia mutations, one from each parent  The clinical severity of the beta thalassemia disease—whether an individual has beta thalassemia intermedia or beta thalassemia major—will depend largely on whether the mutations inherited are beta0 thalassemia or beta + thalassemia mutations  Two beta0 mutations generally lead to beta thalassemia major, and two beta+ thalassemia mutations generally lead to beta thalassemia intermedia  Inheritance of one beta0 and one beta + thalassemia mutation tends to be less predictable 
Although relatively uncommon, there are other thalassemia-like mutations that can affect the beta globin gene  Hemoglobin E is the result of a substitution of a single nucleotide  This change results in a structurally altered hemoglobin that is produced in decreased amounts  Therefore, hemoglobin E is unique in that it is both a quantitative (i e  thalassemia-like) and qualitative trait  When co-inherited with a beta thalassemia trait, it causes a disease that is almost indistinguishable from beta thalassemia disease  Large deletions around and including the beta globin gene can lead to delta/beta thalassemia or hereditary persistence of fetal hemoglobin (HPFH)  Interestingly, delta/beta thalassemia trait behaves very similarly to beta thalassemia trait in its clinical manifestations  However, HPFH trait does not tend to cause hemoglobin disease when co-inherited with a second thalassemia or other beta globin mutation 
ALPHA-THALASSEMIA  Most individuals have four normal copies of the alpha globin gene, two copies on each chromosome 16  These genes make the alpha globin component of normal adult hemoglobin, which is called hemoglobin A  Alpha globin is also a component of fetal hemoglobin and the other major adult hemoglobin called hemoglobin A2  Mutations of the alpha globin genes are usually deletions of the gene, resulting in absent production of alpha globin  Since there are four genes (instead of the usual two) to consider when looking at alpha globin gene inheritance, there are several alpha globin types that are possible 
Absence of one alpha globin gene leads to a condition known as silent alpha thalassemia trait  This condition causes no health problems and can be detected only by special genetic testing  Alpha thalassemia trait occurs when two alpha globin genes are missing  This can occur in two ways  The genes may be deleted from the same chromosome, causing the 'cis' type of alpha thalassemia trait  Alternately, they may be deleted from different chromosomes, causing the 'trans' type of alpha thalassemia trait  In both instances, there are no associated health problems, although the trait status may be detected by more routine blood screening 
Hemoglobin H disease results from the deletion of three alpha globin genes, such that there is only one functioning gene  Typically, this can occur when one parent carries the silent alpha thalassemia trait, and the other parent carries the 'cis' type of the alpha thalassemia trait  In this situation, there is a 25% chance for hemoglobin H disease in each of such a couple's children 
Hemoglobin H disease-like symptoms can also be a part of a unique condition called alpha thalassemia mental retardation syndrome  Alpha thalassemia mental retardation syndrome can be caused by a deletion of a significant amount of chromosome 16, affecting the alpha globin genes  This is usually not inherited, but rather occurs sporadically in the affected individual  Affected individuals have mild hemoglobin H disease, mild-to-moderate mental retardation, and characteristic facial features  This syndrome can also occur as a sex-linked form in which a mutation is inherited in a particular gene on the X-chromosome  This gene influences alpha globin production, as well as various other developmental processes  Individuals affected with this form of the syndrome tend to have more severe mental retardation, delayed development, nearly absent speech, characteristic facial features, and genital-urinary abnormalities  The remaining discussion will focus only on aspects of hemoglobin H disease 
Alpha thalassemia major results from the deletion of all four alpha globin genes, such that there are no functioning alpha globin genes  This can occur when both parents carry the 'cis' type of the alpha thalassemia trait  In this situation, there is a 25% chance for alpha thalassemia major in each of such a couple's children 
Beta thalassemia major is characterized by severe anemia that can begin months after birth  In the United States and other developed countries beta thalassemia is identified and treated early and effectively  Therefore, the following discussion of symptoms applies primarily to affected individuals in the past and unfortunately in some underdeveloped countries now  If untreated, beta thalassemia major can lead to severe lethargy, paleness, and delays in growth and development  The body attempts to compensate by producing more blood, which is made inside the bones in the marrow  However, this is ineffective without the needed genetic instructions to make enough functioning hemoglobin  Instead, obvious bone expansion and changes occur that cause characteristic facial and other changes in appearance, as well as increased risk of fractures  Severe anemia taxes other organs in the body—such as the heart, spleen, and liver—which must work harder than usual  This can lead to heart failure, as well as enlargement and other problems of the liver and spleen  When untreated, beta thalassemia major generally results in childhood death, usually due to heart failure  In 2004, the first known heart attack associated with beta thalassemia major was reported  Fortunately, in developed countries diagnosis is usually made early, often before symptoms have begun  This allows for treatment with blood transfusion therapy, which can prevent most of the complications of the severe anemia caused by beta thalassemia major  Individuals with beta thalassemia intermedia have a more moderate anemia that may only require treatment with transfusion intermittently, such as when infections occur and stress the body  As a person with beta thalassemia intermedia gets older, however, the need for blood transfusions may increase to the point that they are required on a regular basis  When this occurs their disease becomes more similar to beta thalassemia major  Other genetic and environmental factors can influence the course of the disease as well  For example, co-inheritance of one or two alpha thalassemia mutations can tend to ameliorate some of the symptoms of beta thalassemia disease, which result in part from an imbalance in the amount of alpha- and beta-globin present in the red blood cells 
Hemoglobin h disease
Absence of three alpha globin genes causes an imbalance of alpha and beta globin proteins in the red blood cells  The excess beta globin proteins tend to come together to form hemoglobin H, which is unable to release oxygen to the tissues  In addition, hemoglobin H tends to precipitate out in the cells, causing damage to the red blood cell membrane  When affected individuals are exposed to certain drugs and chemicals known to make the membrane more fragile, the cells are thought to become vulnerable to breakdown in large numbers, a complication called hemolytic anemia  Fever and infection are also considered to be triggers of hemolytic anemia in hemoglobin H disease  This can result in fatigue, paleness, and a yellow discoloration of the skin and whites of eyes called jaundice  Usually, the anemia is mild enough not to require treatment  Severe anemia events may require blood transfusion, however, and are usually accompanied by such other symptoms as dark feces or urine and abdominal or back pain  These events are uncommon in hemoglobin H disease, although they occur more frequently in a more serious type of hemoglobin H disease called hemoglobin H/Constant Spring disease  Individuals effected with this type of hemoglobin H disease are also more likely to have enlargement of and other problems with the spleen 
Alpha thalassemia major
Because alpha globin is a necessary component of all major hemoglobins and some minor hemoglobins, absence of all functioning alpha globin genes leads to serious medical consequences that begin even before birth  Affected fetuses develop severe anemia as early as the first trimester of pregnancy  The placenta, heart, liver, spleen, and adrenal glands may all become enlarged  Fluid can begin collecting throughout the body as early as the start of the second trimester, causing damage to developing tissues and organs  Growth retardation is also common  Affected fetuses usually miscarry or die shortly after birth  In addition, women carrying affected fetuses are at increased risk of developing complications of pregnancy and delivery  Up to 80% of such women develop toxemia, a disturbance of metabolism that can potentially lead to convulsions and coma  Other maternal complications include premature delivery and increased rates of delivery by cesarean section, as well as hemorrhage after delivery 
Thalassemia may be suspected if an individual shows signs that are suggestive of the disease  In all cases, however, laboratory diagnosis is essential to confirm the exact diagnosis and to allow for the provision of accurate genetic counseling about recurrence risks and testing options for parents and affected individuals  Screening is likewise recommended to determine trait status for individuals of high-risk ethnic groups 
The following tests are used to screen for thalassemia disease and/or trait:
hemoglobin electrophoresis with quantitative hemoglobin A2 and hemoglobin F
free erythrocyte-protoporphyrin (or ferritin or other studies of serum iron levels)
A complete blood count will identify low levels of hemoglobin, small red blood cells, and other red blood cell abnormalities that are characteristic of a thalassemia diagnosis  Since thalassemia trait can sometimes be difficult to distinguish from iron deficiency, tests to evaluate iron levels are important  A hemoglobin electrophoresis is a test that can help identify the types and quantities of hemoglobin made by an individual  This test uses an electric field applied across a slab of gel-like material  Hemoglobins migrate through this gel at various rates and to specific locations, depending on their size, shape, and electrical charge  Isoelectric focusing and high-performance liquid chromatography (HPLC) use similar principles to separate hemoglobins and can be used instead of or in various combinations with hemoglobin electrophoresis to determine the types and quantities of hemoglobin present  Hemoglobin electrophoresis results are usually within the normal range for all types of alpha thalassemia  However, hemoglobin A2 levels and sometimes hemoglobin F levels are elevated when beta thalassemia disease or trait is present  Hemoglobin electrophoresis can also detect structurally abnormal hemoglobins that may be co-inherited with a thalassemia trait to cause thalassemia disease (i e , hemoglobin E) or other types of hemoglobin disease (i e , sickle hemoglobin)  Sometimes DNA testing is needed in addition to the above screening tests  This can be performed to help confirm the diagnosis and establish the exact genetic type of thalassemia 
Diagnosis of thalassemia can occur under various circumstances and at various ages  Several states offer thalassemia screening as part of the usual battery of blood tests done for newborns  This allows for early identification and treatment  Thalassemia can be identified before birth through the use of prenatal diagnosis  Chorionic villus sampling (CVS) can be offered as early as 10 weeks of pregnancy and involves removing a sample of the placenta made by the baby and testing the cells  CVS carries a risk of causing a miscarriage that is between 0 5%-1%  Amniocentesis is generally offered between 15 and 22 weeks of pregnancy, but can sometimes be offered earlier  Two to three tablespoons of the fluid surrounding the baby is removed  This fluid contains fetal cells that can be tested  The risk of miscarriage associated with amniocentesis ranges from 0 33-0 5%  Pregnant woman and couples may choose prenatal testing in order to prepare for the birth of a baby that may have thalassemia  Alternately, knowing the diagnosis during pregnancy allows for the option of pregnancy termination  Preimplantation genetic diagnosis (PGD) is a relatively new technique that involves in-vitro fertilization followed by genetic testing of one cell from each developing embryo  Only the embryos unaffected by sickle cell disease are transferred back into the uterus  PGD is currently available on a research basis only and is relatively expensive 
Individuals with beta thalassemia major receive regular blood transfusions, usually on a monthly basis  This helps prevent severe anemia and allows for more normal growth and development  Transfusion therapy does have limitations, however  Individuals can develop reactions to certain proteins in the blood—called a transfusion reaction  This can make locating appropriately matched donor blood more difficult  Although blood supplies in the United States are very safe, particularly relative to the past and to other areas of the world, there remains an increased risk of exposure to such blood-borne infections as hepatitis  Additionally, the body is not able to get rid of the excess iron that accompanies each transfusion  An additional medication called desferoxamine is administered, usually five nights per week over a period of several hours, using an automatic pump that can be used during sleep or taken anywhere the person goes  This medication is able to bind to the excess iron, which can then be eliminated through urine  If desferoxamine is not used regularly or is unavailable, iron overload can develop and cause tissue damage and organ damage and failure  The heart, liver, and endocrine organs are particularly vulnerable  Desferoxamine itself may rarely produce allergic or toxic side effects, including hearing damage  Signs of desferoxamine toxicity are screened for and generally develop in individuals who overuse the medication when body iron levels are sufficiently low  Overall, however, transfusion and desferoxamine therapy have increased the life expectancy of individuals with the most severe types of beta thalassemia major to the 4th or 5th decade  This can be expected to improve with time and increased developments in treatment, as well as for those with more mild forms of the disease 
New treatments offer additional options for some individuals with beta thalassemia major  There are various medications that target the production of red blood cells (i e  erythropoeitin) or fetal hemoglobin (i e  hydroxyurea and butyrate)  Their effectiveness in ameliorating the severity of beta thalassemia is currently being investigated  Another promising new treatment is bone marrow transplantation, in which the bone marrow of an affected individual is replaced with the bone marrow of an unaffected donor  If successful, this treatment can provide a cure  However, there is an approximately 10-15% chance the procedure could be unsuccessful (i e  the thalassemia returns); result in complications (i e  graft-versus-host disease); or result in death  The risk for specific individuals depends on current health status, age, and other factors  Because of the risks involved and the fact that beta thalassemia is a treatable condition, transplant physicians require a brother or sister donor who has an identically matched tissue type, called HLA type  HLA type refers to the unique set of proteins present on each individual's cells, which allows the immune system to recognize "self" from "foreign " HLA type is genetically determined, so there is a 25% chance for two siblings to be a match  Transplant physicians and researchers are also investigating ways to improve the safety and effectiveness of bone marrow transplantation  Using newborn sibling umbilical cord blood—the blood from the placenta that is otherwise discarded after birth but contains cells that can go on to make bone marrow—seems to provide a safer and perhaps more effective source of donor cells  Donors and recipients may not have to be perfect HLA matches for a successful transplant using cord blood cells  Trials are also underway to determine the effectiveness of "partial transplants," in which a safer transplant procedure is used to replace only a percentage of the affected individual's bone marrow  Other possible treatments on the horizon may include gene therapy techniques aimed at increasing the amount of normal hemoglobin the body is able to make 
Hemoglobin H disease is a relatively mild form of thalassemia that may go unrecognized  It is not generally considered a condition that will reduce one's life expectancy  Education is an important part of managing the health of an individual with hemoglobin H disease  It is important to be able to recognize the signs of severe anemia that require medical attention  It is also important to be aware of the medications, chemicals, and other exposures to avoid due to the theoretical risk they pose of causing a severe anemia event  When severe anemia occurs, it is treated with blood transfusion therapy  For individuals with hemoglobin H disease, this is rarely required  For those with the hemoglobin H/Constant Spring form of the disease, the need for transfusions may be intermittent or ongoing, perhaps on a monthly basis and requiring desferoxamine treatment  Individuals with this more severe form of the disease may also have an increased chance of requiring removal of an enlarged and/or overactive spleen 
Anemia — A blood condition in which the level of hemoglobin or the number of red blood cells falls below normal values  Common symptoms include paleness, fatigue, and shortness of breath 
Bilirubin — A yellow pigment that is the end result of hemoglobin breakdown  This pigment is metabolized in the liver and excreted from the body through the bile  Bloodstream levels are normally low; however, extensive red cell destruction leads to excessive bilirubin formation and jaundice 
Bone marrow — A spongy tissue located in the hollow centers of certain bones, such as the skull and hip bones  Bone marrow is the site of blood cell generation 
Bone marrow transplantation — A medical procedure used to treat some diseases that arise from defective blood cell formation in the bone marrow  Healthy bone marrow is extracted from a donor to replace the marrow in an ailing individual  Proteins on the surface of bone marrow cells must be identical or very closely matched between a donor and the recipient 
Desferoxamine — The primary drug used in iron chelation therapy  It aids in counteracting the life-threatening buildup of iron in the body associated with long-term blood transfusions 
Globin — One of the component protein molecules found in hemoglobin  Normal adult hemoglobin has a pair each of alpha-globin and beta-globin molecules 
Heme — The iron-containing molecule in hemoglobin that serves as the site for oxygen binding 
Hemoglobin — Protein-iron compound in the blood that carries oxygen to the cells and carries carbon dioxide away from the cells 
Hemoglobin A — Normal adult hemoglobin that contains a heme molecule, two alpha-globin molecules, and two beta-globin molecules 
Hemoglobin electrophoresis — A laboratory test that separates molecules based on their size, shape, or electrical charge 
Hepatomegaly — An abnormally large liver 
HLA type — Refers to the unique set of proteins called human leukocyte antigens  These proteins are present on each individual's cell and allow the immune system to recognize 'self' from 'foreign'  HLA type is particularly important in organ and tissue transplantation 
Hydroxyurea — A drug that has been shown to induce production of fetal hemoglobin  Fetal hemoglobin has a pair of gamma-globin molecules in place of the typical beta-globins of adult hemoglobin  Higher-than-normal levels of fetal hemoglobin can ameliorate some of the symptoms of thalassemia 
Iron overload — A side effect of frequent blood transfusions in which the body accumulates abnormally high levels of iron  Iron deposits can form in organs, particularly the heart, and cause life-threatening damage 
Jaundice — Yellowing of the skin or eyes due to excess of bilirubin in the blood 
Mutation — A permanent change in the genetic material that may alter a trait or characteristic of an individual, or manifest as disease, and can be transmitted to offspring 
Placenta — The organ responsible for oxygen and nutrition exchange between a pregnant mother and her developing baby 
Red blood cell — Hemoglobin-containing blood cells that transport oxygen from the lungs to tissues  In the tissues, the red blood cells exchange their oxygen for carbon dioxide, which is brought back to the lungs to be exhaled 
Screening — Process through which carriers of a trait may be identified within a population 
Splenomegaly — Enlargement of the spleen 
Because alpha thalassemia major is most often a condition that is fatal in the prenatal or newborn period, treatment has previously been focused on identifying affected pregnancies in order to provide appropriate management to reduce potential maternal complications  Pregnancy termination provides one form of management  Increased prenatal surveillance and early treatment of maternal complications is an approach that is appropriate for mothers who wish to continue their pregnancy with the knowledge that the baby will most likely not survive  In recent years, there have been a handful of infants with this condition who have survived long-term  Most of these infants received experimental treatment including transfusions before birth, early delivery, and even bone marrow transplantation before birth, although the latter procedure has not yet been successful  For those infants that survive to delivery, there seems to be an increased risk of developmental problems and physical effects, particularly heart and genital malformations  Otherwise, their medical outlook is similar to a child with beta thalassemia major, with the important exception that ongoing, life-long blood transfusions begin right at birth 
As discussed above, the prognosis for individuals with the most serious types of thalassemia has improved drastically in the last several years following recent medical advances in transfusion, chemo-, and transplantation therapy  Advances continue and promise to improve the life expectancy and quality of life further for affected individuals 
"First Known Heart Attack Associated With Beta-thalassemia Major Reported " Heart Disease Weekly February 22, 2004: 10 
"Novel Alpha-thalassemia Mutations Identified " Hematology Week January 26, 2004: 19 
Children's Blood Foundation  333 East 38th St , Room 830, New York, NY 10016-2745  (212) 297-4336  cfg@nyh med cornell edu 
Cooley's Anemia Foundation, Inc  129-09 26th Ave  #203, Flushing, NY 11354  (800) 522-7222 or (718) 321-2873  http://www thalassemia org 
March of Dimes Birth Defects Foundation  1275 Mamaroneck Ave , White Plains, NY 10605  (888) 663-4637  resourcecenter@modimes org  http://www modimes org 
National Heart, Lung, and Blood Institute  PO Box 30105, Bethseda, MD 20824-0105  (301) 592-8573  nhlbiinfo@rover nhlbi nih gov  http://www nhlbi nih gov 
National Organization for Rare Disorders (NORD)  PO Box 8923, New Fairfield, CT 06812-8923  (203) 746-6518 or (800) 999-6673  Fax: (203) 746-6481  http://www rarediseases org 
Bojanowski J  "Alpha Thalassemia Major: The Possibility of Long-Term Survival " Pamphlet from the Northern California Comprehensive Thalassemia Center  (1999) 
Children's Hospital Oakland, Northern California Comprehensive Thalassemia Center website  http://www thalassemia com 
Cooley's Anemia Foundation, Inc  website  http://www thalassemia org/gohome html 
Joint Center for Sickle Cell and Thalassemic Disorders website  http://cancer mgh harvard edu/medOnc/sickle htm 
[thal″ah-se´me-ah]
a heterogeneous group of hereditary hemolytic anemias marked by a decreased rate of synthesis of one or more hemoglobin polypeptide chains, classified according to the chain involved (α, β, δ); the two major categories are α- and β-thalassemia 
α-thalassemia (alpha-thalassemia) that caused by diminished synthesis of alpha chains of hemoglobin  The homozygous form is incompatible with life, the stillborn infant displaying severe hydrops fetalis  The heterozygous form may be asymptomatic or marked by mild anemia 
β-thalassemia (beta-thalassemia) that caused by diminished synthesis of beta chains of hemoglobin  The homozygous form is called t  major and the heterozygous form is called t  minor 
thalassemia ma´jor the homozygous form of β-thalassemia, in which hemoglobin A is completely absent; it appears in the newborn period and is marked by hemolytic, hypochromic, microcytic anemia; hepatosplenomegaly; skeletal deformation; mongoloid facies; and cardiac enlargement 
thalassemia mi´nor the heterozygous form of β-thalassemia; it is usually asymptomatic, but there may be mild anemia 
sickle cell–thalassemia a hereditary anemia involving simultaneous heterozygosity for hemoglobin S and thalassemia 
thal·as·se·mi·a
, thalassanemia (thal'ă-sē'mē-ă, thă-las-ă-nē'mē-ă),
Any of a group of inherited disorders of hemoglobin metabolism in which there is impaired synthesis of one or more of the polypeptide chains of globin; several genetic types exist, and the corresponding clinical picture may vary from barely detectable hematologic abnormality to severe and fatal anemia 
[G  thalassa, the sea, + haima, blood]
/thal·as·se·mia/ (thal″ah-se´me-ah) a heterogeneous group of hereditary hemolytic anemias marked by a decreased rate of synthesis of one or more hemoglobin polypeptide chains, classified according to the chain involved (α, β, δ); the two major categories are α- and β-thalassemia 
α-thalassemia that caused by diminished synthesis of alpha chains of hemoglobin  The homozygous form is incompatible with life, the stillborn infant displaying severe hydrops fetalis  The heterozygous form may be asymptomatic or marked by mild anemia 
β-thalassemia that caused by diminished synthesis of beta chains of hemoglobin  The homozygous form is called t  major and the heterozygous form is called t  minor 
thalassemia ma´jor the homozygous form of &#x03B2;, in which hemoglobin A is completely absent; it appears in the newborn period and is marked by hemolytic, hypochromic, microcytic anemia, hepatosplenomegaly, skeletal deformation, mongoloid facies, and cardiac enlargement 
thalassemia mi´nor the heterozygous form of &#x03B2;, usually asymptomatic, although there is sometimes mild anemia 
(thăl′ə-sē′mē-ə)
An inherited form of anemia occurring chiefly among people of Mediterranean descent, caused by faulty synthesis of part of the hemoglobin molecule  Also called Mediterranean anemia 
thal′as·se′mic adj 
[thal′əsē′mē·ə]
Etymology: Gk, thalassa, sea, a + haima, without blood
production and hemolytic anemia characterized by microcytic, hypochromic red blood cells  Thalassemia is caused by inherited deficiency of alpha- or beta-globin synthesis  See also hemochromatosis, hemosiderosis 
Beta thalassemia, clinical thalassemia, Cooley's anemia, Mediterranean anemia, thalassemia major Hematology A group of genetic diseases by underproduction of hemoglobin due to mutations in the beta globin gene, which is more common in Mediterraneans Heredity Parents are carriers–heterozygotes; one in 4 children is homozygous for the mutation and thus has full-blown disease Clinical See Anemia  Cf Sickle cell anemia 
α-thalassemia
Hemoglobin Barts Hematology An inherited condition caused by a defect in the synthesis of the Hb α chain; Hb Barts hemoglobinopathy is characterized by the presence of 4 gamma chains; it is more common in southeast Asians; the most severe form of alpha thalassemia causes stillbirth due to hydrops fetalis Heredity Parents are carriers–heterozygotes; one in 4 children is homozygous for the mutation and thus has full-blown disease Clinical Pallor, fatiguability, FTT, fever, infections, diarrhea Management Transfusions
Thalassemia major Hematology A hemoglobinopathy caused by a defect in the synthesis of Hb β chain Clinical Pallor, fatigability, FTT, fever due to infections, diarrhea, bone deformities, hepatosplenomegaly Management Transfusions, but iron overload can damage the heart, liver, and endocrine systems, ergo iron chelation–early use of deferiprone, deferoxamine ↓ transfusion-related iron overload and may protect against DM, cardiac disease, early death
δ-thalassemia
Hematology A condition characterized by a defect of Hb A2–α2δ2; because Hb A2 comprises only 3% of the circulating Hb, even its complete absence; δ-thalassemia has little clinical or hematologic impact
γ-thalassemia
Hematology A condition characterized by a defect of gamma–γ Hb chains found in Hb F–α2γ2; because Hb F is present primarily in the fetus and newborns, it is rarely seen outside of the neonatal period, but may cause transient neonatal hemolytic anemia 
, thalassanemia (thal'ă-sē'mē-ă, -ă-să-nē'mē-ă)
Any of a group of inherited disorders of hemoglobin metabolism in which there is impaired synthesis of one or more of the polypeptide chains of globin; several genetic types exist, and the corresponding clinical picture may vary from barely detectable hematologic abnormality to severe and fatal anemia  People of Mediterranean, extraction are more often affected than others by this type of anemia 
Synonym(s): thalassaemia, thalassanaemia 
Any of a group of inherited disorders of hemoglobin metabolism with impaired synthesis of one or more polypeptide chains of globin; several genetic types exist 
<a href="https://medical-dictionary thefreedictionary com/Thalassaemia+minor">thalassemia</a>
Barts hemoglobin
beta hemoglobinopathy
beta-delta thalassemia
beta-thalassaemia
Bite Cell
black gallstone
I know of a couple, totally unrelated and unbeknownst to them, who are silent carriers of Thalassaemia minor 
Pakistan: Genetic factor: All in the genes
But, unfortunately, when one person with thalassaemia minor carrier happens to marry another with the same diagnosis, there is a strong possibility that their child would be thalassaemia major, as happened in the case of Taneja 
' My life depends upon a monthly blood transfusion '
0] thalassaemia demonstrates variable severity, ranging from a condition similar to [beta] thalassaemia minor to something approaching thalassaemia major 
A retrospective review of homozygous haemoglobin E patients
Thal, Alan P 
thalame
thalamencephalic
thalamencephalon
thalamic
thalamic fasciculus
thalamic nucleus
thalamic pain syndrome
thalamic peduncle
thalamic radiation
thalamo-
thalamocoele
thalamocortical
thalamocortical fibers
thalamogeniculate artery
thalamolenticular
thalamoperforating artery
thalamostriate radiation
thalamotuberal artery
Thalassaemia minor
thalassaemiaor Cooley's disease
thalassemic facies
thalasso-
Thalassobacter
Thalassobacter utilis
thalassoplankton
thalassoposia
thalidomide neuropathy
Thalidomider
thallium poisoning
Thalarctos
THALAS
Thalasaemia
Thalassaemia Association of Malaysia
thalassaemia major
Thalassaemias
thalassaemic
thalassanaemia
Thalassemia Action Group
Thalassemia Clinical Research Network
thalassemia syndrome
Context 2: Paper Info

Title: Environmental variability and network structure determine the optimal plasticity mechanisms in embodied agents
Publish Date: Unkown
Author List: Sina Khajehabdollahi (from Department of Computer Science, University of Tübingen)

Figure

Figure2: An outline of the network controlling the foraging agent The sensor layer receives inputs at each time step (the ingredients of the nearest food), which are processed by the plastic layer in the same way as the static sensory network, Fig 1 The output of that network is given as input to the motor network, along with the distance d and angle α to the nearest food, the current velocity v, and energy E of the agent These signals are processed through two hidden layers to the final output of motor commands as the linear and angular acceleration of the agent
Figure4: The evolved parameters θ = (θ 1 ,      , θ 8 ) of the plasticity rule for the reward prediction (a ) and the decision (b ) tasks, for a variety of parameters (p tr = 0 01, d e ∈ 0, 0 1,      , 1, and σ ∈ 0, 0 1,      , 1 in all 100 combinations) Despite the relatively small difference between the tasks, the evolved learning rules differ considerably For visual guidance, the lines connect θs from the same run 
Figure5: a The trajectory of an agent (blue line) in the 2D environment A well-trained agent will approach and consume food with positive values (green dots) and avoid negative food (red dots) b The learning rate of the plastic sensory network eta p grows with the distance between environments d e c  and decreases with the frequency of environmental change d The fitness of an agent (measured as the total food consumed over its lifetime) increases over generations of the EA for both the scalar and binary readouts in the sensory network e The Pearson correlation coefficient of an evolved agent's weights with the ingredient value vector of the current environment (E 1 -blue, E 2 -red) In this example, the agent's weights are anti-correlated with its environment, which is not an issue for performance since the motor network can interpret the inverted signs of food 

abstract

The evolutionary balance between innate and learned behaviors is highly intricate, and different organisms have found different solutions to this problem  We hypothesize that the emergence and exact form of learning behaviors is naturally connected with the statistics of environmental fluctuations and tasks an organism needs to solve 
Here, we study how different aspects of simulated environments shape an evolved synaptic plasticity rule in static and moving artificial agents  We demonstrate that environmental fluctuation and uncertainty control the reliance of artificial organisms on plasticity  Interestingly, the form of the emerging plasticity rule is additionally determined by the details of the task the artificial organisms are aiming to solve 
Moreover, we show that coevolution between static connectivity and interacting plasticity mechanisms in distinct sub-networks changes the function and form of the emerging plasticity rules in embodied agents performing a foraging task  One of the defining features of living organisms is their ability to adapt to their environment and incorporate new information to modify their behavior 
It is unclear how the ability to learn first evolved , but its utility appears evident  Natural environments are too complex for all the necessary information to be hardcoded genetically and more importantly, they keep changing during an organism's lifetime in ways that cannot be anticipated ;   The link between learning and environmental uncertainty and fluctuation has been extensively demonstrated in both natural ; , and artificial environments  
Nevertheless, the ability to learn does not come without costs  For the capacity to learn to be beneficial in evolutionary terms, a costly nurturing period is often required, a phenomenon observed in both biological , and artificial organisms   Additionally, it has been shown that in some complex environments, hardcoded behaviors may be superior to learned ones given limits in the agent's lifetime and envi-ronmental uncertainty ; ;  
The theoretical investigation of the optimal balance between learned and innate behaviors in natural and artificial systems goes back several decades  However, it has recently found also a wide range of applications in applied AI systems ;   Most AI systems are trained for specific tasks, and have no need for modification after their training has been completed 
Still, technological advances and the necessity to solve broad families of tasks make discussions about life-like AI systems relevant to a wide range of potential application areas  Thus the idea of open-ended AI agents that can continually interact with and adapt to changing environments has become particularly appealing 
Many different approaches for introducing lifelong learning in artificial agents have been proposed  Some of them draw direct inspiration from actual biological systems ;   Among them, the most biologically plausible solution is to equip artificial neural networks with some local neural plasticity , similar to the large variety of synaptic plasticity mechanisms ; ; that performs the bulk of the learning in the brains of living organisms  
The artificial plasticity mechanisms can be optimized to modify the connectivity of the artificial neural networks toward solving a particular task  The optimization can use a variety of approaches, most commonly evolutionary computation  The idea of meta-learning or optimizing synaptic plasticity rules to perform specific functions has been recently established as an engineering tool that can compete with stateof-the-art machine learning algorithms on various complex tasks ; ; Pedersen and Risi (2021);  
Additionally, it can be used to reverse engineer actual plasticity mechanisms found in biological neural networks and uncover their functions ;   Here, we study the effect that different factors (environ-arXiv:2303 06734v1 [q-bio NC] 12 Mar 2023 mental fluctuation and reliability, task complexity) have on the form of evolved functional reward-modulated plasticity rules 
We investigate the evolution of plasticity rules in static, single-layer simple networks  Then we increase the complexity by switching to moving agents performing a complex foraging task  In both cases, we study the impact of different environmental parameters on the form of the evolved plasticity mechanisms and the interaction of learned and static network connectivity 
Interestingly, we find that different environmental conditions and different combinations of static and plastic connectivity have a very large impact on the resulting plasticity rules  We imagine an agent who must forage to survive in an environment presenting various types of complex food particles  Each food particle is composed of various amounts and combinations of N ingredients that can have positive (food) or negative (poison) values 
The value of a food particle is a weighted sum of its ingredients  To predict the reward value of a given resource, the agent must learn the values of these ingredients by interacting with the environment  The priors could be generated by genetic memory, but the exact values are subject to change  To introduce environmental variability, we stochastically change the values of the ingredients 
More precisely, we define two ingredient-value distributions E 1 and E 2 and switch between them, with probability p tr for every time step  We control how (dis)similar the environments are by parametrically setting E 2 = (1 − 2d e )E 1 , with d e ∈ [0, 1] serving as a distance proxy for the environments; when d e = 0, the environment remains unchanged, and when d e = 1 the value of each ingredient fully reverses when the environmental transition happens 
For simplicity, we take values of the ingredients in E 1 equally spaced between -1 and 1 (for the visualization, see Fig  )  The static agent receives passively presented food as a vector of ingredients and can assess its compound value using the linear summation of its sensors with the (learned or evolved) weights, see Fig   
The network consists of N sensory neurons that are projecting to a single post-synaptic neuron  At each time step, an input X t = (x 1 ,       , x N ) is presented, were the value x i , i ∈ {1,       , N } represents the quantity of the ingredient i  We draw x i independently form a uniform distribution on the [0, 1] interval (x i ∼ U (0, 1)) 
The value of each ingredient w c i is determined by the environment (E 1 or E 2 )  The postsynaptic neuron outputs a prediction of the food X t value as y t = g(W X T t )  Throughout the paper, g will be either the identity function, in which case the prediction neuron is linear, or a step-function; however, it could be any other nonlinearity, such as a sigmoid or ReLU 
After outputting the prediction, the neuron receives feedback in the form of the real value of the input R t   The real value is computed as R t = W c X T t + ξ, where W c = (w c 1 ,       , w c N ) is the actual value of the ingredients, and ξ is a term summarizing the noise of reward and sensing system ξ ∼ N (0, σ) 
Figure : An outline of the static agent's network  The sensor layer receives inputs representing the quantity of each ingredient of a given food at each time step  The agent computes the prediction of the food's value y t and is then given the true value R t ; it finally uses this information in the plasticity rule to update the weight matrix 
For the evolutionary adjustment of the agent's parameters, the loss of the static agent is the sum of the mean squared errors (MSE) between its prediction y t and the reward R t over the lifetime of the agent  The agent's initial weights are set to the average of the two ingredient value distributions, which is the optimal initial value for the case of symmetric switching of environments that we consider here 
As a next step, we incorporate the sensory network of static agents into embodied agents that can move around in an environment scattered with food  To this end, we merge the static agent's network with a second, non-plastic motor network that is responsible for controlling the motion of the agent in the environment 
Specifically, the original plastic network now provides the agent with information about the value of the nearest food  The embodied agent has additional sensors for the distance from the nearest food, the angle between the current velocity and the nearest food direction, its own velocity, and its own energy level (sum of consumed food values) 
These inputs are processed by two hidden layers (of 30 and 15 neurons) with tanh activation  The network's outputs are angular and linear acceleration, Fig    The embodied agents spawn in a 2D space with periodic boundary conditions along with a number of food particles that are selected such that the mean of the food value distribution is ∼ 0  An agent can eat food by approaching it sufficiently closely, and each time a food particle is eaten, it is The sensor layer receives inputs at each time step (the ingredients of the nearest food), which are processed by the plastic layer in the same way as the static sensory network, Fig   
The output of that network is given as input to the motor network, along with the distance d and angle α to the nearest food, the current velocity v, and energy E of the agent  These signals are processed through two hidden layers to the final output of motor commands as the linear and angular acceleration of the agent re-spawned with the same value somewhere randomly on the grid (following the setup of ) 
After 5000 time steps, the cumulative reward of the agent (the sum of the values of all the food it consumed) is taken as its fitness  During the evolutionary optimization, the parameters for both the motor network (connections) and plastic network (learning rule parameters) are co-evolved, and so agents must simultaneously learn to move and discriminate good/bad food 
Reward-modulated plasticity is one of the most promising explanations for biological credit assignment   In our network, the plasticity rule that updates the weights of the linear sensor network is a rewardmodulated rule which is parameterized as a linear combination of the input, the output, and the reward at each time step:
Additionally, after each plasticity step, the weights are normalized by mean subtraction, an important step for the stabilization of Hebbian-like plasticity rules   We use a genetic algorithm to optimize the learning rate η p and amplitudes of different terms θ = (θ 1 ,       , θ 8 )  The successful plasticity rule after many food presentations must converge to a weight vector that predicts the correct food values (or allows the agent to correctly decide whether to eat a food or avoid it) 
To have comparable results, we divide θ = (θ 1 ,       , θ 8 ) by We then multiply the learning rate η p with θ max to maintain the rule's evolved form unchanged, η norm p = η p • θ max   In the following, we always use normalized η p and θ, omitting norm   To evolve the plasticity rule and the moving agents' motor networks, we use a simple genetic algorithm with elitism  
The agents' parameters are initialized at random (drawn from a Gaussian distribution), then the sensory network is trained by the plasticity rule and finally, the agents are evaluated  After each generation, the bestperforming agents (top 10 % of the population size) are selected and copied into the next generation 
The remaining 90 % of the generation is repopulated with mutated copies of the best-performing agents  We mutate agents by adding independent Gaussian noise (σ = 0 1) to its parameters  To start with, we consider a static agent whose goal is to identify the value of presented food correctly  The static reward-prediction network quickly evolves the parameters of the learning rule, successfully solving the prediction task 
We first look at the evolved learning rate η p , which determines how fast (if at all) the network's weight vector is updated during the lifetime of the agents  We identify three factors that control the learning rate parameter the EA converges to: the distance between the environments, the noisiness of the reward, and the rate of environmental transition 
The first natural factor is the distance d e between the two environments, with a larger distance requiring a higher learning rate, Fig    This is an expected result since the convergence time to the "correct" weights is highly dependent on the initial conditions  If an agent is born at a point very close to optimality, which naturally happens if the environments are similar, the distance it needs to traverse on the fitness landscape is small 
Therefore it can afford to have a small learning rate, which leads to a more stable convergence and is not affected by noise  A second parameter that impacts the learning rate is the variance of the rewards  The reward an agent receives for the plasticity step contains a noise term ξ that is drawn from a zero mean Gaussian distribution with standard deviation σ 
This parameter controls the unreliability of the agent's sensory system, i e , higher σ means that the information the agent gets about the value of the foods it consumes cannot be fully trusted to reflect the actual value of the foods  As σ increases, the learning rate η p decreases, which means that the more unreliable an environment becomes, the less an agent relies on plasticity to update its weights, Fig   
Indeed for some combinations of relatively small distance d e and high reward variance σ, the EA converges to a learning rate of η p ≈ 0  This means that the agent opts to have no adaptation during its lifetime and remain at the mean of the two environments  It is an optimal solution when the expected loss due to ignoring the environmental transitions is, on average, lower than the loss the plastic network will incur by learning via the (often misleading because of the high σ) environmental cues 
A final factor that affects the learning rate the EA will converge to is the frequency of environmental change during an agent's lifetime  Since the environmental change is modeled as a simple, two-state Markov process (Fig  ), the control parameter is the transition probability p tr   When keeping everything else the same, the learning rate rapidly rises as we increase the transition probability from 0, and after reaching a peak, it begins to decline slowly, eventually reaching zero (Fig  ) 
This means that when environmental transition is very rare, agents opt for a very low learning rate, allowing a slow and stable convergence to an environment-appropriate weight vector that leads to very low losses while the agent remains in that environment  As the rate of environmental transition increases, faster learning is required to speed up convergence in order to exploit the (comparatively shorter) stays in each environment 
Finally, as the environmental transition becomes too fast, the agents opt for slower or even no learning, which keeps them ) and the decision (b ) tasks, for a variety of parameters (p tr = 0 01, d e ∈ 0, 0 1,       , 1, and σ ∈ 0, 0 1,       , 1 in all 100 combinations)  Despite the relatively small difference between the tasks, the evolved learning rules differ considerably 
For visual guidance, the lines connect θs from the same run  near the middle of the two environments, ensuring that the average loss of the two environments is minimal (Fig  )  The form of the evolved learning rule depends on the task: Decision vs  Prediction The plasticity parameters θ = (θ 1 ,       , θ 8 ) for the rewardprediction task converge on approximately the same point, regardless of the environmental parameters (Fig  ) 
In particular, θ 3 → 1, θ 5 → −1, θ i → 0 for all other i, and thus the learning rule converges to: Since by definition y t = g(W t X T t ) = W t X T t (g(x) = x in this experiment) and R t = W c X T t + ξ we get: Thus the distribution of ∆W t converges to a distribution with mean 0 and variance depending on η p and σ and W converges to W c  
So this learning rule will match the agent's weight vector with the vector of ingredient values in the environment  We examine the robustness of the learning rule the EA discovers by considering a slight modification of our task  Instead of predicting the expected food value, the agent now needs to decide whether to eat the presented food or not 
This is done by introducing a step-function nonlinearity (g(x) = 1 if x ≥ 1 and 0 otherwise)  Then the output y(t) is computed as: Instead of the MSE loss between prediction and actual value, the fitness of the agent is now defined as the sum of the food values it chose to consume (by giving y t = 1)  Besides these two changes, the setup of the experiments remains exactly the same 
The qualitative relation between η p and parameters of environment d e , σ and p tr is preserved in the changed experiment  However, the resulting learning rule is significantly different (Fig  )  The evolution converges to the following learning rule: In both cases, the rule has the form ∆W t = η p X t [α y R t + β y ] 
Thus, the ∆W t is positive or negative depending on whether the reward R t is above or below a threshold (γ = −β y /α y ) that depends on the output decision of the network (y t = 0 or 1)  Both learning rules (for the reward-prediction and decision tasks) have a clear Hebbian form (coordination of preand post-synaptic activity) and use the incoming reward signal as a threshold 
These similarities indicate some common organizing principles of reward-modulated learning rules, but their significant differences highlight the sensitivity of the optimization process to task details  We now turn to the moving embodied agents in the 2D environment  To optimize these agents, both the motor network's connections and the sensory network's plasticity parameters evolve simultaneously 
Since the motor network is initially random and the agent has to move to find food, the number of interactions an agent experiences in its lifetime can be small, slowing down the learning  However, having the larger motor network also has benefits for evolution because it allows the output of the plastic network to be read out and transformed in different ways, resulting in a broad set of solutions 
The fitness of an agent (measured as the total food consumed over its lifetime) increases over generations of the EA for both the scalar and binary readouts in the sensory network  e  The Pearson correlation coefficient of an evolved agent's weights with the ingredient value vector of the current environment (E 1 -blue, E 2 -red) 
In this example, the agent's weights are anti-correlated with its environment, which is not an issue for performance since the motor network can interpret the inverted signs of food  The agents can solve the task effectively by evolving a functional motor network and a plasticity rule that converges to interpretable weights (Fig  ) 
After ∼ 100 evolutionary steps (Fig  ), the agents can learn the ingredient value distribution using the plastic network and reliably move towards foods with positive values while avoiding the ones with negative values  We compare the dependence of the moving and the static agents on the parameters of the environment: d e and the state transition probability p tr  
At first, in order to simplify the experiment, we set the transition probability to 0, but fixed the initial weights to be the average of E 1 and E 2 , while the real state is E 2   In this experiment, the distance between states d e indicates twice the distance between the agent's initial weights and the optimal weights (the environment's ingredient values) since the agent is initialized at the mean of the two environment distributions 
Same as for the static agent, the learning rate increases with the distance d e (Fig  )  Then, we examine the effect of the environmental transition probability p tr on the evolved learning rate η p   In order for an agent to get sufficient exposure to each environment, we scale down the probability p tr from the equivalent experiment for the static agents 
We find that as the probability of transition increases, the evolved learning rate η p decreases (Fig  )  This fits with the larger trend for the static agent, although there is a clear difference when it comes to the increase for very small transition probabil-ities that were clearly identifiable in the static but not the moving agents 
This could be due to much sparser data and possibly the insufficiently long lifetime of the moving agent (the necessity of scaling makes direct comparisons difficult)  Nevertheless, overall we see that the associations observed in the static agents between environmental distance d e and transition probability p tr and the evolved learning rate η p are largely maintained in the moving agents 
Still, more data would be needed to make any conclusive assertions about the exact effect of these environmental parameters on the emerging plasticity mechanisms  A crucial difference between the static and the moving agents is the function the plasticity has to perform  While in the static agents, the plasticity has to effectively identify the exact value distribution of the environment in order to produce accurate predictions, in the embodied agents, the plasticity has to merely produce a representation of the environment that the motor network can evolve to interpret adequately enough to make decisions about which food to consume 
To illustrate the difference, we plot the Pearson correlation coefficient between an agent's weights and the ingredient values of the environment it is moving in (Fig  )  We use the correlation instead of the MSE loss (which we used for the static agents in Fig  ) because the amplitude of the vector varies a lot for different agents and meaningful The evolved parameters of moving agents' plasticity rule for the g(s) = x, identity (a ) and the step function (Eq 
4) (b ) sensory networks (the environmental parameters here are d e ∈ [0, 1], σ = 0 and p tr = 0 001)  The step function (binary output) network evolved a more structured plasticity rule (e g , θ 3 > 0 for all realizations) than the linear network  Moreover, the learned weights for the identity network (c ) have higher variance and correlate significantly less with the environment's ingredient distribution compared to the learned weights for the thresholded network (d )
conclusions cannot be drawn from the MSE loss  For many agents, the learned weights are consistently anti-correlated with the actual ingredient values (an example of such an agent is shown in Fig  )  This means that the output of the sensory network will have the opposite sign from the actual food value 
While in the static network, this would lead to very bad predictions and high loss, in the foraging task, these agents perform exactly as well as the ones where the weights and ingredients values are positively correlated, since the motor network can simply learn to move towards food for which it gets a negative instead of a positive sensory input 
This additional step of the output of the plastic network going through the motor network before producing any behavior has a strong effect on the plasticity rules that the embodied agents evolve  Specifically, if we look at the emerging rules the top performing agents have evolved (Fig  ), it becomes clear that, unlike the very well-structured rules of the static agents (Fig  ), there is now virtually no discernible pattern or structure 
The difference becomes even clearer if we look at the learned weights (at the end of a simulation) of the best-performing agents (Fig  )  While there is some correlation with the environment's ingredient value distribution, the variance is very large, and they do not seem to converge on the "correct" values in any way 
This is to some extent expected since, unlike the static agents where the network's output has to be exactly correct, driving the evolution of rules that converge to the precise environmental distribution, in the embodied networks, the bulk of the processing is done by the motor network which can evolve to interpret the scalar value of the sensory network's output in a variety of ways 
Thus, as long as the sensory network's plasticity rule co-evolves with the motor network, any plasticity rule that learns to produce consistent information about the value of encountered food can potentially be selected  To further test this assumption, we introduce a bottleneck of information propagation between the sensory and motor networks by using a step-function nonlinearity on the output of the sensory network (Eq 
4)  Similarly to the decision task of the static network, the output of the sensory network now becomes binary  This effectively reduces the flow of information from the sensory to the motor network, forcing the sensory network to consistently decide whether food should be consumed (with the caveat that the motor network can still interpret the binary sign in either of two ways, either consuming food marked with 1 or the ones marked with 0 by the sensory network) 
The agents perform equally well in this variation of the task as before (Fig  ), but now, the evolved plasticity rules seem to be more structured (Fig  )  Moreover, the variance of the learned weights in the bestperforming agents is significantly reduced (Fig  ), which indicates that the bottleneck in the sensory network is in-creasing selection pressure for rules that learn the environment's food distribution accurately 
We find that different sources of variability have a strong impact on the extent to which evolving agents will develop neuronal plasticity mechanisms for adapting to their environment  A diverse environment, a reliable sensory system, and a rate of environmental change that is neither too large nor too small are necessary conditions for an agent to be able to effectively adapt via synaptic plasticity 
Additionally, we find that minor variations of the task an agent has to solve or the parametrization of the network can give rise to significantly different plasticity rules  Our results partially extend to embodied artificial agents performing a foraging task  We show that environmental variability also pushes the development of plasticity in such agents 
Still, in contrast to the static agents, we find that the interaction of a static motor network with a plastic sensory network gives rise to a much greater variety of wellfunctioning learning rules  We propose a potential cause of this degeneracy; as the relatively complex motor network is allowed to read out and process the outputs from the plastic network, any consistent information coming out of these outputs can be potentially interpreted in a behaviorally useful way 
Reducing the information the motor network can extract from the sensory system significantly limits learning rule variability  Our findings on the effect of environmental variability concur with the findings of previous studies that have identified the constraints that environmental variability places on the evolutionary viability of learning behaviors 
We extend these findings in a mechanistic model which uses a biologically plausible learning mechanism (synaptic plasticity)  We show how a simple evolutionary algorithm can optimize the different parameters of a simple reward-modulated plasticity rule for solving simple prediction and decision tasks 
Reward-modulated plasticity has been extensively studied as a plausible mechanism for credit assignment in the brain ; ; and has found several applications in artificial intelligence and robotics tasks ;   Here, we demonstrate how such rules can be very well-tuned to take into account different environmental parameters and produce optimal behavior in simple systems 
Additionally, we demonstrate how the co-evolution of plasticity and static functional connectivity in different subnetworks fundamentally changes the evolutionary pressures on the resulting plasticity rules, allowing for greater diversity in the form of the learning rule and the resulting learned connectivity 
Several studies have demonstrated how, in biological networks, synaptic plasticity heavily interacts with and is driven by network topology   Moreover, it has been recently demonstrated that biological plasticity mechanisms are highly redundant in the sense that any observed neural connectivity or recorded activity can be achieved with a variety of distinct, unrelated learning rules  
This observed redundancy of learning rules in biological settings complements our results and suggests that the function of plasticity rules cannot be studied independently of the connectivity and topology of the networks they are acting on  The optimization of functional plasticity in neural networks is a promising research direction both as a means to understand biological learning processes and as a tool for building more autonomous artificial systems 
Our results suggest that reward-modulated plasticity is highly adaptable to different environments and can be incorporated into larger systems that solve complex tasks  This work studies a simplified toy model of neural network learning in stochastic environments  Future work could be built on this basic framework to examine more complex reward distributions and sources of environmental variability 
Moreover, a greater degree of biological realism could be added by studying more plausible network architectures (multiple plastic layers, recurrent and feedback connections) and more sophisticated plasticity rule parametrizations  Additionally, our foraging simulations were constrained by limited computational resources and were far from exhaustive 
Further experiments can investigate environments with different constraints, food distributions, multiple seasons, more complex motor control systems and interactions of those systems with different sensory networks as well as the inclusion of plasticity on the motor parts of the artificial organisms 



ground_truth: Severe anemia that begins even before birth.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What factors control the reliance of artificial organisms on plasticity?

Read the following text and answer briefly.

Context 4: \section{Introduction}

Derivate is one of the most important topics not only in mathematics, but also in physics, chemistry, economics and engineering  Every standard Calculus course provides a variety of exercises for the students to learn how to apply the concept of derivative  The types of problems range from finding an equation of the tangent line to the application of differentials and advanced curve sketching  Usually, these exercises heavily rely on such differentiation techniques as Product, Quotient and Chain Rules, Implicit and Logarithmic Differentiation \cite{Stewart2012}  The definition of the derivative is hardly ever applied after the first few classes and its use is not much motivated 

Like many other topics in undergraduate mathematics, derivative gave rise to many misconceptions \cite{Muzangwa2012}, \cite{Gur2007}, \cite{Li2006}  Just when the students seem to learn how to use the differentiation rules for most essential functions, the application of the derivative brings new issues  A common students' error of determining the domain of the derivative from its formula is discussed in \cite{Rivera2013} and some interesting examples of the derivatives, defined at the points where the functions themselves are undefined, are provided  However, the hunt for misconceptions takes another twist for the derivatives undefined at the points where the functions are in fact defined 

The expression of the derivative of the function obtained using differentiation techniques does not necessarily contain the information about the existence or the value of the derivative at the points, where the expression for the derivative is undefined  In this article we discuss a type of continuous functions that have the expression for the derivative undefined at a certain point, while the derivative itself at that point exists  We show, how relying on the formula for the derivative for finding the horizontal tangent line of a function, leads to a false conclusion and consequently to missing a solution  We also provide a simple methodological treatment of similar functions suitable for the classroom 

\section{Calculating the Derivative}

In order to illustrate how deceitful the expression of the derivative can be to a students' eye, let us consider the following problem 

\vspace{12pt}

\fbox{\begin{minipage}{5 25in}

\begin{center}

\begin{minipage}{5 0in}

\vspace{10pt}

\emph{Problem}

\vspace{10pt}

Differentiate the function $f\left(x\right)=\sqrt[3]{x}\sin{\left(x^2\right)}$  For which values of $x$ from the interval $\left[-1,1\right]$ does the graph of $f\left(x\right)$ have a horizontal tangent?

\vspace{10pt}

\end{minipage}

\end{center}

\end{minipage}}

\vspace{12pt}

Problems with similar formulations can be found in many Calculus books \cite{Stewart2012}, \cite{Larson2010}, \cite{Thomas2009}  Following the common procedure, let us find the expression for the derivative of the function $f\left(x\right)$ applying the Product Rule:
\begin{eqnarray}
f'\left(x\right) &=& \left(\sqrt[3]{x}\right)'\sin{\left(x^2\right)}+\left(\sin{\left(x^2\right)}\right)'\sqrt[3]{x} \notag \\ &=& \frac{1}{3\sqrt[3]{x^2}}\sin{\left(x^2\right)}+2x\cos{\left(x^2\right)}\sqrt[3]{x} \notag \\ &=& \frac{6x^2\cos{x^2}+\sin{x^2}}{3\sqrt[3]{x^2}} \label{DerivativeExpression}
\end{eqnarray}

Similar to \cite{Stewart2012}, we find the values of $x$ where the derivative $f'\left(x\right)$ is equal to zero:
\begin{equation}
6x^2\cos{x^2}+\sin{x^2} = 0 
\label{DerivativeEqualZero}
\end{equation}

Since the expression for the derivative (\ref{DerivativeExpression}) is not defined at $x=0$, it is not hard to see that for all values of $x$ from $\left[-1,1\right]$ distinct from zero, the left-hand side of (\ref{DerivativeEqualZero}) is always positive  Hence, we conclude that the function $f\left(x\right)$ does not have horizontal tangent lines on the interval $\left[-1,1\right]$ 

However, a closer look at the graph of the function $f\left(x\right)$ seems to point at a different result: there is a horizontal tangent at $x=0$ (see Figure \ref{fig:FunctionGraph})  

First, note that the function $f\left(x\right)$ is defined in $x=0$  In order to verify if it has a horizontal tangent at this point, let us find the derivative of the function $f\left(x\right)$ using definition:
\begin{eqnarray}
f'\left(0\right) &=& \lim_{h\rightarrow0}{\frac{f\left(0+h\right)-f\left(0\right)}{h}} \notag \\ 
&=& \lim_{h\rightarrow0}{\frac{\sqrt[3]{h}\sin{\left(h^2\right)}}{h}} \notag \\ 
&=& \lim_{h\rightarrow0}{\left(\sqrt[3]{h} \cdot {h} \cdot \frac{\sin{\left(h^2\right)}}{h^2}\right)} \notag \\
&=& \lim_{h\rightarrow0}{\sqrt[3]{h}} \cdot \lim_{h\rightarrow0}{h} \cdot \lim_{h\rightarrow0}{\frac{\sin{\left(h^2\right)}}{h^2}} \notag \\
&=& 0 \cdot 0 \cdot 1 = 0 \notag
\end{eqnarray}
since each of the limits above exists  We see that, indeed, the function $f\left(x\right)$ possesses a horizontal tangent line at the point $x=0$ 

\section{Closer Look at the Expression for the Derivative}

What is the problem with the standard procedure proposed by many textbooks and repeated in every Calculus class? The explanation lies in the following premise: the expression of the derivative of the function does not contain the information as to whether the function is differentiable or not at the points where it is undefined  As it is pointed out in \cite{Rivera2013}, the domain of the derivative is determined \emph{a priori} and therefore should not be obtained from the formula of the derivative itself 

In the example above the Product Law for derivatives requires the existence of the derivatives of both functions at the point of interest  Since the function $\sqrt[3]{x}$ is not differentiable in zero, the Product Rule cannot be applied  

In order to see what exactly happens when we apply the Product Rule, let us find the expression for the derivative using definition of the derivative:
\begin{eqnarray}
f'\left(x\right) &=& \lim_{h\rightarrow0}{\frac{f\left(x+h\right)-f\left(x\right)}{h}} \notag \\ 
&=& \lim_{h\rightarrow0}{\frac{\sqrt[3]{x+h}\sin{\left(x+h\right)^2}-\sqrt[3]{x}\sin{\left(x^2\right)}}{h}} \notag \\ 
&=& \lim_{h\rightarrow0}{\frac{\left(\sqrt[3]{x+h}-\sqrt[3]{x}\right)}{h}\sin{\left(x^2\right)}} + \notag \\
&& \lim_{h\rightarrow0}{\frac{\left(\sin{\left(x+h\right)^2}-\sin{\left(x^2\right)}\right)}{h}\sqrt[3]{x+h}} \notag \\
&=& \lim_{h\rightarrow0}{\frac{\sqrt[3]{x+h}-\sqrt[3]{x}}{h}} \cdot \lim_{h\rightarrow0}{\sin{\left(x^2\right)}} +  \notag \\&& \lim_{h\rightarrow0}{\frac{\sin{\left(x+h\right)^2}-\sin{\left(x^2\right)}}{h}} \cdot \lim_{h\rightarrow0}{\sqrt[3]{x+h}} \notag \\
&=& \frac{1}{3\sqrt[3]{x^2}} \cdot \sin{\left(x^2\right)}+2x\cos{\left(x^2\right)} \cdot \sqrt[3]{x} \notag 
\end{eqnarray}
which seems to be identical to the expression (\ref{DerivativeExpression}) 

Students are expected to develop a skill of deriving similar results and know how to find the derivative of the function using definition of the derivative only  But how `legal' are the performed operations?

\begin{figure}[H]
\begin{center}
	\includegraphics[width=6 0in]{sin pdf}
	\vspace{ 1in}
	\caption{Graph of the function $g\left(x\right)=\sqrt[3]{x}\cos{\left(x^2\right)}$}
	\label{fig:GFunction}
\end{center}
\end{figure}

Let us consider each of the following limits: 
\begin{eqnarray*}
&& \lim_{h\rightarrow0}{\frac{\sqrt[3]{x+h}-\sqrt[3]{x}}{h}} \notag \\
&& \lim_{h\rightarrow0}{\sin{\left(x^2\right)}}\notag \\
&& \lim_{h\rightarrow0}{\frac{\sin{\left(x+h\right)^2}-\sin{\left(x^2\right)}}{h}}\notag \\
&& \lim_{h\rightarrow0}{\sqrt[3]{x+h}} 
\end{eqnarray*}
The last three limits exist for all real values of the variable $x$



ground_truth: Environmental fluctuation and uncertainty control the reliance of artificial organisms on plasticity.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: How do the runtimes and iteration counts of NFPA and FPSA compare to GMRES and DSA in the numerical experiments?

Read the following text and answer briefly.

Context 3: \section{Introduction}\label{sec1}
\setcounter{equation}{0} 

Transport problems with highly forward-peaked scattering are prevalent in a variety of areas, including astrophysics, medical physics, and plasma physics \cite{HGK,aristova,multiphysics} 
For these problems, solutions of the transport equation converge slowly when using conventional methods such as source iteration (SI) \cite{adamslarsen} and the generalized minimal residual method (GMRES) \cite{gmres} 
Moreover, diffusion-based acceleration techniques like diffusion synthetic acceleration (DSA) \cite{alcouffe} and nonlinear diffusion acceleration (NDA) \cite{smithetall} are generally inefficient when tackling these problems, as they only accelerate up to the first moment of the angular flux \cite{JapanFPSA} 
In fact, higher-order moments carry important information in problems with highly forward-peaked scattering and can be used to further accelerate convergence \cite{japanDiss} 

This paper focuses on solution methods for the monoenergetic, steady-state transport equation in homogeneous slab geometry 
Under these conditions, the transport equation is given by
\begin{subequations}\label[pluraleq]{eq1}
\begin{equation}
\label{t1}
\mu\frac{\partial}{\partial x} \psi(x,\mu) + \sigma_t \psi(x,\mu) = \int_{-1}^{1} d\mu' \sigma_s(\mu,\mu') \psi(x,\mu') + Q(x, \mu), \,\,\, x\in [0, X],-1\leq\mu\leq 1  ,\\
\end{equation}
with boundary conditions
\begin{align}
\label{t2}
\psi(0,\mu) &= \psi_L(\mu), \quad \mu > 0,\\
\label{t3}
\psi(X,\mu) &= \psi_R(\mu), \quad \mu < 0 
\end{align}
\end{subequations}
Here, $\psi(x,\mu)$ represents the angular flux at position $x$ and direction $\mu$, $\sigma_t$ is the macroscopic total cross section, $\sigma_s(\mu,\mu')$ is the differential scattering cross section, and $Q$ is an internal source 

New innovations have paved the way to better solve this equation in systems with highly forward-peaked scattering 
For instance, work has been done on modified $P_L$ equations and modified scattering cross section moments to accelerate convergence of anisotropic neutron transport problems \cite{khattab} 
In order to speed up the convergence of radiative transfer in clouds, a quasi-diffusion method has been developed \cite{aristova} 
In addition, the DSA-multigrid method was developed to solve problems in electron transport more efficiently \cite{trucksin} 

One of the most recent convergence methods developed is Fokker-Planck Synthetic Acceleration (FPSA) \cite{JapanFPSA,japanDiss} 
FPSA accelerates up to $N$ moments of the angular flux and has shown significant improvement in the convergence rate for the types of problems described above 
The method returns a speed-up of several orders of magnitude with respect to wall-clock time when compared to DSA  \cite{JapanFPSA} 

In this paper, we introduce a new acceleration technique, called \textit{Nonlinear Fokker-Planck Acceleration} (NFPA) 
This  method  returns  a  modified  Fokker-Planck (FP) equation  that  preserves  the  angular moments of the flux given by the transport  equation 
This preservation of moments is particularly appealing for applications to multiphysics problems \cite{multiphysics}, in which the coupling between the transport physics and the other physics can be done through the (lower-order) FP equation 
To our knowledge, this is the first implementation of a numerical method that returns a Fokker-Planck-like equation that is discretely consistent with the linear Boltzmann equation 

This paper is organized as follows 
\Cref{sec2} starts with a brief description of FPSA 
Then, we derive the NFPA scheme 
In \cref{sec3}, we discuss the discretization schemes used in this work and present numerical results 
These are compared against standard acceleration techniques 
We conclude with a discussion in \cref{sec4} 

\section{Fokker-Planck Acceleration}\label{sec2}
\setcounter{equation}{0} 
In this section we briefly outline the theory behind FPSA, describe NFPA for monoenergetic, steady-state transport problems in slab geometry, and present the numerical methodology behind NFPA 
The theory given here can be easily extended to higher-dimensional problems 
Moreover, extending the method to energy-dependence shall not lead to significant additional theoretical difficulties 

To solve the transport problem given by \cref{eq1} we approximate the in-scattering term in \cref{t1} with a Legendre moment expansion:
\begin{equation}
\label{transport1}
\mu\frac{\partial}{\partial x} \psi(x,\mu) + \sigma_t \psi(x,\mu) = \sum_{l=0}^L \frac{(2l+1)}{2} P_l(\mu) \sigma_{s,l} \phi_l(x) + Q(x, \mu),
\end{equation}
with 
\begin{equation}
\label{transport2}
\phi_l(x) =  \int_{-1}^{1} d\mu P_l(\mu) \psi(x,\mu) 
\end{equation}
Here, $\phi_l$ is the $l^{th}$ Legendre moment of the angular flux, $ \sigma_{s,l}$ is the $l^{th}$ Legendre coefficient of the differential scattering cross section,  and $P_l$ is the $l^{th}$-order Legendre polynomial 
For simplicity, we will drop the notation $(x,\mu)$ in the remainder of this section 

The solution to \cref{transport1} converges asymptotically to the solution of the following Fokker-Planck equation in the forward-peaked limit \cite{pomraning1}:
\begin{equation}
\label{fp1}
\mu\frac{\partial \psi}{\partial x} + \sigma_a \psi = \frac{\sigma_{tr}}{2}\frac{\partial }{\partial \mu} (1-\mu^2) \frac{\partial \psi}{\partial \mu} + Q\,,
\end{equation}
where $\sigma_{tr}= \sigma_{s,0} -\sigma_{s,1}$ is the momentum transfer cross section and $\sigma_a = \sigma_t-\sigma_{s,0}$ is the macroscopic absorption cross section 

Source Iteration \cite{adamslarsen} is generally used to solve \cref{transport1}, which can be rewritten in operator notation:
\begin{equation}
\label{si1}
\mathcal{L} \psi^{m+1} = \mathcal{S} \psi^{m} + Q\,,
\end{equation}
where 
\begin{equation}
\mathcal{L} = \mu \frac{\partial}{\partial x} + \sigma_t,
   \quad
\mathcal{S} = \sum_{l=0}^L \frac{(2l+1)}{2} P_l(\mu) \sigma_{s,l} \int_{-1}^{1}d\mu P_l(\mu) ,
\label{trans1}
\end{equation}
and $m$ is the iteration index 
This equation is solved iteratively until a tolerance criterion is met  The FP approximation shown in \cref{fp1} can be used to accelerate the convergence of \cref{transport1} 

\subsection{FPSA: Fokker-Planck Synthetic Acceleration}\label{FPSA}

In the FPSA scheme \cite{JapanFPSA,japanDiss}, the FP approximation is used as a preconditioner to synthetically accelerate convergence when solving \cref{transport1} (cf  \cite{adamslarsen} for a detailed description of synthetic acceleration) 
When solving \cref{si1}, the angular flux at each iteration $m$ has an error associated with it 
FPSA systematically follows a predict, correct, iterate scheme 
A transport sweep, one iteration in \cref{si1}, is made for a prediction 
The FP approximation is used to correct the error in the prediction, and this iteration is performed until a convergence criterion is met 
The equations used are:
\begin{subequations}
\label{fpsaeq}
\begin{align}
\label{predict}
\mathrm{Predict}&: \mathcal{L} \psi^{m+\frac{1}{2}} = \mathcal{S} \psi^{m} + Q\,,\\
\label{correct}
\mathrm{Correct}&: \psi^{m+1} =  \psi^{m+\frac{1}{2}} + \mathcal{P}^{-1} \mathcal{S} \left( \psi^{m+\frac{1}{2}} -  \psi^{m}\right),
\end{align}
\end{subequations}
where we define $\mathcal{P}$ as
\begin{equation}
\label{FPSAsi1}
\mathcal{P} = \mathcal{A}-\mathcal{F} =\underbrace{\left(\mu\frac{\partial}{\partial x} + \sigma_a\right)}_\mathcal{A} - \underbrace{\left(\frac{\sigma_{tr}}{2}\frac{\partial }{\partial \mu} (1-\mu^2) \frac{\partial }{\partial \mu}\right)}_\mathcal{F},
\end{equation}
In this synthetic acceleration method, the FP approximation is used to correct the error in each iteration of the high-order (HO) equation (\ref{predict})  
Therefore, there is no consistency between the angular moments of the flux in the HO and low-order (LO) equations 

\subsection{NFPA: Nonlinear Fokker-Planck Acceleration}\label{NFPA}

Similar to FPSA, NFPA uses the FP approximation to accelerate the convergence of the solution 
We introduce the additive term $\hat{D}_F$ to \cref{fp1}, obtaining the modified FP equation
\begin{equation}
\label{mfp1}
\mu\frac{\partial \psi}{\partial x} + \sigma_a \psi = \frac{\sigma_{tr}}{2}\frac{\partial }{\partial \mu} (1-\mu^2) \frac{\partial \psi}{\partial \mu} + \hat{D}_F + Q\, 
\end{equation}
The role of $\hat{D}_F$ is to force the transport and modified FP equations to be consistent 
Subtracting \cref{mfp1} from \cref{transport1} and rearranging, we obtain the consistency term
\begin{equation}
\label{dfp}
\hat{D}_F = \sum_{l=0}^L \frac{(2l+1)}{2} P_l \sigma_l \phi_l - \frac{\sigma_{tr}}{2}\frac{\partial}{\partial \mu} (1-\mu^2) \frac{\partial \psi}{\partial \mu} - \sigma_{s,0} \psi\, 
\end{equation}

The NFPA method is given by the following equations:
\begin{subequations}\label[pluraleq]{holocons}
\begin{align}
\label{HO1}
\text{HO}&: \mu\frac{\partial \psi_{HO}}{\partial x} + \sigma_t \psi_{HO} = \sum_{l=0}^L \frac{(2l+1)}{2} P_l \sigma_l \phi_{l, LO} + Q\,,\\
\label{LO11}
\text{LO}&: \mu\frac{\partial \psi_{LO}}{\partial x} + \sigma_a \psi_{LO} = \frac{\sigma_{tr}}{2}\frac{\partial }{\partial \mu} (1-\mu^2) \frac{\partial \psi_{LO}}{\partial \mu} + \hat{D}_F + Q\,,\\
\label{con1}
\text{Consistency term}&: \hat{D}_F = \sum_{l=0}^L \frac{(2l+1)}{2} P_l \sigma_l \phi_{l, HO}^m - \frac{\sigma_{tr}}{2}\frac{\partial }{\partial \mu} (1-\mu^2) \frac{\partial \psi_{HO}}{\partial \mu} - \sigma_{s,0} \psi_{HO}\,,
\end{align}
\end{subequations}
where $\psi_{HO}$ is the angular flux obtained from the HO equation and $\psi_{LO}$ is the angular flux obtained from the LO equation 
The nonlinear HOLO-plus-consistency system given by \cref{holocons} can be solved using any nonlinear solution technique \cite{kelley}  Note that the NFPA scheme returns a FP equation that is consistent with HO transport  
Moreover, this modified FP equation accounts for large-angle scattering which the standard FP equation does not  
The LO equation (\ref{fp1}) can then be integrated into multiphysics models in a similar fashion to standard HOLO schemes \cite{patelFBR}  To solve the HOLO-plus-consistency system above, we use Picard iteration \cite{kelley}:
\begin{subequations}
\begin{align}
\label{H1}
\text{Transport Sweep for HO}&:
\mathcal{L} \psi_{HO}^{k+1} = \mathcal{S} \psi_{LO}^{k} + Q, \\
\label{L1}
\text{Evaluate Consistency Term}&: \hat{D}_F^{k+1} = \left(\mathcal{S} - \mathcal{F} - \sigma_{s,0}\mathcal{I}\right) \psi_{HO}^{k+1}, \\
\label{c1}
\text{Solve LO Equation}&: \psi_{LO}^{k+1} = \mathcal{P}^{-1} \left(\hat{D}_F^{k+1} + Q\right), 
\end{align}
\end{subequations}
where $\mathcal{L}$ and $\mathcal{S}$ are given in \cref{trans1}, $\mathcal{P}$ and $\mathcal{F}$ are given in \cref{FPSAsi1}, $\mathcal{I}$ is the identity operator, and $k$ is the iteration index 
Iteration is done until a convergence criterion is met 

The main advantage of setting up the LO equation in this fashion is that the stiffness matrix for LO needs to be setup and inverted \textit{only once}, just as with FPSA \cite{JapanFPSA, japanDiss}  This has a large impact on the method's performance 
A flowchart of this algorithm is shown in \cref{Nalgorithm} 

\begin{figure}[H]
\centering
\begin{tikzpicture}[node distance = 3cm, auto]
   
    \node [block] (init) {Initial guess of flux moments};
    \node [cloud_HO, right of=init, node distance=4cm] (HOm) {HO};
    \node [cloud_LO, below of=HOm, node distance=2cm] (LOm) {LO};
    \node [HO, below of=init] (transport) {One sweep in transport equation};
    \node [decision, below of=transport,node distance=4cm] (decide) {Flux moments converged?};
    \node [LO, left of=decide, node distance=4cm] (dterm) {Solve for consistency term};
    \node [LO, left of=dterm, node distance=3cm] (MFP) {Solve for FP angular flux};
    \node [LO, above of=MFP, node distance=4cm] (moments) {Convert angular flux to moments};
    \node [block, right of=decide, node distance=4cm] (stop) {Stop};
   
    \path [line] (init) -- (transport);
    \path [line] (transport) -- (decide);
    \path [line] (decide) -- node {no} (dterm);
    \path [line] (dterm) -- (MFP);
    \path [line] (MFP) -- (moments);
    \path [line] (moments) -- (transport);
    \path [line] (decide) -- node {yes}(stop);
\end{tikzpicture}
\caption{NFPA algorithm}
\label{Nalgorithm}
\end{figure}

\section{Numerical Experiments}\label{sec3}

In \cref{sec31} we describe the discretization methods used to implement the algorithms 
In \cref{sec32} we provide numerical results for 2 different choices of source $Q$ and boundary conditions 
For each choice we solve the problem using 3 different scattering kernels, applying 3 different choices of parameters for each kernel 
We provide NFPA numerical results for these 18 cases and compare them against those obtained from FPSA and other standard methods 

All numerical experiments were performed using MATLAB 
Runtime was tracked using the tic-toc functionality \cite{matlab17}, with
only the solver runtime being taken into consideration in the comparisons 
A 2017 MacBook Pro with a 2 8 GHz Quad-Core Intel Core i7 and 16 GB of RAM was used for all simulations 


\subsection{Discretization}\label{sec31}

The Transport and FP equations were discretized using linear discontinuous finite element discretization in space \cite{mpd1}, and discrete ordinates (S$_N$) in angle \cite{landm} 
The Fokker-Planck operator $\mathcal{F}$ was discretized using moment preserving discretization (MPD) \cite{mpd1} 
Details of the derivation of the linear discontinuous finite element discretization can be seen in \cite{japanDiss,martin} 
The finite element discretization for the  Fokker-Planck equation follows the same derivation 

A brief review for the angular discretization used for the FP equation is given below 
First, we use Gauss-Legendre quadrature to discretize the FP equation in angle:
\begin{equation}
\mu_n\frac{\partial \psi_n(x)}{\partial x} + \sigma_a \psi_n(x) - \frac{\sigma_{tr}}{2}\nabla^2_n \psi_n(x) =  Q_n(x),
\end{equation}
for $n=1,  ,N$ 
Here, $\nabla^2_n$ term is the discrete form of the angular Laplacian operator evaluated at angle $n$ 

The MPD scheme is then shown as
\begin{equation}
\nabla^2_n \psi_n = M \psi_n = V^{-1} L V \psi_n,
\end{equation}
where $M$ is the MPD discretized operator defined by
\begin{subequations}
\begin{equation}
V_{i,j} = P_{i-1}(\mu_j)w_j,
\end{equation}
and 
\begin{equation}
L_{i,j} = -i(i-1),
\end{equation}
\end{subequations}
for $i,j=1,   ,N$ 
Here, $P_l(\mu_j)$ are the Legendre polynomials evaluated at each angle $\mu_j$ and $w_j$ are the respective weights 
$M$ is defined as a (N x N) operator for a vector of $N$ angular fluxes $ \psi(x)$, at spatial location $x$  

In summary, if we write the FP equation as
\begin{equation}
\mathcal{H} \frac{\partial \psi}{\partial x}(x) + \sigma_a \psi(x) - \mathcal{F} \psi(x) =  Q(x),
\end{equation}
then $\mathcal{H}$ is Diag$(\mu_n)$ for $n=1,   ,N$, $Q(x)$ is a vector of source terms $Q_n(x)$, and $\mathcal{F}$ is represented by $\frac{\sigma_{tr}}{2}M$ 


\subsection{Numerical Results}\label{sec32}

It is shown that for slowly converging problems, typical convergence methods like $L_\infty$ suffer from false convergence \cite{adamslarsen} 
To work around this issue, the criterion is modified to use information about the current and previous iteration:
\begin{equation}
\label{falseconverge}
\frac{|| \phi^{m}_0(x) - \phi^{m-1}_0(x) ||_2}{1-\frac{|| \phi^{m+1}_0(x) - \phi^{m}_0(x) ||_2}{|| \phi^{m}_0(x) - \phi^{m-1}_0(x) ||_2}} < 10^{-8} 
\end{equation}

Two problems were tested using 200 spatial cells, $X$ = 400, $\sigma_a = 0$, $L$ = 15, and $N$ = 16 
Problem 1 has vacuum boundaries and a homogeneous isotropic source $Q$ for $0 < x < X$ 
Problem 2 has no internal source and an incoming beam at the left boundary  The source and boundary conditions used are shown in \cref{parameters}  
\begin{table}[H]
\begin{center}
\scalebox{0 9}{
\begin{tabular}{c | c | c} \hline 
& Problem 1 & Problem 2 \\ \hline \hline
Q(x) & 0 5 & 0 \\
$\psi_L$ & 0 & $\delta(\mu - \mu_N)$  \\
$\psi_R$ & 0 & 0 \\
\end{tabular}}
\end{center}
\caption{Problem Parameters}
\label{parameters} 
\end{table} 
We consider three scattering kernels in this paper: Screened Rutherford \cite{pomraning1}, Exponential \cite{pomraning2}, and Henyey-Greenstein \cite{HGK} 
Three cases for each kernel were tested 
The results obtained with NFPA are compared with those obtained using GMRES, DSA, and FPSA with the MPD scheme 

\subsubsection{SRK: Screened Rutherford Kernel}

The Screened Rutherford Kernel \cite{pomraning1, JapanFPSA} is a widely used scattering kernel for modeling scattering behavior of electrons \cite{SRK} 
The kernel depends on the parameter $\eta$, such that
\begin{equation}
\sigma^{SRK}_{s,l} = \sigma_s  \int_{-1}^{1} d\mu P_l(\mu) \frac{\eta (\eta+1)}{(1+2\eta-\mu)^2} 
\end{equation}
The SRK has a valid FP limit as $\eta$ approaches 0 \cite{patelFBR}  Three different values of $\eta$ were used to generate the scattering kernels shown in \cref{SRK} 
GMRES, DSA, FPSA, and NFPA all converged to the same solution for problems 1 and 2  \Cref{SRK_plots} shows the solutions for SRK with $\eta = 10^{-7}$ 
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0 1,angle=0]{SRK jpg}
  \caption{Screened Rutherford Kernels}
  \label{SRK}
\end{center}
\end{figure}
\begin{figure}[H]
    \centering
    \subfloat[Problem 1]{{\includegraphics[width=7cm]{s7_iso jpg} }}
    \qquad
    \subfloat[Problem 2]{{\includegraphics[width=7cm]{s7_beam jpg} }}
    \caption{Results for SRK Problems with $\eta = 10^{-7}$}
    \label{SRK_plots}
\end{figure}

\begin{table}[H]
\begin{center}
\scalebox{0 8}{
\begin{tabular}{c || c || c || c} \hline 
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$\eta = 10^{-5}$} & GMRES & 98 8 & 12 \\
& DSA & 2380 & 53585 \\
& FPSA & 1 21 & 26 \\
& NFPA & 1 39 & 26 \\ \hline 
\multirow{4}{*}{$\eta = 10^{-6}$} & GMRES & 208 & 84 \\
& DSA & 3040 & 69156 \\
& FPSA & 0 747 & 16 \\
& NFPA & 0 857 & 16 \\ \hline 
\multirow{4}{*}{$\eta = 10^{-7}$} & GMRES & 174 & 124 \\
& DSA & 3270 & 73940 \\
& FPSA & 0 475 & 10 \\
& NFPA & 0 542 & 10 \\ \hline
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 1 with SRK}
\label{SRKresults1} 
\end{table}
\begin{table}[H]
\begin{center}
\scalebox{0 8}{
\begin{tabular}{c || c || c || c} \hline 
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$\eta = 10^{-5}$} & GMRES & 52 4 & 187 \\
& DSA & 1107 & 25072 \\
& FPSA & 0 953 & 20 \\
& NFPA & 1 14 & 20 \\ \hline 
\multirow{4}{*}{$\eta = 10^{-6}$} & GMRES & 108 & 71 \\
& DSA & 1434 & 32562  \\
& FPSA & 0 730 & 14 \\
& NFPA & 0 857 & 14 \\ \hline 
\multirow{4}{*}{$\eta = 10^{-7}$} & GMRES & 94 1 & 185 \\
& DSA & 1470 & 33246 \\
& FPSA & 0 438 & 8 \\
& NFPA & 0 484 & 8 \\ \hline  
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 2 with SRK}
\label{SRKresults2} 
\end{table}

The results of all solvers are shown in \cref{SRKresults1,SRKresults2} 
We see that NFPA and FPSA tremendously outperform GMRES and DSA in runtime for all cases 
FPSA is a simpler method than NFPA, requiring less calculations per iteration; therefore, it is expected that it outperforms NFPA in runtime 
We see a reduction in runtime and iterations for FPSA and NFPA as the FP limit is approached, with DSA and GMRES requiring many more iterations by comparison as $\eta$ approaches 0 

An advantage that NFPA offers is that the angular moments of the flux in the LO equation will remain consistent with those of the transport equation even as a problem becomes less forward-peaked 
On the other hand, the moments found using only the FP equation and source iteration lose accuracy 
To illustrate this, Problem 1 was tested using different Screened Rutherford Kernels with increasing $\eta$ parameters 
The percent errors (relative to the transport solution) for the scalar flux obtained with the LO equation and with the standard FP equation at the center of the slab are shown in \cref{momcomp} 
It can be seen that the percent relative errors in the scalar flux of the FP solution is orders of magnitude larger than the error produced using the LO equation 
The same trend can be seen when using the exponential and Henyey-Greenstein kernels  

\begin{figure}[H]
\begin{center}
  \includegraphics[scale=0 15,angle=0]{relerrorlog jpg}
  \caption{Log Scale of $\%$ Relative Error vs $\eta$ for Problem 1 at the Center of the Slab with SRK}
  \label{momcomp}
\end{center}
\end{figure}

\subsubsection{EK: Exponential Kernel}

The exponential kernel \cite{pomraning2, JapanFPSA} is a fictitious kernel made for problems that have a valid Fokker-Planck limit \cite{pomraning1} 
The zero$^{\text{th}}$ moment, $\sigma^{EK}_{s,0}$, is chosen arbitrarily; we define $\sigma^{EK}_{s,0}$ as the same zero$^{\text{th}}$ moment from the SRK 
The $\Delta$ parameter determines the kernel: the first and second moments are given by 
\begin{subequations}
\begin{align}
\sigma^{EK}_{s,1} &= \sigma^{EK}_{s,0} (1-\Delta),\\
\sigma^{EK}_{s,2} &= \sigma^{EK}_{s,0} (1-3\Delta+3\Delta^2),
\end{align}
and the relationship for $l\geq 3$ is
\begin{equation}
\sigma^{EK}_{s,l} = \sigma^{EK}_{s,l-2} - \Delta(2l+1) \sigma^{EK}_{s,l-1} 
\end{equation}
\end{subequations}
As $\Delta$ is reduced, the scattering kernel becomes more forward-peaked 

The EK has a valid FP limit as $\Delta$ approaches 0 \cite{patelFBR} 
Three different values of $\Delta$ were used to generate the scattering kernels shown in \cref{EXP} 
The generated scattering kernels are shown in \cref{EXP} 
GMRES, DSA, FPSA, and NFPA all converged to the same solution for problems 1 and 2 
\Cref{EK_plots} shows the solutions for EK with $\Delta = 10^{-7}$ 
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0 1,angle=0]{EXP jpg}
  \caption{Exponential Kernels}
  \label{EXP}
\end{center}
\end{figure}
\begin{figure}[H]
    \centering
    \subfloat[Problem 1]{{\includegraphics[width=7cm]{dta7_iso jpg} }}
    \qquad
    \subfloat[Problem 2]{{\includegraphics[width=7cm]{dta7_beam jpg} }}
    \caption{Results for EK Problems with $\Delta = 10^{-7}$}
    \label{EK_plots}
\end{figure}

The runtimes and iterations for GMRES, DSA, FPSA, and NFPA are shown in \cref{Expresults1,Expresults2} 
We see a similar trend with the EK as seen with SRK 
Smaller $\Delta$ values lead to a reduction in runtime and iterations for NFPA and FPSA, which greatly outperform DSA and GMRES in both categories 

\begin{table}[h]
\begin{center}
\scalebox{0 8}{
\begin{tabular}{c || c || c || c} \hline 
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$\Delta = 10^{-5}$} & GMRES & 196 & 142 \\
& DSA & 3110 & 70140 \\
& FPSA & 0 514 & 11 \\ 
& NFPA & 0 630 & 11 \\\hline 
\multirow{4}{*}{$\Delta = 10^{-6}$} & GMRES & 156 & 132 \\
& DSA & 3120 & 70758 \\
& FPSA & 0 388 & 7 \\ 
& NFPA & 0 393 & 7 \\ \hline 
\multirow{4}{*}{$\Delta = 10^{-7}$} & GMRES & 81 & 127 \\
& DSA & 3120 & 70851  \\
& FPSA & 0 292 & 6 \\ 
& NFPA & 0 318 & 6 \\ \hline
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 1 with EK}
\label{Expresults1} 
\end{table}
\begin{table}[h]
\begin{center}
\scalebox{0 8}{
\begin{tabular}{c || c || c || c} \hline 
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$\Delta = 10^{-5}$} & GMRES & 110 & 73 \\
& DSA & 1455 & 33033 \\
& FPSA & 0 492 & 10 \\ 
& NFPA & 0 613 & 10 \\ \hline 
\multirow{4}{*}{$\Delta = 10^{-6}$} & GMRES & 82 7 & 79 \\
& DSA & 1470 & 33309 \\
& FPSA & 0 358 & 7 \\ 
& NFPA & 0 431 & 7 \\ \hline 
\multirow{4}{*}{$\Delta = 10^{-7}$} & GMRES & 56 8 & 90 \\
& DSA & 1470 & 33339 \\
& FPSA & 0 273 & 5 \\ 
& NFPA & 0 319 & 5 \\ \hline  
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 2 with EK}
\label{Expresults2} 
\end{table}

\subsubsection{HGK: Henyey-Greenstein Kernel}

The Henyey-Greenstein Kernel \cite{HGK,JapanFPSA} is most commonly used in light transport in clouds 
It relies on the anisotropy factor $g$, such that
\begin{equation}
\sigma^{HGK}_{s,l} = \sigma_s g^l 
\end{equation}
As $g$ goes from zero to unity, the scattering shifts from isotropic to highly anisotropic 
\begin{figure}[H]
\begin{center}
  \includegraphics[scale=0 1,angle=0]{HGK jpg}
  \caption{Henyey-Greenstein Kernels}
  \label{HGK}
\end{center}
\end{figure}
\begin{figure}[H]
    \centering
    \subfloat[Problem 1]{{\includegraphics[width=7cm]{g099_iso jpg} }}
    \qquad
    \subfloat[Problem 2]{{\includegraphics[width=7cm]{g099_beam jpg} }}
    \caption{Results for HGK Problems with $g = 0 99$}
    \label{HGK_plots}
\end{figure}


The HGK does not have a valid FP limit \cite{patelFBR} 
The three kernels tested are shown in \cref{HGK} 
GMRES, DSA, FPSA, and NFPA all converged to the same solution for problems 1 and 2 
\Cref{HGK_plots} shows the solutions for HGK with $g = 0 99$ 
The results of each solver are shown in \cref{HGKresults1,HGKresults2}  
\begin{table}[h]
\begin{center}
\scalebox{0 8}{
\begin{tabular}{c || c || c || c} \hline 
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$g=0 9$} & GMRES & 9 88 & 76 \\
& DSA & 24 5 & 554 \\
& FPSA & 1 50 & 32 \\ 
& NFPA & 1 39 & 27 \\ \hline 
\multirow{4}{*}{$g=0 95$} & GMRES & 12 2 & 131 \\
& DSA & 47 7 & 1083 \\
& FPSA & 1 75 & 38 \\ 
& NFPA & 1 83 & 35 \\ \hline 
\multirow{4}{*}{$g=0 99$} & GMRES & 40 0 & 27 \\
& DSA & 243 & 5530  \\
& FPSA & 3 38 & 74 \\ 
& NFPA & 3 93 & 73 \\ \hline
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 1 with HGK}
\label{HGKresults1} 
\end{table}
\begin{table}[h]
\begin{center}
\scalebox{0 8}{
\begin{tabular}{c || c || c || c} \hline 
Parameter & Solver & Runtime (s) & Iterations \\ \hline \hline
\multirow{4}{*}{$g=0 9$} & GMRES & 24 3 & 135 \\
& DSA & 14 8 & 336  \\
& FPSA & 1 15 & 23 \\ 
& NFPA & 1 35 & 24 \\ \hline 
\multirow{4}{*}{$g=0 95$} & GMRES & 31 3 & 107 \\
& DSA & 29 7 & 675 \\
& FPSA & 1 56 & 32 \\ 
& NFPA & 1 90 & 33 \\ \hline 
\multirow{4}{*}{$g=0 99$} & GMRES & 41 4 & 126 \\
& DSA & 146 & 3345 \\
& FPSA & 3 31 & 67 \\ 
& NFPA & 3 99 & 67 \\ \hline  
\end{tabular}}
\end{center}
\caption{Runtime and Iteration Counts for Problem 2 with HGK}
\label{HGKresults2} 
\end{table}

Here we see that NFPA and FPSA do not perform as well compared to their results for the SRK and EK 
Contrary to what happened in those cases, both solvers require more time and iterations as the problem becomes more anisotropic 
This is somewhat expected, due to HGK not having a valid Fokker-Planck limit 
However, both NFPA and FPSA continue to greatly outperform GMRES and DSA 
Moreover, NFPA outperforms FPSA in iteration count for problem 1 


\section{Discussion}\label{sec4}

This paper introduced the Nonlinear Fokker-Planck Acceleration technique for steady-state, monoenergetic transport in homogeneous slab geometry 
To our knowledge, this is the first nonlinear HOLO method that accelerates \textit{all $L$ moments} of the angular flux 
Upon convergence, the LO and HO models are consistent; in other words, the (lower-order) modified Fokker-Planck equation \textit{preserves the same angular moments} of the flux obtained with the (higher-order) transport equation 

NFPA was tested on a homogeneous medium with an isotropic internal source with vacuum boundaries, and in a homogeneous medium with no internal source and an incoming beam boundary 
For both problems, three different scattering kernels were used 
The runtime and iterations of NFPA and FPSA were shown to be similar 
They both vastly outperformed DSA and GMRES for all cases by orders of magnitude 
However, NFPA has the feature of preserving the angular moments of the flux in both the HO and LO equations, which offers the advantage of integrating the LO model into multiphysics models  

In the future, we intend to test NFPA capabilities for a variety of multiphysics problems and analyze its performance 
To apply NFPA to more realistic problems, it needs to be extended to include time and energy dependence  
Additionally, the method needs to be adapted to address geometries with higher-order spatial dimensions 
Finally, for the NFPA method to become mathematically ``complete", a full convergence examination using Fourier analysis must be performed 
However, this is beyond the scope of this paper and must be left for future work 

\section*{Acknowledgements}

The authors acknowledge support under award number NRC-HQ-84-15-G-0024 from the Nuclear Regulatory Commission 
The statements, findings, conclusions, and recommendations are those of the authors and do not necessarily reflect the view of the U S  Nuclear Regulatory Commission 

J ~K  Patel would like to thank Dr ~James Warsa for his wonderful transport class at UNM, as well as his synthetic acceleration codes 
The authors would also like to thank Dr ~Anil Prinja for discussions involving Fokker-Planck acceleration 







ground_truth: NFPA and FPSA greatly outperform GMRES and DSA.