Input: In which electorate was Simon English elected to the New Zealand Parliament?

Context:
Sir Simon William English  (born 30 December 1961) is a New Zealand former National Party politician who served as the 39th prime minister of New Zealand from 2016 to 2017. He had previously served as the 17th deputy prime minister of New Zealand and minister of finance from 2008 to 2016 under John Key and the Fifth National Government.

A farmer and public servant before entering politics, English was elected to the New Zealand Parliament in  as the National Party's candidate in the Wallace electorate. He was elevated to Cabinet in 1996 and in 1999 was made minister of finance, although he served for less than a year due to his party's loss at the 1999 general election. In October 2001, English replaced Jenny Shipley as the leader of the National Party (and consequently as Leader of the Opposition). He led the party to its worst defeat at the 2002 general election, and as a consequence, in October 2003 he was replaced as leader by Don Brash.

In November 2006, after Brash's resignation, English became deputy leader under John Key. After National's victory at the 2008 general election, he became deputy prime minister and was also made minister of finance for the second time. Under English's direction New Zealand's economy maintained steady growth during National's three terms of government. He became a list-only MP after stepping down as an electorate MP at the 2014 general election.

John Key resigned as leader of the National Party and prime minister in December 2016. English won the resulting leadership election unopposed and was sworn in as prime minister on 12 December 2016. His tenure was only ten months, and included a three-month election campaign. In the 2017 general election, National won the largest number of seats but fell short of a majority. The parties holding the balance of power declined to support the existing government, and English was subsequently replaced as prime minister by Jacinda Ardern, leader of the Labour Party. English initially continued on as Leader of the Opposition, but resigned as leader of the National Party on 27 February 2018 and left parliament two weeks later.

Early life
English was born on 30 December 1961 at Lumsden Maternity Centre in Lumsden. He is the eleventh of twelve children of Mervyn English and Norah (née O'Brien) English. His parents purchased Rosedale, a mixed sheep and cropping farm in Dipton, Southland from Mervyn's uncle, Vincent English, a bachelor, in 1944. English was born in the maternity unit at Lumsden.

English attended St Thomas's School in Winton, then boarded at St. Patrick's College in Upper Hutt, where he became head boy. He played in the first XV of the school's rugby team. English went on to study commerce at the University of Otago, where he was a resident at Selwyn College, and then completed an honours degree in English literature at Victoria University of Wellington.

After finishing his studies, English returned to Dipton and farmed for a few years. From 1987 to 1989, he worked in Wellington as a policy analyst for the New Zealand Treasury, at a time when the free market policies favoured by Labour's finance minister Roger Douglas (known collectively as "Rogernomics") were being implemented.

English joined the National Party in 1980, while at Victoria University. He served for a period as chairman of the Southland branch of the Young Nationals, and became a member of the Wallace electorate committee. After moving to Wellington, he served for periods on the Island Bay and Miramar electorate committees, respectively.

Fourth National Government (1990–1999)

At the 1990 general election, English stood as the National candidate in Wallace, replacing the retiring Derek Angus, and was elected with a large majority. He would hold this seat, renamed Clutha-Southland in 1996, until 2014. He and three other newly elected National MPs (Tony Ryall, Nick Smith, and Roger Sowry) were soon identified as rising stars in New Zealand politics, and at various points were dubbed the "brat pack", the "gang of four", and the "young Turks". In his first term in parliament, English chaired a select committee into social services. He was made a parliamentary under-secretary in 1993, serving under the Minister of Health.

First period in cabinet (1996–1999)
In early 1996, English was elevated to cabinet by Prime Minister Jim Bolger, becoming the Minister for Crown Health Enterprises and Associate Minister of Education (to Wyatt Creech). He was 34 at the time, becoming the cabinet's youngest member. After the 1996 general election, the National Party was forced into a coalition with New Zealand First to retain government. In the resulting cabinet reshuffle, English emerged as Minister of Health. However, as a condition of the coalition agreement, NZ First's Neil Kirton (a first-term MP) was made Associate Minister of Health, effectively becoming English's deputy. This arrangement was described in the press as a "shotgun marriage", and there were frequent differences of opinion between the two ministers. After their relationship became unworkable, Kirton was sacked from the role in August 1997, with the agreement of NZ First leader Winston Peters.

As Minister of Health, English was responsible for continuing the reforms to the public health system that National had begun after the 1990 general election. The reforms were unpopular, and health was perceived as one of the government's weaknesses, with the health portfolio consequently being viewed as a challenge. English believed that the unpopularity of the reforms was in part due to a failure in messaging, and encouraged his National colleagues to avoid bureaucratic and money-focused language (such as references to "balance sheets" and "user charges") and instead talk about the improvements to services the government's reforms would bring. He also rejected the idea that public hospitals could be run as commercial enterprises, a view which some of his colleagues had previously promoted.

By early 1997, as dissatisfaction with Bolger's leadership began to grow, English was being touted as a potential successor, along with Jenny Shipley and Doug Graham. His age (35) was viewed as the main impediment to a successful leadership run. National's leadership troubles were resolved in December 1997, when Bolger resigned and Shipley was elected to the leadership unopposed. English had been a supporter of Bolger as leader, but Shipley reappointed him Minister of Health in her new cabinet.

English was promoted to Minister of Finance in a reshuffle in January 1999, a position which was at the time subordinate to the Treasurer, Bill Birch. After a few months, the pair switched positions as part of Birch's transition to retirement, with English assuming the senior portfolio. In early interviews, he emphasised his wish to be seen as a pragmatist rather than an ideologue, and said that the initiatives of some of his predecessors (Roger Douglas's "Rogernomics" and Ruth Richardson's "Ruthanasia") had focused on "fruitless, theoretical debates" when "people just want to see problems solved".

Opposition (1999–2008)

After the National Party lost the 1999 election to Helen Clark's Labour Party, English continued on in the shadow cabinet as National's spokesperson for finance. He was elected deputy leader of the party in February 2001, following the resignation of Wyatt Creech, with Gerry Brownlee being his unsuccessful opponent.

Leader of the Opposition
In October 2001, after months of speculation, Jenny Shipley resigned as leader of the National Party after being told she no longer had the support of the party caucus. English was elected as her replacement unopposed (with Roger Sowry as his deputy), and consequently became Leader of the Opposition. However, he did not openly organise against Shipley, and according to The Southland Times "there was almost an element of 'aw, shucks, I'll do it then' about Mr English's ascension".

Aged 39 when he was elected, English became the second-youngest leader in the National Party's history, after Jim McLay (who was 38 when elected in 1984). He also became only the third Southlander to lead a major New Zealand political party, after Joseph Ward and Adam Hamilton. However, English failed to improve the party's performance. In the 2002 election, National suffered its worst electoral defeat ever, gaining barely more than twenty percent of the vote. English described it as "the worst day of my political life". Both party insiders and the general public were split as to how much to blame him for the loss, but most of the party believed that English would be able to rebuild National's support.

By late 2003, however, National's performance in opinion polls remained poor. The party had briefly increased its popularity in the year following the election, but by October its support had fallen to levels only slightly better than what it achieved in the last ballot. English also appeared in a boxing match for a charity against entertainer Ted Clarke. This did not boost his polling or that of the National party either, with suggestions that it devalued his image as a serious politician. Don Brash, former governor of the Reserve Bank and a relative newcomer to politics, began to build up support to replace English. On 28 October, Brash gained sufficient backing in Caucus to defeat English in a leadership contest.

Shadow cabinet roles and deputy leader
On 2 November 2003, when Brash changed responsibilities for certain MPs, English became National's spokesman for education, ranked at fifth place in the party's parliamentary hierarchy. He remained in parliament after the 2005 election. In his new shadow education portfolio, English performed strongly, and remained a party favourite despite his election defeat as leader in 2002, eventually being returned to the finance portfolio in August 2004 as deputy spokesman (while still retaining responsibility for education).

In November 2006, Brash resigned as leader. English was considered as a potential replacement leader (running against John Key) or deputy leader (against incumbent Gerry Brownlee) in the ensuing leadership election. However, a contest was avoided when the MPs agreed a Key/English ticket would run unopposed in a display of party unity. English took over the deputy leadership and the finance portfolio in the Key shadow cabinet.

Fifth National Government (2008–2017)

Deputy Prime Minister and Minister of Finance (2008–2016)

At the 2008 election, English was re-elected by his electorate, winning by a margin of about 15,500 votes. He became Deputy Prime Minister of New Zealand and Minister of Finance in the fifth National Government, being sworn into office on 19 November 2008 and continued to serve in those roles until becoming Prime Minister on 12 December 2014. He was also made Minister of Infrastructure in National's first term of government and Minister responsible for Housing New Zealand Corporation and minister responsible for the New Zealand flag consideration process in its third.

He was comfortably re-elected in Clutha-Southland in the 2011 election but opted to run as a party-list candidate in 2014.

The pairing of John Key as leader of the National Party and English as his deputy has been compared to that of Bob Hawke and Paul Keating (in Australia) and Tony Blair and Gordon Brown (in the UK).

English acceded to the role of Finance Minister in the continuing wake of the financial crisis. In response to New Zealand's rising debt, English made budget deficit-reduction his main priority. His first budget outlined three focuses in New Zealand's financial recovery: "improving the business environment and removing roadblocks to growth; investment in productive infrastructure; and improving the way government works". One of his first acts was creating the National Infrastructure Unit, charged with formulating a plan for infrastructure projects and investments. He commissioned a government-wide spending review, with an aim to reducing government expenditure—with the exceptions of a two-year stimulus package and long-term increases on infrastructure spending.

In April 2011, the Opposition criticised English for suggesting that New Zealand businesses could use New Zealand's low wages to help it compete with Australia. The National Government campaigned for re-election in 2011 on its economic record. The Government boasted growth for five consecutive quarters up to mid-2010, totalling 1.6% of real GDP.

Strong growth resulted in a surplus of $473 million for the 2015/16 financial year, projected to rise to $8.5 billion by 2020/21. In his 2016 Economic and Fiscal Update address, English stated that reducing debt and tackling the costs of the 2016 Kaikōura earthquake were higher priorities than reducing rates of tax.

Allowances issue
In 2009, the media, including TVNZ and TV3 revealed that English was receiving about NZ$900 a week as part of a living allowance for ministers, to live in his own NZ$1.2 million Wellington home. At the time, English also received $276,200 in his annual salary as Deputy Prime Minister. It was also revealed other ministers with homes in the capital city were also claiming accommodation allowances. On 3 August 2009, Prime Minister John Key started a review of the housing allowances claimed by cabinet ministers. English subsequently paid back $12,000 and only claimed about $24,000 a year in living allowances. The Auditor-General's office said in September 2009 that they were making "preliminary enquiries" into parliamentary housing expenses in response to a letter of complaint from Progressive party leader Jim Anderton. Two days later English stated that he would no longer take up any housing allowance and had paid back all the allowance he had received since the November 2008 election.

Prime Minister (2016–2017)

John Key resigned on 12 December, and endorsed English as his successor in the resulting leadership election. Following the drop-out of both Judith Collins and Jonathan Coleman from the leadership election, English was sworn in as the 39th Prime Minister of New Zealand on 12 December 2016.

English appointed his first cabinet on 18 December. In a reshuffle, he appointed Steven Joyce to succeed him as Finance Minister, while most ministerial portfolios remained the same.

In February 2017, English did not attend Waitangi Day commemorations at the historic treaty grounds, reportedly in response to the Ngāpuhi iwi's decision to stop the Prime Minister from speaking at the marae. Ngāpuhi have protested the Government's negotiation of the Trans Pacific Partnership Agreement (TPPA), which the iwi believe infringes upon Māori sovereignty, and thus does not adhere to the Treaty of Waitangi. English had been invited to attend in an official capacity; his non-attendance was criticised by a Ngāpuhi elder and Opposition leader Andrew Little.

In his first overseas trip as Prime Minister, English travelled to Europe to discuss trade ties, including a prospective New Zealand–European Union free trade agreement. He first travelled to London on 13 January 2017 to meet British Prime Minister Theresa May. Discussing trade relations, English said the two nations were "natural partners" and would "continue to forge ties" after the UK's withdrawal from the EU. He also arranged to meet with London Mayor Sadiq Khan, Belgian Prime Minister Charles Michel and German Chancellor Angela Merkel.  In a meeting with Merkel, English received crucial backing from Germany for a trade deal with the EU. On 16 January, English stated that his government would continue to promote TPPA, despite the United States' decision to withdraw from the agreement. He explained that Southeast Asian countries would now be treated as a priority in negotiations—he also asserted that the United States was ceding influence to China by its rejection of the trade pact.

At a press conference at the Beehive on 1 February 2017, English announced that the 2017 general election would be held on 23 September. The Prime Minister later confirmed that his party would approach ACT, United Future and the Māori Party if confidence and supply agreements were required to form a government following the election. In his second cabinet reshuffle on 24 April, English appointed Gerry Brownlee as his new Foreign Affairs Minister; he also promoted Nikki Kaye to the portfolio of Education Minister, and moved Mark Mitchell into the cabinet to become Defence Minister. The reshuffle was perceived as an election preparation.

On 13 February 2017, English welcomed Australian Prime Minister Malcolm Turnbull to Wellington. The two leaders reaffirmed their shared trade agenda, and discussed changes to the Australian citizenship pathway which will affect permanent residents originating from New Zealand.

On 19 June, it was reported that Todd Barclay, who succeeded English as MP for Clutha-Southland, had clandestinely recorded one of his employee's conversations the previous year, and that John Key's leaders' budget was used to pay a confidential settlement after the employee resigned. English admitted that he had been aware of the illegal recording and the settlement, and thus implicated in the scandal.

During the 2017 National campaign launch, English introduced a $379 million social investment package including digital learning academies for high school students, more resources for mathematics, and boosting support for teaching second languages in schools, and maintaining National Standards in the school curriculum. Prime Minister English also sought to defend National's financial management and economic track record and claimed that the opposition Labour Party would raise taxes. Early opinion polling had forecast a poor showing in the election for the Labour Party, but in early August 37-year-old Jacinda Ardern took over as Labour leader and seemingly energised younger voters.

At the 2017 general election, National won the largest share of the party vote (44.4%) and the largest number of seats (56) in the House Representatives. However, National lacked enough seats to govern alone due to two of the party's support partners, the Māori Party and United Future, losing their parliamentary seats. In response, English stated that the party would be entering into talks to form a coalition with New Zealand First. Following talks with the two largest parties, New Zealand First entered a coalition arrangement with the Labour Party. English was succeeded as prime minister by Jacinda Ardern on 26 October.

Opposition (2017–2018)

Leader of the Opposition
English was re-elected as National Party leader on 24 October 2017. At the time of his re-election, English announced his intention to stay on as leader until the next general election. On 13 February 2018, however, he stood down as National Party leader due to personal reasons, and instructed the party to put into motion the processes to elect a new leader. He also retired from Parliament. English's resignation followed weeks of speculation that he would step aside for a new leader. On 27 February, he was succeeded as party leader by Simon Bridges as the result of the leadership election held that day.

Post-premiership
In 2018, English joined the board of Australian conglomerate, Wesfarmers. English serves in Chairmanships of Mount Cook Alpine Salmon, Impact Lab Ltd and Manawanui Support Ltd. He is also a director of The Instillery, Centre for Independent Studies and The Todd Corporation Limited, and is a member of the Impact Advisory Group of Macquarie Infrastructure and Real Assets.

Political and social views

English is regarded as more socially conservative than his predecessor, John Key. He has stated his opposition to voluntary euthanasia and physician-assisted suicide, same-sex civil unions, and the decriminalisation of prostitution. As Prime Minister he opposed any "liberalisation" of abortion law.

In 2004, English voted against a bill to establish civil unions for both same-sex and opposite-sex couples. In 2005, he voted for the Marriage (Gender Clarification) Amendment Bill, which would have amended the Marriage Act to define marriage as only between a man and a woman. English voted against the Marriage (Definition of Marriage) Amendment Bill, a bill that legalised same-sex marriage in New Zealand. However, in December 2016 he stated, "I'd probably vote differently now on the gay marriage issue. I don't think that gay marriage is a threat to anyone else's marriage".

In 2009, English voted against the Misuse of Drugs (Medicinal Cannabis) Amendment Bill, a bill aimed at amending the Misuse of Drugs Act so that cannabis could be used for medical purposes.

Personal life
English met his future wife, Mary Scanlon, at university. She was studying medicine at the time, and became a general practitioner. Both her parents were immigrants, her father being Samoan and her mother Italian, born on the island of Stromboli. They have six children: a daughter and five sons.

English is a practising Roman Catholic, but has stated that he considers his religious beliefs personal and thus separate from politics.

In June 2002, English took part in TV3's Fight For Life, a celebrity boxing fundraiser to raise money for the Yellow Ribbon anti-youth-suicide campaign, influenced by the death of a teenage nephew in 1997. He lost a split decision to former university colleague Ted Clarke.

Honours
In the 2018 Queen's Birthday Honours, English was appointed a Knight Companion of the New Zealand Order of Merit, for services of over 27 years to the State.

See also

List of New Zealand governments
Politics of New Zealand

References

External links

Profile at National Party
Profile on Parliament.nz
Releases and speeches at Beehive.govt.nz

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1961 births
21st-century New Zealand politicians
Candidates in the 2017 New Zealand general election
Deputy Prime Ministers of New Zealand
Leaders of the Opposition (New Zealand)
Living people
Members of the Cabinet of New Zealand
Members of the New Zealand House of Representatives
New Zealand farmers
New Zealand finance ministers
New Zealand list MPs
New Zealand MPs for South Island electorates
New Zealand National Party MPs
New Zealand National Party leaders
New Zealand Roman Catholics
New Zealand people of Irish descent
People educated at St. Patrick's College, Silverstream
People from Dipton, New Zealand
People from Lumsden, New Zealand
Prime Ministers of New Zealand
University of Otago alumni
Victoria University of Wellington alumni
Knights Companion of the New Zealand Order of Merit
New Zealand politicians awarded knighthoods

Answer: The Wallace electorate.


Input: When was the paper published?

Context:
Paper Info

Title: Interpretable reduced-order modeling with time-scale separation Interpretable reduced-order modeling with time-scale separation
Publish Date: 7 March 2023
Author List: Sebastian Kaltenbach (from CSE-Lab, ETH Zurich, Harvard SEAS), Phaedon-Stelios Koutsourelakis (from CSE-Lab, ETH Zurich, Harvard SEAS), Petros Koumoutsakos (from CSE-Lab, ETH Zurich, Harvard SEAS), Harvard Seas (from CSE-Lab, ETH Zurich, Harvard SEAS)

Figure

FIG. 5. Comparison between the phase-space of the reference solution (left) and the phase-space of the predictions
FIG. 7. Comparison between predictions and reference solutions for a new initial condition fort = 1.25, 3.75, 7.5, 12.5, 20, 30  (from left to right and top to down).We note that with longer prediction time the uncertainty bounds increases.Despite the chaotic nature of the KS equation, the predictive posterior mean is close to the reference solution for t ≤ 12.5

abstract

Partial Differential Equations (PDEs) with high dimensionality are commonly encountered in computational physics and engineering. However, finding solutions for these PDEs can be computationally expensive, making model-order reduction crucial. We propose such a data-driven scheme that automates the identification of the time-scales involved and, can produce stable predictions forward in time as well as under different initial conditions not included in the training data.
To this end, we combine a non-linear autoencoder architecture with a time-continuous model for the latent dynamics in the complex space. It readily allows for the inclusion of sparse and irregularly sampled training data. The learned, latent dynamics are interpretable and reveal the different temporal scales involved.
We show that this data-driven scheme can automatically learn the independent processes that decompose a system of linear ODEs along the eigenvectors of the system's matrix. Apart from this, we demonstrate the applicability of the proposed framework in a hidden Markov Model and the (discretized) Kuramoto-Shivashinsky (KS) equation.
Additionally, we propose a probabilistic version, which captures predictive uncertainties and further improves upon the results of the deterministic framework.

INTRODUCTION

High-fidelity simulations of critical phenomena such as ocean dynamics and epidemics have become essential for decision-making. They are based on physically-motivated PDEs expressing system dynamics that span multiple spatiotemporal scales and which necessitate cumbersome computations . In recent years there is increased attention to the development of data-driven models that can accelerate the solution of these PDEs as well as reveal salient, lower-dimensional features that control the long-term evolution.
In most cases, data-driven reduced-order models are not interpretable. In particular, models based on neural networks despite good predictive capabilities , they offer a black-box description of the system dynamics. A possible remedy is applying a symbolic regression to the learned neural network representation , but this adds additional computational cost due to the two-step procedure.
A number of frameworks such as SINDy allows to learn interpretable dynamics but it relies on the a-priori availability of lower-dimensional descriptors and of time-derivatives which can be very noisy for both simulation and experimental data. Other frameworks are tailored to specific problems such as molecular dynamics .
Here, we present a framework that only needs the value of the observables, and not their derivatives, as training data and is capable of identifying interpretable latent dynamics. The deployment of interpretable latent dynamics ensures that conservation of important properties of that are reflected in the reduced-order model .
The present method is related to approaches based on the Koopman-operator extended Dynamic Mode Decomposition (eDMD) but uses continuous complex-valued latent space dynamics and only requires one scalar variable per latent dimension to describe the latent space dynamics. Therefore we do not have to enforce any parametrizations on the Koopman matrix .
The time-continuous formulation moreover allows to incorporate sparse and irregularly sampled training data and fast generation of predictions after the training phase. By using a complex-valued latent space we can also incorporate harmonic effects and reduce the number of latent variables needed. Linear and non-linear autoencoders are used to map the observed, high-dimensional time-series to the lower-dimensional, latent representation and we identify simultaneously the autoencoder as well as the latent dynamics by optimizing a combined loss function.
Hence the to tasks of dimensionality reduction and discovery of the reduced dynamics are unified while other frameworks treat the two parts separately . Apart from using an architecture based on autoencoders to identify the latent space, projection-based methods could also be employed . We are also proposing a probabilistic version of our algorithm ) that makes use of probabilistic Slow Feature Analysis .
This allows for a latent representation that arart from being time-continuous, can quantify the predictive uncertainty and hierarchically decompose the dynamics into their pertinent scales while promoting the discovery of slow processes that control the system's evolution over long time horizons. The rest of the paper is structured as follows: We introduce the methodological framework as well as algorithmic details in section II.
Particular focus is paid on the interpretability of the inferred lower-dimensional dynamics. In section III we present three numerical illustrations, i.e. a system of linear ODEs, a hidden Markov Model and the discretized KS-equation. We then present in section IV the probabilistic extension of the framework and apply it to the KS-equation.
We conclude with a summary and a short discussion about possible next steps. We introduce the autoencoders deployed in this work, followed by the interpretable latent space dynamic and discuss the training process. We consider data from high-dimensional time series x n ∈ R f with n = 1, ..., T . We remark that the intervals between the different states do not need to be uniformly spaced.

Autoencoder

A core assumption of the method is that each high-dimensional state x n can be compressed to a lower-dimensional representation z n ∈ C c with c << f . We identify this lower-dimensional representation by an autoencoder consisiting of a parameterized encoder and decoder. The encoder maps the high-dimensional representation to the latent space as:
The latent space is complex-valued. The decoder reconstructs the high-dimensional representation based on the latent variables as: We denote the parameters of the encoder as well as the decoder by θ. As discussed later in Section II C, both set of parameters are optimized simultaneously during training and therefore there is no need for differentiating them.

Interpretable Latent Space Dynamics

We employ a propagator in the latent space to capture the reduced-order dynamics of the system. In contrast to other time-extended variational autoencoder frameworks, our representation uses complex valued latent variables. In addition the latent variables are treated independently. The latter feature enables us to have an interpretable latent dynamics as well as a model that is especially suitable for being trained in the Small Data regime due to the small number of required parameters.
This is in contrast to temporal propagators such as LSTMs . For each dimension i of the latent variable z we are using the following continuous ODE in the complex plane: By solving this ODE, we can define the operator: Interpretable reduced-order modeling with time-scale separation Here, λ is a vector containing all the individual λ's and ∆t n indicates the time-step between the latent states.
The symbol is used to indicate a component-wise multiplication. We remark that the latent variables and the parameter governing the temporal evolution are complex numbers and their role in describing the system dynamics is similar to that of an eigenvalue. The real part is associated with growth and decay whereas the imaginary part is representing the periodic component.
This approach has similarities with the Koopman-operator based methods and the extended dynamic mode decomposition . In contrast to the methods mentioned before we are using a continuous formulation in the latent space that allows us to incorporate scarce and irregularly sampled training data and directly rely on complex numbers in the latent space.

Training and Predictions

We optimize a loss function that combines both a reconstruction loss as well as a loss associated with the error of our learned propagator in the latent space: (5) We note that we could directly incorporate mini-batch training by only taking the summation over a subset of the N available training data.
For new predictions of unseen states, we use the encoder to generate a latent representations which is then advanced in time by the learned propagator. At a designated time step we are using the decoder to reconstruct the high-dimensional solution. We applied our algorithm to three systems. First, we show that the algorithm is capable of exactly reproducing the solution of a linear ODE and to identify its eigenvalues.
Afterwards we are applying the framework to a high-dimensional process generated by a complex latent dynamics, which is correctly identified. As a final test case, we are applying the algorithm to a Kuramoto Shivashinski (KS) equation. Interpretable reduced-order modeling with time-scale separation

Linear ODE

We are considering a two-dimensional ODE system for x = y 1 y 2 : Based on the obtained training data we run our algorithm using a linear encoder and decoder structure as well as two latent variables z. The loss function was optimized using the Adam algorithm . As we consider a linear ODE we can analytically compute the eigenvalues involved and compare it with the parameters λ identified by our algorithm.
We observe in Figure that the algorithm was able to recover the correct values, i.e. the eigenvalues 7 and 3 of the given linear ODE. The system does not have a periodic component and the two imaginary parts correctly go to zero, whereas the real parts converge to the reference value. Moreover we are also able to identify for the linear mapping between our latent variables z and the training data a matrix consisting of a multiple of the eigenvectors (1,1) and (1,-1) and thus the correct solution.
This example was chosen to show that the algorithm is able to quickly identify the exact solution of a linear ODE in terms of its linearly independent components.

Hidden multiscale dynamics

We consider eight-dimensional synthetic time series data produced by an underlying twodimensional complex valued process. In particular, the data points x are generated by first solving for the temporal evolution for the two complex-valued processes p 1 and p 2 and than mapping to the eight-dimensional space by using a randomly sampled linear mapping W .
One of the two processes used to generate the data is chosen to be much slower than the other one and both processes have a periodic component. dp 2 dt = (−0.9 + 1.5i)p 2 (8) As training data we consider 40 time series with 150 data points each, obtained by simulating the described processes for a maximum of t = 15 s and then sampling from the obtained data points.
Hence the training data consists of: • 40 time-series • with each consisting 150 observations of the x at a uniform time-step ∆t = 0.0025 The autoencoder obtained consists of one linear layer for both the decoder as well as the encoder. The model is trained for 5000 iterations using the Adam optimizer and a learning rate of 10 −3 .
The results for the convergence of the parameters λ 1 and λ 2 can be found in Figure . We note that the process which is slower decaying and thus more responsible for the long-term evolution of the system has a higher convergence rate than the faster process. With the obtained parameters λ as well as the trained autoencoder, we compute predictions based on the last time step used for training, i.e. we apply the encoder to obtain our latent representation and than use the latent dynamics to advance the latent representation in time.
Afterwards, we employ the decoder to reconstruct the full high-dimensional system. The results can be found in Figure and show very good agreement between predictions and reference data. This example shows that our model is successfully able to carry out dimensionality reduction and moreover indicates that the convergence rate between latent processes can be different.
The latter is relevant when training models as for accurate predictions all latent processes and their dynamics should be converged.

Kuramoto-Sivashinsky

Finally, we applied our algorithm to the KS equation and aim to identify a reduced-order model for the solution u(y, t): We employed periodic boundary conditions, µ = 1 and a domain size y ∈ [0, 22]. For this domain-size, the KS-equation exhibits a structurally stable chaotic attractor as discussed in The black lines divides the area for which training data was given from the area without raining data.
; . The equation is discretized in space using a discretization step of 22 64 resulting in a state vector x of dimension 64 and a nonlinear system of coupled ODEs. This is solved using a stiff fourth-order solver  We employed a non-linear encoder and decoder with four fully-connected layers each and ReLU-activation functions as well as Dropout Layers between the fully-connected layers.
We trained the model for 200000 iterations using Adam and a learning rate of 5 • 10 4 and assuming a five-dimensional latent space. We obtained the λ's in Figure . Four latent variables have λ's close to zero and thus a slow temporal dynamic that is responsible for the long-term evolution whereas one latent variable is quickly decaying.
Based on the obtained parameters, we do predictions based on an unseen initial condition not contained in the training data. We are able to reconstruct the correct phase space based on our predictions despite only using a very limited amount of training data. The results for the phase space can be seen in Figure .
Although the small-scale fluctuations in the temporal dynamics are not well captured, the model identifies the correct manifold which has a good accuracy compared to the reference solution. All phase-spaces were obtained by using a finite-difference operator on the data or predictions. These results are in accordance Interpretable reduced-order modeling with time-scale separation with whose LSTM-based temporal dynamic model was also able to find the correct phase space but not to track the actual dynamics for long-term predictions.
Our model is not able to account for noise in the temporal evolution and thus dealing with chaotic, small-scale fluctuations is challenging. We believe that a probabilistic version of our algorithm could be advantageous here. This section contains a fully probabilistic formulation for the deterministic model discussed before.
We replace the Autoencoder with a Variational Autoencoder and the ODE in the latent space with a SDE. The loss function which we optimize is the Evidence-Lower Bound (ELBO).

Model Structure

We postulate the following relations for our probabilistic model using an Ornstein-Uhlenbeck (OU) for each dimension i of the latent space and a Wiener process W t in the latent space: We again assume that the latent variables z t are complex-valued and a priori independent. Complex variables were chosen as their evolution includes a harmonic components which are observed in many physical systems.
We assume an initial conditions z 0,i ∼ CN (0, σ 2 0,i ). The total parameters associated with the latent space dynamics of our model are thus {σ 2 0,i , σ 2 i , λ i } c i=1 and will be denoted by θ together with all parameters responsible for the decoder mapping G (see next section). These parameters along with the state variables z t have to be inferred from the data x t .
Based on probabilistic Slow Feature Analysis (SFA) , we set σ 2 i = 2; (λ j ) and σ 2 0,i = 1. As a consequence, a priori, the latent dynamics are stationary. A derivation and reasoning for this choice can be found in Appendix A. Hence the only independent parameters are the λ i , the imaginary part of which can account for periodic effects in the latent dynamics.

Variational Autoencoder

We employ a variational autoencoder to account for a probabilistic mappings from the lower-dimensional representation z n to the high-dimensional system x n . In particular we are employing a probabilistic decoder The encoder is used to infer the state variables z based on the given data and thus defined in the inference and learning section.

Inference and Learning

Given the probabilistic relations , our goal is to infer the latent variables z 0:T as well as all model parameters θ. We follow a hybrid Bayesian approach in which the posterior of the state variables is approximated using amortized Variational Inference and Maximum-A-Posteriori (MAP) point-estimates for θ are computed.
The application of Bayes' rule for each data sequence x 0:T leads to the following posterior: where p(θ) denotes the prior on the model parameters. In the context of variational inference, we use the following factorization of the approximate posterior i.e. we infer only the mean µ and variance σ for each state variable based on the given data points.
This conditional density used for inference is the encoder-counterpart to the probabilistic decoder defined in the section before. It can be readily shown that the optimal parameter values are found by maximizing the Evidence Lower Bound (ELBO) F(q φ (z 0:T ), θ) which is derived in Appendix B. We compute Monte Carlo estimates of the gradient of the ELBO with respect to φ and θ with the help of the reparametrization trick and carry out stochastic optimization with the ADAM algorithm .

Results for the probabilistic extension

We applied our probabilistic version to the KS-equation. We used the same settings as for the deterministic approach but considered up to 10 complex latent variables. The obtained λ's are in Figure . The probabilistic model allows us to quantify the uncertainty in predictions. In Figure predictions for various time-steps and the respective uncertainty bounds are shown for an unseen initial condition.
Due to the chaotic nature of the KS-equation and the small amount of training data, the underlying linear dynamic of our model is only able to capture the full dynamics for a limited time horizon. Fortunately, due to the probabilistic approach the model is capable of capturing chaotic fluctuations with increasingly wide uncertainty bounds.
We also computed the phase space representation for the KS-equation based on the predictions obtained by our model and compare it with the reference solution. The probabilistic model identifies the correct manifold with a better accuracy than the deterministic model. As some of the small-scale fluctuations are accounted as noise, the resulting manifold is more concentrated at the origin and the obtained values are slightly smaller than the reference manifold although their shape is very similar.

Answer: The paper was published on 7 March 2023.


Input: When was Weep Not, Child first published?

Context:
Weep Not, Child is a 1964 novel by Kenyan author Ngũgĩ wa Thiong'o. It was his first novel, published in 1964 under the name James Ngugi. It was among the African Writers Series. It was the first English language|English novel to be published by an East African. Thiong'o's works deal with the relationship between Africans and white settlers in colonial Kenya, and are heavily critical of colonial rule. Specifically, Weep Not, Child deals with the Mau Mau Uprising, and "the bewildering dispossession of an entire people from their ancestral land." Ngũgĩ wrote the novel while he was a student at Makerere University.

The book is divided into two parts and eighteen chapters. Part one deals mostly with the education of Njoroge, while part two deals with the rising Mau Mau movement.

Plot summary

Njoroge, a little boy, is urged to attend school by his mother. He is the first one of his family able to go to school. His family lives on the land of Jacobo, an African made rich by his dealings with white settlers, namely Mr. Howlands, the most powerful land owner in the area. Njoroge's brother Kamau works as an apprentice to a carpenter, while Boro, the eldest living son, is troubled by his experiences while in forced service during World War II, including witnessing the death of his elder brother. Ngotho, Njoroge's father and a respected man in the surrounding area, tends Mr. Howlands' crops, but is motivated by his passion to preserve his ancestral land, rather than for any compensation or loyalty.

One day, black workers call for a strike to obtain higher wages. Ngotho is ambivalent about participating in the strike because he fears he will lose his job. However, he decides to go to the gathering, even though his two wives do not agree. At the demonstration, there are calls for higher wages. Suddenly, the white police inspector brings Jacobo to the gathering to pacify the native people. Jacobo tries to put an end to the strike. Ngotho attacks Jacobo, and the result is a riot where two people are killed. Jacobo survives and swears revenge. Ngotho loses his job and Njoroge’s family is forced to move. Njoroge’s brothers fund his education and seem to lose respect for their father.

Mwihaki, Jacobo's daughter and Njoroge's best friend, enters a girls' only boarding school, leaving Njoroge relatively alone. He reflects upon her leaving, and realizes that he was embarrassed by his father's actions towards Jacobo. For this reason, Njoroge is not upset by her exit and their separation. Njoroge switches to another school.

For a time, everyone's attention is focused on the upcoming trial of Jomo Kenyatta – a revered leader of the movement. Many blacks think that he is going to bring forth Kenya’s independence. But Jomo loses the trial and is imprisoned. This results in further protests and greater suppression of the black population.

Jacobo and a white landowner, Mr. Howlands, fight against the rising activities of the Mau Mau, an organization striving for Kenyan economic, political, and cultural independence. Jacobo accuses Ngotho of being the leader of the Mau Mau and tries to imprison the whole family. Meanwhile, the situation in the country is deteriorating. Six black men are taken out of their houses and executed in the woods.

One day Njoroge meets Mwihaki again, who has returned from boarding school. Although Njoroge had planned to avoid her due to the conflict between their fathers, their friendship is unaffected. Njoroge passes an important exam that allows him to advance to High School. His village is proud of him, and collects money to pay Njoroge's High School tuition.

Several months later, Jacobo is murdered in his office by a member of the Mau Mau. Mr. Howlands has Njoroge removed from school for questioning. Both father and son are brutally beaten before release and Ngotho is left barely alive. Although there doesn't seem to be a connection between Njoroge's family and the murder, it is eventually revealed that Njoroge's brothers are behind the assassination, and that Boro is the real leader of the Mau Mau. Ngotho soon dies from his injuries and Njoroge finds out that his father was protecting his brothers. Kamau has been imprisoned for life. Only Njoroge and his two mothers remain free, and Njoroge is left as the sole provider of his two mothers. Njoroge fears that he cannot make ends meet; he gives up hope of continuing in school and loses faith in God.

Njoroge asks Mwihaki's for support, but she is angry because of her father’s death. When he finally pledges his love to her, she refuses to leave with him, realizing her obligation to Kenya and her mother. Njoroge decides to leave town and makes an attempt at suicide; however, he fails when his mothers find him before he is able to hang himself. The novel closes with Njoroge feeling hopeless, and ashamed of cowardice.

Characters in Weep Not, Child
 Njoroge: the main character of the book whose main goal throughout the book is to become as educated as possible.
 Ngotho: Njoroge's father. He works for Mr.Howlands and is respected by him until he attacks Jacobo at a workers strike. He is fired and the family is forced to move to another section of the country. Over the course of the book his position as the central power of the family weakened, to the point where his self-realization that he has spent his whole life waiting for the prophecy (that proclaims the blacks will be returned their land) to come true rather than fighting for Kenyan independence, leads to his depression.
 Nyokabi and Njeri: the two wives of Ngotho. Njeri is Ngotho's first wife, and mother of Boro, Kamau, and Kori. Nyokabi is his second wife, and the mother of Njoroge and Mwangi.
 Njoroge has four brothers: Boro, Kamau, Kori and Mwangi (who is Njoroge's only full brother, who died in World War II).
 Boro: Son of Njeri who fights for the Allies in World War II. Upon returning his anger against the colonial government is compounded by their confiscation of the his land. Boro's anger and position as eldest son leads him to question and ridicule Ngotho, which eventually defeats their father's will (upon realizing his life was wasted waiting and not acting). It is eventually revealed that Boro is the leader of the Mau Mau (earlier alluded to as "entering politics") and murders Mr.Howlands. He is caught by police immediately after and is scheduled to be executed by the book's end. It is highly likely that it is also Boro who kills Jacobo.
 Mwihaki: Njoroge's best friend (and later develops into his love interest). Daughter of Jacobo. When it is revealed that his family killed Jacobo (most likely Boro), Mwihaki distances herself from Njoroge, asking for time to mourn her father and care for her mother.
 Jacobo: Mwihaki's father and an important landowner. Chief of the village.
 Mr. Howlands: A white settler who emigrated to colonial Kenya and now owns a farm made up of land that originally belonged to Ngotho's ancestors. Has three children: Peter who died in World War II before the book's beginning, a daughter who becomes a missionary, and Stephen who met Njoroge while the two were in high school.

Themes and motifs
Weep Not, Child integrates Gikuyu mythology and the ideology of nationalism that serves as catalyst for much of the novel's action. The novel explores the negative aspects of colonial rule over Kenya. Njoroge's aspiration to attend university is frustrated by both the violence of the Mau Mau rebels and the violent response of the colonial government. This disappointment leads to his alienation from his family and ultimately his suicide attempt.

The novel also ponders the role of saviours and salvation. The author notes in his The River Between: "Salvation shall come from the hills. From the blood that flows in me, I say from the same tree, a son shall rise. And his duty shall be to lead and save the people." Jomo Kenyatta, the first prime minister of Kenya, is immortalised in Weep Not, Child. The author says, "Jomo had been his (Ngotho's) hope. Ngotho had come to think that it was Jomo who would drive away the white man. To him, Jomo stood for custom and traditions purified by grace of learning and much travel." Njoroge comes to view Jomo as a messiah who will win the struggle against the colonial government.

See also

Things Fall Apart
Death and the King's Horseman

References

External links
Official homepage of Ngũgĩ wa Thiong'o
BBC profile of Ngũgĩ wa Thiong'o
Weep Not, Child at Google Books

British Empire in fiction
Novels set in colonial Africa
Historical novels
Kenyan English-language novels
Novels by Ngũgĩ wa Thiong'o
Novels set in Kenya
1964 novels
Heinemann (publisher) books
Postcolonial novels
African Writers Series
1964 debut novels

Answer: Weep Not, Child was first published in 1964.


Input: How is the vacuum processing system configured in terms of the arrangement of the vacuum processing apparatus?

Context:
PROBLEM TO BE SOLVED: To provide a vacuum processor which can suppress the rise of manufacture cost while coping with the enlargement of diameter of a sample, and besides is excellent in maintainability. SOLUTION: This vacuum processor comprises an air loader 1 which is equipped with a plurality of juxtaposed cassette stands 2a and 2b and a carrier 13 for carrying a wafer 20 from or to the cassette stands, a vacuum loader 7 which is equipped with etching treatment chambers 11 and 11b for processing the wafer and a vacuum carriage chamber 16 connected to them through gate valves 15a and 15b, and a lock device 6 consisting of a load lock chamber 6b and an unload lock chamber 6b equipped with gate valves 12a, 12b, 12c, and 12d for connecting the said carrier 13 with the vacuum carriage chamber 16. For the etching treatment chambers, two are provided only on the opposite side of the vacuum carriage chamber symmetrically about the axis line A passing the center, and besides the arrangement positions of the two etching treatment chambers are at an acute angle on the opposite side of the vacuum carriage chamber.
エッチング，ＣＶＤ（化学的気相成長），スパッタリング，アッシング，リンサ（水洗）等の枚葉処理をするのに好適な真空処理装置とそれを用いて半導体デバイスを製造する半導体製造ラインに関するものである。 Those etching, CVD (chemical vapor deposition), sputtering, ashing, to rinser (water washing) suitable vacuum processing apparatus for single wafer processing, such as a semiconductor manufacturing line for manufacturing a semiconductor device using the same .
【従来の技術】試料を処理する真空処理装置は、大別すると、カセットブロックと真空処理ブロックから構成されており、カセットブロックは、半導体製造ラインのベイ通路に面して長手方向に伸びるフロントを有し、試料用のカセットや試料のオリエンテーションを合わせるアライメントユニットと、大気ロボットがある。 BACKGROUND OF THE INVENTION Vacuum processing apparatus for processing a sample, the roughly is constituted by a cassette block and a vacuum processing block, the cassette block, a front extending in the longitudinal direction facing the bay aisle of a semiconductor manufacturing line a, an alignment unit for aligning the cassette and sample orientation for sample, there is an atmospheric robot. 真空処理ブロックには、ロード側ロードロック室，アンロード側ロードロック室，真空処理室，後真空処理室，真空ポンプ及び真空ロボット等が設けられている。 The vacuum processing block, the load-side load lock chamber, the unload side load lock chamber, the vacuum processing chamber, a rear vacuum processing chamber, a vacuum pump and a vacuum robot and the like.
【０００３】これらの真空処理装置では、カセットブロックのカセットから取り出された試料が、大気ロボットにより真空処理ブロックのロードロック室まで搬送される。  In these vacuum processing apparatus, a sample taken from the cassette in the cassette block is transported to the load lock chamber in the vacuum processing block by the atmospheric robot. ロードロック室から真空ロボットによりさらに処理室に搬送され、電極構造体上にセットされた試料は、プラズマエッチング等の処理がなされる。 Is conveyed from the load lock chamber to the further processing chamber by the vacuum robot, the sample is set on an electrode structure, processing such as plasma etching is performed. その後、必要に応じて後真空処理室に搬送，処理される。 Thereafter, the conveyance to the rear vacuum processing chamber as necessary and processed. 処理済みの試料は、真空ロボット及び大気ロボットによりカセットブロックのカセットに搬送される。 Processed sample is conveyed to the cassette of the cassette block by the vacuum robot and the atmospheric robot.
【０００４】試料をプラズマエッチング処理する真空処理装置の例としては、例えば特公昭61−8153号公報，特開昭63−133532号公報，特公平6−30369号公報，特開平  Examples of the sample vacuum processing apparatus for plasma etching treatment, for example Japanese Patent Publication 61-8153, JP-Sho 63-133532 and JP Kokoku 6-30369, JP-A No.
6−314729号公報，特開平6−314730号公報，米国特許第 6-314729, JP-A No. 6-314730, JP-U.S. Patent No.
5,314,509号明細書および5,784,799号明細書に記載されたようなものがある。 There are such as described in Pat and 5,784,799 Pat 5,314,509.
509号明細書に記載された装置は、真空処理ブロックの中央付近に真空ロボット、その周囲に３個の処理室が同心状に配置され、真空ロボットとカセットブロックの間に、ロード側ロードロック室，アンロード側ロードロック室が設けられている。 Device described in 509 Pat are vacuum robot in the vicinity of the center of the vacuum processing block, three process chambers around it are arranged concentrically, between the vacuum robot and the cassette block, the load-side load-lock chamber , unload side load lock chamber is provided. これらの装置では、大気ロボットや真空ロボットの搬送アームの回転角度が大きく従って装置全体の必要床面積が大きいという問題がある。 In these devices, there is a problem that needs floor space for the entire rotation angle is large therefore device of the transfer arm of the atmospheric robot and the vacuum robot is large.
【０００６】一方、真空処理装置の真空処理ブロック内の処理室や真空ポンプその他各種の配管機器については、定期，不定期に点検修理等のメンテナンスを行うことが必要である。  On the other hand, the processing chamber and the vacuum pump and other various piping components of the vacuum processing block of the vacuum processing apparatus, periodically, it is necessary to perform maintenance such as inspection and repair irregularly. そのため、一般に、真空処理ブロックの周囲には、扉が設けられており、この扉を開けることにより、ロードロック室，アンロードロック室，処理室，真空ロボット及び各種の配管機器の点検修理ができるようになっている。 Therefore, in general, around the vacuum processing block, the door is provided by opening the door, load lock chambers, unload lock chambers, processing chambers, the servicing of the vacuum robot and various piping devices It has become way.
キャリアポッドが必要となるために、約３５０mm程度と大きくなり、複数のキャリアポッドを収納するカセットブロックの幅も大きくなる。 For carrier pod required, large as about 350 mm, the greater width of the cassette block for housing a plurality of carrier pods. この幅に合わせて真空処理ブロックの幅を決定すると、真空処理装置全体が大きなスペースを必要とすることになる。 When determining the width of the vacuum processing block in accordance with the this width, the entire vacuum processing apparatus requires a large space. 一例として、４個のキャリアポッドを収納するカセットブロックについて考えると、試料の直径ｄが８インチから１２インチになった場合、カセットの幅は少なくとも約４０cm以上大きくならざるを得ない。 As an example, considering the cassette block for accommodating four carriers pods, if the diameter d of the sample was a 12-inch 8 inch wide cassette inevitably increases at least about 40cm or more.
【０００８】一方、試料に各種の処理を行いながら大量の処理を行うために、一般の半導体製造ラインでは、同じ処理を行う複数の真空処理装置を同じベイに集め、各ベイ間の搬送を自動またはマニュアルで行っている。 On the other hand, in order to perform a lot of processing while performing various processes in the sample, in a general semiconductor manufacturing lines, gathering a plurality of vacuum processing apparatus for performing the same processing in the same bay, automatic conveyance between the bays or it is carried out manually. このような半導体製造ラインは、高いクリーン度を必要とするため、半導体製造ライン全体が大きなクリーンルーム内に設置される。 Such a semiconductor manufacturing line, requires a high degree of cleanliness, the whole semiconductor manufacturing line is placed in a large clean room. 試料の大口径化に伴う真空処理装置の大型化は、クリーンルーム占有面積の大型化を伴うが、これはもともと建設コストの高いクリーンルームの建設コストを一層増加させることになる。 Size of the vacuum processing apparatus due to the large diameter of the sample is accompanied by a large clean room area occupied, which will be further increased construction costs of the high construction cost clean room originally. もし、同じ面積のクリーンルームに占有面積の大きな真空処理装置を設置するとすれば、真空処理装置の全体の台数を減らすか、あるいは各真空処理装置間の間隔を狭くせざるを得ない。 If, if the clean room of the same area to install a large vacuum processing apparatus of the occupied area, reduce the overall number of the vacuum processing apparatus, or interval narrower forced between the vacuum processing apparatus. 同じ面積のクリーンルームにおける真空処理装置の設置台数減少は、必然的に半導体の製造ラインの生産性の低下ひいては半導体の製造コストの上昇を伴う。 Installed base reduction in the vacuum processing apparatus in a clean room having the same area is accompanied inevitably rise of the semiconductor decrease and thus the semiconductor manufacturing cost of productivity of the production line. 他方、各真空処理装置間の間隔を狭くすることは、点検修理のためのメンテナンススペースが不足し、真空処理装置のメンテナンス性を著しく阻害する。 On the other hand, to reduce the distance between each of the vacuum processing apparatus, the maintenance space for inspection and repair is insufficient to significantly inhibit the maintenance of the vacuum processing apparatus.
【０００９】本発明の目的は、試料の大口径化に対応しつつ、製造コストの上昇を抑えることのできる真空処理装置を提供することにある。 An object of the present invention, while corresponding to the large diameter of the sample, is to provide a vacuum processing apparatus capable of suppressing an increase in manufacturing cost.
【００１０】本発明の他の目的は、試料の大口径化に対応しつつ、メンテナンス性に優れた真空処理装置を提供することにある。 Another object of the present invention, while corresponding to the large diameter of the sample is to provide a vacuum processing apparatus having excellent maintainability.
【００１１】本発明の他の目的は、試料の大口径化に対応しつつ、真空処理装置の必要設置台数を確保して製造コストの上昇を抑え、かつ、メンテナンス性も損なわない半導体製造ラインを提供することにある。 Another object of the present invention, while corresponding to the large diameter of the sample, to ensure the necessary number of installed vacuum processing apparatus suppressing an increase in manufacturing cost, and a semiconductor manufacturing line is not impaired maintainability It is to provide.
【課題を解決するための手段】本発明は、並設した複数のカセット台およびカセット台から、あるいはカセット台へウエハを搬送するための搬送装置を備えた大気ローダと、ウエハを処理するための真空処理室およびこれにゲート弁を介して連接された真空搬送室を備えた真空ローダと、前記搬送装置と前記真空搬送室とを連接するためのゲート弁を備えたロードロック室およびアンロードロック室からなるロック装置とを含んで構成される真空処理装置において、ウエハを処理するための真空処理室は、有磁場ＵＨＦ帯電磁波放射放電方式リアクタ（以下、ＵＨＦ−ＥＣＲリアクタという。）によって形成される真空処理室であり、該真空処理室には、分解可能な側壁インナーユニットおよびアンテナが設けられ、該真空処理室は、真空搬 The present invention SUMMARY OF THE INVENTION from a plurality of cassette tables and cassette stand juxtaposed, or the atmosphere loader having a conveying device for conveying the wafer to the cassette table, for processing a wafer the load lock chamber and the unload lock having a gate valve for connecting the vacuum loader vacuum processing chamber and having a vacuum transfer chamber which is connected via a gate valve to the said vacuum transfer chamber and the conveying device in the vacuum processing apparatus configured to include a lock device comprising a chamber, a vacuum processing chamber for processing a wafer it is formed by a magnetic field UHF band electromagnetic wave radiation discharge type reactor (hereinafter. referred UHF-ECR reactor) that a vacuum processing chamber, the vacuum processing chamber, degradable sidewall inner unit and an antenna are provided, the vacuum processing chamber, a vacuum transportable 室およびロック装置の中央を通る軸線に対して対称にして、かつ真空搬送室を中心にしてロック装置の反対側のみに２つ設けられ、かつ真空搬送室に対して前記２つの真空処理室の配置位置は鋭角をなしている真空処理装置を提供する。 And symmetrically with respect to the axis passing through the center of the chamber and the locking device, and the opposite side only two provided for to lock device around the vacuum transfer chamber, and the two vacuum processing chamber to the vacuum transfer chamber position is to provide a vacuum processing apparatus which forms an acute angle.
ＣＲリアクタによって形成される真空処理室であり、該真空処理室は、真空搬送室およびロック装置の中央を通る軸線に対して対称にして、かつ真空搬送室を中心にしてロック装置の反対側のみに２つ設けられ、かつ真空搬送室に対して前記２つの真空処理室の配置位置は鋭角をなしており、ＵＨＦ−ＥＣＲのアンテナは、前記軸線に対して平行で、かつ前記真空搬送室とは反対側に開放される真空処理装置を提供する。 A vacuum processing chamber formed by a CR reactor, vacuum processing chamber, and symmetrically with respect to the axis passing through the center of the vacuum transfer chamber and the locking device, and the opposite side of the locking device around the vacuum transfer chamber only two provided, and positions of the two vacuum processing chamber to the vacuum transfer chamber is an acute angle, the UHF-ECR antennas, and parallel, and the vacuum transfer chamber with respect to said axis to provides a vacuum processing apparatus is opened to the opposite side.
かつ真空搬送室を中心にしてロック装置の反対側のみに２つ設けられ、かつ真空搬送室に対して前記２つの真空処理室の配置位置は鋭角をなしており、大気ローダ，真空ローダおよびロック装置はＴ字配置とされた真空処理方法を提供する。 And around the vacuum transfer chamber opposite only two provided a locking device, and positions of the two vacuum processing chamber to the vacuum transfer chamber is an acute angle, atmospheric loader, vacuum loader and locking apparatus to provide a vacuum processing method which is a T-arrangement.
【００１５】本発明は、並設した複数のカセット台およびカセット台から、あるいはカセット台へウエハを搬送するための搬送装置を備えた大気ローダと、ウエハを処理するための真空処理室およびこれにゲート弁を介して連接された真空搬送室を備えた真空ローダと、前記搬送装置と前記真空搬送室とを連接するためのゲート弁を備えたロードロック室およびアンロードロック室からなるロック装置とを含んで構成される真空処理装置が平行に複数台並設された真空処理システムにおいて、ウエハを処理するための真空処理室は、ＵＨＦ−ＥＣＲリアクタによって形成される真空処理室であり、該真空処理室は、真空搬送室およびロック装置の中央を通る軸線に対して対称にして、かつ真空搬送室を中心にしてロック装置の反対側のみに２ The present invention, a plurality of cassette tables and cassette stand juxtaposed, or the atmosphere loader having a conveying device for conveying the wafer to the cassette base, the vacuum processing chamber for processing the wafer and to a vacuum loader having a vacuum transfer chamber which is connected via a gate valve, the locking consisting of the conveying device and the load lock chamber and the unload lock chamber with gate valves for connecting the said vacuum transfer chamber apparatus and in the vacuum processing system vacuum processing apparatus is constituted with a plurality Tainami set in parallel include vacuum processing chamber for processing a wafer is vacuum processing chamber formed by the UHF-ECR reactor, vacuum processing chamber, and symmetrically with respect to the axis passing through the center of the vacuum transfer chamber and the locking device, and 2 only on the opposite side of the locking device around the vacuum transfer chamber 設けられ、かつ真空搬送室に対して前記２つの真空処理室の配置位置は鋭角をなしており、並設された複数の真空処理装置のすべての真空処理室に一直線上に配列される真空処理システムを提供する。 Provided, and positions of the two vacuum processing chamber to the vacuum transfer chamber is an acute angle, a vacuum process that is arranged in alignment with all of the vacuum processing chamber of a plurality of vacuum processing apparatus are arranged in parallel to provide a system.
【発明の実施の形態】以下、本発明にかかる一実施例を図面に基づいて説明する。 BEST MODE FOR CARRYING OUT THE INVENTION Hereinafter, will be explained based on an embodiment according to the present invention with reference to the accompanying drawings.
１０Ｂ，１０Ｃで示す。 10B, shown by 10C.
【００１８】図１に示す真空処理システムを説明する前に、図２から図４に基づいて真空処理装置を説明する。  Before describing the vacuum processing system shown in FIG. 1, illustrating a vacuum processing apparatus on the basis of FIGS. 2-4.
５ｂを介して連接された真空搬送室１６を備えた真空ローダ７と、前記搬送装置と前記真空搬送室とを連接するためのゲート弁を備えたロードロック室６ａおよびアンロードロック室６ｂからなるロック装置６とを含んで構成される。 5b a vacuum loader 7 having a vacuum transfer chamber 16 which is connected via consist load lock chamber 6a and the unload lock chamber 6b has a gate valve for connecting and said vacuum transfer chamber and the conveying device configured to include a locking device 6.
平行形に配置され、その位置および姿勢を変えることなく、装置への導入／払出しが可能な位置、すなわち、カセット１ａないし１ｃを略水平の平面上で常に一定の位置に固定される。 Arranged parallel type, without changing its position and orientation, the introduction / dispensing of the locations of the device, i.e., always fixed to a constant position to free the cassette 1a 1c on a substantially horizontal plane. カセット台２ａおよび２ｂは、平行に隣合わせて配置してある。 Cassette tables 2a and 2b, are arranged side by side in parallel. カセット台２ｃは、最右端に配置してある。 Cassette table 2c is, are arranged on the top right edge. カセット１ａおよび１ｂは、処理を行うための末処理ウエハを収納したり、処理済みのウエハを回収するためのもので、複数枚（通常２５枚）の被処理基板であるウエハ２０が収納可能となっている。 Cassettes 1a and 1b, or storing the processed wafers end for processing, for recovering the processed wafer, the wafer 20 is a substrate to be processed in a plurality (25 pieces Normal) can be stored going on. カセット１ｃは、この場合、プラズマを用いたドライクリーニング（以下、「プラズマクリーニング」という。）を行うためのダミーウエハを収納したり、プラズマクリーニング後のダミーウエハを回収するためのもので、複数枚（通常２５枚）のダミーウエハ３０が収納可能となっている。 Cassette 1c in this case, dry cleaning using plasma (hereinafter, "plasma cleaning" hereinafter.) Or housing a dummy wafer for performing, for recovering the dummy wafers after plasma cleaning, a plurality (usually dummy wafer 30 of 25 sheets) has become can be stored.
とカセット１ａ，１ｂとの間でウエハ２０を、そしてロードロック室６ａおよびアンロードロック室６ｂとカセット１ｃとの間でダミーウエハ３０を授受可能に動作する。 A cassette 1a, the wafer 20 with the 1b, and operates to allow exchanging dummy wafer 30 between the load lock chamber 6a and the unload lock chamber 6b and the cassette 1c.
ａ，１５ｂを介して真空処理室であるエッチング処理室１１ａ，１１ｂが設けてある。 a, etching chamber is a vacuum processing chamber through a 15b 11a, 11b are provided. 以下、エッチング処理室を例に取って説明する。 Hereinafter will be described taking the etching chamber as an example. 真空搬送室１６内には、ロードロック室６ａ，アンロード室６ｂおよびエッチング処理室１１ａ，１１ｂとの間でウエハ２０またはダミーウエハ３０を授受可能に動作する搬送装置１４が設けてある。 The vacuum transfer chamber 16, load lock chambers 6a, unload chamber 6b and etching chambers 11a, conveying device 14 is provided that operates to allow transferring the wafer 20 or dummy wafer 30 between 11b. 真空搬送室１６は、独立に真空排気可能な真空排気装置１７を装備している。 Vacuum transfer chamber 16 is equipped with a vacuum evacuable vacuum evacuation device 17 independently.
【００２２】ＵＨＦ−ＥＣＲリアクタのエッチング処理室１１ａ，１１ｂは、この場合、同一の構成で対称配置とされてエッチング処理が行われるようになっている。 The UHF-ECR reactor etching chambers 11a, 11b is in this case are symmetrically arranged in the same configuration so that the etching process is performed.
エッチング処理室１１ａを例に説明する。 The etching chamber 11a will be described as an example. エッチング処理室１１ａは、ウエハ２０を配置するための試料台を有し、試料台８ａの上部に放電部７ａを形成するように放電室が設けてある。 Etching chamber 11a has a sample stage for placing the wafer 20, the discharge chamber so as to form a discharge portion 7a at the top of the sample table 8a is provided. エッチング処理室１１ａは、放電部７ａへの処理ガス供給のためのガス導入装置１０ａを有するとともに、エッチング処理室１１ａ内を所定圧力に減圧排気する真空排気装置９ａを有し、放電部７ａの処理ガスをプラズマ化するための、この場合、ＵＨＦ波と磁場の発生手段を有している。 The etching chambers 11a, which has a gas introduction device 10a for processing the gas supply to the discharge portion 7a, having a vacuum exhaust device 9a for evacuating the etching chamber 11a to a predetermined pressure, the process of the discharge portion 7a for plasma gas, in this case, it has a generating means of the UHF wave and magnetic field.
９は、センサ１８からの計測値を所定値と比較して、エッチング処理室内のクリーニング時期を判断する。 9, the measured value from the sensor 18 is compared with a predetermined value to determine the cleaning time of the etching chamber. また、制御装置１９は、真空搬送装置１３および１４を制御して、ダミーウエハ３０をカセット１ｃおよびエッチング処理室１１ａないし１１ｂの間で搬送制御する。 The control device 19 controls the vacuum transfer apparatus 13 and 14, to the dummy wafer 30 to the cassette 1c and the etching process chamber 11a carrying controlled between 11b.
いずれかの方法によりウエハ処理またはプラズマクリーニングを実行する。 It executes the wafer processing or plasma cleaning by any method.
によって、カセット１ａ内のウエハ２０を下から順にエッチング処理室１１ａ，１１ｂに搬入し、それぞれのウエハ２０をエッチング処理する。 Accordingly, the wafer 20 to an etching treatment chamber in order from the bottom 11a of the cassette 1a, and carried into 11b, and each of the wafer 20 is etched. 処理されたそれぞれのウエハ２０は、真空搬送装置１４および搬送装置１３によって、カセット１ａ内の元の位置に収納する。 Each wafer 20 processed by vacuum transfer apparatus 14 and the carrier 13 is housed in its original position in the cassette 1a. この場合、運転開始から終了に至る間、カセットの位置および姿勢を変えることなく未処理のウエハを取り出し、そして処理済みのウエハを未処理のウエハが収納されていた元の位置に戻して収納する。 In this case, while leading to completion of the start of the operation, it takes out an unprocessed wafer without changing the position and posture of the cassettes, and the processed wafer unprocessed wafer is accommodated back into position which is housed . このようにすることで、生産ラインの自動化への対応が容易で、且つ、ゴミの発生によるウエハの汚染を低減でき、高い生産効率と高い製品歩留まりを実現できる。 In this way, it is easy to respond to automation of production lines, and can reduce the contamination of the wafer due to the generation of dust can achieve high production efficiency and high product yield.
の処理が全て終り次のカセットの内のウエハ処理に移る前でも良い。 Of the process is good, even before moving on to the wafer processing of all end next cassette.
【００２７】プラズマクリーニングの実施にあたっては、次の順序で行われる。 The practice of plasma cleaning is performed in the following order. この場合、カセット１ｃに収納されたダミーウエハ３０（この

Answer: Multiple vacuum processing apparatuses are arranged in parallel.


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

Context:
\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.

Answer: NFPA and FPSA greatly outperform GMRES and DSA.