experimental study of ilu preconditioners for indefinite matrices incomplete lu factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices however their failure rate is still too high for them to be useful as black box library software for general matrices besides fatal breakdowns due to zero pivots the major causes of failure are inaccuracy and instability of the triangular solves when there are small pivots both these problems can occur but these problems can also occur without small pivots through examples from actual problems this paper shows how these problems evince themselves how these problems can be detected and how these problems can sometimes be circumvented through pivoting reordering scaling perturbing diagonal elements and preserving symmetric structure the goal of this paper is to gain a better practical understanding of ilu preconditioners and help improve their reliability

generalized simulated annealing we discuss and illustrate a new stochastic algorithm generalized simulated annealing for computationally finding the global minimum of a given not necessarily convex energy cost function defined in a continuous d dimensional space this algorithm recovers as particular cases the so called classical boltzmann machine and fast cauchy machine simulated annealings and turns out to be quicker than both

on the stability of relaxed incomplete lu factorizations when solving large linear systems of equations arising from the discretization of elliptic boundary value problems a combination of iterative methods and preconditioners based on incomplete lu factorizations is frequently used given a model problem with variable coefficients we investigate a class of incomplete lu factorizations depending on a relaxation parameter we show that the associated preconditioner and the factorization itself both are numerically stable the theoretical results are complemented by numerical experiments

bilum block versions of multielimination and multilevel ilu preconditioner for general sparse linear systems we introduce block versions of the multielimination incomplete lu ilum factorization preconditioning technique for solving general sparse unstructured linear systems these preconditioners have a multilevel structure and for certain types of problems may exhibit properties that are typically enjoyed by multigrid methods several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed the advantages of block ilum over point ilum include increased robustness and efficiency we compare several versions of the block ilum point ilum and the dual threshold based ilut preconditioners in particular tests with some convection diffusion problems show that it may be possible to obtain convergence that is nearly independent of the reynolds number as well as of the grid size

the markov process admits a consistent steady state thermodynamic formalism the search for a unified formulation for describing various non equilibrium processes is a central task of modern non equilibrium thermodynamics in this paper a novel steady state thermodynamic formalism was established for general markov processes described by the chapman kolmogorov equation furthermore corresponding formalisms of steady state thermodynamics for the master equation and fokker planck equation could be rigorously derived in mathematics to be concrete we proved that 1 in the limit of continuous time the steady state thermodynamic formalism for the chapman kolmogorov equation fully agrees with that for the master equation 2 a similar one to one correspondence could be established rigorously between the master equation and fokker planck equation in the limit of large system size 3 when a markov process is restrained to one step jump the steady state thermodynamic formalism for the fokker planck equation with discrete state variables also goes to that for master equations as t

weighted matrix ordering and parallel banded preconditioners for iterative linear system solvers the emergence of multicore architectures and highly scalable platforms motivates the development of novel algorithms and techniques that emphasize concurrency and are tolerant of deep memory hierarchies as opposed to minimizing raw flop counts while direct solvers are reliable they are often slow and memory intensive for large problems iterative solvers on the other hand are more efficient but in the absence of robust preconditioners lack reliability while preconditioners based on incomplete factorizations whenever they exist are effective for many problems their parallel scalability is generally limited in this paper we advocate the use of banded preconditioners instead and introduce a reordering strategy that enables their extraction in contrast to traditional bandwidth reduction techniques our reordering strategy takes into account the magnitude of the matrix entries bringing the heaviest elements closer to the diagonal thus enabling the use of banded preconditioners when used with effective banded solvers in our case the spike solver we show that banded preconditioners i are more robust compared to the broad class of incomplete factorization based preconditioners ii deliver higher processor performance resulting in faster time to solution and iii scale to larger parallel configurations we demonstrate these results experimentally on a large class of problems selected from diverse application domains

refinements of pinsker s inequality let v and d denote respectively total variation and divergence we study lower bounds of d with v fixed the theoretically best i e largest lower bound determines a function l l v vajda s 1970 tight lower bound the main result is an exact parametrization of l this leads to taylor polynomials which are lower bounds for l and thereby to extensions of the classical pinsker 1960 inequality which has numerous applications cf pinsker and followers

a scalable parallel algorithm for incomplete factor preconditioning we describe a parallel algorithm for computing incomplete factor ilu preconditioners the algorithm attains a high degree of parallelism through graph partitioning and a two level ordering strategy both the subdomains and the nodes within each subdomain are ordered to preserve concurrency we show through an algorithmic analysis and through computational results that this algorithm is scalable experimental results include timings on three parallel platforms for problems with up to 20 million unknowns running on up to 216 processors the resulting preconditioned krylov solvers have the desirable property that the number of iterations required for convergence is insensitive to the number of processors

regular incomplete factorizations of real positive definite matrices abstract we study the classes of matrices which admit a regular incomplete factorization with respect to any graph set we extend the construction to the class of real positive definite matrices the convergence of the basic iteration associated with the splittings we have is discussed

the earth mover s distance as a metric for image retrieval we investigate the properties of a metric between two distributions the earth mover s distance emd for content based image retrieval the emd is based on the minimal cost that must be paid to transform one distribution into the other in a precise sense and was first proposed for certain vision problems by peleg werman and rom for image retrieval we combine this idea with a representation scheme for distributions that is based on vector quantization this combination leads to an image comparison framework that often accounts for perceptual similarity better than other previously proposed methods the emd is based on a solution to the transportation problem from linear optimization for which efficient algorithms are available and also allows naturally for partial matching it is more robust than histogram matching techniques in that it can operate on variable length representations of the distributions that avoid quantization and other binning problems typical of histograms when used to compare distributions with the same overall mass the emd is a true metric in this paper we focus on applications to color and texture and we compare the retrieval performance of the emd with that of other distances

on the rate of convergence of the preconditioned conjugate gradient method we derive new estimates for the rate of convergence of the conjugate gradient method by utilizing isolated eigenvalues of parts of the spectrum we present a new generalized version of an incomplete factorization method and compare the derived estimates of the number of iterations with the number actually found for some elliptic difference equations and for a similar problem with a model empirical distribution function

robust approximate inverse preconditioning for the conjugate gradient method we present a variant of the ainv factorized sparse approximate inverse algorithm which is applicable to any symmetric positive definite matrix the new preconditioner is breakdown free and when used in conjunction with the conjugate gradient method results in a reliable solver for highly ill conditioned linear systems we also investigate an alternative approach to a stable approximate inverse algorithm based on the idea of diagonally compensated reduction of matrix entries the results of numerical tests on challenging linear systems arising from finite element modeling of elasticity and diffusion problems are presented

iterative solution methods for certain sparse linear systems with a non symmetric matrix arising from pde problems abstract in this paper methods are described for the solution of certain sparse linear systems with a non symmetric matrix the power of these methods is demonstrated by extensive numerical experiments application of the methods is limited to problems where the matrix has only eigenvalues with positive real part an important class of this type of matrix arises from discretisation of second order partial differential equations with first order derivative terms

directional agglomeration multigrid techniques for high reynolds number viscous flows a preconditioned directional implicit agglomeration algorithm is developed for solving two and three dimensional viscous flows on highly anisotropic unstructured meshes of mixed element types the multigrid smoother consists of a pre conditioned point or line implicit solver which operates on lines constructed in the unstructured mesh using a weighted graph algorithm directional coarsening or agglomeration is achieved using a similar weighted graph algorithm a tight coupling of the line construction and directional agglomeration algorithms enables the use of aggressive coarsening ratios in the multigrid algorithm which in turn reduces the cost of a multigrid cycle convergence rates which are independent of the degree of grid stretching are demonstrated in both two and three dimensions further improvement of the three dimensional convergence rates through a gmres technique is also demonstrated

multiscale finite element method for linear elastic geomechanics the demand for accurate and efficient simulation of geomechanical effects is widely increasing in the geoscience community high resolution characterizations of the mechanical properties of subsurface formations are essential for improving modeling predictions such detailed descriptions impose severe computational challenges and motivate the development of multiscale solution strategies we propose a multiscale solution framework for the geomechanical equilibrium problem of heterogeneous porous media based on the finite element method after imposing a coarse scale grid on the given fine scale problem the coarse scale basis functions are obtained by solving local equilibrium problems within coarse elements these basis functions form the restriction and prolongation operators used to obtain the coarse scale system for the displacement vector then a two stage preconditioner that couples the multiscale system with a smoother is derived for the iterative solution of the fine scale linear system various numerical experiments are presented to demonstrate accuracy and robustness of the method

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

preconditioned iterative methods for the nine point approximation to the convection diffusion equation iterative methods preconditioned by incomplete factorizations and sparse approximate inverses are considered for solving linear systems arising from fourth order finite difference schemes for convection diffusion problems simple recurrences for implementing the ilu 0 factorization of the nine point scheme are derived different sparsity patterns are considered for computing approximate inverses for the coefficient matrix and the quality of the preconditioner is studied in terms of plots of the field of values of the preconditioned matrices in terms of algebraic properties of the preconditioned matrices our experimental results show that incomplete factorizations give a preconditioner of better quality than approximate inverses comparison of the convergence rates of gmres applied to the preconditioned linear systems is done with respect to the field of values ritz and harmonic ritz values of the preconditioned matrices numerical results show that the gmres residual norm decreases rapidly when the difference between the ritz and harmonic ritz values becomes small we also describe the results of experiments when some well known krylov subspace methods are used to solve the linear system arising from the compact fourth order discretizations

the effect of ordering on preconditioned gmres algorithm for solving the compressible navier stokes equations we investigate the effect of the ordering of the blocks of unknowns on the rate of convergence of a preconditioned non linear gmres algorithm for solving the navier stokes equations for compressible flows using finite element methods on unstructured grids the gmres algorithm is preconditioned by an incomplete ldu block factorization of the jacobian matrix associated with the non linear problem to solve we examine a wide range of ordering methods including minimum degree reverse cuthill mckee and snake and consider preconditionings without fill in we show empirically that there can be a significant difference in the number of iterations required by the preconditioned non linear gmres method and suggest a criterion for choosing a good ordering algorithm according to the problem to solve we also consider the effect of orderings when an incomplete factorization which allows some fill in is performed we consider the effect of automatically controlling the sparsity of the incomplete factorization through the level of fill in finally following the principal ideas of non linear gmres algorithm we suggest other inexact newton methods

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

nested grids ilu decomposition ngilu a preconditioning technique is described which shows in many cases grid independent convergence this technique only requires an ordering of the unknowns based on the different levels of multigrid and an incomplete lu decomposition based on a drop tolerance the method is demonstrated on a variety of well known elliptic test problems including strongly varying coefficients advective terms and grid refinement

preconditioners for nonsymmetric indefinite linear systems abstract in this paper we develop algorithms for solving nonsymmetric indefinite linear systems by considering the augmented linear systems resulting from a weighted linear least squares problem even though the augmented system is more ill conditioned than the original linear system one can construct preconditioned gmres methods for solving these augmented systems capable of obtaining reasonable approximation of the solution in fewer iterations than the classical ilu preconditioned gmres method for solving the original linear system more specifically we present two different preconditioners for these augmented systems examine the spectral properties of these preconditioned augmented systems and report numerical results to illustrate the effectiveness of these preconditioners

an interesting class of partial differential equations this paper presents an observation that under reasonable conditions many partial differential equations from mathematical physics possess three structural properties one of them can be understand as a variant of the celebrated onsager reciprocal relation in modern thermodynamics it displays a direct relation of irreversible processes to the entropy change we show that the properties imply various entropy dissipation conditions for hyperbolic relaxation problems as an application of the observation we propose an approximation method to solve relaxation problems moreover the observation is interpreted physically and verified with eight sets of systems from different fields

fundamental properties of tsallis relative entropy fundamental properties for the tsallis relative entropy in both classical and quantum systems are studied as one of our main results we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by hiai and petz the monotonicity of the quantum tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible the generalized tsallis relative entropy is defined and its subadditivity is shown by its joint convexity moreover the generalized peierls bogoliubov inequality is also proven

on alternative q weibull and q extreme value distributions properties and applications abstract tsallis statistics and tsallis distributions have been attracting a significant amount of research work in recent years importantly the tsallis statistics q distributions have been applied in different disciplines yet a relationship between some existing q weibull distributions and q extreme value distributions that is parallel to the well established relationship between the conventional weibull and extreme value distributions through a logarithmic transformation has not be established in this paper we proposed an alternative q weibull distribution that leads to a q extreme value distribution via the q logarithm transformation some important properties of the proposed q weibull and q extreme value distributions are studied maximum likelihood and least squares estimation methods are used to estimate the parameters of q weibull distribution and their performances are investigated through a monte carlo simulation study the methodologies and the usefulness of the proposed distributions are illustrated by fitting the 2014 traffic fatalities data from the national highway traffic safety administration

incomplete lu preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems we consider the preconditioned iterative solution of large dense linear systems where the coefficient matrix is a complex valued matrix arising from discretizing the integral equation of electromagnetic scattering for some scattering structures this matrix can be poorly conditioned the main purpose of this study is to evaluate the efficiency of a class of incomplete lu ilu factorization preconditioners for solving this type of matrices we solve the electromagnetic wave equations using the bicg method with an ilu preconditioner in the context of a multilevel fast multipole algorithm mlfma the novelty of this work is that the ilu preconditioner as constructed using the near part block diagonal submatrices generated from the mlfma experimental results show that the ilu preconditioner reduces the number of bicg iterations substantially compared to the block diagonal preconditioner the preconditioned iteration scheme also maintains the computational complexity of the mlfma and consequently reduces the total cpu time

parallel jacobian free newton krylov solution of the discrete ordinates method with flux limiters for 3d radiative transfer the present study introduces a parallel jacobian free newton krylov jfnk general minimal residual gmres solution for the discretized radiative transfer equation rte in 3d absorbing emitting and scattering media for the angular and spatial discretization of the rte the discrete ordinates method dom and the finite volume method fvm including flux limiters are employed respectively instead of forming and storing a large jacobian matrix jfnk methods allow for large memory savings as the required jacobian vector products are rather approximated by semiexact and numerical formulations for which convergence and computational times are presented parallelization of the gmres solution is introduced in a combined memory shared memory distributed formulation that takes advantage of the fact that only large vector arrays remain in the jfnk process results are presented for 3d test cases including a simple homogeneous isotropic medium and a more complex non homogeneous non isothermal absorbing emitting and anisotropic scattering medium with collimated intensities additionally convergence and stability of gram schmidt and householder orthogonalizations for the arnoldi process in the parallel gmres algorithms are discussed and analyzed overall the introduction of jfnk methods results in a parallel yet scalable to the tested 2048 processors and memory affordable solution to 3d radiative transfer problems without compromising the accuracy and convergence of a newton like solution

upper continuity bounds on the relative q entropy for q 1 generalized entropies and relative entropies are the subject of active research similar to the standard relative entropy the relative q entropy is generally unbounded for q 1 upper bounds on the quantum relative q entropy in terms of norm distances between its arguments are obtained in finite dimensional context these bounds characterize a continuity property in the sense of fannes

preconditioning techniques for nonsymmetric and indefinite linear systems abstract the standard preconditioning techniques for conjugate gradient methods often fail for matrices that are indefinite and or strongly nonsymmetric the most common alterative considered for these cases are either to use expensive direct solvers or to resort to one of many techniques based on the normal equations this paper examines several such alternatives and compares them in particular an incomplete lq factorization is proposed and some of its implementation details are described a number of experiments are reported to compare these methods

continuity bounds on the quantum relative entropy ii the quantum relative entropy is frequently used as a distance measure between two quantum states and inequalities relating it to other distance measures are important mathematical tools in many areas of quantum information theory we have derived many such inequalities in previous work the present paper is a follow up on this and provides a sharp upper bound on the relative entropy in terms of the trace norm distance and of the smallest eigenvalues of both states concerned the result obtained here is more general than the corresponding one from our previous work as a corollary we obtain a sharp upper bound on the regularised relative entropy introduced by lendi farhadmotamed and van wonderen

general existence theorems for unilateral problems in continuum mechanics the problem of minimizing a possibly non convex and non coercive functional is studied either necessary or sufficient conditions for the existence of solutions are given involving a generalized recession functional whose properties are discussed thoroughly the abstract results are applied to find existence of equilibrium configurations of a deformable body subject to a system of applied forces and partially constrained to lie inside a possibly unbounded region

on the eigenvalue distribution of a class of preconditioning methods a class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationax b wherea is a symmetric positive definite matrix the methods are based on an incomplete factorization of the matrixa and include both pointwise and blockwise factorization we study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofc 1a wherec is the preconditioning matrix we also show graphic representations of the eigenvalues and present numerical tests of the methods

generalization of the kullback leibler divergence in the tsallis statistics abstract in this paper we present a generalization of the kullerback leibler kl divergence in form of the tsallis statistics in parallel with the classical kl divergence several important properties of this new generalization including the pseudo additivity positivity and monotonicity are shown moreover some strengthened estimates on the positivity of the new divergence and the information loss during transformations are obtained

fast unsteady flow computations with a jacobian free newton krylov algorithm despite the advances in computer power and numerical algorithms over the last decades solutions to unsteady flow problems remain computing time intensive especially for large reynolds number flows nonlinear multigrid which is commonly used to solve the nonlinear systems of equations converges slowly the stiffness induced by the large aspect ratio cells and turbulence is not tackled well by this solution method in previous work we showed that a jacobian free newton krylov jfnk algorithm preconditioned with an approximate factorization of the jacobian that approximately matches the target residual operator enables a speed up of a factor of 10 compared to standard nonlinear multigrid for two dimensional large reynolds number unsteady flow computations the goal of this paper is to demonstrate that the jfnk algorithm is also suited to tackle the stiffness induced by the maximum aspect ratio the grid density the physical time step and the reynolds number compared to standard nonlinear multigrid speed ups up to a factor of 25 are achieved

on pinsker s and vajda s type inequalities for csiszar s f divergences let d and v denote respectively information divergence and total variation distance pinsker s and vajda s inequalities are respectively d 1 2 v2 and d log 2 v 2 v 2v 2 v in this paper several generalizations and improvements of these inequalities are established for wide classes of f divergences first conditions on f are determined under which an f divergence df will satisfy df cf v2 or df c2 f v2 c4 f v4 where the constants cf c2 f and c4 f are best possible as a consequence lower bounds in terms of v are obtained for many well known distance and divergence measures including the 2 and hellinger s discrimination and the families of tsallis and renyi s divergences for instance if d p q 1 1 p q1 d 1 and p q 1 1 log p q1 d are respectively the relative information of type and the renyi s information gain of order it is shown that d 1 2 v2 1 72 1 2 v4 whenever 1 2 0 1 and that 2 v2 1 36 1 5 5 2 v4 for 0 1 in a somewhat different direction and motivated by the fact that these pinsker s type lower bounds are accurate only for small variation v close to zero lower bounds for df which are accurate for both small and large variation v close to two are also obtained in the special case of the information divergence they imply that d log 2 2 v 2 v 2 log 2 v 2 which uniformly improves vajda s inequality

guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems abstract this paper presents incomplete decompositions for various types of matrices as they occur in the implicit discretisation of practical problems a review is given of methods for the usual five point discretisation of a self adjoint elliptic second order partial differential equation in two dimensions on a square the matrices which occur in this type of problem are symmetric m matrices of very regular structure the convergence behaviour of the different decompositions for this case is demonstrated by numerical experiments the paper also gives decompositions for the following type of matrices i symmetric m matrices of a different structure ii symmetric positive definite matrices iii non symmetric matrices

convergence of simulated annealing using the generalized transition probability we prove weak ergodicity of the inhomogeneous markov process generated by the generalized transition probability of tsallis and stariolo under power law decay of the temperature we thus have a mathematical foundation to conjecture convergence of simulated annealing processes with the generalized transition probability to the minimum of the cost function an explicitly solvable example in one dimension is analyzed in which the generalized transition probability leads to a fast convergence of the cost function to the optimal value we also investigate how far our arguments depend upon the specific form of the generalized transition probability proposed by tsallis and stariolo it is shown that a few requirements on analyticity of the transition probability are sufficient to assure fast convergence in the case of the solvable model in one dimension

a stability analysis of incomplete lu factorizations abstract the combination of iterative methods with preconditionings based on incomplete lu factorizations constitutes an effective class of methods for solving the sparse linear systems arising from the discretization of elliptic partial differential equations in this paper we show that there are some settings in which the incomplete lu preconditioners are not effective and we demonstrate that their poor performance is due to numerical instability our analysis consists of an analytic and numerical study of a sample two dimensional non self adjoint elliptic problem discretized by several finite difference schemes author

a two phase preconditioning strategy of sparse approximate inverse for indefinite matrices a two phase preconditioning strategy based on a factored sparse approximate inverse is proposed for solving sparse indefinite matrices in each phase the strategy first makes the original matrix diagonally dominant to enhance the stability by a shifting method and constructs an inverse approximation of the shifted matrix by utilizing a factored sparse approximate inverse preconditioner the two inverse approximation matrices produced from each phase are then combined to be used as a preconditioner experimental results show that the presented strategy improves the accuracy and the stability of the preconditioner on solving indefinite sparse matrices furthermore the strategy ensures that convergence rate of the preconditioned iterations of the two phase preconditioning strategy is much better than that of the standard sparse approximate inverse ones for solving indefinite matrices

analysis of incomplete factorizations for a nine point approximation to a convection diffusion model problem we study the stability of zero fill incomplete lu factorizations of a nine point coefficient matrix arising from a high order compact discretisation of a two dimensional constant coefficient convection diffusion problem nonlinear recurrences for computing entries of the lower and upper triangular matrices are derived and we show that the sequence of diagonal entries of the lower triangular factor is unconditionally convergent a theoretical estimate of the limiting value is derived and we show that this estimate is a good predictor of the computed value the unconditional convergence of the diagonal sequence of the lower triangular factor to a positive limit implies that the incomplete factorization process never encounters a zero pivot and that the other diagonal sequences are also convergent the characteristic polynomials associated with the lower and upper triangular solves that occur during the preconditioning step are studied and conditions for the stability of the triangular solves are derived in terms of the entries of the tridiagonal matrices appearing in the lower and upper subdiagonals of the block triangular system matrix and a triplet of parameters which completely determines the solution of the nonlinear recursions results of ilu preconditioned gmres iterations and the effects of orderings on their convergence are also described

stochastic preconditioning for diagonally dominant matrices this paper presents a new stochastic preconditioning approach for large sparse matrices for the class of matrices that are rowwise and columnwise irreducibly diagonally dominant we prove that an incomplete rm ldl t factorization in a symmetric case or an incomplete ldu factorization in an asymmetric case can be obtained from random walks and used as a preconditioner it is argued that our factor matrices have better quality i e better accuracy size trade offs than preconditioners produced by existing incomplete factorization methods therefore a resulting preconditioned krylov subspace iterative solver requires less computation than traditional methods to solve a set of linear equations with the same error tolerance the advantage increases for larger and denser matrices these claims are verified by numerical tests and we provide techniques that can potentially extend the theory to non diagonally dominant matrices

iterative solutions to the steady state density matrix for optomechanical systems we present a sparse matrix permutation from graph theory that gives stable incomplete lower upper preconditioners necessary for iterative solutions to the steady state density matrix for quantum optomechanical systems this reordering is efficient adding little overhead to the computation and results in a marked reduction in both memory and runtime requirements compared to other solution methods with performance gains increasing with system size either of these benchmarks can be tuned via the preconditioner accuracy and solution tolerance this reordering optimizes the condition number of the approximate inverse and is the only method found to be stable at large hilbert space dimensions this allows for steady state solutions to otherwise intractable quantum optomechanical systems

entropy and global existence for hyperbolic balance laws this paper presents a general result on the existence of global smooth solutions to hyperbolic systems of balance laws in several space variables we propose an entropy dissipation condition and prove the existence of global smooth solutions under initial data close to a constant equilibrium state in addition we show that a system of balance laws satisfies a kawashima condition if and only if its first order approximation namely the hyperbolic parabolic system derived through the chapman enskog expansion satisfies the corresponding kawashima condition the result is then applied to bouchut s discrete velocity bgk models approximating hyperbolic systems of conservation laws

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators

efficient preconditioning of sequences of nonsymmetric linear systems we present a new approach for approximate updates of factorized nonsymmetric preconditioners for solving sequences of linear algebraic systems this approach is algebraic and it is theoretically motivated it generalizes diagonal updates introduced by benzi and bertaccini bit 43 2003 pp 231 244 and bertaccini electron trans numer anal 18 2004 pp 49 64 it is shown experimentally that this approach can be very beneficial for example it is successful in significantly decreasing the number of iterations of a preconditioned iterative method for solving subsequent systems of a sequence when compared with freezing the preconditioner from the first system of the sequence in some cases the updated preconditioners offer a rate of convergence similar to or even higher than the rate obtained when preconditioning with recomputed preconditioners since the updates are typically cheap and straightforward their use is of practical interest they can replace recomputing preconditioners which is often expensive especially in parallel and matrix free environments