compressed graphs and the minimum degree algorithm the minimum degree algorithm is well known and widely used to find a permutation matrix p such that a sparse symmetric positive definite matrix a can be factored as pap t ll t to yield a lower triangular matrix l with relatively few nonzero entries in this paper we show that by compressing the graph of a when possible the run time of the minimum degree algorithm can be significantly reduced some empirical results for a collection of test matrices are presented

parallel preconditioners for monolithic solution of shear bands shear bands are one of the most fascinating instabilities in metals that occur under high strain rates they describe narrow regions on the order of micron scales where plastic deformations and significant heating are localized which eventually leads to fracture nucleation and failure of the component in this work shear bands are described by a set of four strongly coupled thermo mechanical equations discretized by a mixed finite element formulation a thermo viscoplastic flow rule is used to model the inelastic constitutive law and finite thermal conductivity is prescribed which gives rise to an inherent physical length scale governed by competition of shear heating and thermal diffusion the residual equations are solved monolithically by a newton type method at every time step and have been shown to yield mesh insensitive result the jacobian of the system is sparse and has a nonsymmetric block structure that varies with the different stages of shear bands formation the aim of the current work is to develop robust parallel preconditioners to gmres in order to solve the resulting jacobian systems efficiently the main idea is to design schur complements tailored to the specific block structure of the system and that account for the varying stages of shear bands we develop multipurpose preconditioners that apply to standard irreducible discretizations as well as our recent work on isogeometric discretizations of shear bands the proposed preconditioners are tested on benchmark examples and compared to standard state of practice solvers such as gmres ilu and lu direct solvers nonlinear and linear iterations counts as well as cpu times and computational speedups are reported and it is shown that the proposed preconditioners are robust efficient and outperform traditional state of the art solvers

least squares based lattice boltzmann method a meshless approach for simulation of flows with complex geometry a version of lattice boltzmann method lbm is presented in this work which is derived from the standard lbm by using taylor series expansion and optimized by the least squares method the method is basically meshless and can be applied to any complex geometry and nonuniform grids it can also be applied to different lattice models the proposed method explicitly updates the distribution functions at mesh points by an algebraic formulation in which the relevant coefficients are precomputed from the coordinates of mesh points we have successfully applied this method to simulate many two dimensional incompressible viscous flows the numerical results are very accurate and the computational time needed is much less as compared with other existing methods in this paper we mainly show the method

a direct solver for elliptic pdes in three dimensions based on hierarchical merging of poincare steklov operators a numerical method for variable coefficient elliptic pdes on three dimensional domains is described the method is designed for problems with smooth solutions and is based on a multidomain spectral collocation discretization scheme the resulting system of linear equations can very efficiently be solved using a nested dissection style direct as opposed to iterative solver this makes the scheme particularly well suited to solving problems for which iterative solvers struggle in particular for problems with oscillatory solutions a principal feature of the scheme is that once the solution operator has been constructed the actual solve is extremely fast an upper bound on the asymptotic cost of the build stage of o n 4 3 is proved for the case where the pde is held fixed as n increases the solve stage has close to linear complexity the scheme requires a relatively large amount of storage per degree of freedom but since it is a high order scheme a small number of degrees of freedom is sufficient to achieve high accuracy the method is presented for the case where there is no body load present but it can with little difficulty be generalized to the non homogeneous case numerical experiments demonstrate that the scheme is capable of solving helmholtz type equations on a domain of size 20 20 20 wavelengths to three correct digits on a modest personal workstation with n 2 10 6

lattice boltzmann simulations on irregular grids introduction of the natrium library the lattice boltzmann method is a modern approach to simulate fluid flow in its original formulation it is restricted to regular grids second order discretizations and a unity cfl number this paper describes our new off lattice boltzmann solver natrium an extensible and parallel c code to perform lattice boltzmann simulations on irregular grids natrium also allows high order spatial discretizations and non unity cfl numbers to be used we demonstrate how these features can efficiently decrease the number of grid points required in a simulation and thus reduce the computational time compared to the standard lattice boltzmann method we detail the implementation of a recently proposed semi lagrangian lattice boltzmann method and prove its efficiency in comparisons to other state of the art off lattice boltzmann schemes

efficient structured multifrontal factorization for general large sparse matrices rank structures provide an opportunity to develop new efficient numerical methods for practical problems when the off diagonal blocks of certain dense intermediate matrices have small numerical

a geometrical formulation of fermionic integrable systems a fermionic hamiltonian system is formulated on a supermanifold it is shown that if the system possesses a bi hamiltonian structure one can naturally define a lax equation associated with a 1 1 tensor on this supermanifold and this allows one to construct a set of conserved quantities furthermore if the corresponding nijenhuis tensor vanishes it is shown that all these conserved quantities would be in involution which is a sufficient condition for integrability of the system the fermionic extension of the kdv equation with a bi hamiltonian structure within this geometrical approach is studied

efficient preconditioning of hp fem matrix sequences with slowly varying coefficients an application to topology optimization abstract we previously introduced a preconditioner that has proven effective for h p fem discretizations of various challenging elliptic and hyperbolic problems the construction is inspired by standard nested dissection and relies on the assumption that the schur complements can be approximated to high precision by hierarchically semi separable matrices the preconditioner is built as an approximate l d m t factorization through a divide and conquer approach this implies an enhanced flexibility which allows to handle unstructured geometric meshes anisotropies and discontinuities we build on our previous numerical experiments and develop a preconditioner update strategy that allows us to handle matrix sequences arising from problems with slowly varying coefficients we investigate the performance of the preconditioner along with the update strategy in context of topology optimization of an acoustic cavity

data sparse approximation to a class of operator valued functions in earlier papers we developed a method for the data sparse approximation of the solution operators for elliptic parabolic and hyperbolic pdes based on the dunford cauchy representation to the operator valued functions of interest combined with the hierarchical matrix approximation of the operator resolvents in the present paper we discuss how these techniques can be applied to approximate a hierarchy of the operator valued functions generated by an elliptic operator

the approximation of parabolic equations involving fractional powers of elliptic operators we study the numerical approximation of a time dependent equation involving fractional powers of an elliptic operator l defined to be the unbounded operator associated with a hermitian coercive and bounded sesquilinear form on h 0 1 the time dependent solution u x t is represented as a dunford taylor integral along a contour in the complex plane the contour integrals are approximated using sinc quadratures in the case of homogeneous right hand sides and initial value v the approximation results in a linear combination of functions z q i l 1 v h 0 1 for a finite number of quadrature points z q lying along the contour in turn these quantities are approximated using complex valued continuous piecewise linear finite elements our main result provides l 2 error estimates between the solution u t and its final approximation numerical results illustrating the behavior of the algorithms are provided

decay bounds for the numerical quasiseparable preservation in matrix functions abstract given matrices a and b such that b f a where f z is a holomorphic function we analyze the relation between the singular values of the off diagonal submatrices of a and b we provide a family of bounds which depend on the interplay between the spectrum of the argument a and the singularities of the function in particular these bounds guarantee the numerical preservation of quasiseparable structures under mild hypotheses we extend the dunford cauchy integral formula to the case in which some poles are contained inside the contour of integration we use this tool together with the technology of hierarchical matrices h matrices for the effective computation of matrix functions with quasiseparable arguments

hypergraph partitioning based fill reducing ordering for symmetric matrices a typical first step of a direct solver for the linear system mx b is reordering of the symmetric matrix m to improve execution time and space requirements of the solution process in this work we propose a novel nested dissection based ordering approach that utilizes hypergraph partitioning our approach is based on the formulation of graph partitioning by vertex separator gpvs problem as a hypergraph partitioning problem this new formulation is immune to deficiency of gpvs in a multilevel framework and hence enables better orderings in matrix terms our method relies on the existence of a structural factorization of the input m matrix in the form of m aa t or m ad 2a t we show that the partitioning of the row net hypergraph representation of the rectangular matrix a induces a gpvs of the standard graph representation of matrix m in the absence of such factorization we also propose simple yet effective structural factorization techniques that are based on finding an edge clique cover of the standard graph representation of matrix m and hence applicable to any arbitrary symmetric matrix m our experimental evaluation has shown that the proposed method achieves better ordering in comparison to state of the art graph based ordering tools even for symmetric matrices where structural m aa t factorization is not provided as an input for matrices coming from linear programming problems our method enables even faster and better orderings

exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in banach spaces new exponentially convergent algorithms for the operator exponential generated by a strongly positive operator a in a banach space x are proposed these algorithms are based on representations by a dunford cauchy integral along paths enveloping the spectrum of a combined with a proper quadrature involving a short sum of resolvents where the choice of the integration path dramatically affects desired features of the algorithms a parabola and a hyperbola are analyzed as the integration paths and scales of estimates of dependence on the smoothness of initial data i e of the initial vector and of the inhomogeneous right hand side are obtained one of the algorithms possesses an exponential convergence rate for the operator exponential e at for all t ge 0 including the initial point this allows one to construct an exponentially convergent algorithm for inhomogeneous initial value problems the algorithm is parallelizable it turns out that the resolvent must be modified in order to get numerically stable algorithms near the initial point the efficiency of the proposed method is demonstrated by numerical examples

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

improving the run time and quality of nested dissection ordering when performing sparse matrix factorization the ordering of matrix rows and columns has a dramatic impact on the factorization time this paper describes an approach to the reordering problem that produces significantly better orderings than prior methods the algorithm is a hybrid of nested dissection and minimum degree ordering and combines an assortment of different algorithmic advances new or improved algorithms are described for graph compression multilevel partitioning and separator improvement when these techniques are combined the resulting orderings average 39 better than minimum degree over a suite of test matrices while requiring roughly 2 7 times the run time of liu s multiple minimum degree

on the exponential of semi infinite quasi toeplitz matrices let a z sum i in mathbb z a iz i be a complex valued function defined for z 1 such that sum i in mathbb z a i infty define t a t i j i j in mathbb z t i j a j i for i j in mathbb z the semi infinite toeplitz matrix associated with the symbol a z let e e i j i j in mathbb z be a compact operator in ell p with 1 le p le infty a semi infinite matrix of the kind a t a e is said quasi toeplitz qt the problem of the computation of exp a or exp a v with a quasi toeplitz and v a vector arises in many applications we prove that the exponential of a qt matrix a is qt that is exp a t exp a f where f is a compact operator in ell p this property allows the design of an algorithm for computing exp a and exp a v up to any precision the case of families of n times n matrices obtained by truncating infinite qt matrices to finite size is also considered numerical experiments show the effectiveness of this approach

algorithms without accuracy saturation for evolution equations in hilbert and banach spaces we consider the cauchy problem for the first and the second order differential equations in banach and hilbert spaces with an operator coefficient a t depending on the parameter t we develop discretization methods with high parallelism level and without accuracy saturation i e the accuracy adapts automatically to the smoothness of the solution for analytical solutions the rate of convergence is exponential these results can be viewed as a development of parallel approximations of the operator exponential e ta and of the operator cosine family cos at with a constant operator a possessing exponential accuracy and based on the sinc quadrature approximations of the corresponding dunford cauchy integral representations of solutions or the solution operators

reduction techniques for infinite dimensional hamiltonian systems some ideas and applications in the language of tensor analysis on differentiable manifolds we present a reduction method of integrability structures and apply it to recover some well known hierarchies of integrable nonlinear evolution equations

mathcal h h matrix approximability of the inverses of fem matrices we study the question of approximability for the inverse of the fem stiffness matrix for scalar second order elliptic boundary value problems by blockwise low rank matrices such as those given by the mathcal h h matrix format introduced by hackbusch computing 62 2 89 108 1999 we show that exponential convergence in the local block rank r r can be achieved we also show that exponentially accurate lu lu decompositions in the mathcal h h matrix format are possible for the stiffness matrices arising in the fem our analysis avoids any coupling of the block rank r r to the mesh width h h we also cover fairly general boundary conditions of mixed dirichlet neumann robin boundary conditions

domain decomposition based ℌ lu preconditioners hierarchical matrices provide a data sparse way to approximate fully populated matrices the two basic steps in the construction of an h matrix are a the hierarchical construction of a matrix block partition and b the blockwise approxi mation of matrix data by low rank matrices in this paper we develop a new approach to construct the necessary partition based on domain decomposition compared to standard geometric bisection based h matrices this new approach yields h lu fac torizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements these rigorously proven and numerically verified improvements result from an h matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an lu factorization we provide numerical results in which a domain decom position based h lu factorization is used as a preconditioner in the iterative solution of the discrete three dimensional convection diffusion equation

continuous integrable systems with multi poisson brackets for discrete finite systems it is known that the zero nijenhuis tensor condition can be used to explicitly construct conserved quantities in involution although the same method is not directly generalizable for continuous systems because of the divergence problem the difficulty can be overcome with some minor modifications in this way various cases of kdv nonlocal kdv continuous toda lattice kac moody hierarchy and a model based upon a w3 algebra systematically by the same technique can be discussed

lattice boltzmann method for groundwater flow in non orthogonal structured lattices the efficiency of the lattice boltzmann method lbm in modeling isotropic groundwater flow in domains of arbitrary geometry has been investigated the poisson equation was transformed in general curvilinear coordinates the corresponding equilibrium function for the d2q9 square lattice based on metric function between the physical and the computational domain has been established the resulting lbm was checked on examples having higher generality flows in confined and unconfined aquifers in vertical and horizontal plane have been considered in addition the phreatic water table representing upper boundary in the vertical plane was determined by the dynamic stretching approach not requiring complex concepts for dealing with the free surface like the volume of fluid method the accuracy and stability of the model is controlled by the adaptive mesh concept this allows application of higher density grid in critical areas with high pressure and velocity gradients and vice versa the number of computation points is significantly reduced without loosing accuracy the basic characteristics of the lbm including features like parallelization and simplicity are maintained the advantages of the proposed curvilinear lbm in modeling groundwater flow in domains of complex shape over the former published methods is demonstrated by three examples

exponentially convergent duhamel like algorithms for differential equations with an operator coefficient possessing a variable domain in a banach space a suitable abstract setting of an initial value problem for the first order operator differential equation is introduced for the case when both the unbounded operator coefficient and its domain can depend on the differentiation variable t a new exponentially convergent algorithm is proposed the algorithm is based on a generalization of the duhamel integral for vector valued functions the duhamel like technique makes it possible to translate the initial problem to an integral equation and then approximate it with exponential accuracy the theoretical results are illustrated by examples associated with the heat transfer boundary value problems

mathcal h matrix based second moment analysis for rough random fields and finite element discretizations we consider the efficient solution of strongly elliptic partial differential equations with random load based on the finite element method the solution s two point correlation can efficiently be approximated by means of an mathcal h matrix in particular if the correlation length is rather short or the correlation kernel is nonsmooth since the inverses of the finite element matrices which correspond to the differential operator under consideration can likewise efficiently be approximated in the mathcal h matrix format we can solve the correspondent mathcal h matrix equation in essentially linear time by using the mathcal h matrix arithmetic numerical experiments for three dimensional finite element discretizations for several correlation lengths and different smoothness are provided they validate the presented method and demonstrate that the computation times do not increase for nonsmooth or shortly correlated data

parallel accelerated cyclic reduction preconditioner for three dimensional elliptic pdes with variable coefficients abstract we present a robust and scalable preconditioner for the solution of large scale linear systems that arise from the discretization of elliptic pdes amenable to rank compression the preconditioner is based on hierarchical low rank approximations and the cyclic reduction method the setup and application phases of the preconditioner achieve log linear complexity in memory footprint and number of operations and numerical experiments exhibit good weak and strong scalability at large processor counts in a distributed memory environment numerical experiments with linear systems that feature symmetry and nonsymmetry definiteness and indefiniteness constant and variable coefficients demonstrate the preconditioner applicability and robustness furthermore it is possible to control the number of iterations via the accuracy threshold of the hierarchical matrix approximations and their arithmetic operations and the tuning of the admissibility condition parameter together these parameters allow for optimization of the memory requirements and performance of the preconditioner

a fully implicit method for lattice boltzmann equations existing approaches for solving the lattice boltzmann equations with finite difference methods are explicit and semi implicit both have certain stability constraints on the time step size in this work a fully implicit second order finite difference scheme is developed we focus on a parallel highly scalable newton krylov ras algorithm for the solution of a large sparse nonlinear system of equations arising at each time step here ras is a restricted additive schwarz preconditioner based on a first order spatial discretization we show numerically that by using the fully implicit method the time step size is no longer constrained by the cfl condition and the newton krylov ras algorithm is scalable on a supercomputer with more than ten thousand processors moreover to calculate the steady state solution we investigate an adaptive time stepping strategy the total compute time required by the implicit method with adaptive time stepping is much smaller than that of an explicit method for several t

h matrix approximation for the operator exponential with applications we develop a data sparse and accurate approximation to parabolic solution operators in the case of a rather general elliptic part given by a strongly p positive operator 4

multi core parallel robust structured multifrontal factorization method for large discretized pdes in this paper based on the current mainstream multi core architecture of parallel computer and the robust structured multifrontal factorization in brief rsmf method we propose a multi core parallelization of rsmf in brief mrsmf method mrsmf method parallelizes the nested dissection ordering symbolic decomposition and numerical decomposition of rsmf method which aims to implement these operations on the multi core parallel machine the multi core parallelization of symbolic decomposition and numerical decomposition are based on the binary elimination tree numerical experiments show that the mrsmf method is effective

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

a parallel method for time discretization of parabolic problems based on contour integral representation and quadrature we treat the time discretization of an initial value problem for a homogeneous abstract parabolic equation by first using a representation of the solution as an integral along the boundary of a sector in the right half of the complex plane then transforming this into a real integral on the finite interval and finally applying a standard quadrature formula to this integral the method requires the solution of a finite set of elliptic problems with complex coefficients which are independent and may therefore be done in parallel the method is combined with spatial discretization by finite elements

existence of ℋ matrix approximants to the inverse fe matrix of elliptic operators with l coefficients this article deals with the existence of blockwise low rank approximants so called matrices to inverses of fem matrices in the case of uniformly elliptic operators with l coefficients unlike operators arising from boundary element methods for which the matrix theory has been extensively developed the inverses of these operators do not benefit from the smoothness of the kernel function however it will be shown that the corresponding green functions can be approximated by degenerate functions giving rise to the existence of blockwise low rank approximants of fem inverses numerical examples confirm the correctness of our estimates as a side product we analyse the matrix property of the inverse of the fe mass matrix

error analysis and iterative solvers for navier stokes projection methods with standard and sparse grad div stabilization abstract this paper shows that use of a recently introduced sparse grad div stabilization can increase the accuracy of projection methods for solving the navier stokes equations sparse grad div stabilization has recently been introduced as an alternative to standard grad div stabilization which has a sparser matrix representation for both sparse and standard grad div stabilized projection methods we prove error estimates and provide numerical experiments which reveal that both stabilizations can cause a significant decrease in the error we then compare iterative solvers for the linear systems of equations arising from the use of both of the stabilizations a theoretical analysis of a simplified model problem as well as numerical tests show that iterative solvers perform better for systems arising from sparse grad div compared to standard grad div stabilized systems

an alternate characterization of integrability this paper will show that the existence of at least three independent symplectic forms related in a simple way on the phase space of a dynamical system is a sufficient condition for the integrability of the system

approximate iterations for structured matrices important matrix valued functions f a are e g the inverse a 1 the square root sqrt a and the sign function their evaluation for large matrices arising from pdes is not an easy task and needs techniques exploiting appropriate structures of the matrices a and f a often f a possesses this structure only approximately however intermediate matrices arising during the evaluation may lose the structure of the initial matrix this would make the computations inefficient and even infeasible however the main result of this paper is that an iterative fixed point like process for the evaluation of f a can be transformed under certain general assumptions into another process which preserves the convergence rate and benefits from the underlying structure it is shown how this result applies to matrices in a tensor format with a bounded tensor rank and to the structure of the hierarchical matrix technique we demonstrate our results by verifying all requirements in the case of the iterative computation of a 1 and sqrt a

an exponentially convergent algorithm for nonlinear differential equations in banach spaces an exponentially convergent approximation to the solution of a nonlinear first order differential equation with an operator coefficient in banach space is proposed the algorithm is based on an equivalent volterra integral equation including the operator exponential generated by the operator coefficient the operator exponential is represented by a dunford cauchy integral along a hyperbola enveloping the spectrum of the operator coefficient and then the integrals involved are approximated using the chebyshev interpolation and an appropriate sinc quadrature numerical examples are given which confirm theoretical results

a fast direct solver for elliptic problems on general meshes in 2d we present a fast direct algorithm for solutions to linear systems arising from 2d elliptic equations we follow the approach in xia et al 2009 on combining the multifrontal method with hierarchical matrices we present a variant of that approach with additional hierarchical structure extend it to quasi uniform meshes and detail an adaptive decomposition procedure for general meshes linear time complexity is shown for a quasi regular grid and demonstrated via numerical results for the adaptive algorithm

distributed lagrange multiplier fictitious domain method in the framework of lattice boltzmann method for fluid structure interactions a new implementation of the lattice boltzmann method lbm for fluid structure interactions is presented the idea of the distributed lagrange multiplier fictitious domain method dlm fd is introduced in the framework of the lattice boltzmann algorithm this implementation employs a fixed mesh for the solution of the fluid problem and a lagrangian formulation for the solid problem the main advantage of the method is that the re meshing procedure normally required in the ale method is circumvented numerical examples are provided to verify the algorithm and illustrate the capacity of the method to deal with the fluid elastic solid interactions

domain decomposition based mathcal h lu preconditioning hierarchical matrices provide a data sparse way to approximate fully populated matrices the two basic steps in the construction of an mathcal h matrix are a the hierarchical construction of a matrix block partition and b the blockwise approximation of matrix data by low rank matrices in this paper we develop a new approach to construct the necessary partition based on domain decomposition compared to standard geometric bisection based mathcal h matrices this new approach yields mathcal h lu factorizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements these rigorously proven and numerically verified improvements result from an mathcal h matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an lu factorization we provide numerical results in which a domain decomposition based mathcal h lu factorization is used as a preconditioner in the iterative solution of the discrete three dimensional convection diffusion equation

lattice boltzmann method for conjugate natural convection with heat generation on non uniform meshes abstract in this paper the taylor series expansion and least square based lattice boltzmann method tllbm is combined with a local lattice boltzmann lb conjugate heat transfer scheme to simulate different conjugate natural convection problems in a differentially heated enclosure on non uniform meshes four case studies are designed not only to verify the developed basic tllbm fortran code but also to validate the results of the tllbm for different conjugate natural convection problems the convergence study of the presented tllbm is also carried out and the order of accuracy of the method is reported in different mesh regions it is concluded that the results of streamlines isotherms temperature distributions along a test line and average nusselt numbers calculated on the cold and hot walls of the enclosure are compared well with those available in the literature from conventional numerical methods as well as the standard lbm the computational efficiency of the tllbm is also examined through comparing the results of simulations on uniform standard lbm and non uniform tllbm meshes in terms of the accuracy and cpu time needed for solutions to converge it is shown that the tllbm is able to give accurate results with lower cpu time and mesh sizes than the standard lbm

integrable dynamical systems with hierarchy i formulation it is shown that any model with the zero generalized nijenhuis tensor under some additional assumptions will automatically satisfy the hierarchy equation with an infinite number of conserved quantities in involution especially if there is a dual symplectic structure with zero nijenhuis tensor then there exist infinite numbers of poisson brackets and of lagrangians giving the same equation of motion toda lattice is one such example

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

approximation of functions of large matrices with kronecker structure we consider the numerical approximation of f mathcal a b f a b where b in mathbb r n b rn and mathcal a a is the sum of kronecker products that is mathcal a m 2 otimes i i otimes m 1 in mathbb r n times n a m2 i i m1 rn n here f is a regular function such that f mathcal a f a is well defined we derive a computational strategy that significantly lowers the memory requirements and computational efforts of the standard approximations with special emphasis on the exponential and on completely monotonic functions for which the new procedure becomes particularly advantageous our findings are illustrated by numerical experiments with typical functions used in applications

laplace transform method for parabolic problems with time dependent coefficients the laplace transform method has proven to be very efficient for dealing with parabolic problems whose coefficients are time independent and it is easily parallelizable however the method has not been proven to be applicable to linear problems whose coefficients are time dependent the reason is that the laplace transform of two time dependent functions leads to a convolution of the laplace transformed functions in the dual variable in this paper we propose a laplace transform method to linear parabolic problems with time dependent coefficients which is as efficient as the method for parabolic problems with time independent coefficients several numerical results are provided which support the efficiency of the proposed scheme

iterative methods for nearly singular linear systems iterative methods are developed and studied for near singular linear systems cx b our approach called the transformed minimal residual algorithm tmres is derived from any convergent iterative scheme sxk 1 txk b associated with a splitting c s t in each step of tmres the transformed residual s 1 b cx is minimized over a krylov space generated by s 1 t the original iterative scheme typically converges slowly when c is nearly singular while a krylov space generated by s 1 t often contains a much better approximation to a solution tmres is algebraically equivalent to the generalized minimal residual algorithm gmres preconditioned by s 1 although there are numerical differences since a different matrix s 1 c is used to generate the krylov space in preconditioned gmres special attention is given to sparsity and convergence issues related to linear systems of the form bf aa tr sigma bf i bf x bf b where sigma ge 0

minimizing the profile of a symmetric matrix new approaches are developed for minimizing the profile of a sparse symmetric matrix the heuristic approaches seek to minimize the profile growth either absolutely or in a weighted sense the exchange methods make a series of permutations in an initial ordering to strictly improve the profile comparisons with the spectral algorithm a level structure method and a wave front method are presented

exact and approximate solutions of some operator equations based on the cayley transform abstract we consider the operator equation sx j 1 m u j xv j y where u j v j are some communicative sets of operators but in general u j need not compute with v j particular cases of this equation are the sylvester and ljapunov equations we give a new representation and an approximation of the solution which is suitable to perform it algorithmically error estimates are given which show exponential covergence for bounded operators and polynomial convergence for unbounded ones based on these considerations we construct an iterative process and give an existence theorem for the operator equation z 2 a 1 z a 2 0 arising for example when solving an abstract second order differential equation with non commutative coefficients

a fast nested dissection solver for cartesian 3d elliptic problems using hierarchical matrices we present a fast algorithm for solutions to linear systems arising from three dimensional elliptic problems on a regular cartesian mesh we follow the approach of schmitz and ying 2012 on combining the nested dissection matrix factorization method with hierarchical matrices in two dimensions and extend it to the three dimensional case a theoretical linear time complexity is derived and a more practical variant with slightly worse scaling is demonstrated

hypergraph based unsymmetric nested dissection ordering for sparse lu factorization in this paper we discuss a hypergraph based unsymmetric nested dissection hund ordering for reducing the fill in incurred during gaussian elimination it has several important properties it takes a global perspective of the entire matrix as opposed to local heuristics it takes into account the asymmetry of the input matrix by using a hypergraph to represent its structure it is suitable for performing gaussian elimination in parallel with partial pivoting this is possible because the row permutations performed due to partial pivoting do not destroy the column separators identified by the nested dissection approach the hypergraph nested dissection approach is essentially equivalent to graph nested dissection on the matrix a t a but we need only the original matrix a and never form the usually denser matrix a t a the usage of hypergraphs in our approach is fairly standard and hund can be implemented by calling an existing hypergraph partitioner that uses recursive bisection our implementation uses local reordering constrained column approximate minimum degree ccolamd to further improve the ordering we also explain how weighted matching hsl routine mc64 can be used in this context experimental results on 27 medium and large size matrices with highly unsymmetric structures compare our approach to four other well known reordering algorithms the results show that it provides a robust reordering algorithm in the sense that it is the best or close to the best often within 10 of all the other methods in particular on matrices with highly unsymmetric structures

investigation of stability and hydrodynamics of different lattice boltzmann models stability and hydrodynamic behaviors of different lattice boltzmann models including the lattice boltzmann equation lbe the differential lattice boltzmann equation dlbe the interpolation supplemented lattice boltzmann method islbm and the taylor series expansion and least square based lattice boltzmann method tllbm are studied in detail our work is based on the von neumann linearized stability analysis under a uniform flow condition the local stability and hydrodynamic dissipation behaviors are studied by solving the evolution operator of the linearized lattice boltzmann equations numerically our investigation shows that the lbe schemes with interpolations such as dlbe islbm and tllbm improve the numerical stability by increasing hyper viscosities at large wave numbers small scales it was found that these interpolated lbe schemes with the upwind interpolations are more stable than those with central interpolations because of much larger hyper viscosities

a sparse h matrix arithmetic general complexity estimates in a preceding paper hackbusch computing 62 1999 89 108 a class of matrices h matrices has been introduced which are data sparse and allow an approximate matrix arithmetic of almost linear complexity several types of h matrices have been analysed in hackbusch computing 62 1999 89 108 and hackbusch and khoromskij preprint mpi no 22 leipzig 1999 computing 64 2000 21 47 which are able to approximate integral nonlocal operators in fem and bem applications in the case of quasi uniform unstructured meshes in the present paper the general construction of h matrices on rectangular and triangular meshes is proposed and analysed first the reliability of h matrices in bem is discussed then we prove the optimal complexity of storage and matrix vector multiplication in the case of rather arbitrary admissibility parameters 1 and for nite elements up to the order 1 dened on quasi uniform rectangular triangular meshes in r d d 1 2 3 the almost linear complexity of the matrix addition multiplication and inversion of h matrices is also veried c 2000 elsevier science b v all rights reserved msc 65f05 65f30 65f50

hierarchical interpolative factorization for elliptic operators differential equations this paper introduces the hierarchical interpolative factorization for integral equations hif ie associated with elliptic problems in two and three dimensions this factorization takes the form of an approximate generalized lu decomposition that permits the efficient application of the discretized operator and its inverse hif ie is based on the recursive skeletonization algorithm but incorporates a novel combination of two key features 1 a matrix factorization framework for sparsifying structured dense matrices and 2 a recursive dimensional reduction strategy to decrease the cost thus higher dimensional problems are effectively mapped to one dimension and we conjecture that constructing applying and inverting the factorization all have linear or quasilinear complexity numerical experiments support this claim and further demonstrate the performance of our algorithm as a generalized fast multipole method direct solver and preconditioner hif ie is compatible with geometric adaptivity and can handle both boundary and volume problems matlab codes are freely available 2015 wiley periodicals inc

a direct solver with o n complexity for variable coefficient elliptic pdes discretized via a high order composite spectral collocation method a numerical method for solving elliptic pdes with variable coefficients on two dimensional domains is presented the method is based on high order composite spectral approximations and is designed for problems with smooth solutions the resulting system of linear equations is solved using a direct as opposed to iterative solver that has optimal o n complexity for all stages of the computation when applied to problems with nonoscillatory solutions such as the laplace and the stokes equations numerical examples demonstrate that the scheme is capable of computing solutions with a relative accuracy of 10 10 or better for challenging problems such as highly oscillatory helmholtz problems and convection dominated convection diffusion equations in terms of speed it is demonstrated that a problem with a nonoscillatory solution that was discretized using 10 8 nodes can be solved in 115 minutes on a personal workstation with two quad core 3 3 ghz cpus since the solver is direct and the solution

a parallel shared memory implementation of a high order accurate solution technique for variable coefficient helmholtz problems abstract the recently developed hierarchical poincare steklov hps method is a high order discretization technique that comes with a direct solver results from previous papers demonstrate the method s ability to solve helmholtz problems to high accuracy without the so called pollution effect while the asymptotic scaling of the direct solver s computational cost is the same as the nested dissection method serial implementations of the solution technique are not practical for large scale numerical simulations this manuscript presents the first parallel implementation of the hps method specifically we introduce an approach for a shared memory implementation of the solution technique utilizing parallel linear algebra this approach is the foundation for future large scale simulations on supercomputers and clusters with large memory nodes performance results on a desktop computer resembling a large memory node are presented

bosonic reduction of susy generalized harry dym equation in this paper we construct the two component supersymmetric generalized harry dym equation which is integrable and study various properties of this model in the bosonic limit in particular in the bosonic limit we obtain a new integrable system which under a hodograph transformation reduces to a coupled three component system we show how the hamiltonian structure transforms under a hodograph transformation and study the properties of the model under a further reduction to a two component system we find a third hamiltonian structure for this system which has been shown earlier to be a bi hamiltonian system making this a genuinely tri hamiltonian system the connection of this system to the modified dispersive water wave equation is clarified we also study various properties in the dispersionless limit of our model

stability and regularization of three level difference schemes with unbounded operator coefficients in banach spaces the problem of stability of difference schemes for second order evolution problems is considered difference schemes are treated as abstract cauchy problems for difference equations with operator coefficients in a banach or hilbert space to construct stable difference schemes the regularization principle is employed i e one starts from any simple scheme possibly unstable and derives absolutely stable schemes by perturbing the operator coefficients the main result of this paper is the following for the first time sufficient conditions are pointed out under which an unstable three level difference scheme with unbounded operator coefficients in a banach space can be regularized to a stable scheme the principal stability condition is the strong p positivity of the unbounded operator coefficients

strongly p positive operators and explicit representations of the solutions of initial value problems for second order differential equations in banach space abstract the initial value problems for two second order differential equations with an unbounded operator coefficient a in a banach space are considered using a linear fractional transform the cayley transform of the operator a we give explicit formulas for the solution of these problems if the spectrum of a is situated inside of a parabola these formulas also provide the algorithmic representations of the operator cosine function and of the operator bessel function with the generator a being a basis for approximate solutions for which error estimates are given one of the important properties of our approach is the following the accuracy of the approximate solutions depends automatically on the regularity of the initial data

a neumann neumann preconditioned iterative substructuring approach for computing solutions to poisson s equation with prescribed jumps on an embedded boundary in multifluid problems with surface tension the fluid pressure and its derivative are discontinuous at fluid interfaces we present a cartesian grid embedded boundary method for numerically resolving these discontinuities in which we use neumann neumann preconditioned iterative substructuring to solve the governing equations we validate this method by computing several well known poisson problems with discontinuous coefficients and we compare its performance to an approach based on simple iteration by analogy with the conjugate gradient method we hypothesize that the scaling of the neumann neumann preconditioned iterative substructuring is o h dlnh 1 where h is the cell size and d 2 3 is the dimensionality of the problem in contrast we show that the simple iterative procedure scales like o h d 1 and is slower by a factor of 4000 for a small i e 64x64 cell model calculation with physical parameters corresponding to a 1 5mm air bubble in water we present an analytical model to explain the scaling of this iterative procedure

data sparse approximation to the operator valued functions of elliptic operator llin previous papers the arithmetic of hierarchical matrices has been described which allows us to compute the inverse for instance of finite element stiffness matrices discretising an elliptic operator l the required computing time is up to logarithmic factors linear in the dimension of the matrix in particular this technique can be used for the computation of the discrete analogue of a resolvent zi l 1 z c in the present paper we consider various operator functions the operator exponential e tl negative fractional powers l the cosine operator function cos t l l k and finally the solution operator of the lyapunov equation using the dunford cauchy representation we get integrals which can be discretised by a quadrature formula which involves the resolvents z k i l 1 mentioned above we give error estimates which are partly exponentially partly polynomially decreasing

the evolution of the minimum degree ordering algorithm over the past fifteen years the implementation of the minimum degree algorithm has received much study and many important enhancements have been made to it this paper describes these various enhancements their historical development and some experiments showing how very effective they are in improving the execution time of the algorithm a shortcoming is also presented that exists in all of the widely used implementations of the algorithm namely that the quality of the ordering provided by the implementations is surprisingly sensitive to the initial ordering for example changing the input ordering can lead to an increase or decrease of as much as a factor of three in the cost of the subsequent numerical factorization this sensitivity is caused by the lack of an effective tie breaking strategy and the authors experiments illustrate the importance of developing such a strategy