a zero delay sequential scheme for lossy coding of individual sequences we consider adaptive sequential lossy coding of bounded individual sequences when the performance is measured by the sequentially accumulated mean squared distortion the encoder and the decoder are connected via a noiseless channel of capacity r and both are assumed to have zero delay no probabilistic assumptions are made on how the sequence to be encoded is generated for any bounded sequence of length n the distortion redundancy is defined as the normalized cumulative distortion of the sequential scheme minus the normalized cumulative distortion of the best scalar quantizer of rate r which is matched to this particular sequence we demonstrate the existence of a zero delay sequential scheme which uses common randomization in the encoder and the decoder such that the normalized maximum distortion redundancy converges to zero at a rate n sup 1 5 log n as the length of the encoded sequence n increases without bound

an extrapolation method with step size control for nonlinear volterra integral equations in 6 it has been shown that the midpoint rule applied to second kind volterra integral equations possesses an asymptotic expansion in even powers of the stepsizeh in this paper we describe an extrapolation method based on the midpoint rule together with a mechanism of step size control

recycling krylov subspaces for efficient large scale electrical impedance tomography electrical impedance tomography eit captures images of internal features of a body electrodes are attached to the boundary of the body low intensity alternating currents are applied and the resulting electric potentials are measured then based on the measurements an estimation algorithm obtains the three dimensional internal admittivity distribution that corresponds to the image one of the main goals of medical eit is to achieve high resolution and an accurate result at low computational cost however when the finite element method fem is employed and the corresponding mesh is refined to increase resolution and accuracy the computational cost increases substantially especially in the estimation of absolute admittivity distributions therefore we consider in this work a fast iterative solver for the forward problem which was previously reported in the context of structural optimization we propose several improvements to this solver to increase its performance in the eit context the solver is based on the recycling of approximate invariant subspaces and it is applied to reduce the eit computation time for a constant and high resolution finite element mesh in addition we consider a powerful preconditioner and provide a detailed pseudocode for the improved iterative solver the numerical results show the effectiveness of our approach the proposed algorithm is faster than the preconditioned conjugate gradient cg algorithm the results also show that even on a standard pc without parallelization a high mesh resolution more than 150 000 degrees of freedom can be used for image estimation at a relatively low computational cost

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

on optimal finite state digital transmission systems the digital transmission of signals by transmitters and receivers that are time invariant finite state machines were investigated in general form an average distortion criterion is used some general structure theorems are proved and some examples given that show improvement in performance as the memories of the transmitter and receiver are increased many conjectures and unsolved problems are mentioned and areas for further study are indicated

source coding with limited look ahead side information at the decoder we characterize the rate distortion function for the source coding with decoder side information setting when the ith reconstruction symbol is allowed to depend only on the first i lscr side information symbols for some finite look ahead lscr in addition to the index from the encoder for the case of causal side information i e lscr 0 we find that the penalty of causality is the omission of the subtracted mutual information term in the wyner ziv rate distortion function for lscr 0 we derive a computable infinite letter expression for the rate distortion function when specialized to the near lossless case our results characterize the best achievable rate for the slepian wolf source coding problem with finite side information looka head and have some surprising implications we find that side information is useless for any fixed lscr when the joint probability mass function pmf of the source and side information satisfies the positivity condition p x y 0 for all x y more generally the optimal rate depends on the distribution of the pair x y only through the distribution of x and the bipartite graph whose edges represent the pairs x y for which p x y 0 on the other hand if side information look ahead is allowed to grow faster than logarithmic in the block length then h x y is achievable finally we apply our approach to derive a computable expression for channel capacity when state information is available at the encoder with limited look ahead

robust parallel smoothing for multigrid via sparse approximate inverses sparse approximate inverses are considered as smoothers for multigrid they are based on the spai algorithm m j grote and t huckle siam j sci comput 18 1997 pp 838 853 which constructs a sparse approximate inverse m of a matrix a by minimizing i ma in the frobenius norm this yields a new hierarchy of smoothers spai 0 spai 1 spai varepsilon advantages of spai smoothers over classical smoothers are inherent parallelism possible local adaptivity and improved robustness the simplest smoother spai 0 is based on a diagonal matrix m it is shown to satisfy the smoothing property for symmetric positive definite problems numerical experiments show that spai 0 smoothing is usually preferable to damped jacobi smoothing for the spai 1 smoother the sparsity pattern of m is that of a its performance is typically comparable to that of gauss seidel smoothing however both the computation and the application of the smoother remain inherently parallel in more difficult situations where the simpler spai 0 and spai 1 smoothers are not adequate the spai varepsilon smoother provides a natural procedure for improvement where needed numerical examples illustrate the usefulness of spai smoothing

large scale topology optimization using preconditioned krylov subspace methods with recycling summary the computational bottleneck of topology optimization is the solution of a large number of linear systems arising in the finite element analysis we propose fast iterative solvers for large threedimensional topology optimization problems to address this problem since the linear systems in the sequence of optimization steps change slowly from one step to the next we can significantly reduce the number of iterations and the runtime of the linear solver by recycling selected search spaces from previous linear systems in addition we introduce a minres minimum residual method version with recycling and a short term recurrence to make recycling more efficient for symmetric problems furthermore we discuss preconditioning to ensure fast convergence we show that a proper rescaling of the linear systems reduces the huge condition numbers that typically occur in topology optimization to roughly those arising for a problem with constant density we demonstrate the effectiveness of our solvers by solving a topology optimization problem with more than a million unknowns on a fast pc

on the structure of optimal entropy constrained scalar quantizers the nearest neighbor condition implies that when searching for a mean square optimal fixed rate quantizer it is enough to consider the class of regular quantizers i e quantizers having convex cells and codepoints which lie inside the associated cells in contrast quantizer regularity can preclude optimality in entropy constrained quantization this can be seen by exhibiting a simple discrete scalar source for which the mean square optimal entropy constrained scalar quantizer ecsq has disconnected and hence nonconvex cells at certain rates in this work new results concerning the structure and existence of optimal ecsqs are presented one main result shows that for continuous sources and distortion measures of the form d x y spl rho x y where spl rho is a nondecreasing convex function any finite level ecsq can be regularized so that the resulting regular quantizer has the same entropy and equal or less distortion regarding the existence of optimal ecsqs we prove that under rather general conditions there exists an almost regular optimal ecsq for any entropy constraint for the squared error distortion measure and sources with piecewise monotone and continuous densities the existence of a regular optimal ecsq is shown

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

multilevel sparse approximate inverse preconditioners for adaptive mesh refinement abstract we present an efficient and effective preconditioning method for time dependent simulations with dynamic adaptive mesh refinement and implicit time integration adaptive mesh refinement greatly improves the efficiency of simulations where the solution develops steep gradients in small regions of the computational domain that change over time unfortunately adaptive mesh refinement also introduces a number of problems for preconditioning in parallel iterative linear solvers as the changes in the mesh lead to structural changes in the linear systems we must solve hence we may need to compute a new preconditioner at every time step since this would be expensive we propose preconditioners that are cheap to adapt for dynamic changes to the mesh more specifically we propose preconditioners that require only localized changes to the preconditioner for localized changes in the mesh our preconditioners combine sparse approximate inverses and multilevel techniques we demonstrate significant improvements in convergence rates of krylov subspace methods and significant reductions of the overall runtime furthermore we demonstrate experimentally level independent convergence rates for various problems

a priori sparsity patterns for parallel sparse approximate inverse preconditioners parallel algorithms for computing sparse approximations to the inverse of a sparse matrix either use a prescribed sparsity pattern for the approximate inverse or attempt to generate a good pattern as part of the algorithm this paper demonstrates that for pde problems the patterns of powers of sparsified matrices psms can be used a priori as effective approximate inverse patterns and that the additional effort of adaptive sparsity pattern calculations may not be required psm patterns are related to various other approximate inverse sparsity patterns through matrix graph theory and heuristics concerning the pde s green s function a parallel implementation shows that psm patterned approximate inverses are significantly faster to construct than approximate inverses constructed adaptively while often giving preconditioners of comparable quality

multiresolution approximate inverse preconditioners we introduce a new preconditioner for elliptic pdes on unstructured meshes using a wavelet inspired basis we compress the inverse of the matrix allowing an effective sparse approximate inverse by solving the sparsity vs accuracy conflict the key issue in this compression is to use second generation wavelets which can be adapted to the unstructured mesh the true boundary conditions and even the pde coefficients we also show how this gives a new perspective on multiresolution algorithms such as multigrid interpreting the new preconditioner as a variation on node nested multigrid in particular we hope the new preconditioner will combine the best of both worlds fast convergence when multilevel methods can succeed but with robust performance for more difficult problems the rest of the paper discusses the core issues for the preconditioner ordering and construction of a factored approximate inverse in the multiresolution basis robust interpolation on unstructured meshes automatic mesh coarsening and purely algebraic alternatives some exploratory numerical experiments suggest the superiority of the new basis over the standard basis for several tough problems including discontinuous anisotropic coefficients strong convection and indefinite reaction problems on unstructured meshes with scalability like hierarchical basis methods achieved

collocation and iterated collocation methods for a class of weakly singular volterra integral equations we discuss the convergence properties of spline collocation and iterated collocation methods for a weakly singular volterra integral equation associated with certain heat conduction problems this work completes the previous studies of numerical methods for this type of equations with noncompact kernel in particular a global convergence result is obtained and it is shown that discrete superconvergence can be achieved with the iterated collocation if the exact solution belongs to some appropriate spaces some numerical examples illustrate the theoretical results

krylov subspace acceleration of nonlinear multigrid with application to recirculating flows this paper deals with the combination of two solution methods multigrid and gmres siam j sci comput 14 1993 pp 856 869 the generality and parallelizabili nonlinear pdes as the preconditioner for a nonlinear krylov subspace method we use the full approximation storage fas scheme math comp 31 1977 pp 333 390 a nonlinear multigrid method the nonlinear krylov acceleration is applied also on coarse grids so that recirculating incompressible flow problems discretized with a higher order upwind scheme can be solved efficiently

improving the accuracy of collocation solutions of volterra integral equations of the first kind by local interpolation we study the numerical solution of volterra integral equations of the first kind by collocation methods using piecewise polynomials of degreep we show that when superconvergence occurs the error in the collocation solution at the node points alternates in sign independent from the kernel or the exact solution evaluation at certain points of interpolating polynomials of degreep 2 annihilates the leading term in the asymptotic expansion of the error thus yielding convergence of orderp 3 the derivation of the asymptotic expansion of the error for the general equation is reduced to the special case of numerical differentiation by the use of an asymptotic continuity result for the projectors associated with the collocation method under compact perturbations we present some numerical results on the improved order of convergence forp 2 and 3

superconvergence of collocation methods for volterra and abel integral equations of the second kind this paper deals with the question of the attainable order of convergence in the numerical solution of volterra and abel integral equations by collocation methods in certain piecewise polynomial spaces and which are based on suitable interpolatory quadrature for the resulting moment integrals the use of a nonlinear variation of constants formula for the representation of the error function in terms of the defect allows for a unified treatment of equations with continuous and weakly singular kernels

optimal causal coding decoding problems the symbols produced by a finite markov source are causally encoded so as to be transmitted through a noisy memoryless channel the encoder is assumed to have channel feedback information and the decoder to be causal the feedback information is shown to be useful in general separation results are derived and used to prove that encoding is useless for a class of symmetric channels

approximation properties of quadrature methods for volterra integral equations of the first kind we present a unifying analysis of quadrature methods for volterra integral equa tions of the first kind that are zero stable and have an asymptotic repetition factor we show that such methods are essentially collocation projection methods with underlying subspaces that have nice approximation properties and which are stable as projection methods this is used to derive asymptotically optimal error estimates under minimal smoothness conditions the class of quadrature methods covered includes the cyclic linear multistep and the reducible quadrature methods but not really runge kutta methods 1 introduction we study approximation properties of quadrature methods for the numerical solution of volterra integral equations of the first kind 1 1 f x x9 x y f y dy g x xe 0 l o i e methods of the form i

efficient adaptive algorithms and minimax bounds for zero delay lossy source coding zero delay lossy source coding schemes are considered for both individual sequences and random sources performance is measured by the distortion redundancy which is defined as the difference between the normalized cumulative mean squared distortion of the scheme and the normalized cumulative distortion of the best scalar quantizer of the same rate that is matched to the entire sequence to be encoded by improving and generalizing a scheme of linder and lugosi weissman and merhav showed the existence of a randomized scheme that for any bounded individual sequence of length n achieves a distortion redundancy o n sup 1 3 logn however both schemes have prohibitive complexity both space and time which makes practical implementation infeasible in this paper we present an algorithm that computes weissman and merhav s scheme efficiently in particular we introduce an algorithm with encoding complexity o n sup 4 3 and distortion redundancy o n sup 1 3 logn the complexity can be made linear in the sequence length n at the price of increasing the distortion redundancy to o n sup 1 4 spl radic logn we also consider the problem of minimax distortion redundancy in zero delay lossy coding of random sources by introducing a simplistic scheme and proving a lower bound we show that for the class of bounded memoryless sources the minimax expected distortion redundancy is upper and lower bounded by constant multiples of n sup 1 2

collocation for volterra integral equations of the first kind with iterated kernel we apply collocation methods using piecewise polynomials to the numerical solution of volterra integral equations of the first kind with iterated kernel we view these methods as projection methods and derive necessary and sufficient conditions on the collocation points guaranteeing the boundedness of the projectors asymptotically optimal rates of convergence for the collocation method and a related quadrature method are proven and verified experimentally

a sparse approximate inverse preconditioner for the conjugate gradient method a method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix a is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations it is proved that in exact arithmetic the preconditioner is well defined if a is an h matrix the results of numerical experiments are presented

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

order of convergence of one step methods for volterra integral equations of the second kind nice proofs of convergence and asymptotic expansions are known for one step methods for ordinary differential equations it is shown that these proofs can be generalized in a natural way to extended one step methods for volterra integral equations of the second kind furthermore the convergence of mixed one step methods is investigated for both types general volterra runge kutta methods are considered as examples

natural continuous extensions of runge kutta methods for volterra integral equations of the second kind and their applications we consider a very general class of runge kutta methods for the numerical solution of volterra integral equations of the second kind which includes as special cases all the more important methods which have been considered in the literature the main purpose of this paper is to define and prove the existence of the natural continuous extensions nce s of runge kutta methods i e piecewise polynomial functions which extend the approximation at the grid points to the whole interval of integration the particular properties required of the nce s allow us to construct the tail approximations which are quite efficient in terms of kernel evaluations

a survey of recent advances in the numerical treatment of volterra integral and integro differential equations abstract the present survey paper samples recent advances in the numerical analysis of volterra integral equations of the first and second kind and of integro differential equations including equations with weakly singular kernels except for some important earlier references the discussion focuses on the development which has taken place during the last dozen years a fairly extensive bibliography selected to be representative rather than comprehensive complements the paper

combination of the boundary integral equation method and the extrapolation method abstract many numerical methods including the boundary integral equation method start with division of the domain of calculation into intervals the accuracy of their results can be improved considerably by extrapolation to be able to apply the extrapolation method it is necessary to know the asymptotic expansion of the error in this paper the principle of the extrapolation method and subjects important for its application are described above all it is shown how to determine the asymptotic expansions numerically by trial and error in the first sections the matter is explained in a general manner to encourage users of various numerical methods among them users of the finite element method to try to extrapolate their results then the investigations are exemplified in detail by the boundary integral equation method the accuracy of approximate solutions of integral equations for plane elastostatic problems with prescribed boundary tractions and displacements is improved by extrapolation particular attention is paid to boundary tractions and displacements with discontinuous derivatives to induce also practically orientated readers without specialized mathematical knowledge to think about applying the extrapolation method the basic topics are represented in an extensive manner and illustrated by simple examples for a survey of this paper see end of section 1

real time coding with limited lookahead a real time coding system with lookahead consists of a memoryless source a memoryless channel an encoder which encodes the source symbols sequentially with knowledge of future source symbols up to a fixed finite lookahead d with or without feedback of the past channel output symbols and a decoder which sequentially constructs the source symbols using the channel output the objective is to minimize the expected per symbol distortion for a fixed finite lookahead d 1 we invoke the theory of controlled markov chains to obtain an average cost optimality equation acoe the solution of which denoted by d d is the minimum expected per symbol distortion with increasing d d d bridges the gap between causal encoding d 0 where symbol by symbol encoding decoding is optimal and the infinite lookahead case d where shannon theoretic arguments show that separation is optimal we extend the analysis to a system with finite state decoders with or without noise free feedback for a bernoulli source and binary symmetric channel under hamming loss we compute the optimal distortion for various source and channel parameters and thus obtain computable bounds on d d we also identify regions of source and channel parameters where symbol by symbol encoding decoding is suboptimal finally we demonstrate the wide applicability of our approach by applying it in additional coding scenarios such as the case where the sequential decoder can take cost constrained actions affecting the quality or availability of side information about the source

on optimal causal coding of partially observed markov sources in single and multiterminal settings the optimal causal zero delay coding of a partially observed markov process is studied where the cost to be minimized is a bounded nonnegative additive measurable single letter function of the source and the receiver output a structural result is obtained extending witsenhausen s and walrand varaiya s structural results on optimal causal coders to more general state spaces and to a partially observed setting the decentralized multiterminal setup is also considered for the case where the source is an i i d process it is shown that an optimal solution to the decentralized causal coding of correlated observations problem is memoryless for markov sources a counterexample to a natural separation conjecture is presented

approximate inverse preconditioners via sparse sparse iterations the standard incomplete lu ilu preconditioners often fail for general sparse indefinite matrices because they give rise to unstable factors l and u in such cases it may be attractive to approximate the inverse of the matrix directly this paper focuses on approximate inverse preconditioners based on minimizing i am f where am is the preconditioned matrix an iterative descent type method is used to approximate each column of the inverse for this approach to be efficient the iteration must be done in sparse mode i e with sparse matrix by sparse vector operations numerical dropping is applied to maintain sparsity compared to previous methods this is a natural way to determine the sparsity pattern of the approximate inverse this paper describes newton global and column oriented algorithms and discusses options for initial guesses self preconditioning and dropping strategies some limited theoretical results on the properties and convergence of approximate inverses are derived numerical tests on problems from the harwell boeing collection and the fidap fluid dynamics analysis package show the strengths and limitations of approximate inverses finally some ideas and experiments with practical variations and applications are presented

a reliable technique for solving the weakly singular second kind volterra type integral equations abstract in this paper the adomian decomposition method and the phenomenon of the self canceling noise terms are used for solving the weakly singular volterra type linear and nonlinear integral equations the solution obtained is in the form of a convergent power series with elegantly computable terms comparing this scheme with many collocation type methods that use the nonpolynomial basis shows that the present approach is effective and powerful many test modeling problems from mathematical physics linear and nonlinear are discussed to illustrate the effectiveness and the performance of the decomposition method

on causal source codes with side information we study the effect of the introduction of side information into the causal source coding setting of neuhoff and gilbert we find that the spirit of their result namely the sufficiency of time sharing scalar quantizers followed by appropriate lossless coding for attaining optimum performance within the family of causal source codes extends to many scenarios involving availability of side information at both encoder and decoder or only on one side for example in the case where side information is available at both encoder and decoder we find that time sharing side information dependent scalar quantizers at most two for each side information symbol attains optimum performance this remains true even when the reproduction sequence is allowed noncausal dependence on the side information and even for the case where the source and the side information rather than consisting of independent and identically distributed i i d pairs form respectively the output of a memoryless channel and its stationary ergodic input

the second generation feti methods and their application to the parallel solution of large scale linear and geometrically non linear structural analysis problems the feti algorithms are a family of numerically scalable domain decomposition methods they have been designed in the early 1990s for solving iteratively and on parallel machines large scale systems of equations arising from the finite element discretization of solid mechanics structural engineering structural dynamics and acoustic scattering problems and for analyzing complex structures obtained from the assembly of substructures with incompatible discrete interfaces in this paper we present the second generation of these methods that operate more efficiently on large numbers of subdomains offer greater robustness better performance and more flexibility for implementation on a wider variety of computational platforms we also report on the application and performance of these methods for the solution of geometrically non linear structural analysis problems we discuss key aspects of their implementation on shared and distributed memory parallel processors benchmark them against optimized direct sparse solvers and highlight their potential with the solution of large scale structural mechanics problems with several million degrees of freedom

on optimal zero delay coding of vector markov sources optimal zero delay coding quantization of a vector valued markov source driven by a noise process is consid ered using a stochastic control problem formulation the exis tence and structure of optimal quantization policies are studied for a finite horizon problem with bounded per stage distortion measure the existence of an optimal zero delay quantization policy is shown provided that the quantizers allowed are ones with convex codecells the bounded distortion assumption is relaxed to cover cases that include the linear quadratic gaussian problem for the infinite horizon problem and a stationary markov source the optimality of deterministic markov coding policies is shown the existence of optimal stationary markov quantization policies is also shown provided randomization that is shared by the encoder and the decoder is allowed

a sparse approximate inverse preconditioner for nonsymmetric linear systems this paper is concerned with a new approach to preconditioning for large sparse linear systems a procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient type methods some theoretical properties of the preconditioner are discussed and numerical experiments on test matrices from the harwell boeing collection and from tim davis s collection are presented our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners furthermore the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners

an evaluation of parallel multigrid as a solver and a preconditioner for singularly perturbed problems in this paper we try to achieve h independent convergence with preconditioned gmres y saad and m h schultz siam j sci comput 7 1986 pp 856 869 and bicgstab h a van der vorst siam j sci comput 13 1992 pp 63 644 for two dimensional 2d singularly perturbed equations three recently developed multigrid methods are adopted as a preconditioner they are also used as solution methods in order to compare the performance of the methods as solvers and as preconditioners two of the multigrid methods differ only in the transfer operators one uses standard matrix dependent prolongation operators from j e dendy jr j comput phys 48 1982 pp 366 386 w hackbusch multi grid methods and applications springer berlin 1985 the second uses upwind prolongation operators developed in p m de zeeuw j comput appl math 33 1990 pp 1 27 both employ the galerkin coarse grid approximation and an alternating zebra line gauss seidel smoother the third method is based on the block lu decomposition of a matrix and on an approximate schur complement this multigrid variant is presented in a reusken a multigrid method based on incomplete gaussian elimination university of eindhoven the netherlands 1995 all three multigrid algorithms are algebraic methods the eigenvalue spectra of the three multigrid iteration matrices are analyzed for the equations solved in order to understand the convergence of the three algorithms furthermore the construction of the search directions for the multigrid preconditioned gmres solvers is investigated by the calculation and solution of the minimal residual polynomials for poisson and convection diffusion problems all solution methods are investigated and evaluated for finite volume discretizations on fine grids the methods have been parallelized with a grid partitioning technique and are compared on an mimd machine

approximate inverse techniques for block partitioned matrices this paper proposes some preconditioning options when the system matrix is in block partitioned form this form may arise naturally for example from the incompressible navier stokes equations or may be imposed after a domain decomposition reordering approximate inverse techniques are used to generate sparse approximate solutions whenever these are needed in forming the preconditioner the storage requirements for these preconditioners may be much less than for incomplete lu factorization ilu preconditioners for tough large scale computational fluid dynamics cfd problems the numerical experiments show that these preconditioners can help solve difficult linear systems whose coefficient matrices are highly indefinite

optimized schwarz methods optimized schwarz methods are a new class of schwarz methods with greatly enhanced convergence properties they converge uniformly faster than classical schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical schwarz methods if the overlap is of the order of the mesh parameter which is often the case in practical applications they achieve this performance by using new transmission conditions between subdomains which greatly enhance the information exchange between subdomains and are motivated by the physics of the underlying problem we analyze in this paper these new methods for symmetric positive definite problems and show their relation to other modern domain decomposition methods like the new finite element tearing and interconnect feti variants

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

modified multilag methods for volterra functional equations linear multistep methods for ordinary differential equations in conjunction with a family of computationally efficient quadrature rules are employed to define a class of so called multilag methods for the solution of volterra integral and integro differential equations in addition modified multilag methods are proposed which have the property that the stability behavior is independent of the choice of the quadrature rules high order convergence of the methods is established in particular a special class of high order convergent methods is presented for the efficient solution of first kind volterra equations numerical experiments are reported

numerical solution of a nonuniquely solvable volterra integral equation using extrapolation methods in this work the numerical solution of a volterra integral equation with a certain weakly singular kernel depending on a real parameter is considered although for certain values of this equation possesses an infinite set of solutions we have been able to prove that euler s method converges to a particular solution it is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize so that general extrapolation algorithms like the e algorithm can be applied to improve the numerical results this is illustrated by means of some examples

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations

numerical solution of a class of integro differential equations by the tau method with an error estimation the tau method by construction produces approximate polynomial solutions of differential equations the purpose of this paper is to extend the tau method to the integro differential equations an efficient error estimation for the tau method is also introduced details of this method are presented and some numerical results along with estimated errors are given to clarify the method and its error estimator

parallel preconditioning with sparse approximate inverses a parallel preconditioner is presented for the solution of general sparse linear systems of equations a sparse approximate inverse is computed explicitly and then applied as a preconditioner to an iterative method the computation of the preconditioner is inherently parallel and its application only requires a matrix vector product the sparsity pattern of the approximate inverse is not imposed a priori but captured automatically this keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system and the proximity of the approximate to the true inverse is estimated an extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach

structure theorems for real time variable rate coding with and without side information the output of a discrete markov source is to be encoded instantaneously by a variable rate encoder and decoded by a finite state decoder our performance measure is a linear combination of the distortion and the instantaneous rate structure theorems pertaining to the encoder and next state functions are derived for every given finite state decoder which can have access to side information

optimal entropy constrained scalar quantization of a uniform source optimal scalar quantization subject to an entropy constraint is studied for a wide class of difference distortion measures including rth power distortions with r 0 it is proved that if the source is uniformly distributed over an interval then for any entropy constraint r in nats an optimal quantizer has n e sup r interval cells such that n 1 cells have equal length d and one cell has length c spl les d the cell lengths are uniquely determined by the requirement that the entropy constraint is satisfied with equality based on this result a parametric representation of the minimum achievable distortion d sub h r as a function of the entropy constraint r is obtained for a uniform source the d sub h r curve turns out to be nonconvex in general moreover for the squared error distortion it is shown that d sub h r is a piecewise concave function and that a scalar quantizer achieving the lower convex hull of d sub h r exists only at rates r log n where n is a positive integer

superconvergence of collocation methods for a class of weakly singular volterra integral equations we discuss the application of spline collocation methods to a certain class of weakly singular volterra integral equations it will be shown that by a special choice of the collocation parameters superconvergence properties can be obtained if the exact solution satisfies certain conditions this is in contrast with the theory of collocation methods for abel type equations several numerical examples are given which illustrate the theoretical results

product integration methods for second kind abel integral equations the construction and convergence of high order product integration methods for the second kind abel equation are discussed and the results of de hoog and weiss are generalised backward difference methods are introduced and numerical results are presented which verify the theoretical rates of convergence