a generalization of the optimal diagonal approximate inverse preconditioner the classical optimal in the frobenius sense diagonal preconditioner for large sparse linear systems ax b is generalized and improved the new proposed approximate inverse preconditioner n is based on the minimization of the frobenius norm of the residual matrix am i where m runs over a certain linear subspace of nxn real matrices defined by a prescribed sparsity pattern the number of nonzero entries of the nxn preconditioning matrix n is less than or equal to 2n and n of them are selected as the optimal positions in each of the n columns of matrix n all theoretical results are justified in detail in particular the comparison between the proposed preconditioner n and the optimal diagonal one is theoretically analyzed finally numerical experiments reported confirm the theory and illustrate that our generalization of the optimal diagonal preconditioner improves in general its efficiency when they do not coincide

an efficient wavelet preconditioner for iterative solution of three dimensional electromagnetic integral equations a wavelet based preconditioning method is proposed to facilitate the iterative solution of three dimensional 3 d electromagnetic integral equations the preconditioner is derived from the wavelet transform of the moment matrix it is based on the observation that both the moment matrix and its inverse exhibit a sparse multilevel finger structure a method based on the forbenius norm minimization is used to solve the inverse of the matrix under the multilevel finger structure numerical results on a 3 d cavity show that the iteration numbers are significantly reduced with the wavelet preconditioned system the computational cost of the preconditioner is kept under o nlogn

an extension of the conjugate residual method to nonsymmetric linear systems the conjugate gradient cg method and the conjugate residual cr method are krylov subspace methods for solving symmetric positive definite linear systems to solve nonsymmetric linear systems the bi conjugate gradient bi cg method has been proposed as an extension of cg bi cg has attractive short term recurrences and it is the basis for the successful variants such as bi cgstab in this paper we extend cr to nonsymmetric linear systems with the aim of finding an alternative basic solver numerical experiments show that the resulting algorithm with short term recurrences often gives smoother convergence behavior than bi cg hence it may take the place of bi cg for the successful variants

advances in residual vector quantization a review advances in residual vector quantization rvq are surveyed definitions of joint encoder optimality and joint decoder optimality are discussed design techniques for rvqs with large numbers of stages and generally different encoder and decoder codebooks are elaborated and extended fixed rate rvqs and variable rate rvqs that employ entropy coding are examined predictive and finite state rvqs designed and integrated into neural network based source coding structures are revisited successive approximation rvqs that achieve embedded and refinable coding are reviewed a new type of successive approximation rvq that varies the instantaneous block rate by using different numbers of stages on different blocks is introduced and applied to image waveforms and a scalar version of the new residual quantizer is applied to image subbands in an embedded wavelet transform coding system

decay rates for inverses of band matrices spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices the rate of decay of a 1 can be bounded in terms of the essential spectrum of aa for general a and in terms of the essential spectrum of a for positive definite a in the positive definite case the bound can be attained these results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices we also establish decay rates for certain generalized inverses

sparse matrix approximations for multigrid methods abstract we discuss the application of sparse matrix approximations for two grid and v cycle multigrid methods sparse approximate inverses can be used as smoothers further the galerkin coarse matrix can be sparsified by sparse approximation techniques also the projection can be defined by combining sparse approximation with side conditions related to high frequency components numerical results are given which demonstrate the efficiency and accuracy of the proposed strategies

a transformation approach that makes spai psai and rsai procedures efficient for large double irregular nonsymmetric sparse linear systems abstract a sparse matrix is called double irregular sparse if it has at least one relatively dense column and row and it is double regular sparse if all the columns and rows of it are sparse the sparse approximate inverse preconditioning procedures spai psai t o l and rsai t o l are costly and even impractical to construct preconditioners for a large sparse nonsymmetric linear system with the coefficient matrix being double irregular sparse but they are efficient for double regular sparse problems double irregular sparse linear systems have a wide range of applications and 24 4 of the nonsymmetric matrices in the florida university collection are double irregular sparse for this class of problems we propose a transformation approach which consists of four steps i transform a given double irregular sparse problem into a small number of double regular sparse ones with the same coefficient matrix a ii use spai psai t o l and rsai t o l to construct sparse approximate inverses m of a iii solve the preconditioned double regular sparse linear systems by krylov solvers and iv recover an approximate solution of the original problem with a prescribed accuracy from those of the double regular sparse ones a number of theoretical and practical issues are considered on the transformation approach numerical experiments on a number of real world problems confirm the very sharp superiority of the transformation approach to the standard approach that preconditions the original double irregular sparse problem by spai psai t o l or rsai t o l and solves the resulting preconditioned system by krylov solvers

optimal discrete transmission conditions for a nonoverlapping domain decomposition method for the helmholtz equation this paper is dedicated to recent developments of a two lagrange multipliers domain decomposition method for the helmholtz equation c farhat et al comput methods appl mech engrg 184 2000 pp 213 240 m j gander f magoules and f nataf siam j sci comput 24 2002 pp 38 60 involving an additional augmented operator along the interface between the subdomains most methods for optimizing the augmented interface operator are based on the discretization of approximations of the continuous transparent operator b despres proceedings of the second international conference on mathematical and numerical aspects of wave propagation r kleinman et al eds siam philadelphia 1993 pp 197 206 j d benamou and b despres j comput phys 136 1997 pp 68 82 p chevalier and f nataf domain decomposition methods 10 ams providence ri 1998 pp 400 407 m j gander proceedings of the 12th international conference on domain decomposition methods chiba japan ddm org 2000 pp 247 254 m j gander f magoules and f nataf siam j sci comput 24 2002 pp 38 60 at the discrete level the optimal operator can be proved to be equal to the schur complement of the outer domain this schur complement can be directly approximated using purely algebraic techniques like sparse approximate inverse methods or incomplete factorization the main advantage of such an algebraic approach is that it is much easier to implement in existing code without any information on the geometry of the interface and the finite element formulation used convergence results and parallel efficiency of several algebraic optimization techniques of an interface operator for acoustic analysis applications will be presented

parallel and explicit finite element time domain method for maxwell s equations we construct a parallel and explicit finite element time domain fetd algorithm for maxwell s equations in simplicial meshes based on a mixed e b discretization and a sparse approximation for the inverse mass matrix the sparsity pattern of the approximate inverse is obtained from edge adjacency information which is naturally encoded by the sparsity pattern of successive powers of the mass matrix each column of the approximate inverse is computed independently allowing for different processors to be used with no communication costs and hence linear ideal speedup in parallel processors the convergence of the approximate inverse matrix to the actual inverse full matrix is investigated numerically and shown to exhibit exponential convergence versus the density of the approximate inverse matrix the resulting fetd time stepping is explicit is the sense that it does not require a linear solve at every time step akin to the finite difference time domain fdtd method

diagonally striped matrices and approximate inverse preconditioners the inverse of a banded matrix is in general dense if the structure of the original banded matrix is striped that is the non zero diagonals are separated by one or more zero diagonals the inverse may exhibit a similar striped structure the motivation for studying inverses of striped matrices is to obtain efficient preconditioners for systems arising from radiation transport equations whose matrices include dominant values along diagonal stripes linear systems whose system matrix has a striped inverse lend themselves to the use of a sparse approximate inverse spai preconditioner whose structure is derived from that of the actual inverse

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

wavelet based ilu preconditioners for the numerical solution by pufem of high frequency elastic wave scattering daubechies family of wavelets combined to the incomplete lower upper ilu factorization are considered as preconditioners for a block sparse linear system arising from the approximation of the time harmonic elastic wave equations by the partition of unity finite element method pufem after applying the discrete wavelet transform dwt to each dense block in the final matrix and the known right hand side due to the local enrichment by pressure p and shear s plane waves the resulting linear system is solved by the restarted generalized minimum residual method gmres with ilu preconditioners allowing fill in elements in the l and u matrix factors a reordering algorithm of the vertices based on gibbs method is also introduced it leads to a significant reduction of the bandwidth of the wave based finite element fe matrix and enables the ilu preconditioners to be more effective to study the performance of the proposed preconditioners a problem of a horizontal s plane wave scattered by a rigid circular body in an infinite elastic medium is considered the numerical tests show the good performance of the dwt based ilu preconditioners in improving the rate of convergence of gmres for high numbers of approximating p and s plane waves on coarse mesh grids containing multi wavelength sized elements moreover the haar dwt enhances the scalability with respect to the problem size when the number of the nodal points increases of the ilu preconditioner which uses the threshold strategy in the control of the fill in elements despite the high level of the conditioning a good accuracy may be achieved for a discretization level of around 1 9 degrees of freedom per s wavelength which is far below the rule of thumb of 10 nodal points per wavelength adopted for the conventional fe

low entropy image pyramids for efficient lossless coding an efficient image source coding technique gives good compression performance at low computational complexity this research introduces an efficient coding technique based on pyramid coding that involves transforming an image into an equivalent lower entropy form prior to lossless coding the proposed method is also a multiresolution technique that facilitates progressive image transmission

a splitting technique of higher order for the navier stokes equations this article presents a splitting technique for solving the time dependent incompressible navier stokes equations using nested finite element spaces which can be interpreted as a postprocessing step the splitting method is of more than second order accuracy in time the integration of adaptive methods in space and time in the splitting are discussed in this algorithm a gradient recovery technique is used to compute boundary conditions for the pressure and to achieve a higher convergence order for the gradient at different points of the algorithm results on the flow around a cylinder s and the driven cavity s problem are presented

differential forms galerkin duality and sparse inverse approximations in finite element solutions of maxwell equations we identify primal and dual formulations in the finite element method fem solution of the vector wave equation using a geometric discretization based on differential forms these two formulations entail a mathematical duality denoted as galerkin duality galerkin dual fem formulations yield identical nonzero dynamical eigenvalues up to machine precision but have static zero eigenvalue solution spaces of different dimensions algebraic relationships among the degrees of freedom of primal and dual formulations are explained using a deep rooted connection between the hodge helmholtz decomposition of differential forms and descartes euler polyhedral formula and verified numerically in order to tackle the fullness of dual formulation algebraic and topological thresholdings are proposed to approximate inverse mass matrices by sparse matrices

encoding and progressive transmission of still pictures in ntsc composite format using transform domain methods coding schemes suitable for progressive transmission of still pictures in ntsc composite format are studied transform coding methods based on the discrete cosine transform dct and the walshhadamard transform wht are used the transform coefficients are segmented into groups having similar properties variable blocklength to variable length codes are used to encode the quantized coefficients for each of the groups progressive transmission is achieved by sending the coefficients belonging to particular groups in successive passes over the image results are presented for coding the ntsc signal sampled at four times the color subcarrier frequency with an orthogonal structure and at twice the subcarrier frequency with a hexagonal structure

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

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

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

preconditioning sparse nonsymmetric linear systems with the sherman morrison formula let ax b be a large sparse nonsymmetric system of linear equations a new sparse approximate inverse preconditioning technique for such a class of systems is proposed we show how the matrix a0 1 a 1 where a0 is a nonsingular matrix whose inverse is known or easy to compute can be factorized in the form u omega v t using the sherman morrison formula when this factorization process is done incompletely an approximate factorization may be obtained and used as a preconditioner for krylov iterative methods for a0 sin where in is the identity matrix and s is a positive scalar the existence of the preconditioner for m matrices is proved in addition some numerical experiments obtained for a representative set of matrices are presented results show that our approach is comparable with other existing approximate inverse techniques

iterative solution of linear systems in the 20th century this paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century although iterative methods for solving linear systems nd their origin in the early 19th century work by gauss the eld has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences the past ve decades have been particularly rich in new developments ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientic and industrial computational models as in any other scientic area research in iterative methods has been a journey characterized by a chain of contributions building on each other it is the aim of this paper not only to sketch the most signicant of these contributions during the past century but also to relate them to one another c 2000 elsevier science b v all rights reserved

uniform fuzzy relations and fuzzy functions in this paper we introduce and study the concepts of a uniform fuzzy relation and a partially uniform f function we give various characterizations and constructions of uniform fuzzy relations and uniform f functions we show that the usual composition of fuzzy relations is not convenient for f functions so we introduce another kind of composition and we establish a mutual correspondence between uniform f functions and fuzzy equivalences we also give some applications of uniform fuzzy relations in approximate reasoning especially in fuzzy control and we show that uniform fuzzy relations are closely related to the defuzzification problem

hypergraph partitioning for the parallel computing of fuzzy differential equations in our previous work fuzzy differential equations have been used to describe service composition and the solutions have been used for the performance analysis this paper presents a method to compute these fuzzy differential equations in parallel we first map the underlying petri net that generates fuzzy differential equations to a hypergraph and then partition the hypergraph to minimize inter processor communication while maintaining a good load balance based on the partition result we divide the fuzzy differential equations into several blocks finally we design parallel computing algorithm to compute these equations software hmetis is used to partition the hypergraph and software sundials is used to support the parallel computing of fuzzy differential equations online shop problem from ibm web site has been used to demonstrate the benefit of our method

imf an incomplete multifrontal lu factorization for element structured sparse linear systems we propose an incomplete multifrontal lu factorization imf preconditioner that extends supernodal multifrontal methods to incomplete factorizations it can be used as a preconditioner in a krylov subspace method to solve large scale sparse linear systems with an element structure e g those arising from a finite element discretization of a partial differential equation the fact that the element matrices are dense is exploited to increase the computational performance and the robustness of the factorization through efficient partial pivoting imf is compared with the multilevel arms2 the level of fill in ilu and the threshold based ilutp preconditioners our experiments indicate that imf is competitive with arms2 on saddle point problems arising in the solution of the steady state navier stokes equation experiments with element structured matrices arising from structural engineering applications found in the university of florida sparse matrix collection illustrate the robustness of imf final

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

multilevel block ilu preconditioner for sparse nonsymmetric m matrices we develop in this paper a multilevel block ilu preconditioner for solving sparse nonsymmetric m matrices arising from convection diffusion equations discretized by finite difference scheme this preconditioner is based on recursive block ilu factorization where the schur complement matrices are successively approached by some sparse matrices satisfying some hypothesis we study the preconditioned gmres iterative method and multigrid method for solving linear systems arising from this scheme

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

orderings for factorized sparse approximate inverse preconditioners the influence of reorderings on the performance of factorized sparse approximate inverse preconditioners is considered some theoretical results on the effect of orderings on the fill in and decay behavior of the inverse factors of a sparse matrix are presented it is shown experimentally that certain reorderings like minimum degree and nested dissection can be very beneficial the benefit consists of a reduction in the storage and time required for constructing the preconditioner and of faster convergence of the preconditioned iteration in many cases of practical interest

an efficient preconditioner for electromagnetic integral equations using predefined wavelet packet basis an approximate inverse preconditioner based on the predefined wavelet packet pwp basis is proposed for the fast iterative solution of electromagnetic integral equations the pwp basis is designed to achieve a sparse representation of the moment matrix and the preconditioner is constructed by inverting the block diagonal approximation of the pwp based moment matrix and transforming the results into the space domain numerical results show that the pwp preconditioner is effective in accelerating the convergence rate of iterative solution to moment equations it is also demonstrated that by properly designing the block diagonal matrix and computing the matrix elements the total computational complexity and memory costs for the preconditioner can be kept to o nlogn

on iterative techniques for computing flow in large two dimensional discrete fracture networks computation of flow in discrete fracture networks often involves solving for hydraulic head values at all intersection points of a large number of stochastically generated fractures inside a bounded domain for large systems this approach leads to the generation of problems involving highly sparse matrices which must be solved iteratively distributions of fracture lengths spanning over several orders of magnitude and the randomness of fracture orientations and locations lead to coefficient matrices that are devoid of any regular structure in the sparsity pattern in addition to the rapid increase in computational effort with increase in the size of the fracture network the spread in the distribution of fracture parameters such as length and transmissivity dramatically influences the convergence behavior of the system of linear equations an overview of the discrete fracture network dfn methodology for computation of flow is presented along with a comparative study of various krylov subspace iterative methods for the resulting class of sparse matrices the rate of convergence of the iterative techniques is found to exhibit a systematic pattern with respect to changes in statistical parameters of the stochastically generated fracture networks salient features of the observed trends in the convergence pattern are discussed and guidelines for design of dfn algorithms are provided

multiresolution multiresource progressive image transmission this paper presents a new progressive image transmission pit design algorithm in which the resolution and resources rate or distortion and storage size at each transmission stage are allowed to be prespecified this algorithm uses the wavelet transform and tree structured vector quantizer tsvq techniques the wavelet transform is used to obtain a pyramid structure representation of an image the vector quantizer technique is used to design a tsvq for each subimage so that all the subimages that constitute the image at the current stage can be successively refined according to the resources available at that stage the resources assigned to each subimage for the successive refinement is determined to optimize the performance at the current stage under the resource constraints normally the resource constraints at each stage are determined by the specification of the transmission time or distortion for image data and the storage complexity of the tsvq the resolution at each stage is determined specified according to the application or as part of the design process to optimize the visual effect

the fast multipole method we study integral methods applied to the resolution of the maxwell equations where the linear system is solved using an iterative method which requires only matrix vector products the fast multipole method fmm is one of the most efficient methods used to perform matrix vector products and accelerate the resolution of the linear system a problem involving n degrees of freedom may be solved in cniternlogn floating operations where c is a constant depending on the implementation of the method in this article several techniques allowing one to reduce the constant c are analyzed this reduction implies a lower total cpu time and a larger range of application of the fmm in particular new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms several numerical tests are also described these confirm the efficiency and the theoretical complexity of the fmm

factored approximate inverse preconditioners with dynamic sparsity patterns abstract we propose two sparsity pattern selection algorithms for factored approximate inverse preconditioners to solve general sparse matrices the sparsity pattern is adaptively updated in the construction phase by using combined information of the inverse and original triangular factors of the original matrix in order to determine the sparsity pattern our first algorithm uses the norm of the inverse factors multiplied by the largest absolute value of the original factors and the second employs the norm of the inverse factors divided by the norm of the original factors experimental results show that these algorithms improve the robustness of the preconditioners to solve general sparse matrices

block preconditioners based on approximate commutators this paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible navier stokes equations we consider the pressure convection diffusion preconditioners proposed by kay loghin and wathen siam j sci comput 24 2002 pp 237 256 and silvester elman kay and wathen j comput appl math 128 2001 pp 261 279 numerous theoretical and numerical studies have demonstrated mesh independent convergence on several problems and the overall efficacy of this methodology a drawback however is that it requires the construction of a convection diffusion operator denoted f p projected onto the discrete pressure space this means that integration of this idea into a code that models incompressible flow requires a sophisticated understanding of the discretization and other implementation issues something often held only by the developers of the model as an alternative we consider automatic ways of computing f p based on purely algebraic considerations the new methods are closely related to the bfbt preconditioner of elman siam j sci comput 20 1999 pp 1299 1316 we use the fact that the preconditioner is derived from considerations of commutativity between the gradient and convection diffusion operators together with methods for computing sparse approximate inverses to generate the required matrix f p automatically we demonstrate that with this strategy the favorable convergence properties of the preconditioning methodology are retained

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

image compression using adaptive vector quantization the paper demonstrates the application of adaptive vector quantization techniques to the coding of monochromatic and color pictures codebooks of representative vectors are generated for different portions of the image by means of extensive simulations the performance of the coder for monochrome images is estimated as a function of the various coding parameters vector dimensionality number of representative vectors and degree of adaptivity it is shown that for a vector size of dimension 4 there is a zone of operation ranging from 1 0 to 1 5 bits pixel where adaptive vector quantization is advantageous despite the simplicity of the decoder the performance of the adaptive vector quantizer is found to be comparable with block transform coding schemes for both monochrome and color pictures

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

progressive image transmission using vector quantization on images in pyramid form a progressive image transmission scheme in which vector quantization is applied to images represented by pyramids is proposed a mean pyramid representation of an image is first built up by forming a sequence of reduced size images by averaging over blocks of 2 2 pixels a difference pyramid is then built up by taking the differences between successive levels in the mean pyramid progressive transmission is achieved by sending all the nodes in the difference pyramid starting from the top level and ending at the bottom level the kth approximate image can be formed by adding the information of level k to the previously reproduced k 1 st approximation to gain efficiency vector quantization is applied to the difference pyramid of the image on a level by level basis if the errors due to quantization at level k are properly delivered and included in the next level k 1 then it is demonstrated that the original image can be reconstructed an entropy coder is used to encode the final residual error image losslessly thus ensuring perfect reproduction of the original image the experiments demonstrate that it is possible to achieve simultaneously lossless and progressive transmission with compression at the intermediate level the use of vector quantization results in a coding gain over that obtained using only a huffman coder excellent reproduction is achieved at a bit rate of only 0 06 bits pixel

image sequence coding using vector quantization in this paper a new interframe coding technique based upon vector quantization is presented this algorithm has as its basis twodimensional block vector quantization at the frame level onto which is grafted the concept of adaptive codebook replenishment and frame replenishment this algorithm includes three strategies label replenishment label replenishment with mean shift and label replenishment with mean shift and selective codeword replacement the last two strategies efficiently update the codebook to track the changes in local statistics on a frame basis compared with three dimensional block vector quantization 16 for bit rates between 0 5 and 0 65 the normalized mean squared error nmse is reduced by a factor ranging between 2 and 2 5 by the use of replenishment

partitioning rectangular and structurally unsymmetric sparse matrices for parallel processing a common operation in scientific computing is the multiplication of a sparse rectangular or structurally unsymmetric matrix and a vector in many applications the matrix transpose vector product is also required this paper addresses the efficient parallelization of these operations we show that the problem can be expressed in terms of partitioning bipartite graphs we then introduce several algorithms for this partitioning problem and compare their performance on a set of test matrices

semidefinite programming and eigenvalue bounds for the graph partition problem the graph partition problem is the problem of partitioning the vertex set of a graph into a fixed number of sets of given sizes such that the sum of weights of edges joining different sets is optimized in this paper we simplify a known matrix lifting semidefinite programming relaxation of the graph partition problem for several classes of graphs and also show how to aggregate additional triangle and independent set constraints for graphs with symmetry we present an eigenvalue bound for the graph partition problem of a strongly regular graph extending a similar result for the equipartition problem we also derive a linear programming bound of the graph partition problem for certain johnson and kneser graphs using what we call the laplacian algebra of a graph we derive an eigenvalue bound for the graph partition problem that is the first known closed form bound that is applicable to any graph thereby extending a well known result in spectral graph theory finally we strengthen a known semidefinite programming relaxation of a specific quadratic assignment problem and the above mentioned matrix lifting semidefinite programming relaxation by adding two constraints that correspond to assigning two vertices of the graph to different parts of the partition this strengthening performs well on highly symmetric graphs when other relaxations provide weak or trivial bounds

finite element implementation of the generalized lorenz gauged a phi formulation for low frequency circuit modeling the a phi formulation with generalized lorenz gauge is free of catastrophic breakdown in low frequency regime in the formulation a and phi are completely separated and maxwell s equations are reduced into two independent equations pertinent to a and phi this however leads to more complicated equations in contrast to the traditional e formulation the numerical dicretization of the equations is challenging especially for the equation pertinent to a by virtue of the differential forms theory and whitney elements the direct action of divergence operator on a is bypassed thus the equations can be discretized compatibly using regular finite element method the condition of the resultant matrix system is much better than that of the e formulation as frequency becomes low and even approaches to zero the generalized lorenz gauged a phi formulation is verified to be accurate and efficient for low frequency circuit problems

on an adaptive multigrid solver for convection dominated problems the objective of this paper is to develop and describe an adaptive multigrid scheme for the numerical solution of convection diffusion reaction equations with dominating convection or reaction term specifically we propose a systematic construction of problem and scale dependent restriction prolongation and smoothing operators in combination with an adaptive mesh refinement strategy in particular adaptivity is not only to reduce computational complexity but also to stabilize standard galerkin discretizations so that adding artificial viscosity can be avoided the approach is suited for any choice of underlying multiresolution sequences in particular for standard finite element discretizations on nested but possibly nonuniform meshes moreover all concepts are in principle independent of the spatial dimension and avoid any special flow dependent enumeration of unknowns the performance of the scheme is illustrated by various examples in one and two spatial dimensions they are chosen so as to exhibit possibly many types of adverse or singular behavior in connection with constant and varying convection or the interplay between convective and reactive terms in the light of these tests those issues are identified that require further research

preconditioning with direct approximate factoring of the inverse to precondition a large and sparse linear system two direct methods for approximate factoring of the inverse are devised the algorithms are fully parallelizable and appear to be more robust than the iterative methods suggested for the task a method to compute one of the matrix subspaces optimally is derived possessing a considerable amount of flexibility these approaches extend the approximate inverse preconditioning techniques in several natural ways numerical experiments are given to illustrate the performance of the preconditioners on a number of challenging benchmark linear systems

hypergraph partitioning based models and methods for exploiting cache locality in sparse matrix vector multiplication sparse matrix vector multiplication spmxv is a kernel operation widely used in iterative linear solvers the same sparse matrix is multiplied by a dense vector repeatedly in these solvers matric

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

fast computational methods for locating fold points for the power flow equations voltage collapse in a power system can occur following a progressive decline in voltage magnitude at the system buses and mathematically this phenomenon is associated with a fold bifurcation point occurring in the nonlinear algebraic equations used to model the power system in this paper we first discuss some of the methods used to speed up the process of detecting a fold bifurcation focussing on designing test function methods to predict a performance index we then discuss some iterative methods that can be used to improve the newton iterations in particular we present new and efficient preconditioners of the two level type for the jacobian matrix numerical results are given using standard ieee test bus systems

gradation of being fuzzy function abstract for crisp sets x and y considering two fuzzy equalities e x e y on x and y respectively the concept of fuzzy function and several types of it were studied in m demirci fuzzy sets and systems 106 1999 239 246 in this paper for a fuzzy relation f on x y considering the fuzzy equalities e x on x and e y on y the degree of f of being fuzzy function being surjective being injective and being bijective is defined then setting the connections between several types of degree of being fuzzy function and various types of fuzzy function the essential properties of these degrees are investigated

progressive image transmission by transform coefficient residual error quantization a progressive image transmission scheme based on iterative transform coding structure is proposed for application in interactive image communication over low bandwidth channels the scheme not only provides progressive transmission but also guarantees lossless reproduction combined with a degree of compression the image to be transmitted undergoes an orthogonal transform and the transform coefficients are quantized scalar or vector before transmission the novelty is that the residual error array due to quantization is iteratively fedback and requantized scalar or vector the coded residual error information is progressively transmitted and utilized in reconstructing the successive approximations it is shown that the average reconstruction error variance converges to zero as the number of iterative stages approaches infinity in practice lossless reproduction can be achieved with a small number of iterations by using an entropy coder on the final residual error image computer simulation results demonstrate the effectiveness of the technique

comparative performance of pyramid data structures for progressive image transmission a review of various pyramid data structures for progressive image transmission focusing on the steps involved in forming the pyramid and the performance comparisons is presented the simple mean pyramid its variants including the truncated mean reduced sum difference reduced difference and s transform pyramids and the more general laplacian pyramid are discussed it is also shown that the use of an interpolation function improves the quality of the full resolution approximations both objectively and subjectively

inverse free recursive multiresolution algorithms for a data approximation problem we present inverse free recursive multiresolution algorithms for data approximation problems based on energy functionals minimization during the multiresolution process a linear system needs to be solved at each different resolution level which can be solved with direct or iterative methods numerical results are reported using the sparse cholesky factorization for two applications one concerning the localization of regions in which the energy of a given surface is mostly concentrated and another one regarding noise reduction of a given dataset in addition for large scale data approximation problems that require a very fine resolution we discuss the use of the preconditioned conjugate gradient pcg iterative method coupled with a specialized monolithic preconditioner for which one preconditioner is built for the highest resolution level and then the corresponding blocks of that preconditioner are used as preconditioners for the forthcoming lower levels

fuzzy functions and their fundamental properties in this paper it is noticed that the definition of fuzzy equality of sasaki 1993 is not concordant with that of es and coker 1995 and sostak 1985 1988 furthermore the construction of fuzzy equality and fuzzy function is far from the way of natural fuzzification for this purpose the fuzzy equality of two points of a crisp set which is more convenient than that in es and coker 1995 and sostak 1985 1988 and the fuzzy function having weaker conditions than that is given in sasaki 1993 are redefined in much more natural way then various types of fuzzy functions are introduced and their fundamental properties are established

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

multilevel ilu with reorderings for diagonal dominance this paper presents a preconditioning method based on combining two sided permutations with a multilevel approach the nonsymmetric permutation exploits a greedy strategy to put large entries of the matrix in the diagonal of the upper leading submatrix the method can be regarded as a complete pivoting version of the incomplete lu factorization this leads to an effective incomplete factorization preconditioner for general nonsymmetric irregularly structured sparse linear systems

orthogonal projections of the identity spectral analysis and applications to approximate inverse preconditioning many strategies for constructing different structures of sparse approximate inverse preconditioners for large linear systems have been proposed in the literature in a more general framework this paper analyzes the theoretical effectiveness of the optimal preconditioner in the frobenius norm of a linear system over an arbitrary subspace of m n left mathbb r right for this purpose the spectral analysis of the frobenius orthogonal projections of the identity matrix onto the linear subspaces of m n left mathbb r right is performed this analysis leads to a simple general criterion the effectiveness of the optimal approximate inverse preconditioners parametrized by any vectorial structure improves at the same time as the smallest singular value or the smallest eigenvalue s modulus of the corresponding preconditioned matrices increases to 1

sparse approximate inverse preconditioners using norm minimization techniques we investigate the use of sparse approximate inverse preconditioners for the iterative solution of unsymmetric linear systems of equations we consider the approximations proposed by cosgrove diaz and griewank internat j comput math 44 1992 pp 91 110 and huckle and grote a new approach to parallel preconditioning with sparse approximate inverses tech report sccm 94 03 stanford university 1994 which are based on norm minimization techniques such methods are of particular interest because of the considerable scope for parallelization we propose a number of enhancements which may improve their performance when run in a sequential environment these methods can perform unfavorably when compared with other techniques however they can be successful when other methods fail and simulations indicate that they can be competitive when considered in a parallel environment

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

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

on the gpgpu parallelization issues of finite element approximate inverse preconditioning during the past decades explicit finite element approximate inverse preconditioning methods have been extensively used for efficiently solving sparse linear systems on multiprocessor systems the effectiveness of explicit approximate inverse preconditioning schemes relies on the use of efficient preconditioners that are close approximants to the coefficient matrix and are fast to compute in parallel new parallel computational techniques are proposed for the parallelization of the optimized banded generalized approximate inverse finite element matrix obgaifem algorithm based on the concept of the fish bone computational approach and for the explicit preconditioned conjugate gradient type methods on a general purpose graphics processing unit gpgpu the proposed parallel methods have been implemented using compute unified device architecture cuda developed by nvidia finally numerical results for the performance of the finite element explicit approximate inverse preconditioning for solving characteristic two dimensional boundary value problems on a massive multiprocessor interface on a gpu are presented the cuda implementation issues of the proposed methods are also discussed

a hierarchical low rank schur complement preconditioner for indefinite linear systems nonsymmetric and highly indefinite linear systems can be quite difficult to solve by iterative methods this paper combines ideas from the multilevel schur low rank preconditioner developed by y x

a remark on the single scattering preconditioner applied to boundary integral equations this article deals with boundary integral equation preconditioning for the multiple scattering problem the focus is put on the single scattering preconditioner corresponding to the diagonal part of the integral operator for which two results are proved indeed after applying this geometric preconditioner it appears that firstly every direct integral equations become identical to each other and secondly that the indirect integral equation of brakhage werner becomes equal to the direct integral equations up to a change of basis these properties imply in particular that the convergence rate of a krylov subspaces solver will be exactly the same for every preconditioned integral equations to illustrate this some numerical simulations are provided at the end of the paper

robust multigrid methods for nonsmooth coefficient elliptic linear systems we survey the literature on robust multigrid methods which have been developed in recent years for solving second order elliptic pdes with nonsmooth coefficients we highlight the key ideas of designing robust multigrid methods which are able to recover the usual multigrid efficiency for nonsmooth coefficient pdes on structured or unstructured grids in particular we shall describe various approaches for constructing the interpolation and the smoothing operators and the coarse grid points selections

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

deterministic prediction in progressive coding deterministic prediction in progressive coding of images is investigated progressive coding first creates a sequence of resolution layers by beginning with an original image and reducing its resolution several times by factors of some natural number m the resultant layers are losslessly encoded beginning with the lowest resolution layer and then encoding each higher resolution image incrementally upon the previous one coding efficiency may be improved if knowledge of the rules which produced the lower resolution image of each pair is used to deterministically predict pixels of the higher so they need not be encoded given reduction rules expressing each low resolution pixel as a function of nearby high resolution pixels and previously generated low resolution pixels it is shown that finding a complete set of rules each of which deterministically predicts the value of a high resolution pixel when certain values are found in nearby low resolution pixels and previously coded high resolution pixels is np complete a recursive algorithm for solving the problem in optimal time as a depth first tree search is proposed and the characteristics of the resultant prediction process are studied

motivations and realizations of krylov subspace methods for large sparse linear systems we briefly introduce typical and important direct and iterative methods for solving systems of linear equations concretely describe their fundamental characteristics in viewpoints of both theory and applications and clearly clarify the substantial differences among these methods in particular the motivations of searching the solution of a linear system in a krylov subspace are described and the algorithmic realizations of the generalized minimal residual gmres method are shown and several classes of state of the art algebraic preconditioners are briefly reviewed all this is useful for correctly deeply and completely understanding the application scopes theoretical properties and numerical behaviors of these methods and is also helpful in designing new methods for solving systems of linear equations

normalized explicit approximate inverse preconditioning for solving 3d boundary value problems on uniprocessor and distributed systems normalized explicit approximate inverse matrix techniques based on normalized approximate factorization procedures for solving sparse linear systems resulting from the finite difference discretization of partial differential equations in three space variables are introduced normalized explicit preconditioned conjugate gradient schemes in conjunction with normalized approximate inverse matrix techniques are presented for solving sparse linear systems the convergence analysis with theoretical estimates on the rate of convergence and computational complexity of the normalized explicit preconditioned conjugate gradient method are also derived a parallel normalized explicit preconditioned conjugate gradient method for distributed memory systems using message passing interface mpi communication library is also given along with theoretical estimates on speedups efficiency and computational complexity application of the proposed method on a three dimensional boundary value problem is discussed and numerical results are given for uniprocessor and multicomputer systems copyright 2005 john wiley sons ltd

combining kronecker product approximation with discrete wavelet transforms to solve dense function related linear systems a new solution technique is proposed for linear systems with large dense matrices of a certain class including those that come from typical integral equations of potential theory this technique combines kronecker product approximation and wavelet sparsification for the kronecker product factors the user is only required to supply a procedure for computation of each entry of the given matrix the main sources of efficiency are the incomplete cross approximation procedure adapted from the mosaic skeleton method of the second author and data sparse preconditioners the incomplete lu decomposition with dynamic choice of the fill in structure with a prescribed threshold and the inverse kronecker product preconditioner constructed for the sum of kronecker products of sparsified finger like matrices computed by the discrete wavelet transform in some model but quite representative examples the new technique allowed us to solve dense systems with more than 1 million unknowns in a few minutes on a personal computer with 1 gbyte operative memory

optimal block tridiagonalization of matrices for coherent charge transport numerical quantum transport calculations are commonly based on a tight binding formulation a wide class of quantum transport algorithms require the tight binding hamiltonian to be in the form of a block tridiagonal matrix here we develop a matrix reordering algorithm based on graph partitioning techniques that yields the optimal block tridiagonal form for quantum transport the reordered hamiltonian can lead to significant performance gains in transport calculations and allows to apply conventional two terminal algorithms to arbitrarily complex geometries including multi terminal structures the block tridiagonalization algorithm can thus be the foundation for a generic quantum transport code applicable to arbitrary tight binding systems we demonstrate the power of this approach by applying the block tridiagonalization algorithm together with the recursive green s function algorithm to various examples of mesoscopic transport in two dimensional electron gases in semiconductors and graphene

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

a two dimensional data distribution method for parallel sparse matrix vector multiplication a new method is presented for distributing data in sparse matrix vector multiplication the method is two dimensional tries to minimize the true communication volume and also tries to spread the computation and communication work evenly over the processors the method starts with a recursive bipartitioning of the sparse matrix each time splitting a rectangular matrix into two parts with a nearly equal number of nonzeros the communication volume caused by the split is minimized after the matrix partitioning the input and output vectors are partitioned with the objective of minimizing the maximum communication volume per processor experimental results of our implementation mondriaan for a set of sparse test matrices show a reduction in communication volume compared to one dimensional methods and in general a good balance in the communication work experimental timings of an actual parallel sparse matrix vector multiplication on an sgi origin 3800 computer show that a sufficiently large reduction in communication volume leads to savings in execution time

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

improving approximate inverses based on frobenius norm minimization approximate inverses based on frobenius norm minimization of real nonsingular matrices are analyzed from a purely theoretical point of view in this context this paper provides several sufficient conditions that assure us the possibility of improving in the sense of the frobenius norm some given approximate inverses moreover the optimal approximate inverses of matrix a r n x n among all matrices belonging to certain subspaces of r n x n are obtained particularly a natural generalization of the classical normal equations of the system ax b is given when searching for approximate inverses n a t such that an is symmetric and an i f

a graded notion of functionality abstract we will introduce the notion of functionality as a graded property of a fuzzy relation we will extend known results relating to functional fuzzy relations to their graded counterparts and provide new properties w r t various kinds of relational compositions as a contribution to the theory of fuzzy control we will moreover provide an example of use of graded properties in establishing implicative models of fuzzy rule bases where the non graded property fails

parallel preconditioned conjugate gradient optimization of the rayleigh quotient for the solution of sparse eigenproblems a parallel algorithm based on the multidimensional minimization of the rayleigh quotient is proposed to evaluate the leftmost eigenpairs of the generalized symmetric positive definite eigenproblem the minimization is performed via a conjugate gradient like procedure accelerated by a factorized approximate inverse preconditioner fsai and by a number of block preconditioners the resulting code obtains a high level of parallel efficiency and proves to be comparable with the parpack package on a set of large matrices arising from various discretizations of pdes of elliptic parabolic type