possible line sums for a qualitative matrix abstract given m ordered row sums r 1 r m and n ordered column sums c 1 c n we characterize the sign patterns for which there is a real m by n matrix b with row sums r 1 r m and column sums c 1 c n situations in which there is a symmetric solution or in which there is a skew symmetric solution are characterized and circumstances in which there is a solution when s or s are relaxed to 0 are also determined most generally our principal result may be used to characterize those sign patterns for which for given vectors x v r n and y u r m there is a real m by n matrix b such that bx y and v t u t b

restarted gmres preconditioned by deflation this paper presents a new preconditioning technique for the restarted gmres algorithm it is based on an invariant subspace approximation which is updated at each cycle numerical examples show that this deflation technique gives a more robust scheme than the restarted algorithm at a low cost of operations and memory

the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

on the construction of deflation based preconditioners in this article we introduce new bounds on the effective condition number of deflated and preconditioned deflated symmetric positive definite linear systems for the case of a subdomain deflation such as that of nicolaides siam j numer anal 24 1987 pp 355 365 these theorems can provide direction in choosing a proper decomposition into subdomains if grid refinement is performed keeping the subdomain grid resolution fixed the condition number is insensitive to the grid size subdomain deflation is very easy to implement and has been parallelized on a distributed memory system with only a small amount of additional communication numerical experiments for a steady state convection diffusion problem are included

deflation of conjugate gradients with applications to boundary value problems a method for improving the convergence of the standard conjugate gradient method is given this method involves the use of auxiliary subspaces it is shown how such subspaces may be constructed for boundary value problems and an analysis of convergence for second order problems is presented

component averaged row projections a robust block parallel scheme for sparse linear systems a new method for the parallel solution of large sparse linear systems is introduced it proceeds by dividing the equations into blocks and operating in block parallel iterative mode i e all the blocks are processed in parallel and the partial results are merged to form the next iterate the new scheme performs kaczmarz row projections within the blocks and merges the results by certain component averaging operations hence it is called component averaged row projections or carp the system matrix can be general nonsymmetric and ill conditioned and the division into blocks is unrestricted for partial differential equations pdes if the blocks are domain based then only variables at the boundaries between domains are averaged thereby minimizing data transfer between processors carp is very robust its application to test cases of linear systems derived from pdes shows that it converges in difficult cases where state of the art methods fail it is also very memory efficient and exhibits an almost linear speedup ratio with efficiency greater than unity in some cases a formal proof of convergence is presented it is shown that the component averaging operations are equivalent to row projections in a certain superspace so the convergence properties of carp are identical to those of kaczmarz s algorithm in the superspace carp and its convergence proof also apply to the consistent convex feasibility problem

preconditioned conjugate gradients for solving singular systems abstract in this paper the preconditioned conjugate gradient method is used to solve the system of linear equations ax b where a is a singular symmetric positive semi definite matrix the method diverges if b is not exactly in the range r a of a if the null space n a of a is explicitly known then this divergence can be avoided by subtracting from b its orthogonal projection onto n a as well as analysing this subtraction conditions necessary for the existence of a nonsingular incomplete cholesky decomposition are given finally the theory is applied to the discretized semi definite neumann problem

review of options for structural design sensitivity analysis part 1 linear systems the paper reviews options for structural design sensitivity analysis including global finite differences continuum derivatives discrete derivatives and computational or automated differentiation the objective is to put these different approaches to design sensitivity analysis in the context of accuracy and consistency computational cost and implementation options and effort linear static analysis and transient dynamic analysis are reviewed in a separate appendix special attention is paid to the semi analytical method a future paper will address design sensitivity analysis in nonlinear structural problems

a stabilized mixed finite element method for darcy flow we develop new stabilized mixed finite element methods for darcy flow stability and an a priori error estimate in the stability norm are established a wide variety of convergent finite elements present themselves unlike the classical galerkin formulation which requires highly specialized elements an interesting feature of the formulation is that there are no mesh dependent parameters numerical tests confirm the theoretical results 2002 elsevier science b v all rights reserved

the numerical solution of diffusion problems in strongly heterogeneous non isotropic materials a new second order finite difference algorithm for the numerical solution of diffusion problems in strongly heterogeneous and non isotropic media is constructed on problems with rough coefficients or highly nonuniform grids the new algorithm is superior to all other algorithms we have compared it with for problems with smooth coefficients on smooth grids the method is comparable with other second order methods the new algorithm is formulated for logically rectangular grids and is derived using thesupport operators method a key idea in deriving the method was to replace the usual inner product of vector functions by an inner product weighted by the inverse of the material properties tensor and to use the flux operator defined as the material properties tensor times the gradient rather than the gradient as one of the basic first order operators in the support operators method the discrete analog of the flux operator must also be the negative adjoint of the discrete divergence in an inner product that is a discrete analog of the continuum inner product the resulting method is conservative and the discrete analog of the variable coefficient laplacian is symmetric and negative definite on nonuniform grids in addition on any grid the discrete divergence is zero on constant vectors the null space for the gradient is the constant functions and when the material properties are piecewise constant the discrete flux operator is exact for piecewise linear functions we compare the methods on some of the most difficult examples to be found in the literature

comparison of fully implicit and impes formulations for simulation of water injection in fractured and unfractured media in this work we compare the fully implicit fi and implicit pressure explicit saturation impes formulations for the simulation of water injection in fractured media the system of partial differential equations is discretized within the discrete fracture framework using a control volume method a unique feature of the methodology is that there is no need for the computation of matrix fracture transfer terms the non linear system of equations resulting from the fi formulation is solved with state of the art newton and tensor methods direct and krylov iterative methods are employed to solve the system resulting from the newton linearization the performance of the fi and impes formulations is compared with numerical testing results show that the contrast between matrix and fracture properties affects the performance of both impes and fi formulations and that the tensor method outperforms all the newton solvers for the near singular jacobian matrix copyright 2006 john wiley sons ltd

scaling symmetric positive definite matrices to prescribed row sums we give a constructive proof of a theorem of marshall and olkin that any real symmetric positive definite matrix can be symmetrically scaled by a positive diagonal matrix to have arbitrary positive row sums we give a slight extension of the result showing that given a sign pattern there is a unique diagonal scaling with that sign pattern and we give upper and lower bounds on the entries of the scaling matrix

the parallel subdomain levelset deflation method in reservoir simulation extreme and isolated eigenvalues are known to be harmful to the convergence of an iterative solver these eigenvalues can be produced by strong heterogeneity in the underlying physics we can improve the quality of the spectrum by deflating the harmful eigenvalues in this work deflation is applied to linear systems in reservoir simulation in particular large sudden differences in the permeability produce extreme eigenvalues the number and magnitude of these eigenvalues is linked to the number and magnitude of the permeability jumps two deflation methods are discussed firstly we state that harmonic ritz eigenvector deflation which computes the deflation vectors from the information produced by the linear solver is unfeasible in modern reservoir simulation due to high costs and lack of parallelism secondly we test a physics based subdomain levelset deflation algorithm that constructs the deflation vectors a priori numerical experiments show that both methods can improve the performance of the linear solver we highlight the fact that subdomain levelset deflation is particularly suitable for a parallel implementation for cases with well defined permeability jumps of a factor 104 or higher parallel physics based deflation has potential in commercial applications in particular the good scalability of parallel subdomain levelset deflation combined with the robust parallel preconditioner for deflated system suggests the use of this method as an alternative for amg

preconditioning via a schur complement method an application in state estimation in this work the schur complement is analyzed and applied for preconditioning symmetric matrices in particular we eliminate the in modulo extreme eigenvalues by computing the schur complement with respect to a coarse subspace which captures the corresponding eigenvectors the potential of this method is presented as an application in state estimation where we employ a multiscale approach based on a wavelet galerkin discretization the hierarchy of arising optimization problems is solved in the framework of nested iterations due to the wavelet approach for all scales we can use one coarse adapted subspace in the schur complement method typically this provides not only a reduced condition number but also large gaps in the arising eigenvalue distribution which is favorable for several iterative methods numerical investigations show the enormous reduction in the condition number the necessary cg iteration steps and consequently in the computation time

superconvergence of the velocity in mimetic finite difference methods on quadrilaterals superconvergence of the velocity is established for mimetic finite difference approximations of second order elliptic problems over h 2 uniform quadrilateral meshes the superconvergence result holds for a full tensor coefficient the analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space the theoretical results are confirmed by numerical experiments

approximation of boundary conditions for mimetic finite difference methods abstract the numerical solution of partial differential equations solved with finite difference approximations that mimic the symmetry properties of the continuum differential operators and satisfy discrete versions of the appropriate integral identities are more likely to produce physically faithful results furthermore those properties are often needed when using the energy method to prove convergence and stability of a particular difference approximation unless special care is taken mimetic difference approximations derived for the interior grid points will fail to preserve the symmetries and identities between the gradient curl and divergence operators at the computational boundary in this paper we describe how to incorporate boundary conditions into finite difference methods so the resulting approximations mimic the identities for the differential operators of vector and tensor calculus the approach is valid for a wide class of partial differential equations of mathematical physics and will be described for poisson s equation with dirichlet neumann and robin boundary conditions we prove that the resulting difference approximation is symmetric and positive definite for each of these boundary conditions

efficient computational schemes of the conjugate gradient method for solving linear systems the technique of the initial guess calculation for the conjugate gradient method is proposed computational schemes of the linear system solution with symmetrical positive definite matrices are constructed on its basis their efficient modifications for systems with five diagonal matrices are proposed the investigation of the developed methods using the problem of two dimensional numerical simulation of bipolar transistors has been carried out experimental evidence of the proposed method s efficiency has been obtained

accelerating the shifted laplace preconditioner for the helmholtz equation by multilevel deflation many important physical phenomena can be described by the helmholtz equation we investigate to what extent the convergence of the shifted laplacian preconditioner for the helmholtz equation can be accelerated using deflation with multigrid vectors we therefore present a unified framework for two published algorithms the first deflates the preconditioned operator and requires no further preconditioning the second deflates the original operator and combines deflation and preconditioning in a multiplicative fashion we pursue two scientific contributions first we show using a model problem analysis that both algorithms cluster the eigenvalues the new and key insight here is that the near kernel of the coarse grid operator causes a limited set of eigenvalues to shift away from the center of the cluster with a distance proportional to the wave number this effect is less pronounced in the first algorithmic variant at the expense of a higher computational cost in the second contribution we quantify for the first time the large amount of reduction in cpu time that results from the clustering of eigenvalues and the reduction in iteration count we report to this end on the findings of an implementation in petsc on two and three dimensional problems with constant and variable wave number

domain decomposition methods for large linearly elliptic three dimensional problems abstract the idea of solving large problems using domain decomposition technique appears particularly attractive on present day large scale parallel computers but the performance of such techniques used on a parallel computer depends on both the numerical efficiency of the proposed algorithm and the efficiency of its parallel implementation the approach proposed herein splits the computational domain in unstructured subdomains of arbitrary shape and solves for unknown on the interface using the associated trace operator the steklov poincare operator on the continuous level or the schur complement matrix after a finite element discretization and a preconditioned conjugate gradient method this algorithm involves the solution of dirichlet and of neumann problems defined on each subdomain and which can be solved in parallel this method has been implemented on a cray 2 computer using multitasking and on an intel hypercube it was tested on a large scale industrial ill conditioned three dimensional linear elasticity problem which gives a fair indication of its performance in a real life environment in such situations the proposed method appears operational and competitive on both machines compared to standard techniques it yields faster results with far less memory requirements

on pod based deflation vectors for dpcg applied to porous media problems we study fast and robust iterative solvers for large systems of linear equations resulting from simulation of flow trough strongly heterogeneous porous media we propose the use of preconditioning and deflation techniques based on information obtained frfrom the system to reduce the time spent in the solution of the linear system an important question when using deflation techniques is how to find good deflation vectors which lead to a decrease in the number of iterations and a small increase in the required computing time per iteration in this paper we propose the use of deflation vectors based on a pod reduced set of snapshots we investigate convergence and the properties of the resulting methods finally we illustrate these theoretical results with numerical experiments we consider compressible and incompressible single phase flow in a layered model with variations in the permeability layers up to 10 3 and the spe 10 benchmark model with a contrast in permeability coefficients of 10 7 using deflation for the incompressible problem we reduce the number of iterations to 1 or 2 iterations with deflation for the compressible problem we reduce up to 80 the number of iterations when compared with the only preconditioned solver

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

solving groundwater flow problems by conjugate gradient methods and the strongly implicit procedure the performance of the preconditioned conjugate gradient method with three preconditioners is compared with the strongly implicit procedure sip using a scalar computer the preconditioners considered are the incomplete cholesky iccg and the modified incomplete cholesky miccg which require the same computer storage as sip as programmed for a problem with a symmetric matrix and a polynomial preconditioner polcg which requires less computer storage than sip although polcg is usually used on vector computers it is included here because of its small storage requirements in this paper published comparisons of the solvers are evaluated all four solvers are compared for the first time and new test cases are presented to provide a more complete basis by which the solvers can be judged for typical groundwater flow problems based on nine test cases the following conclusions are reached 1 sip is actually as efficient as iccg for some of the published linear two dimensional test cases that were reportedly solved much more efficiently by iccg 2 sip is more efficient than other published comparisons would indicate when common convergence criteria are used and 3 for problems that are three dimensional nonlinear or both and for which common convergence criteria are used sip is often more efficient than iccg and is sometimes more efficient than miccg

parallel multigrid preconditioning of the conjugate gradient method for systems of subsurface hydrology parallel preconditioners are considered for improving the convergence rate of the conjugate gradient method for solving sparse symmetric positive definite systems generated by finite element models of subsurface flow the difficulties of adapting effective sequential preconditioners to the parallel environment are illustrated by our treatment of incomplete cholesky preconditioning these difficulties are avoided with multigrid preconditioning which can be extended naturally to many processors so that the preconditioner remains global and effective the coarse grid correction which defines the multigrid preconditioner is outlined and its parallel implementation with the distributed finite element data structure is presented along with some examples of its use as a parallel preconditioner

deflation and projection methods applied to symmetric positive semi definite systems linear systems with a singular symmetric positive semi definite matrix appear frequently in practice this usually does not lead to difficulties for cg methods as long as these systems are consistent however the construction of a preconditioner especially the construction of two level and multilevel methods becomes more complicated since singular coarse grid matrices or galerkin matrices may occur here we continue the work started in 21 22 where deflation is used for some special singular coefficient matrices here we show that deflation and other projection type preconditioners can be applied to arbitrary singular problems without any difficulties in each of these methods a two level preconditioner is involved where coarse grid systems based on a singular galerkin matrix should be solved we prove that each projection operator consisting of a singular galerkin matrix can be written as an operator with a nonsingular galerkin matrix therefore many results that hold for nonsingular galerkin matrices are also valid for problems with singular galerkin matrices

deflated preconditioned conjugate gradient solvers for the pressure poisson equation a deflated preconditioned conjugate gradient technique has been developed for the solution of the pressure poisson equation within an incompressible flow solver the deflation is done using a region based decomposition of the unknowns making it extremely simple to implement the procedure has shown a considerable reduction in the number of iterations for grids with large graph depth the savings exceed an order of magnitude furthermore the technique has shown a remarkable insensitivity to the number of groups regions chosen and to the way the groups are formed

multilevel upscaling through variational coarsening 1 a new efficient multilevel upscaling procedure for single phase saturated flow in porous media is presented while traditional approaches to this problem have focused on the computation of an upscaled hydraulic conductivity here the coarse scale model is created explicitly from the fine scale model through the application of operator induced variational coarsening this technique which originated with robust multigrid solvers has been shown to accurately capture the influence of fine scale heterogeneous structure over the complete hierarchy of coarse scale models that it generates moreover implicit in this hierarchy is the construction of interpolation operators that provide a natural and complete multiscale basis for the fine scale problem thus this new multilevel upscaling methodology is similar to the multiscale finite element method and indeed it attains similar accuracy in computations of the fine scale hydraulic head and coarse scale normal flux on a variety of problems yet it is an order of magnitude faster on the examples considered here

ilum a multi elimination ilu preconditioner for general sparse matrices standard preconditioning techniques based on incomplete lu ilu factorizations offer a limited degree of parallelism in general a few of the alternatives advocated so far consist of either using some form of polynomial preconditioning or applying the usual ilu factorization to a matrix obtained from a multicolor ordering in this paper we present an incomplete factorization technique based on independent set orderings and multicoloring we note that in order to improve robustness it is necessary to allow the preconditioner to have an arbitrarily high accuracy as is done with ilus based on threshold techniques the ilum factorization described in this paper is in this category it can be viewed as a multifrontal version of a gaussian elimination procedure with threshold dropping which has a high degree of potential parallelism the emphasis is on methods that deal specifically with general unstructured sparse matrices such as those arising from finite element methods on unstructured meshes

optimized interface conditions in domain decomposition methods for problems with extreme contrasts in the coefficients when the coefficients of a problem have jumps of several orders of magnitude and are anisotropic many preconditioners and domain decomposition methods ddm suffer from plateaus in the convergence due to the presence of very small isolated eigenvalues in the spectrum of the preconditioned linear system one way to improve the preconditioner is to use a linear algebra technique called deflation or very similarly coarse grid corrections in both cases it is necessary to identify and compute at least approximately all the eigenvectors corresponding to the bad eigenvalues in the framework of ddm we propose a way to design interface conditions so that convergence is fast and does not have any plateau the method relies only on the knowledge of the smallest and largest eigenvalues of an auxiliary matrix the eigenvectors are not used the method relies on van der sluis result on a quasi optimal diagonal preconditioner for a symmetric positive definite matrix it is then possible to design robin interface conditions using only one real parameter for the entire interface by adding a second real parameter and more general interface conditions it is possible to take into account highly heterogeneous and anisotropic media numerical results are given and compared with other approaches

support operator finite difference algorithms for general elliptic problems abstract an algorithm is developed for discretizing boundary value problems given by a general linear elliptic second order partial differential equation with general mixed or robin boundary conditions in general logically rectangular grids the continuum problem can be written as an operator equation where the operator is self adjoint and positive definite the discrete approximations have the same property consequently the matrices for the discrete problem are symmetric and positive definite also the scheme has a nearest neighbor stencil consequently the most powerful linear solvers can be applied in smooth grids the algorithm produces second order accurate solutions it is the generality of the problem general matrix coefficients general boundary conditions general logically rectangular grids that makes finding such an algorithm difficult the algorithm which is a combination of the method of support operators and the mapping method overcomes certain difficulties of the individual methods producing a high quality algorithm for solving general elliptic problems

accelerating iterative solution methods using reduced order models as solution predictors we propose the use of reduced order models to accelerate the solution of systems of equations using iterative solvers in time stepping schemes for large scale numerical simulation the acceleration is achieved by determining an improved initial guess for the iterative process based on information in the solution vectors from previous time steps the algorithm basically consists of two projection steps 1 projecting the governing equations onto a subspace spanned by a low number of global empirical basis functions extracted from previous time step solutions and 2 solving the governing equations in this reduced space and projecting the solution back on the original high dimensional one we applied the algorithm to numerical models for simulation of two phase flow through heterogeneous porous media in particular we considered implicit pressure explicit saturation impes schemes and investigated the scope to accelerate the iterative solution of the pressure equation which is by far the most time consuming part of any impes scheme we achieved a substantial reduction in the number of iterations and an associated acceleration of the solution our largest test problem involved 93 500 variables in which case we obtained a maximum reduction in computing time of 67 the method is particularly attractive for problems with time varying parameters or source terms copyright 2006 john wiley sons ltd

testing iterative logic arrays for sequential faults with a constant number of patterns shows that a constant number of test vectors are sufficient for fully testing a k dimensional ila for sequential faults if the cell function is bijective the authors then present an efficient algorithm to obtain such a test sequence by extending the concept of c testability and m testability to sequential faults the constant length test sequence can be obtained a pipelined array multiplier is shown to be c testable with only 53 test vectors for exhaustively testing the sequential faults

deflated preconditioned conjugate gradient solvers for the pressure poisson equation extensions and improvements extensions and improvements to a deflated preconditioned conjugate gradient technique for the solution of the pressure poisson equation within an incompressible flow solver are described in particular the use of the technique for embedded grids for cases where volume of fluid or level set schemes are required and its implementation on parallel machines are considered several examples are included that demonstrate a considerable reduction in the number of iterations and a remarkable insensitivity to the number of groups regions chosen and or to the way the groups are formed copyright 2010 john wiley sons ltd

an iterative perturbation method for the pressure equation in the simulation of miscible displacement in porous media the miscible displacement problem in porous media is modeled by a nonlinear coupled system of two partial differential equations the pressure velocity equation and the concentration equation an iterative perturbation procedure is proposed and analyzed for the pressure velocity equation which is capable of producing as accurate a velocity approximation as the mixed finite element method and which requires the solution of symmetric positive definite linear systems only the velocity variable is involved in the linear systems and the pressure variable is obtained by substitution trivially applying perturbation methods can only give an error o epsilon while our iterative scheme can improve the error to o epsilon m at the m th iteration level where epsilon is a small positive number thus the convergence rate of our iterative procedure is o epsilon and consequently a small number of iterations is required theoretical convergence analysis and numerical experiments are presented to show the efficiency and accuracy of our method

an efficient preconditioned cg method for the solution of a class of layered problems with extreme contrasts in the coefficients knowledge of fluid pressure is important to predict the presence of oil and gas in reservoirs a mathematical model for the prediction of fluid pressures is given by a time dependent diffusion equation application of the finite element method leads to a system of linear equations a complication is that the underground consists of layers with very large differences in permeability this implies that the symmetric and positive definite coefficient matrix has a very large condition number bad convergence behavior of the cg method has been observed moreover a classical termination criterion is not valid in this problem after diagonal scaling of the matrix the number of extreme eigenvalues is reduced and it is proved to be equal to the number of layers with a high permeability for the ic preconditioner the same behavior is observed to annihilate the effect of the extreme eigenvalues a deflated cg method is used the convergence rate improves considerably and the termination criterion becomes again reliable finally a cheap approximation of the eigenvectors is proposed

a comparison of deflation and coarse grid correction applied to porous media flow in this paper we compare various preconditioners for the numerical solution of partial differential equations we compare a coarse grid correction preconditioner used in domain decomposition methods with a so called deflation preconditioner we prove that the effective condition number of the deflated preconditioned system is always for all deflation vectors and all restrictions and prolongations below the condition number of the system preconditioned by the coarse grid correction this implies that the conjugate gradient method applied to the deflated preconditioned system is expected always to converge faster than the conjugate gradient method applied to the system preconditioned by the coarse grid correction numerical results for porous media flows emphasize the theoretical results

parallel computation of flow in heterogeneous media modelled by mixed finite elements abstract in this paper we describe a fast parallel method for solving highly ill conditioned saddle point systems arising from mixed finite element simulations of stochastic partial differential equations pdes modelling flow in heterogeneous media each realisation of these stochastic pdes requires the solution of the linear first order velocity pressure system comprising darcy s law coupled with an incompressibility constraint the chief difficulty is that the permeability may be highly variable especially when the statistical model has a large variance and a small correlation length for reasonable accuracy the discretisation has to be extremely fine we solve these problems by first reducing the saddle point formulation to a symmetric positive definite spd problem using a suitable basis for the space of divergence free velocities the reduced problem is solved using parallel conjugate gradients preconditioned with an algebraically determined additive schwarz domain decomposition preconditioner the result is a solver which exhibits a good degree of robustness with respect to the mesh size as well as to the variance and to physically relevant values of the correlation length of the underlying permeability field numerical experiments exhibit almost optimal levels of parallel efficiency the domain decomposition solver doug http www maths bath ac uk parsoft used here not only is applicable to this problem but can be used to solve general unstructured finite element systems on a wide range of parallel architectures

high order mimetic finite difference method for diffusion problems on polygonal meshes the mimetic finite difference mfd methods mimic important properties of physical and mathematical models as a result conservation laws solution symmetries and the fundamental identities of the vector and tensor calculus are held for discrete models the mfd methods retain these attractive properties for full tensor coefficients and arbitrary polygonal meshes which may include non convex and degenerate elements the existing mfd methods for solving diffusion type problems are second order accurate for the conservative variable temperature pressure energy etc and only first order accurate for its flux we developed new high order mfd methods which are second order accurate for both scalar and vector variables the second order convergence rates are demonstrated with a few numerical examples on randomly perturbed quadrilateral and polygonal meshes

a comparison of deflation and the balancing preconditioner in this paper we compare various preconditioners for the numerical solution of partial differential equations we compare the well known balancing preconditioner used in domain decomposition methods with a so called deflation preconditioner we prove that the effective condition number of the deflated preconditioned system is always i e for all deflation vectors and all restrictions and prolongations below the condition number of the system preconditioned by the balancing preconditioner even more we establish that both preconditioners lead to almost the same spectra the zero eigenvalues of the deflation preconditioned system are replaced by eigenvalues which are one if the balancing preconditioner is used moreover we prove that the a norm of the errors of the iterates built by the deflation preconditioner is always below the a norm of the errors of the iterates built by the balancing preconditioner depending on the implementation of the balancing preconditioner the amount of work of one iteration of the deflation preconditioned system is less than or equal to the amount of work of one iteration of the balancing preconditioned system if the amount of work is equal both preconditioners are sensitive with respect to inexact computations finally we establish that the deflation preconditioner and the balancing preconditioner produce the same iterates if one uses certain starting vectors numerical results for porous media flows emphasize the theoretical results

preconditioning in h div and applications we consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator i grad div the natural setting for such problems is in the hilbert space h div and the variational formulation is based on the inner product in h div we show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques these preconditioners are shown to be spectrally equivalent to the inverse of the operator as a consequence they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations with the number independent of the mesh discretization we describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems

unstructured additive schwarz conjugate gradient method for elliptic problems with highly discontinuous coefficients this paper concerns the iterative solution of symmetric elliptic problems with piecewise constant coefficients in two or three space dimensions discretized by linear finite element methods on unstructured triangular or tetrahedral meshes the effect of the discontinuous coefficients is studied by first postulating that there are d fixed regions of the domain where the coefficient takes constant positive values a a1 ad and then considering certain positive coefficient sequences a m in which some of these values approach 0 or infty as m rightarrow infty we consider the performance of additive schwarz domain decomposition preconditioners constructed from local solves on automatically generated subdomains together with a global solve on some coarser grid assuming no relationship between the regions on which the coefficient function is constant and either the subdomains or the coarse grid we show that the preconditioned conjugate gradient method converges in both the energy and the euclidean norms with a number of iterations which grows only logarithmically in the size of the maximum jump cal j m max a k m a l m k l 1 ldots d as m rightarrow infty the result is obtained by a careful analysis of the preconditioned matrix it cannot be obtained by the usual procedure of estimating condition numbers a simple example is given in which the condition number of both the original and the preconditioned stiffness matrices degrade linearly in cal j m recent results of chan smith and zou have shown that by using this preconditioner together with the conjugate gradient method the number of iterations can be bounded independently of the mesh diameter provided the subdomains have overlap commensurate with the size of the coarse mesh our results now show that this method is also highly resilient to discontinuous coefficients even if no attention is paid to the coefficient discontinuity in the construction of the solver

multiple right hand side techniques for the numerical simulation of quasistatic electric and magnetic fields the simulation of slowly varying transient electric high voltage fields and magnetic fields requires the repeated and successive solution of high dimensional linear algebraic systems of equations with identical or near identical system matrices and different right hand side vectors for these solution processes which are required within implicit time integration schemes and nonlinear quasi newton raphson methods an iterative multiple right hand side mrhs scheme is used which recycles vector subspaces resulting from previous preconditioned conjugate gradient iteration runs the combination of this scheme with a subspace projection extrapolation start value generation scheme is discussed numerical results for three dimensional electric and magnetic field simulations are presented and the efficiency of the new schemes re using eigenvector information from previous iteration processes with different tolerance criteria are compared to those of standard conjugate gradient iterations

a minimum test set for multiple fault detection in ripple carry adders previous papers have shown that a ripple carry adder composed of several full adder cells can be completely tested by a minimum test set of size 8 independent of the number of cells in the ripple carry adder under single faulty cell assumption the fault model assumed is that faults in a cell can change the cell behavior in any arbitrary way as long as the cell remains a combinational circuit in this paper we assume that any number of cells can be faulty at any time a minimum test set of size 11 which can detect arbitrary length ripple carry adders under this fault model is presented for general n p adders in which each cell is a p bit adder a minimum test set of size 3 22p 1 is also presented

a deflation technique for linear systems of equations iterative methods for solving linear systems of equations can be very efficient if the structure of the coefficient matrix can be exploited to accelerate the convergence of the iterative process however for classes of problems for which suitable preconditioners cannot be found or for which the iteration scheme does not converge iterative techniques may be inappropriate this paper proposes a technique for deflating the eigenvalues and associated eigenvectors of the iteration matrix which either slow down convergence or cause divergence this process is completely general and works by approximating the eigenspace bbb p corresponding to the unstable or slowly converging modes and then applying a coupled iteration scheme on bbb p and its orthogonal complement bbb q

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

easily testable iterative logic arrays iterative logic arrays ilas are studied with respect to two testing problems first a variety of conditions is presented meeting these conditions guarantees an upper bound on the size of the test set for the ila under consideration second techniques for designing optimally testable ilas are presented the arrays treated are in some cases more general than those that have been reported by other researchers they include multidimensional and inhomogeneous arrays octagonally connected arrays and bilateral arrays are also discussed the results indicate that the characteristics of the individual cell functions e g whether they are bijective are a good guide to the test complexity of the overall array matrix multiplication as an example is shown to have several different optimally testable implementations the results are useful for combinational and pipelined arrays and for certain systolic arrays

testability of convergent tree circuits the testing properties of a class of regular circuits called convergent trees are investigated convergent trees include such practical circuits as comparators multiplexers and carry lookahead adders the conditions for the testability of these tree circuits are derived for a functional fault model the notion of l testability is introduced where the number of tests for a p level tree is directly proportional to p rather than exponential in p convergent trees that are c testable testable with a fixed number of tests regardless of the tree s size are also characterized two design techniques are also introduced that modify arbitrary tree modules in order to achieve land c testability finally we apply these techniques to the design of a large carry lookahead adder

analysis of feti methods for multiscale pdes in this paper we study a variant of the finite element tearing and interconnecting feti method which is suitable for elliptic pdes with highly heterogeneous multiscale coefficients x in particular coefficients with strong variation within subdomains and or jumps that are not aligned with the subdomain interfaces using energy minimisation and cut off arguments we can show rigorously that for an arbitrary positive coefficient function alpha in l infty omega the condition number of the preconditioned feti system can be bounded by c 1 log h h 2 where h is the subdomain diameter and h is the mesh size and where the function c depends only on the coefficient variation in the vicinity of subdomain interfaces in particular if alpha omega i varies only mildly in a layer i of width near the boundary of each of the subdomains i then c alpha mathcal o h eta 2 independent of the variation of in the remainder i i of each subdomain and independent of any jumps of across subdomain interfaces the quadratic dependence of c on h can be relaxed to a linear dependence under stronger assumptions on the behaviour of in the interior of the subdomains our theoretical findings are confirmed in numerical tests

multiscale linear solvers for very large systems derived from pdes we present a novel linear solver that works well for large systems obtained from discretizing pdes it is robust and for the examples we studied the computational effort scales linearly with the number of equations the algorithm is based on a wavelength decomposition that combines conjugate gradient multiscaling and iterative splitting methods into a single approach on the surface the algorithm is a simple preconditioned conjugate gradient with all the sophistication of the algorithm in the choice of the preconditioning matrix the preconditioner is a very good approximate inverse of the linear operator it is constructed from the inverse of the coarse grained linear operator and from smoothing operators that are based on an operator splitting on the fine grid the coarse graining captures the long wavelength behavior of the inverse operator while the smoothing operator captures the short wavelength behavior the conjugate gradient iteration accounts for the coupling between long and short wavelengths the coarse grained operator corresponds to a lower resolution approximation to the pdes while the coarse grained inverse is not known explicitly the algorithm requires only that the preconditioner can be a applied to a vector the coarse inverse applied to a vector can be obtained as the solution of another preconditioned conjugate gradient solver that applies the same algorithm to the smaller problem thus the method is naturally recursive the recursion ends when the matrix is sufficiently small for a solution to be obtained efficiently with a standard solver the local feedback provided by the conjugate gradient step at every level makes the algorithm very robust in spite of the effort required for the coarse inverse the algorithm is efficient because the increased quality of the approximate inverse greatly reduces the number of times the preconditioner needs to be evaluated a feature of the algorithm is that the transition between coarse grids is determined dynamically by the accuracy requirement of the conjugate gradient solver at each level typically later iterations on the finer scales need fewer iterations on the coarser scales and the computational effort is proportional to n rather than n log n where n is the number of equations we have tested our solver on the porous flow equation on a workstation we have solved problems on grids ranging in dimension over three orders of magnitude from 103 to 106 and found that the linear scaling holds the algorithm works well even when the permeability tensor has spatial variations exceeding a factor of 109

stabilization of unstable procedures the recursive projection method fixed point iterative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes the recursive projection method rpm which stabilizes such procedures by computing a projection onto the unstable subspace is presented on this subspace a newton or special newton iteration is performed and the fixed point iteration is used on the complement as continuation in the parameter proceeds the projection is efficiently updated possibly increasing or decreasing the dimension of the unstable subspace the method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system convergence proofs are given and pseudo arclength continuation on the unstable subspace is introduced to allow continuation past folds examples are presented for an important application of the rpm in which a black box time integration scheme is stabilized enabling it to compute unstable steady states the rpm can also be used to accelerate iterative procedures when slow convergence is due to a few slowly decaying modes

a new solver for the elastic normal contact problem using conjugate gradients deflation and an fft based preconditioner this paper presents our new solver bccg fai for solving elastic normal contact problems this is a comprehensible approach that is based on the conjugate gradients cg algorithm and that uses ffts a first novel aspect is the definition of the fft based approximate inverse preconditioner the underlying idea is that the inverse matrix can be approximated well using a toeplitz or block toeplitz form which can be computed using the fft of the original matrix elements this preconditioner makes the total number of cg iterations effectively constant in 2d and very slowly increasing in 3d problems a second novelty is how we deal with a prescribed total force this uses a deflation technique in such a way that cgs convergence and finite termination properties are maintained numerical results show that this solver is more effective than existing cg based strategies such that it can compete with multi grid strategies over a much larger problem range in our opinion it could be the new method of choice because of its simple structure and elegant theory and because robust performance is achieved independently of any problem specific parameters we present a new cg and fft based solver for the elastic normal contact problem using ffts we construct an effective preconditioner with block toeplitz form o 1 iterations are needed in 2d problems as predicted by theory a prescribed total force is dealt with by ignoring the average using deflation the performance of the resulting algorithm is competitive with multi grid strategies

multilevel preconditioners for discontinuous galerkin approximations of elliptic problems with jump coefficients in this article we develop and analyze two level and multi level methods for the family of interior penalty ip discontinuous galerkin dg discretizations of second order elliptic problems with rough coecients exhibiting large jumps across interfaces in the domain these methods are based on a decomposition of the dg nite element space that inherently hinges on the diusion coecient of the problem our analysis of the proposed preconditioners is presented for both symmetric and non symmetric ip schemes and we establish both robustness with respect to the jump in the coecient and near optimality with respect to the mesh size following the analysis we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods

on the testability of one dimensional ilas for multiple sequential faults it is shown that one dimensional unilateral iterative logic arrays ilas of combinational cells are c testable for multiple sequential faults provided the fault free cell functions satisfy appropriate conditions the test sequence is of length o m sup 2 n sup 2 mn sup 3 k where n resp m is the number of signal values that can be applied to the horizontal resp vertical cell input and k

overlapping schwarz methods on unstructured meshes using non matching coarse grids we consider two level overlapping schwarz domain decomposition methods for solving the finite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions standard finite element interpolation from the coarse to the fine grid may be used our theory requires no assumption on the substructures that constitute the whole domain so the substructures can be of arbitrary shape and of different size the global coarse mesh is allowed to be non nested to the fine grid on which the discrete problem is to be solved and neither the coarse mesh nor the fine mesh need be quasi uniform in addition the domains defined by the fine and coarse grid need not be identical the one important constraint is that the closure of the coarse grid must cover any portion of the fine grid boundary for which neumann boundary conditions are given in this general setting our algorithms have the same optimal convergence rate as the usual two level overlapping domain decomposition methods on structured meshes the condition number of the preconditioned system depends only on the possibly small overlap of the substructures and the size of the coarse grid but is independent of the sizes of the subdomains

parallel preconditioned conjugate gradient algorithm on gpu we propose a parallel implementation of the preconditioned conjugate gradient algorithm on a gpu platform the preconditioning matrix is an approximate inverse derived from the ssor preconditioner used through sparse matrix vector multiplication the proposed preconditioner is well suited for the massively parallel gpu architecture as compared to cpu implementation of the conjugate gradient algorithm our gpu preconditioned conjugate gradient implementation is up to 10 times faster 8 times faster at worst

scalings of matrices which have prespecified row sums and column sums via optimization abstract the problem of scaling a matrix so that it has given row and column sums is transformed into a convex minimization problem in particular we use this transformation to characterize the existence of such scaling or corresponding approximations we obtain new results as well as new streamlined proofs of known results

analysis of feti methods for multiscale pdes part ii interface variation in this article we give a new rigorous condition number estimate of the finite element tearing and interconnecting feti method and a variant thereof all floating feti we consider a scalar elliptic equation in a two or three dimensional domain with a highly heterogeneous multiscale diffusion coefficient this coefficient is allowed to have large jumps not only across but also along subdomain interfaces and in the interior of the subdomains in other words the subdomain partitioning does not need to resolve any jumps in the coefficient under suitable assumptions we derive bounds for the condition numbers of one level and all floating feti that are robust with respect to strong variations in the contrast in the coefficient and that are explicit in some geometric parameters associated with the coefficient variation in particular robustness holds for face edge and vertex islands in high contrast media as a central tool we prove and use new weighted poincare and discrete sobolev type inequalities that are explicit in the weight our theoretical findings are confirmed in a series of numerical experiments

a testable design of iterative logic arrays testable design of unilateral iterative logic arrays ila of combinational cells under the assumption of a single cell failure is considered the concepts of one step testability and one step c testability are introduced methods to modify the basic cell flow table so as to facilitate fault detection and location are given it is shown that if no directly observable outputs from each cell are available then it is possible to augment the cell flow table by the addition of a fixed number 4 of columns and a row so that a faulty cell can be located by a test of length proportional to log2p where p is the number of cells in the array however if directly observable outputs are available from each cell then the test length is shown to be independent of the array length to locate a faulty cell

a coarse space construction based on local dirichlet to neumann maps coarse grid correction is a key ingredient of scalable domain decomposition methods in this work we construct coarse grid space using the low frequency modes of the subdomain dirichlet to neumann maps and apply the obtained two level preconditioners to the extended or the original linear system arising from an overlapping domain decomposition our method is suitable for parallel implementation and its efficiency is demonstrated by numerical examples on problems with large heterogeneities for both manual and automatic partitionings

parallel solution of high frequency helmholtz equations using high order finite difference schemes abstract high frequency helmholtz equations pose a challenging computational task especially at high frequencies this work examines the parallel solution of such problems using 2nd 4th and 6th order finite difference schemes the examples include problems with known analytic solutions enabling error evaluation of the different schemes on various grids including problems on l shaped domains the resulting linear systems are solved with the block parallel carp cg algorithm which succeeded in lowering the relative residual indicating that it is a robust and reliable parallel solver of the resulting linear systems however lowering the error of the solution to reasonable levels with practical mesh sizes is obtained only with the higher order schemes these results corroborate the known limitations of the 2nd order scheme at modeling the helmholtz equation at high frequencies and they indicate that carp cg can also be used effectively with high order finite difference schemes furthermore the parallel scalability of carp cg improves when the wave number is increased on a fixed grid or when for a fixed wave number the grid size is decreased

two level algebraic domain decomposition preconditioners using jacobi schwarz smoother and adaptive coarse grid corrections we investigate two level preconditioners on the extended linear system arising from the domain decomposition method the additive schwarz method is used as a smoother and the coarse grid space is constructed by using the ritz vectors obtained in the arnoldi process the coarse grid space can be improved adaptively as the ritz vectors become a better approximation of the eigenvectors numerical tests on the model problem demonstrate the efficiency

a low numerical dissipation immersed interface method for the compressible navier stokes equations a numerical method to solve the compressible navier stokes equations around objects of arbitrary shape using cartesian grids is described the approach considered here uses an embedded geometry representation of the objects and approximate the governing equations with a low numerical dissipation centered finite difference discretization the method is suitable for compressible flows without shocks and can be classified as an immersed interface method the objects are sharply captured by the cartesian mesh by appropriately adapting the discretization stencils around the irregular grid nodes located around the boundary in contrast with available methods no jump conditions are used or explicitly derived from the boundary conditions although a number of elements are adopted from previous immersed interface approaches a new element in the present approach is the use of the summation by parts formalism to develop stable non stiff first order derivative approximations at the irregular grid points second order derivative approximations as those appearing in the transport terms can be stiff when irregular grid points are located too close to the boundary this is addressed using a semi implicit time integration method moreover it is shown that the resulting implicit equations can be solved explicitly in the case of constant transport properties convergence studies are performed for a rotating cylinder and vortex shedding behind objects of varying shapes at different mach and reynolds numbers

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

a preconditioned multilevel mimetic finite difference method for solving the pressure velocity system abstract a mimetic finite difference mfd implementation is proposed to solve the pressure velocity system iteratively the recursive procedure only solves for the velocity variable while the pressure is calculated by direct substitution simplifying the problem the symmetric positive definite system obtained is solved with an innovative implementation of a pre conditioned multilevel conjugate gradient algorithm where coarse algebraic objects are easily computed from the corresponding fine level ones based on fundamental geometric properties of the mfd technology a succinct eigen analysis shows second order convergence on the velocity variable for cartesian grids which is corroborated numerically also results consider different topologies and different medium properties showing the applicability of the method and optimal convergence rates for the velocity on cartesian and h 2 uniform curved grids

a numerical study on neumann neumann and feti methods for hp approximations on geometrically refined boundary layer meshes in two dimensions abstract we present extensive numerical tests showing the performance and robustness of certain balancing neumann neumann and one level feti methods for the solution of algebraic linear systems arising from hp finite element approximations of scalar elliptic problems on geometrically refined boundary layer meshes in two dimensions the numerical results confirm the theoretical results derived in a toselli x vasseur technical report 02 15 seminar fur angewandte mathematik eth zurich september 2002 the condition numbers are independent of the aspect ratio of the mesh and of potentially large jumps of the coefficients in addition they only grow polylogarithmically with the polynomial degree as in the case of p approximations on shape regular meshes

multilevel schwarz methods for elliptic problems with discontinuous coefficients in three dimensions multilevel schwarz methods are developed for a conforming finite element approximation of second order elliptic problems we focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions we establish a condition number estimate for the iterative operator which is independent of the coefficients and grows at most as the square of the number of levels we also characterize a class of distributions of the coefficients called quasi monotone for which the weighted l 2 projection is stable and for which we can use the standard piecewise linear functions as a coarse space in this case we obtain optimal methods i e bounds which are independent of the number of levels and subregions we also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas we consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods

spectral properties and conservation laws in mimetic finite difference methods for pdes the mimetic finite difference mfd methods for pdes mimic crucial properties of mathematical systems duality and self adjointness of differential operators conservation laws and properties of the solution on general polytopal meshes in this article the structure and the spectral properties of the linear systems derived by the spatial discretization of diffusion problem are analysed in addition the numerical approximation of parabolic equations is discussed where the mfd approach is used in the space discretization while implicit method and explicit runge kutta chebyshev schemes are used in time discretization moreover we will show how the numerical solution preserves certain conservation laws of the theoretical solution

a generalized unit system for concise electromagnetic formulation and accurate numerical solutions we present in this paper a novel generalized unit system gus for electromagnetic formulation and for improving the numerical behavior of the solution procedure by decreasing the condition number of the finite element fe matrices the goal is to express in the gus the governing electromagnetic equations in a form that does not involve any factors such as 4 and those related to the physical properties of vacuum this goal is achieved by scaling all physical quantities the scaling factors are not all free parameters but are constrained by specific relations we prove that there are only two free parameters but only one is significant for minimizing the condition number of the fe matrices numerical examples are presented to illustrate the essential features of the proposed gus including the minimization of the condition number

fast and robust solvers for pressure correction in bubbly flow problems we consider the numerical simulation of two phase fluid flow where bubbles or droplets of one phase move against a background of the other phase such flows are governed by the navier stokes equations the solution of which may be approximated using a pressure correction approach within such an approach the computational cost is often dominated by the solution of a linear system corresponding to a discrete poisson equation with discontinuous coefficients in this paper we explore the efficient solution of these linear systems using robust multilevel solvers such as deflated variants of the preconditioned conjugate gradient method or robust multigrid techniques we consider these families of methods in more detail and compare their performance in the simulation of bubbly flows some of these methods turn out to be very effective and reduce the amount of work to solve the pressure correction system substantially resulting in efficient calculations for two phase flows on highly resolved grids

row scaling as a preconditioner for some nonsymmetric linear systems with discontinuous coefficients linear systems with large differences between the coefficients called discontinuous coefficients often arise when physical phenomena in heterogeneous media are modeled by partial differential equations pdes such problems are usually solved by domain decomposition techniques but these can be difficult to implement when subdomain boundaries are complicated or the grid is unstructured it is known that for such systems diagonal scaling can sometimes improve the eigenvalue distribution and the convergence properties of some algorithm preconditioner combinations however there seems to be no study outlining both the usefulness and limitations of this approach it is shown that l 2 scaling of the equations is a generally useful preconditioner for such problems when the system matrices are nonsymmetric but only when the off diagonal elements are small to moderate tests were carried out on several nonsymmetric linear systems with discontinuous coefficients derived from convection diffusion elliptic pdes with small to moderate convection terms it is shown that l 2 scaling improved the eigenvalue distribution of the system matrix by reducing their concentration around the origin very significantly furthermore such scaling improved the convergence properties of restarted gmres and bi cgstab with and without the ilu 0 preconditioner since ilu 0 is theoretically oblivious to diagonal scaling these results indicate that l 2 scaling also improves the runtime numerical stability

a boundary condition capturing multigrid approach to irregular boundary problems we propose a geometric multigrid method for solving linear systems arising from irregular boundary problems involving multiple interfaces in two and three dimensions in this method we adopt a matrix free approach i e we do not form the fine grid matrix explicitly and we never form nor store the coarse grid matrices as many other robust multigrid methods do the main idea is to construct an accurate interpolation which captures the correct boundary conditions at the interfaces via a level set function numerical results are given to compare our multigrid method with black box and algebraic multigrid methods

robust and highly scalable parallel solution of the helmholtz equation with large wave numbers numerical solution of the helmholtz equation is a challenging computational task particularly when the wave number is large for two dimensional problems direct methods provide satisfactory solutions but large three dimensional problems become unmanageable in this work the block parallel carp cg algorithm parallel computing 36 2010 is applied to the helmholtz equation with large wave numbers the effectiveness of this algorithm is shown both theoretically and practically with numerical experiments on two and three dimensional domains including heterogeneous cases and a wide range of wave numbers a second order finite difference discretization scheme is used leading to a complex nonsymmetric and indefinite linear system carp cg is both robust and efficient on the tested problems on a fixed grid its scalability improves as the wave number increases additionally when the number of grid points per wavelength is fixed the number of iterations increases linearly with the wave number convergence rates for heterogeneous cases are similar to those of homogeneous cases carp cg also outperforms at all wave numbers one of the leading methods based on the shifted laplacian preconditioner with a complex shift and solved with a multigrid

testability conditions for bilateral arrays of combinational cells two sets of conditions are derived that make one dimensional bilateral arrays of combinational cells testable for single faulty cells the test sequences are preset and in the worst case grow quadratically with the size of the array conditions for testability in linear time are also derived the basic cell can operate at the bit or at the word level an implementation of fir filters using systolic one dimensional bilateral arrays of cells which can be considered combinational at the word level is presented as an example a straightforward generalization for the two dimensional case is made a systolic array used for matrix multiplication is presented as an example for this case

domain decomposition for multiscale pdes we consider additive schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic pdes with highly variable coefficients in contrast to standard analyses we do not assume that the coefficients can be resolved by a coarse mesh this situation arises often in practice for example in the computation of flows in heterogeneous porous media in both the deterministic and monte carlo simulated stochastic cases we consider preconditioners which combine local solves on general overlapping subdomains together with a global solve on a general coarse space of functions on a coarse grid we perform a new analysis of the preconditioned matrix which shows rather explicitly how its condition number depends on the variable coefficient in the pde as well as on the coarse mesh and overlap parameters the classical estimates for this preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse grid elements by contrast our new results show that with a good choice of subdomains and coarse space basis functions the preconditioner can still be robust even for large coefficient variation inside domains when the classical method fails to be robust in particular our estimates prove very precisely the previously made empirical observation that the use of low energy coarse spaces can lead to robust preconditioners we go on to consider coarse spaces constructed from multiscale finite elements and prove that preconditioners using this type of coarsening lead to robust preconditioners for a variety of binary i e two scale media model problems moreover numerical experiments show that the new preconditioner has greatly improved performance over standard preconditioners even in the random coefficient case we show also how the analysis extends in a straightforward way to multiplicative versions of the schwarz method

the construction of projection vectors for a deflated iccg method applied to problems with extreme contrasts in the coefficients abstract to predict the presence of oil and natural gas in a reservoir it is important to know the fluid pressure in the rock formations a mathematical model for the prediction of the fluid pressure history is given by a time dependent diffusion equation application of the finite element method leads to systems of linear equations a complication is that the underground consists of layers with very large contrasts in permeability this implies that the symmetric and positive definite coefficient matrix has a very large condition number bad convergence behavior of the iccg method has been observed and a classical termination criterion is not valid in this problem in 19 we have shown that the number of small eigenvalues of the diagonally scaled matrix is equal to the number of high permeability domains which are not connected to a dirichlet boundary in this paper the proof is extended to an incomplete cholesky decomposition to annihilate the bad effect of these small eigenvalues on the convergence the deflated iccg method is used in 19 we have shown how to construct a deflation subspace for the case of a set of more or less parallel layers that subspace proved to be a good approximation of the span of the small eigenvectors as a result of this the convergence of diccg is independent of the contrasts in the permeabilities in this paper it is shown how to construct deflation vectors even in the case of very irregular shaped layers and layers with so called inclusions a theoretical investigation and numerical experiments show that the diccg method is not sensitive to small perturbations of the deflation vectors the efficiency of the diccg method is illustrated by numerical experiments

detection of multiple faults in two dimensional ilas we provide test sets proportional to the sum of the two dimensions of the array for a large class of cells which allow us to test rows or columns of cells of the array independently constant length test sets for array multipliers have been found under the single faulty cell model if the array is modified and otherwise test sets are proportional to the number of cells we can verify the full adder array of a combinational n spl times m multiplier in o n m tests under the multiple faulty cell mfc model the entire multiplier including the and gates which generate the summands can be verified after applying the same modifications which make the multiplier c testable under the single faulty cell model