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

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

domain decomposition method for harmonic wave propagation a general presentation in this paper we give a general presentation of non overlapping domain decomposition methods for harmonic wave propagation models our abstract framework leads to concise convergence proofs and contains some recent developments about the use of non local transmission conditions it also includes applications to acoustic electromagnetic or elastic waves as well as the treatment of space discretization

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

optimized schwarz methods without overlap for the helmholtz equation the classical schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel convergence is achieved through overlap of the subdomains we study in this paper a variant of the schwarz method which converges without overlap for the helmholtz equation we show that the key ingredients for such an algorithm are the transmission conditions we derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps these conditions are however nonlocal in nature and we introduce local approximations which we optimize for performance of the schwarz method this leads to an algorithm in the class of optimized schwarz methods we present an asymptotic analysis of the optimized schwarz method for two types of transmission conditions robin conditions and transmission conditions with second order tangential derivatives numerical results illustrate the effectiveness of the optimized schwarz method on a model problem and on a problem from industry

two level mortar domain decomposition preconditioners for heterogeneous elliptic problems abstract we consider a second order elliptic problem with a heterogeneous coefficient which models for example single phase flow through a porous medium we write this problem in mixed form and approximate it for parallel computation using the multiscale mortar domain decomposition mixed finite element method which gives rise to a saddle point linear system we use a relatively fine mortar space which allows us to enforce continuity of the normal velocity flux or nearly so in the case of nonmatching meshes to solve the schur complement linear system for the mortar unknowns we propose a two level preconditioner based on the interfaces between subdomains the coarse preconditioner also uses the multiscale mortar domain decomposition method but with instead a very coarse mortar space we show that the prolongation operator of the coarse mortar to the fine is defined uniquely by the condition that the l 2 projection of a coarse mortar agrees with its projection onto the space of normal velocity fluxes i e no energy is introduced when changing mortar scales the local smoothing preconditioner is based on block jacobi using blocks defined by the interfaces we use restrictive smoothing domains that are smaller normal to the interfaces and overlapping in the directions tangential to the interfaces in the simplest case the condition number of the preconditioned interface operator is bounded by a multiple of log 1 h h 2 we show several numerical examples involving strongly heterogeneous porous media to demonstrate the efficiency and robustness of the preconditioner we see that it is often desirable and sometimes necessary to use a piecewise linear or higher order coarse mortar space to achieve good convergence for heterogeneous problems

simultaneous feti and block feti robust domain decomposition with multiple search directions domain decomposition methods often exhibit very poor performance when applied to engineering problems with large heterogeneities in particular for heterogeneities along domain interfaces the iterative techniques to solve the interface problem are lacking an efficient preconditioner recently a robust approach named feti geneo was proposed where troublesome modes are precomputed and deflated from the interface problem the cost of the feti geneo is however high we propose in this paper techniques that share similar ideas with feti geneo but where no pre processing is needed and that can be easily and efficiently implemented as an alternative to standard domain decomposition methods in the block iterative approaches presented in this paper the search space at every iteration on the interface problem contains as many directions as there are domains in the decomposition those search directions originate either from the domain wise preconditioner in the simultaneous feti method or from the block structure of the right hand side of the interface problem block feti we show on 2d structural examples that both methods are robust and provide good convergence in the presence of high heterogeneities even when the interface is jagged or when the domains have a bad aspect ratio the simultaneous feti was also efficiently implemented in an optimized parallel code and exhibited excellent performance compared to the regular feti method

strengthened h sup spl infin control via state feedback a majorization approach using algebraic riccati inequalities we extend the algebraic riccati equation approach to h sup spl infin h sup 2 state feedback design to incorporate additional constraints on the other nonmaximum singular values it is shown that a state feedback control law that satisfies a singular values singular vectors constraint exists if and only if a corresponding algebraic riccati inequality admits a stabilizing and positive definite solution the results are applied to devise a majorization approach for constructing state feedback controllers that strive to optimize some obvious design objectives

domain decomposition for linear elasticity with dg jumps and mortars in this article we present a domain decomposition method for solving a linear elasticity system data are transmitted by jumps as in a discontinuous galerkin method and mortars are introduced at the interfaces to dissociate the computation between neighboring subdomains a decoupling algorithm condenses the unknowns on the interface the matrix of the system is constructed by parallel computation numerical experiments confirming the theory are presented

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

the dual schur complement method with well posed local neumann problems regularization with a perturbed lagrangian formulation the dual schur complement dsc domain decomposition dd method introduced by farhat and roux is an efficient and practical algorithm for the parallel solution of self adjoint elliptic partial dif

absence of renormalization group pathologies near the critical temperature two examples we consider real space renormalization group transformations for ising type systems which are formally defined by exp left h sigma right sum limits sigma t sigma sigma exp left h sigma right wheret is a probability kernel i e t 1 for every configuration for each choice of the block spin configuration let let be the measure on spin configurations which is formally given by taking the probability of to be proportional tot exp h we give a condition which is sufficient to imply that the renormalized hamiltonianh is defined roughly speaking the condition is that the collection of measures is in the high temperature phase uniformly in the block spin configuration the proof of this result uses methods of olivieri and picco we use our theorem to prove that the first iteration of the renormalization group transformation is defined in the following two examples decimation with spacingb 2 on the square lattice with 1 36 c and the kadanoff transformation with parameterp on the trian gular lattice in a subset of the p plane that includes values of greater than c

balancing domain decomposition for problems with large jumps in coefficients the balancing domain decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent the abstract theory introduced recently by the first named author is used to develop condition number bounds for conforming linear elements in two and three dimensions the bounds are independent of arbitrary coefficient jumps between subdomains and of the number of subdomains and grow only as the squared logarithm of the mesh size h computational experiments for two and three dimensional problems confirm the theory

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

non overlapping schwarz methods with optimized transmission conditions for the helmholtz equation abstract the development of efficient transmission conditions for the schwarz method without overlap is an important area of current research the techniques developed in recent years mainly consist of algebraic approximations of the discrete transparent operator or optimized approximations of the continuous transparent operator these optimized approximations are currently based on an analytical approach involving a minimization problem over all the available frequencies and using the l norm in this paper new numerical optimization techniques are investigated these original approaches allow selection of the frequency range for the minimization problem which is based on the l p norm new numerical experiments performed on a model problem demonstrate the performance and the robustness of the proposed optimization techniques in the particular cases p 1 and p applications to industrial automotive problems confirm the efficiency of these techniques when using arbitrary mesh partitions

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 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

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

overlapping schwarz methods with adaptive coarse spaces for multiscale problems in 3d we propose two variants of the overlapping additive schwarz method for the finite element discretization of scalar elliptic problems in 3d with highly heterogeneous coefficients the methods are efficient and simple to construct using the abstract framework of the additive schwarz method and an idea of adaptive coarse spaces in one variant the coarse space consists of finite element functions associated with the wire basket nodes and functions based on solving some generalized eigenvalue problems on the faces in the other variant it contains functions associated with the vertex nodes with functions based on solving some generalized eigenvalue problems on subdomain faces and subdomain edges the functions that constitute the coarse spaces are chosen adaptively and they correspond to the eigenvalues that are smaller than a given threshold the convergence rate of the preconditioned conjugate gradients method in both cases is shown to be independent of the variations in the coefficients for the sufficient number of eigenfunctions in the coarse space numerical results are given to support the theory

overlapping schwarz methods for mixed linear elasticity and stokes problems overlapping schwarz preconditioners are introduced and studied for saddle point problems with a penalty term such as stokes equations and mixed formulations of linear elasticity these preconditioners are based on the solution of local saddle point problems on overlapping subdomains and the solution of a coarse saddle point problem numerical experiments show that these are parallel and scalable preconditioners since the rate of convergence of the preconditioned operator is independent of the mesh size h the number of subdomains n and the penalty parameter

financial networks based on granger causality a case study connectivity analysis is performed on a long financial record of 21 international stock indices employing a linear and a nonlinear causality measure the conditional granger causality index cgci and the partial mutual information on mixed embedding pmime respectively both measures aim to specify the direction of the interrelationships among the international stock indexes and portray the links of the resulting networks by the presence of direct couplings between variables exploiting all available information however their differences are assessed due to the presence of nonlinearity the weighted networks formed with respect to the causality measures are transformed to binary ones using a significance test the financial networks are formed on sliding windows in order to examine the network characteristics and trace changes in the connectivity structure subsequently two statistical network quantities are calculated the average degree and the average shortest path length the empirical findings reveal interesting time varying properties of the constructed network which are clearly dependent on the nature of the financial cycle

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

state observers of a vascular fluid structure interaction model through measurements in the solid we analyze the performances of two types of luenberger observers namely the so called direct velocity feedback and schur displacement feedback procedures originally devised for elasto dynamics to estimate the state of a fluid structure interaction model for hemodynamics when the measurements are assumed to be restricted to displacements or velocities in the solid we first assess the observers using hemodynamics inspired test problems with the complete model including the navier stokes equations in arbitrary lagrangian eulerian formulation in particular then in order to obtain more detailed insight we consider several well chosen simplified models each of which allowing a thorough analysis emphasizing spectral considerations while illustrating a major phenomenon of interest for the observer performance namely the added mass effect for the structure the coupling with a lumped parameter boundary condition model for the fluid flow and the fluid dynamics effect per se whereas improvements can be sought when additional measurements are available in the fluid domain in order to more effectively deal with strong uncertainties in the fluid state in the present framework this establishes luenberger observer methods as very attractive strategies compared e g to classical variational techniques to perform state estimation and more generally for uncertainty estimation since other observer procedures can be conveniently combined to estimate uncertain parameters

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

an algebraic approach to h output feedback control problems abstract we derive an output feedback controller which stabilizes a system and satisfies a prescribed h norm bound of the closed loop transfer function the proposed design method is available for any system i e there are no restrictions on d12 and d21 this approach utilizes only algebraic operations thus proofs are simple and clear

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

abstract robust coarse spaces for systems of pdes via generalized eigenproblems in the overlaps coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations pdes however it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the pde coefficients especially for systems of pdes here we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities we achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence we prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous pdes and systems of pdes that confirm this property

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

ill defined block spin transformations at arbitrarily high temperatures examples are presented of block spin transformations which map the gibbs measures of the ising model in two or more dimensions at temperature intervals extending to arbitrarily high temperatures onto non gibbsian measures in this way we provide the first example of this kind of pathology for very high temperatures and as a corollary also the first example of such a pathology happening at a critical point

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

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

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

absorbing boundary conditions for the discretization schemes of the one dimensional wave equation when computing a partial differential equation it is often necessary to introduce artificial boundaries here we explain a systematic method to obtain boundary conditions for the wave equation in one dimension fitting to the discretization scheme and stable moreover we give error estimates on the reflected part

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

hybrid domain decomposition algorithms for compressible and almost incompressible elasticity overlapping schwarz methods are considered for mixed finite element approximations of linear elasticity with discontinuous pressure spaces as well as for compressible elasticity approximated by standard conforming finite elements the coarse components of the preconditioners are based on spaces with a number of degrees of freedom per subdomain which are uniformly bounded which are similar to those previously developed for scalar elliptic problems and domain decomposition methods of iterative substructuring type i e methods based on nonoverlapping decompositions of the domain the local components of the new preconditioners are based on solvers on a set of overlapping subdomains in the current study the dimension of the coarse spaces is smaller than in recently developed algorithms in the compressible case all independent face degrees of freedom have been eliminated while in the almost incompressible case five out of six are not needed in many cases this will result in a reduction of the dimension of the coarse space by about one half compared with that of the algorithm previously considered in addition in spite of using overlapping subdomains to define the local components of the preconditioner values of the residual and the approximate solution need only to be retained on the interface between the subdomains in the iteration of the new hybrid schwarz algorithm the use of discontinuous pressures makes it possible to work exclusively with symmetric positive definite problems and the standard preconditioned conjugate gradient method bounds are established for the condition number of the preconditioned operators the bound for the almost incompressible case grows in proportion to the square of the logarithm of the number of degrees of freedom of individual subdomains and the third power of the relative overlap between the overlapping subdomains and it is independent of the poisson ratio as well as jumps in the lame parameters across the interface between the subdomains numerical results illustrate the findings copyright 2009 john wiley sons ltd

an overlapping schwarz algorithm for almost incompressible elasticity overlapping schwarz methods are extended to mixed finite element approximations of linear elasticity which use discontinuous pressure spaces the coarse component of the preconditioner is based on a low dimensional space previously developed for scalar elliptic problems and a domain decomposition method of iterative substructuring type i e a method based on nonoverlapping decompositions of the domain while the local components of the preconditioner are based on solvers on a set of overlapping subdomains a bound is established for the condition number of the algorithm which grows in proportion to the logarithm of the number of degrees of freedom in individual subdomains and essentially to the third power of the relative overlap between the overlapping subdomains and which is independent of the poisson ratio as well as jumps in the lame parameters across the interface between the subdomains a positive definite reformulation of the discrete problem makes the use of the standard preconditioned conjugate gradient method straightforward numerical results which include a comparison with problems of compressible elasticity illustrate the findings

automatic spectral coarse spaces for robust finite element tearing and interconnecting and balanced domain decomposition algorithms we introduce spectral coarse spaces for the balanced domain decomposition and the finite element tearing and interconnecting methods these coarse spaces are specifically designed for the two level methods to be scalable and robust with respect to the coefficients in the equation and the choice of the decomposition we achieve this by solving generalized eigenvalue problems on the interfaces between subdomains to identify the modes that slow down convergence theoretical bounds for the condition numbers of the preconditioned operators which depend only on a chosen threshold and the maximal number of neighbors of a subdomain are presented and proved for the finite element tearing and interconnecting method there are two versions of the two level method one based on the full dirichlet preconditioner and the other on the cheaper lumped preconditioner some numerical tests confirm these results copyright 2013 john wiley sons ltd

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

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

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

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

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

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

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

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

algebraic approximation of dirichlet to neumann maps for the equations of linear elasticity abstract the absorbing boundary conditions defined on the interface between the sub domains are of major importance for the convergence of domain decomposition methods in linear elasticity optimal absorbing boundary conditions can be derived and are associated with a dirichlet to neumann map in this paper several original algebraic techniques of approximation of this dirichlet to neumann map are investigated asymptotic spectral and numerical analysis of these techniques are successively presented for linear elasticity problems various numerical experiments illustrate the convergence properties of these original techniques

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

calculation of effective hamiltonians for renormalized or non hamiltonian systems we develop a method for calculating renormalized hamiltonians based on the brandt ron representation of renormalization group transformations our approach allows us to make a stable calculation for larger sets of renormalized coupling constants than either the swendsen or the gupta cordery methods thus reducing the effects of truncation in renormalization group calculations the generality of the brandt ron representation also makes it suitable for analyzing a broader class of models including driven diffusive systems another advantage of our approach is that it contains internal criteria that provide information on the error involved in truncation approximations

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

a two scale approximation of the schur complement and its use for non intrusive coupling this paper presents a two scale approximation of the schur complement of a subdomain s stiffness matrix obtained by combining local ie element strips and global ie homogenized contributions this approximation is used in the context of a coupling strategy that is designed to embed local plasticity and geometric details into a small region of a large linear elastic structure the strategy consists in creating a local model which contains the desired features of the concerned region and then substituting it into the global problem by the means of a non intrusive solver coupling technique adapted from domain decomposition methods using the two scale approximation of the schur complement as a robin condition on the local model enables to reach high efficiency examples include a large 3d problem provided by our industrial partner snecma

dual primal feti methods for linear elasticity dual primal feti methods are nonoverlapping domain decomposition methods where some of the continuity constraints across subdomain boundaries are required to hold throughout the iterations as in primal iterative substructuring methods while most of the constraints are enforced by lagrange multipliers as in one level feti methods these methods are used to solve the large algebraic systems of equations that arise in elliptic finite element problems the purpose of this article is to develop strategies for selecting these constraints which are enforced throughout the iterations such that good convergence bounds are obtained that are independent of even large changes in the stiffness of the subdomains across the interface between them the algorithms are described in terms of a change of basis that has proven to be quite robust in practice a theoretical analysis is provided for the case of linear elasticity and condition number bounds are established that are uniform with respect to arbitrarily large jumps in the young s modulus of the material and otherwise depend only polylogarithmically on the number of unknowns of a single subdomain the strategies have already proven quite successful in large scale implementations of these iterative methods 2006 wiley periodicals inc

optimized multiplicative additive and restricted additive schwarz preconditioning we demonstrate that a small modification of the multiplicative additive and restricted additive schwarz preconditioner at the algebraic level motivated by optimized schwarz methods defined at the continuous level leads to a significant reduction in the iteration count of the iterative solver numerical experiments using finite difference and spectral element discretizations of the positive definite helmholtz problem and an idealized climate simulation illustrate the effectiveness of the new approach

two level domain decomposition methods with lagrange multipliers for the fast iterative solution of acoustic scattering problems we present two different but related lagrange multiplier based domain decomposition dd methods for solving iteratively large scale systems of equations arising from the finite element discretization of high frequency exterior helmholtz problems the proposed methods are essentially two distinct extensions of the regularized finite element tearing and interconnecting feti method to indefinite or complex problems the first method employs a single lagrange multiplier field to glue the local solutions at the subdomain interface boundaries the second method employs two lagrange multiplier fields for that purpose the key ingredients of both of these feti methods are the regularization of each subdomain matrix by a complex lumped mass matrix defined on the subdomain interface boundary and the preconditioning of the global interface problem by a coarse second level problem constructed with planar waves we show numerically that both methods are scalable with respect to the mesh size the subdomain size and the wavenumber but that the feti method with a single lagrange multiplier field labeled feti h h for helmholtz in this paper delivers superior computational performances we apply the feti h method to the parallel solution on a 24 processor origin 2000 of an acoustic scattering problem with a submarine shaped obstacle and report performance results that highlight the unique efficiency of this dd method for the solution of high frequency acoustic scattering problems

comments on strengthened h infinity control via state feedback a majorization approach using algebraic riccati inequalities for original paper see y k foo ibid vol 49 no 5 p 824 27 2004 theorem 3 3 of the above paper contains some errors which are discussed in this note consequently the necessary corrections are presented

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

asynchronous optimized schwarz methods with and without overlap an asynchronous version of the optimized schwarz method for the solution of differential equations on a parallel computational environment is studied in a one way subdivision of the computational domain with and without overlap the method is shown to converge when the optimal artificial interface conditions are used convergence is also proved for the laplacian operator under very mild conditions on the size of the subdomains when approximate non optimal interface conditions are utilized numerical results are presented on a large three dimensional problem on modern parallel clusters and supercomputers illustrating the efficiency of the asynchronous approach

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 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

a mortar method based on nurbs for curved interfaces abstract to tackle general sub domain problems in geomechanics we present an mfem scheme on curved interfaces based on nurbs curves and surfaces the goal is to have a more robust geometrical representation for mortar spaces which allows gluing non conforming interfaces on realistic geometries the resulting mortar saddle point problem is decoupled using standard domain decomposition techniques such as dirichlet neumann to exploit current parallel machine architectures two and three dimensional examples ranging from near wellbore applications to field level subsidence computations show that the proposed scheme can handle problems of practical interest

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

computer simulations at the fixed point using an inverse renormalization group transformation abstract following brandt and ron s suggestion of inverting the renormalization group transformation used in monte carlo renormalization it is shown that efficient computer simulations of the fixed point of the transformation can be carried out on very large systems without critical slowing down we illustrate the new method with calculations of critical exponents for the two and three dimensional ising models based on several different transformations

the geometry of the first non zero stekloff eigenvalue let mn g be a compact riemannian manifold with boundary and dimensionn 2 in this paper we discuss the first non zero eigenvalue problem begin align delta varphi 0 qquad on quad m frac partial varphi partial eta u 1 varphi qquad on quad partial m end align eqno 1 problem 1 is known as the stekloff problem because it was introduced by him in 1902 for bounded domains of the plane we discuss estimates of the eigenvalue 1in terms of the geometry of the manifold mn g in the two dimensional case we generalize payne s theorem p for bounded domains in the plane to non negative curvature manifolds in this case we show that 1 k0 wherekg k0andkgrepresents the geodesic curvature of the boundary in higher dimensionsn 3 for non negative ricci curvature manifolds we show that 1 k0 2 wherek0is a lower bound for any eigenvalue of the second fundamental form of the boundary we introduce an isoperimetric constant and prove a cheeger s type inequality for the stekloff eigenvalue

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

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

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

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

renormalization group maps for ising models in lattice gas variables real space renormalization group maps e g the majority rule transformation map ising type models to ising type models on a coarser lattice we show that each coefficient in the renormalized hamiltonian in the lattice gas variables depends on only a finite number of values of the renormalized hamiltonian we introduce a method which computes the values of the renormalized hamiltonian with high accuracy and so computes the coefficients in the lattice gas variables with high accuracy for the critical nearest neighbor ising model on the square lattice with the majority rule transformation we compute over 1 000 different coefficients in the lattice gas variable representation of the renormalized hamiltonian and study the decay of these coefficients we find that they decay exponentially in some sense but with a slow decay rate we also show that the coefficients in the spin variables are sensitive to the truncation method used to compute them

a coarse space for heterogeneous helmholtz problems based on the dirichlet to neumann operator the helmholtz equation governing wave propagation and scattering phenomena is difficult to solve numerically its discretization with piecewise linear finite elements results in typically large linear systems of equations the inherently parallel domain decomposition methods constitute hence a promising class of preconditioners an essential element of these methods is a good coarse space here the helmholtz equation presents a particular challenge as even slight deviations from the optimal choice can be devastating in this paper we present a coarse space that is based on local eigenproblems involving the dirichlet to neumann operator our construction is completely automatic ensuring good convergence rates without the need for parameter tuning moreover it naturally respects local variations in the wave number and is hence suited also for heterogeneous helmholtz problems the resulting method is parallel by design and its efficiency is demonstrated on 2d homogeneous and heterogeneous numerical examples

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

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

a domain decomposition method for the helmholtz equation and related optimal control problems we present an iterative domain decomposition method to solve the helmholtz equation and related optimal control problems the proof of convergence of this method relies on energy techniques this method leads to efficient algorithms for the numerical resolution of harmonic wave propagation problems in homogeneous and heterogeneous media

algebraic multilevel krylov methods in erlangga and nabben siam j sci comput 30 2008 pp 1572 1595 we developed a new type of multilevel method called the multilevel krylov mk method to solve linear systems of equations the basic idea of this type of method is to shift small eigenvalues that are responsible for slow convergence of krylov methods to an a priori fixed constant this shifting of the eigenvalues is similar to projection type methods and is achieved via the solution of subspace or coarse level systems numerical results show that mk works very well for the two dimensional 2d poisson and convection diffusion equation i e the convergence can be made almost independent of the grid size h and the physical parameter involved in a follow up paper we have used mk in the context of the preconditioned helmholtz equation for this problem we show that the convergence can be made only mildly dependent on the wavenumber even though the coarse level system of mk is algebraically related to that of multigrid the const

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

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

balancing neumann neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity balancing neumann neumann methods are extented to mixed formulations of the linear elasticity system with discontinuous coefficients discretized with mixed finite or spectral elements with discontinuous pressures these domain decomposition methods implicitly eliminate the degrees of freedom associated with the interior of each subdomain and solve iteratively the resulting saddle point schur complement using a hybrid preconditioner based on a coarse mixed elasticity problem and local mixed elasticity problems with natural and essential boundary conditions a polylogarithmic bound in the local number of degrees of freedom is proven for the condition number of the preconditioned operator in the constant coefficient case parallel and serial numerical experiments confirm the theoretical results indicate that they still hold for systems with discontinuous coefficients and show that our algorithm is scalable parallel and robust with respect to material heterogeneities the results on heterogeneous general problems are also supported in part by our theory

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

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

algebraic domain decomposition methods for highly heterogeneous problems we consider the solving of linear systems arising from porous media flow simulations with high heterogeneities using a newton algorithm to handle the non linearity leads to the solving of a sequence of linear systems with different but similar matrices and right hand sides the parallel solver is a schwarz domain decomposition method the unknowns are partitioned with a criterion based on the entries of the input matrix this leads to substantial gains compared to a partition based only on the adjacency graph of the matrix from the information generated during the solving of the first linear system it is possible to build a coarse space for a two level domain decomposition algorithm that leads to an acceleration of the convergence of the subsequent linear systems we compare two coarse spaces a classical approach and a new one adapted to parallel implementation

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