Block‐triangular preconditioners for PDE‐constrained optimization

In this paper we investigate the possibility of using a block‐triangular preconditioner for saddle point problems arising in PDE‐constrained optimization. In particular, we focus on a conjugate gradient‐type method introduced by Bramble and Pasciak that uses self‐adjointness of the preconditioned system in a non‐standard inner product. We show when the Chebyshev semi‐iteration is used as a preconditioner for the relevant matrix blocks involving the finite element mass matrix that the main drawback of the Bramble–Pasciak method—the appropriate scaling of the preconditioners—is easily overcome. We present an eigenvalue analysis for the block‐triangular preconditioners that gives convergence bounds in the non‐standard inner product and illustrates their competitiveness on a number of computed examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Preconditioner updates for solving sequences of linear systems in matrix‐free environment

We present two new ways of preconditioning sequences of nonsymmetric linear systems in the special case where the implementation is matrix free. Both approaches are fully algebraic, they are based on the general updates of incomplete LU decompositions recently introduced in (SIAM J. Sci. Comput. 2007; 29 (5):1918–1941), and they may be directly embedded into nonlinear algebraic solvers. The first of the approaches uses a new model of partial matrix estimation to compute the updates. The second approach exploits separability of function components to apply the updated factorized preconditioner via function evaluations with the discretized operator. Experiments with matrix‐free implementations of test problems show that both new techniques offer useful, robust and black‐box solution strategies. In addition, they show that the new techniques are often more efficient in matrix‐free environment than either recomputing the preconditioner from scratch for every linear system of the sequence or than freezing the preconditioner throughout the whole sequence. Copyright © 2010 John Wiley & Sons, Ltd.     

A preconditioned GMRES for complex dense linear systems from electromagnetic wave scattering problems

We consider the numerical solution of linear systems arising from the discretization of the electric field integral equation (EFIE). For some geometries the associated matrix can be poorly conditioned making the use of a preconditioner mandatory to obtain convergence. The electromagnetic scattering problem is here solved by means of a preconditioned GMRES in the context of the multilevel fast multipole method (MLFMM). The novelty of this work is the construction of an approximate hierarchically semiseparable (HSS) representation of the near-field matrix, the part of the matrix capturing interactions among nearby groups in the MLFMM, as preconditioner for the GMRES iterations. As experience shows, the efficiency of an ILU preconditioning for such systems essentially depends on a sufficient fill-in, which apparently sacrifices the sparsity of the near-field matrix. In the light of this experience we propose a multilevel near-field matrix and its corresponding HSS representation as a hierarchical preconditioner in order to substantially reduce the number of iterations in the solution of the resulting system of equations.     

Modified Hermitian and skew‐Hermitian splitting methods for non‐Hermitian positive‐definite linear systems

To further study the Hermitian and non‐Hermitian splitting methods for a non‐Hermitian and positive‐definite matrix, we introduce a so‐called lopsided Hermitian and skew‐Hermitian splitting and then establish a class of lopsided Hermitian/skew‐Hermitian (LHSS) methods to solve the non‐Hermitian and positive‐definite systems of linear equations. These methods include a two‐step LHSS iteration and its inexact version, the inexact Hermitian/skew‐Hermitian (ILHSS) iteration, which employs some Krylov subspace methods as its inner process. We theoretically prove that the LHSS method converges to the unique solution of the linear system for a loose restriction on the parameter α. Moreover, the contraction factor of the LHSS iteration is derived. The presented numerical examples illustrate the effectiveness of both LHSS and ILHSS iterations. Copyright © 2007 John Wiley & Sons, Ltd.     

On the convergence of general stationary iterative methods for range‐Hermitian singular linear systems

General stationary iterative methods with a singular matrix M for solving range‐Hermitian singular linear systems are presented, some convergence conditions and the representation of the solution are also given. It can be verified that the general Ortega–Plemmons theorem and Keller theorem for the singular matrix M still hold. Furthermore, the singular matrix M can act as a good preconditioner for solving range‐Hermitian linear systems. Numerical results have demonstrated the effectiveness of the general stationary iterations and the singular preconditioner M. Copyright © 2009 John Wiley & Sons, Ltd.     

BCCB preconditioners for systems of BVM‐based numerical integrators

Boundary value methods (BVMs) for ordinary differential equations require the solution of non‐symmetric, large and sparse linear systems. In this paper, these systems are solved by using the generalized minimal residual (GMRES) method. A block‐circulant preconditioner with circulant blocks (BCCB preconditioner) is proposed to speed up the convergence rate of the GMRES method. The BCCB preconditioner is shown to be invertible when the BVM is A_k1,k2‐stable. The spectrum of the preconditioned matrix is clustered and therefore, the preconditioned GMRES method converges fast. Moreover, the operation cost in each iteration of the preconditioned GMRES method by using our BCCB preconditioner is less than that required by using block‐circulant preconditioners proposed earlier. In numerical experiments, we compare the number of iterations of various preconditioners. Copyright © 2003 John Wiley & Sons, Ltd.     

Generalized skew‐Hermitian triangular splitting iteration methods for saddle‐point linear systems

A generalized skew‐Hermitian triangular splitting iteration method is presented for solving non‐Hermitian linear systems with strong skew‐Hermitian parts. We study the convergence of the generalized skew‐Hermitian triangular splitting iteration methods for non‐Hermitian positive definite linear systems, as well as spectrum distribution of the preconditioned matrix with respect to the preconditioner induced from the generalized skew‐Hermitian triangular splitting. Then the generalized skew‐Hermitian triangular splitting iteration method is applied to non‐Hermitian positive semidefinite saddle‐point linear systems, and we prove its convergence under suitable restrictions on the iteration parameters. By specially choosing the values of the iteration parameters, we obtain a few of the existing iteration methods in the literature. Numerical results show that the generalized skew‐Hermitian triangular splitting iteration methods are effective for solving non‐Hermitian saddle‐point linear systems with strong skew‐Hermitian parts. Copyright © 2013 John Wiley & Sons, Ltd.     

Right preconditioned MINRES for singular systems†

We consider solving large sparse symmetric singular linear systems. We first introduce an algorithm for right preconditioned minimum residual (MINRES) and prove that its iterates converge to the preconditioner weighted least squares solution without breakdown for an arbitrary right‐hand‐side vector and an arbitrary initial vector even if the linear system is singular and inconsistent. For the special case when the system is consistent, we prove that the iterates converge to a min‐norm solution with respect to the preconditioner if the initial vector is in the range space of the right preconditioned coefficient matrix. Furthermore, we propose a right preconditioned MINRES using symmetric successive over‐relaxation (SSOR) with Eisenstat's trick. Some numerical experiments on semidefinite systems in electromagnetic analysis and so forth indicate that the method is efficient and robust. Finally, we show that the residual norm can be further reduced by restarting the iterations.     

AMLI preconditioner for the p‐version of the FEM

From the literature it is known that the conjugate gradient method with domain decomposition preconditioners is one of the most efficient methods for solving systems of linear algebraic equations resulting from p‐version finite element discretizations of elliptic boundary value problems. One ingredient of such a preconditioner is a preconditioner related to the Dirichlet problems. In the case of Poisson's equation, we present a preconditioner for the Dirichlet problems which can be interpreted as the stiffness matrix K_h,k resulting from the h‐version finite element discretization of a special degenerated problem. We construct an AMLI preconditioner C_h,k for the matrix K_h,k and show that the condition number of C K_h,k is independent of the discretization parameter. This proof is based on the strengthened Cauchy inequality. The theoretical result is confirmed by numerical examples. Copyright © 2003 John Wiley & Sons, Ltd.     

A robust algebraic multilevel preconditioner for non‐symmetric M‐matrices

Stable finite difference approximations of convection‐diffusion equations lead to large sparse linear systems of equations whose coefficient matrix is an M‐matrix, which is highly non‐symmetric when the convection dominates. For an efficient iterative solution of such systems, it is proposed to consider in the non‐symmetric case an algebraic multilevel preconditioning method formerly proposed for pure diffusion problems, and for which theoretical results prove grid independent convergence in this context. These results are supplemented here by a Fourier analysis that applies to constant coefficient problems with periodic boundary conditions whenever using an ‘idealized’ version of the two‐level preconditioner. Within this setting, it is proved that any eigenvalue λ of the preconditioned system satisfies for some real constant c such that . This result holds independently of the grid size and uniformly with respect to the ratio between convection and diffusion. Extensive numerical experiments are conducted to assess the convergence of practical two‐ and multi‐level schemes. These experiments, which include problems with highly variable and rotating convective flow, indicate that the convergence is grid independent. It deteriorates moderately as the convection becomes increasingly dominating, but the convergence factor remains uniformly bounded. This conclusion is supported for both uniform and some non‐uniform (stretched) grids. Copyright © 2000 John Wiley & Sons, Ltd.     

Fast solvers with block‐diagonal preconditioners for linear FEM–BEM coupling

The purpose of this paper is to present optimal preconditioned iterative methods to solve indefinite linear systems of equations arising from symmetric coupling of finite elements and boundary elements. This is a block‐diagonal preconditioner together with a conjugate residual method and a preconditioned inner–outer iteration. We prove the efficiency of these methods by showing that the number of iterations to preserve a given accuracy is bounded independent of the number of unknowns. Numerical examples underline the efficiency of these methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Multicolor low‐rank preconditioner for general sparse linear systems

This article presents a multilevel parallel preconditioning technique for solving general large sparse linear systems of equations. Subdomain coloring is invoked to reorder the coefficient matrix by multicoloring the adjacency graph of the subdomains, resulting in a two‐level block diagonal structure. A full binary tree structure $\mathrm{??}$ is then built to facilitate the construction of the preconditioner. A key property that is exploited is the observation that the difference between the inverse of the original matrix and that of its block diagonal approximation is often well approximated by a low‐rank matrix. This property and the block diagonal structure of the reordered matrix lead to a multicolor low‐rank (MCLR) preconditioner. The construction procedure of the MCLR preconditioner follows a bottom‐up traversal of the tree $\mathrm{??}$. All irregular matrix computations, such as ILU factorizations and related triangular solves, are restricted to leaf nodes where these operations can be performed independently. Computations in nonleaf nodes only involve easy‐to‐optimize dense matrix operations. In order to further reduce the number of iteration of the Preconditioned Krylov subspace procedure, we combine MCLR with a few classical block‐relaxation techniques. Numerical experiments on various test problems are proposed to illustrate the robustness and efficiency of the proposed approach for solving large sparse symmetric and nonsymmetric linear systems.     

Variable-step multilevel preconditioning methods,I: Self-adjoint and positive definite elliptic problems

When solving large size systems of equations by preconditioned iterative solution methods, one normally uses a fixed preconditioner which may be defined by some eigenvalue information, such as in a Chebyshev iteration method. In many problems, however, it may be more effective to use variable preconditioners, in particular when the eigenvalue information is not available. In the present paper, a recursive way of constructing variable-step of, in general, nonlinear multilevel preconditioners for selfadjoint and coercive second-order elliptic problems, discretized by the finite element method is proposed. The preconditioner is constructed recursively from the coarsest to finer and finer levels. Each preconditioning step requires only block-diagonal solvers at all levels except at every k₀, k₀ ≥ 1 level where we perform a sufficient number ν, ν ≥ 1 of GCG-type variable-step iterations that involve the use again of a variable-step preconditioning for that level. It turns out that for any sufficiently large value of k₀ and, asymptotically, for ν sufficiently large, but not too large, the method has both an optimal rate of convergence and an optimal order of computational complexity, both for two and three space dimensional problem domains. The method requires no parameter estimates and the convergence results do not depend on the regularity of the elliptic problem.     

One‐level Newton–Krylov–Schwarz algorithm for unsteady non‐linear radiation diffusion problem

In this paper, we present a parallel Newton–Krylov–Schwarz (NKS)‐based non‐linearly implicit algorithm for the numerical solution of the unsteady non‐linear multimaterial radiation diffusion problem in two‐dimensional space. A robust solver technology is required for handling the high non‐linearity and large jumps in material coefficients typically associated with simulations of radiation diffusion phenomena. We show numerically that NKS converges well even with rather large inflow flux boundary conditions. We observe that the approach is non‐linearly scalable, but not linearly scalable in terms of iteration numbers. However, CPU time is more important than the iteration numbers, and our numerical experiments show that the algorithm is CPU‐time‐scalable even without a coarse space given that the mesh is fine enough. This makes the algorithm potentially more attractive than multilevel methods, especially on unstructured grids, where course grids are often not easy to construct. Copyright © 2004 John Wiley & Sons, Ltd.     

Multistep matrix splitting iteration preconditioning for singular linear systems

Multistep matrix splitting iterations serve as preconditioning for Krylov subspace methods for solving singular linear systems. The preconditioner is applied to the generalized minimal residual (GMRES) method and the flexible GMRES (FGMRES) method. We present theoretical and practical justifications for using this approach. Numerical experiments show that the multistep generalized shifted splitting (GSS) and Hermitian and skew-Hermitian splitting (HSS) iteration preconditioning are more robust and efficient compared to standard preconditioners for some test problems of large sparse singular linear systems.     

Subspace‐by‐subspace preconditioners for structured linear systems

We consider the iterative solution of symmetric positive‐definite linear systems whose coefficient matrix may be expressed as the outer product of low‐rank terms. We derive suitable preconditioners for such systems, and demonstrate their effectiveness on a number of test examples. We also consider combining these methods with existing techniques to cope with the commonly‐occuring case where the coefficient matrix is the linear sum of elements, some of which are of very low rank. Copyright © 1999 John Wiley & Sons, Ltd.     

Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods

Use of the stochastic Galerkin finite element methods leads to large systems of linear equations obtained by the discretization of tensor product solution spaces along their spatial and stochastic dimensions. These systems are typically solved iteratively by a Krylov subspace method. We propose a preconditioner, which takes an advantage of the recursive hierarchy in the structure of the global matrices. In particular, the matrices posses a recursive hierarchical two‐by‐two structure, with one of the submatrices block diagonal. Each of the diagonal blocks in this submatrix is closely related to the deterministic mean‐value problem, and the action of its inverse is in the implementation approximated by inner loops of Krylov iterations. Thus, our hierarchical Schur complement preconditioner combines, on each level in the approximation of the hierarchical structure of the global matrix, the idea of Schur complement with loops for a number of mutually independent inner Krylov iterations, and several matrix–vector multiplications for the off‐diagonal blocks. Neither the global matrix nor the matrix of the preconditioner need to be formed explicitly. The ingredients include only the number of stiffness matrices from the truncated Karhunen–Loève expansion and a good preconditioned for the mean‐value deterministic problem. We provide a condition number bound for a model elliptic problem, and the performance of the method is illustrated by numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Algebraic multilevel preconditioning of finite element matrices using local Schur complements

We consider an algebraic multilevel preconditioning technique for SPD matrices arising from finite element discretization of elliptic PDEs. In particular, we address the case of non‐M matrices. The method is based on element agglomeration and assumes access to the individual element matrices. The left upper block of the considered multiplicative two‐level preconditioner is approximated using incomplete factorization techniques. The coarse‐grid element matrices are simply Schur complements computed from local neighbourhood matrices, i.e. small collections of element matrices. Assembling these local Schur complements results in a global Schur complement approximation that can be analysed by regarding (local) macro elements. These components, when combined in the framework of an algebraic multilevel iteration, yield a robust and efficient linear solver. The presented numerical experiments include also the Lamé differential equation for the displacements in the two‐dimensional plane‐stress elasticity problem. Copyright © 2005 John Wiley & Sons, Ltd.     

Parallel Jacobi–Davidson with block FSAI preconditioning and controlled inner iterations

The Jacobi–Davidson (JD) algorithm is considered one of the most efficient eigensolvers currently available for non‐Hermitian problems. It can be viewed as a coupled inner‐outer iteration, where the inner one expands the search subspace and the outer one reduces the eigenpair residual. One of the difficulties in the JD efficient use stems from the definition of the most appropriate inner tolerance, so as to avoid useless extra work and keep the number of outer iterations under control. To this aim, the use of an efficient preconditioner for the inner iterative solver is of paramount importance. The present paper describes a fresh implementation of the JD algorithm with controlled inner iterations and block factorized sparse approximate inverse preconditioning for non‐Hermitian eigenproblems in a parallel computational environment. The algorithm performance is investigated by comparison with a freely available software package such as SLEPc. The results show that combining the inner tolerance control with an efficient preconditioning technique can allow for a significant improvement of the JD performance, preserving a good scalability. Copyright © 2016 John Wiley & Sons, Ltd.     

Robust Multiscale Iterative Solvers for Nonlinear Flows in Highly Heterogeneous Media

In this paper, we study robust iterative solvers for finite element systems resulting in approximation of steady-state Richards' equation in porous media with highly heterogeneous conductivity fields. It is known that in such cases the contrast, ratio between the highest and lowest values of the conductivity, can adversely affect the performance of the preconditioners and, consequently, a design of robust preconditioners is important for many practical applications. The proposed iterative solvers consist of two kinds of iterations, outer and inner iterations. Outer iterations are designed to handle nonlinearities by linearizing the equation around the previous solution state. As a result of the linearization, a large-scale linear system needs to be solved. This linear system is solved iteratively (called inner iterations), and since it can have large variations in the coefficients, a robust preconditioner is needed. First, we show that under some assumptions the number of outer iterations is independent of the contrast. Second, based on the recently developed iterative methods, we construct a class of preconditioners that yields convergence rate that is independent of the contrast. Thus, the proposed iterative solvers are optimal with respect to the large variation in the physical parameters. Since the same preconditioner can be reused in every outer iteration, this provides an additional computational savings in the overall solution process. Numerical tests are presented to confirm the theoretical results.