This paper is devoted to the computation of transmission eigenvalues in the inverse acoustic scattering theory. This problem is first reformulated as a two by two boundary system of boundary integral equations. Next, utilizing the Schur complement technique, we develop a Schur complement operator with regularization to obtain a reduced system of boundary integral equations. The Nyström discretization is then used to obtain an eigenvalue problem for a matrix. We employ the recursive integral method for the numerical computation of the matrix eigenvalue. Numerical results show that the proposed method is efficient and reduces computational costs.
An effective power based parallel preconditioner is proposed for general large sparse linear systems. The preconditioner combines a power series expansion method with some low-rank correction techniques, where the Sherman-Morrison-Woodbury formula is utilized. A matrix splitting of the Schur complement is proposed to expand the power series. The number of terms used in the power series expansion can control the approximation accuracy of the preconditioner to the inverse of the Schur complement. To construct the preconditioner, graph partitioning is invoked to reorder the original coefficient m…
A nested Schur complement solver is proposed for iterative solution of linear systems arising in exponential and implicit time integration of the Maxwell equations with perfectly matched layer (PML) nonreflecting boundary conditions. These linear systems are the so-called double saddle point systems whose structure is handled by the Schur complement solver in a nested, two-level fashion. The solver is demonstrated to have a mesh-independent convergence at the outer level, whereas the inner level system is of elliptic type and thus can be treated efficiently by a variety of solvers.
We propose a novel preconditioned inexact primal-dual interior point method for constrained convex quadratic programming problems. The algorithm we describe invokes the preconditioned conjugate gradient method on a new reduced Schur complement KKT system, in implicit form. In contrast to standard approaches, the Schur complement formulation we consider enables reuse of the factorization of the KKT matrix with rows and columns corresponding to inequality constraints excluded, across all interior point iterations. Further, two new preconditioners are presented for the resulting reduced system, t…
Parameter-free inexact block Schur complement preconditioning for linear poroelasticity under a hybrid Bernardi-Raugel and weak Galerkin finite element discretization
For 2x2 block matrices, it is well-known that block-triangular or block-LDU preconditioners with an exact Schur complement (inverse) converge in at most two iterations for fixed-point or minimal-residual methods. Similarly, for saddle-point matrices with a zero (2,2)-block, block-diagonal preconditioners converge in at most three iterations for minimal-residual methods, although they may diverge for fixed-point iterations. But, what happens for non-saddle-point matrices and block-diagonal preconditioners with an exact Schur complement? This note proves that minimal-residual methods applied to …
In this paper, a robust and effective preconditioner for the fast Method of Moments(MoM) based Hierarchal Electric Field Integral Equation(EFIE) solver is proposed using symmetric near-field Schur's complement method. In this preconditioner, near-field blocks are scaled to a diagonal block matrix and these near-field blocks are replaced with the scaled diagonal block matrix which reduces the near-field storage memory and the overall matrix vector product time. Scaled diagonal block matrix is further used as a preconditioner and due to the block diagonal form of the final preconditioner, no add…
This paper discusses parGeMSLR, a C++/MPI software library for the solution of sparse systems of linear algebraic equations via preconditioned Krylov subspace methods in distributed-memory computing environments. The preconditioner implemented in parGeMSLR is based on algebraic domain decomposition and partitions the symmetrized adjacency graph recursively into several non-overlapping partitions via a p-way vertex separator, where p is an integer multiple of the total number of MPI processes. From a numerical perspective, parGeMSLR builds a Schur complement approximate inverse preconditioner a…
GPU-Accelerated Nested Schur-Complement Solvers for Atmospheric Flow Simulations Using Horizontally-Explicit, Vertically-Implicit Schemes and Discontinuous Galerkin Methods
Johannes Gutenberg University Mainz（约翰内斯·古腾堡美因茨大学）；Universidad Politécnica de Madrid（马德里理工大学）；Leibniz Institute for Tropospheric Research (TROPOS)（莱布尼茨对流层研究所）；University of Cologne（科隆大学）
Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods
University of Southern California（南加州大学）；Institute of Thermomechanics, Academy of Sciences of the Czech Republic（捷克科学院热力学研究所）；Sandia National Laboratories（桑迪亚国家实验室）
Schur complement solver for Quantum Monte-Carlo simulations of strongly interacting fermions
Institute of Theoretical Physics, Würzburg University（维尔茨堡大学理论物理研究所）；School of Mathematics and Natural Sciences, University of Wuppertal（伍珀塔尔大学数学与自然科学学院）；Institute of Theoretical Physics, Regensburg University（雷根斯堡大学理论物理研究所）
Generalization of the Sherman-Morrison-Woodbury formula involving the Schur complement
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）
We consider a linear kinetic transport equation under a diffusive scaling, that converges to a diffusion equation as the Knudsen number $\varepsilon\rightarrow0$. In [3, 21], to achieve the asymptotic preserving (AP) property and unconditional stability in the diffusive regime with $\varepsilon\ll 1$, numerical schemes are developed based on an additional reformulation of the even-odd or micro-macro decomposed version of the equation. The key of the reformulation is to add a weighted diffusive term on both sides of one equation in the decomposed system. The choice of the weight function, howev…
Efficient and robust iterative solvers for strong anisotropic elliptic equations are very challenging. In this paper a block preconditioning method is introduced to solve the linear algebraic systems of a class of micro-macro asymptotic-preserving (MMAP) scheme. MMAP method was developed by Degond {\it et al.} in 2012 where the discrete matrix has a $2\times2$ block structure. By the approximate Schur complement a series of block preconditioners are constructed. We first analyze a natural approximate Schur complement that is the coefficient matrix of the original non-AP discretization. However…
We derive bounds on the eigenvalues of a generic form of double saddle-point matrices. The bounds are expressed in terms of extremal eigenvalues and singular values of the associated block matrices. Inertia and algebraic multiplicity of eigenvalues are considered as well. The analysis includes bounds for preconditioned matrices based on block diagonal preconditioners using Schur complements, and it is shown that in this case the eigenvalues are clustered within a few intervals bounded away from zero. Analysis for approximations of Schur complements is included. Some numerical experiments valid…
Deflation-accelerated preconditioning of the Poisson-Neumann Schur problem on long domains with a high-order discontinuous element-based collocation method
Cornell University（康奈尔大学）；Applied Research Laboratories（应用研究实验室）
Randomized methods are becoming increasingly popular in numerical linear algebra. However, few attempts have been made to use them in developing preconditioners. Our interest lies in solving large-scale sparse symmetric positive definite linear systems of equations where the system matrix is preordered to doubly bordered block diagonal form (for example, using a nested dissection ordering). We investigate the use of randomized methods to construct high quality preconditioners. In particular, we propose a new and efficient approach that employs Nyström's method for computing low rank approximat…
We present a weighted BFBT approximation (w-BFBT) to the inverse Schur complement of a Stokes system with highly heterogeneous viscosity. When used as part of a Schur complement-based Stokes preconditioner, we observe robust fast convergence for Stokes problems with smooth but highly varying (up to 10 orders of magnitude) viscosities, optimal algorithmic scalability with respect to mesh refinement, and only a mild dependence on the polynomial order of high-order finite element discretizations ($Q_k \times P_{k-1}^{disc}$, order $k \ge 2$). For certain difficult problems, we demonstrate numeric…
We derive and analyse a new variant of the iteratively regularized Landweber iteration, for solving linear and nonlinear ill-posed inverse problems. The method takes into account training data, which are used to estimate the interior of a black box, which is used to define the iteration process. We prove convergence and stability for the scheme in infinite dimensional Hilbert spaces. These theoretical results are complemented by several numerical experiments for solving linear inverse problems for the Radon transform and a nonlinear inverse problem for Schlieren tomography.
We present augmented Lagrangian Schur complement preconditioners and robust multigrid methods for incompressible Stokes problems with extreme viscosity variations. Such Stokes systems arise, for instance, upon linearization of nonlinear viscous flow problems, and they can have severely inhomogeneous and anisotropic coefficients. Using an augmented Lagrangian formulation for the incompressibility constraint makes the Schur complement easier to approximate, but results in a nearly singular (1,1)-block in the Stokes system. We present eigenvalue estimates for the quality of the Schur complement a…
This is the third part in a series on a mass conserving, high order, mixed finite element method for Stokes flow. In this part, we study a block-diagonal preconditioner for the indefinite Schur complement system arising from the discretization of the Stokes equations using these elements. The underlying finite element method is uniformly stable in both the mesh size h and polynomial order p, and we prove bounds on the eigenvalues of the preconditioned system which are independent of h and grow modestly in p. The analysis relates the Schur complement system to an appropriate variational setting…
We develop a novel iterative solution method for the incompressible Navier-Stokes equations with boundary conditions coupled with reduced models. The iterative algorithm is designed based on the variational multiscale formulation and the generalized-$α$ scheme. The spatiotemporal discretization leads to a block structure of the resulting consistent tangent matrix in the Newton-Raphson procedure. As a generalization of the conventional block preconditioners, a three-level nested block preconditioner is introduced to attain a better representation of the Schur complement, which plays a key role …
