Symmetric Positive Semidefinite System Research Articles

Symmetric Positive Semidefinite FDTD Subgridding Algorithms for Arbitrary Grid Ratios Without Compromising Accuracy

Instability has been a major problem in finite-difference time-domain (FDTD) subgridding methods. Reciprocity has been proposed to overcome the problem but with limited success in producing a symmetric positive semidefinite (SPD) system without compromising accuracy. In this paper, we algebraically derive both 2- and 3-D FDTD subgridding operators, which are SPD by construction, and independent of the grid ratio. Such operators have only nonnegative real eigenvalues, and hence the stability of the resulting explicit time marching is guaranteed. The 3-D operator, the algorithm of which is also applicable to 2-D analysis, further permits the use of a local time step, thus achieving a natural subgridding in both space and time. In addition, the interpretation of the proposed operators in the original FDTD formulation is also provided. Interestingly, not only interface unknowns but also subgrid unknowns are generated in a different way, as compared to the original FDTD, to simultaneously obtain an SPD system and to ensure accuracy. Extensive numerical simulations have demonstrated the accuracy, stability, and efficiency of the proposed new subgridding algorithms.

Solving the Stokes Problem on a Massively Parallel Computer

Abstract We describe a numerical procedure for solving the stationary two‐dimensional Stokes problem based on piecewise linear finite element approximations for both velocity and pressure, a regularization technique for stability, and a defect‐correction technique for improving accuracy. Eliminating the velocity unknowns from the algebraic system yields a symmetric positive semidefinite system for pressure which is solved by an inner‐outer iteration. The outer iterations consist of the unpreconditioned conjugate gradient method. The inner iterations, each of which corresponds to solving an elliptic boundary value problem for each velocity component, are solved by the conjugate gradient method with a preconditioning based on the algebraic multi‐level iteration (AMLI) technique. The velocity is found from the computed pressure. The method is optimal in the sense that the computational work is proportional to the number of unknowns. Further, it is designed to exploit a massively parallel computer with distri...

Avoiding singular coarse grid systems

Here we consider the iterative solution of linear systems of equations with a symmetric positive semidefinite system matrix. If multilevel methods in combination with Krylov subspace methods are used to find the solution of these systems, often singular subsystems or coarse grid systems have to be solved. Then the Moore–Penrose inverse of the coarse grid systems can be used. Here, we establish some theoretical techniques how to avoid the singularity of the coarse grid system, while the resulting operator remains the same as we would have used the Moore–Penrose inverse. One option is to delete specific columns of the restriction and prolongation operator. The other option is to perturb the system matrix.

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.

SOLVING THE STOKES PROBLEM ON A MASSIVELY PARALLEL COMPUTER

We describe a numerical procedure for solving the stationary two-dimensional Stokes problem based on piecewise linear finite element approximations for both velocity and pressure, a regularization technique for stability, and a defect‐correction technique for improving accuracy. Eliminating the velocity unknowns from the algebraic system yields a symmetric positive semidefinite system for pressure which is solved by an inner‐outer iteration. The outer iterations consist of the unpreconditioned conjugate gradient method. The inner iterations, each of which corresponds to solving an elliptic boundary value problem for each velocity component, are solved by the conjugate gradient method with a preconditioning based on the algebraic multi‐level iteration (AMLI) technique. The velocity is found from the computed pressure. The method is optimal in the sense that the computational work is proportional to the number of unknowns. Further, it is designed to exploit a massively parallel computer with distributed memory architecture. Numerical experiments on a Cray T3E computer illustrate the parallel performance of the method.

Convergence of two-stage iterative methods for singular symmetric positive semidefinite systems*

Convergence of two-stage iterative methods for singular symmetric positive semidefinite (spsd) systems is studied. The main tool we used to derive the iterative methods and to analyze their convergence is the extended diagonally compensated reduction

The Convergence of Linear Stationary Iterative Processes for Solving Singular Unstructured Systems of Linear Equations

This paper reviews the convergence properties of linear stationary iterative processes for solving large unstructured systems of linear equations. Special attention is given to the case where the system to be solved is singular and possibly inconsistent. The review starts by investigating the iteration ${\bf x}_{k + 1} = H{\bf x}_k + {\bf f},\, k = 0,1,2, \cdots $, when the system $(I - H){\bf x} = {\bf f}$ is inconsistent. In this case the sequence $\left\{ {\bf x_k } \right\}$ always diverges. Nevertheless, when H is “convergent” there exists a vector ${\bf q} \in \,{\text{Null}}(I - H)$ such that the sequence $\hat {\bf x}_k = {\bf x}_k - k{\bf q}$ converges. The limit of this sequence, $\hat {\bf x}$, solves a certain least-squares problem, and the sequence of residual vectors, ${\bf r}_k = (I - H){\bf x}_k - {\bf f}$, converges to $(I - H)\hat {\bf x} - {\bf f}$. Furthermore, if H is similar to a real diagonal matrix then the corresponding norms of ${\bf r}_k $ and $\hat {\bf x}_k - \hat {\bf x}$ decreases monotonically. More can be said when ${\bf x}_0 \in {\text{Range}}(I - H)$. There are several iterative methods for solving a symmetric positive semidefinite system of linear equations, $G{\bf x} = {\bf h}$, that have the form ${\bf x}_{k + 1} = Q^{ - 1} (Q - G){\bf x}_k + Q^{ - 1} {\bf h}$. Here the iteration matrix, $H = Q^{ - 1} (Q - G)$, is convergent whenever the matrix $P = Q + Q^T - G$ is positive definite. In this case we obtain the surprising result that the sequence of residuals, $\{ G{\bf x}_k - {\bf h}\} $, converges even when the system $G{\bf x} = {\bf h}$ is inconsistent. The developed theory reveals interesting properties of the “basic” iterative methods (e.g., Richardson’s method, Jacobi’s method, SOR, and SSOR) with minimal assumptions on the structure of G. For example, if H is convergent and symmetric as with Richardson’s method, then the sequences $\{ ||\hat {\bf x}_k - \hat {\bf x}||\} $ and $\{ ||G{\bf x}_k - {\bf h}||\} $ decrease monotonically, and the sequence $\{ G{\bf x}_k - {\bf h}\} $ converges to $G\hat {\bf x} - {\bf h}$, where $\hat {\bf x}$ solves the least-squares problem: minimize $||{\bf Gx}_k - {\bf h}||^2 $. Moreover, if ${\bf x}_0 \in {\text{Range}}(G)$ and the system $(I - H){\bf x} = {\bf f}$ is solvable, then the sequence $\{ {\bf x}_k \} $ converges to the minimum-norm solution of this system. The paper ends with a discussion of iterative methods for solving a general linear system, $A{\bf x} = {\bf b}$, where A is large, sparse, and unstructured. These methods can be divided into two classes. One is aimed at solving the normal equations $A^T A{\bf x} = A^T {\bf b}$ (e.g., column SOR or Cimmino’s method), and one is aimed at solving the Björck–Elfving equations ${\bf x} = A^T {\bf y}$ and $AA^T {\bf y} = {\bf b}$ (e.g., Kaczmarz’s method). It is shown that in both cases the theory developed for symmetric positive semidefinite systems enables us to characterize the behavior of the methods and to derive simplified proofs of convergence.

A note on the convergence of linear stationary iterative processes

This note investigates the convergence of a linear stationary iterative process of the form x k+1=H x k+ f . It is shown that if H is a weakly convergent matrix, then the sequence {(I−H) x k} converges for any choice of x 0 and f. There are relaxation schemes for solving a symmetric positive semidefinite system of linear equations, G x= b whose iteration matrix is weakly convergent (e.g. the SOR method). In such a case the above result implies that the sequence {G x k− b} always converges. In other words, the sequence of residuals converges even when the system G x= b is inconsistent. Special attention is given to the case when H is weakly convergent and symmetric. Here the sequence {‖G x k− b‖} is monotonic decreasing, and the sequence { G x k } converges to G x̂, where x̂ is a least squares solution. Furthermore, if x 0 belongs to the column space of G and the system G x= b is solvable, then the sequence { x k } converges to the minimum norm solution.