Tarek P. Mathew

The interface probing technique in domain decomposition

The interface probing technique, which was developed and used by Chan and Resasco and Keyes and Gropp, is an algebraic technique for constructing interface preconditioners in domain decomposition algorithms. The basic technique is to approximate interface matrices by matrices having a specified sparsity pattern. The construction involves only matrix-vector products, and thus the interface matrix need not be known explicitly. A special feature is that the approximations adapt to the variations in the coefficients of the equations and the aspect ratios of the subdomains. This preconditioner can then be used in conjunction with many standard iterative methods, such as conjugate gradient methods.In this paper, some old results are summarized and new ones are presented, both algebraic and analytic, about the interface probing technique and its applications to interface operators. Comparisons are made with some optimal preconditioners.

Domain Decomposition Operator Splittings for the Solution of Parabolic Equations

We study domain decomposition counterparts of the classical alternating direction implicit (ADI) and fractional step (FS) methods for solving the large linear systems arising from the implicit time stepping of parabolic equations. In the classical ADI and FS methods for parabolic equations, the elliptic operator is split along coordinate axes; they yield tridiagonal linear systems whenever a uniform grid is used and when mixed derivative terms are not present in the differential equation. Unlike coordinate-axes-based splittings, we employ domain decomposition splittings based on a partition of unity. Such splittings are applicable to problems on nonuniform meshes and even when mixed derivative terms are present in the differential equation and they require the solution of one problem on each subdomain per time step, without iteration. However, the truncation error in our proposed method deteriorates with smaller overlap amongst the subdomains unless a smaller time step is chosen. Estimates are presented for the asymptotic truncation error, along with computational results comparing the standard Crank--Nicolson method with the proposed method.

Analysis of Block Parareal Preconditioners for Parabolic Optimal Control Problems

In this paper, we describe block matrix algorithms for the iterative solution of a large-scale linear-quadratic optimal control problem involving a parabolic partial differential equation over a finite control horizon. We consider an “all at once” discretization of the problem and formulate three iterative algorithms. The first algorithm is based on preconditioning a symmetric positive definite reduced linear system involving only the unknown control variables; however inner-outer iterations are required. The second algorithm modifies the first algorithm to avoid inner-outer iterations by introducing an auxiliary variable. It yields a symmetric indefinite system with a positive definite block preconditioner. The third algorithm is the central focus of this paper. It modifies the preconditioner in the second algorithm by a parallel-in-time preconditioner based on the parareal algorithm. Theoretical results show that the preconditioned algorithms have optimal convergence properties and parallel scalability. Numerical experiments confirm the theoretical results.

Maximum Norm Analysis of Overlapping Nonmatching Grid Discretizations of Elliptic Equations

In this paper, we provide a maximum norm analysis of a finite difference scheme defined on overlapping nonmatching grids for second order elliptic equations. We consider a domain which is the union of p overlapping subdomains where each subdomain has its own independently generated grid. The grid points on the subdomain boundaries need not match the grid points from adjacent subdomains. To obtain a global finite difference discretization of the elliptic problem, we employ standard stable finite difference discretizations within each of the overlapping subdomains and the different subproblems are coupled by enforcing continuity of the solutions across the boundary of each subdomain, by interpolating the discrete solution on adjacent subdomains. If the subdomain finite difference schemes satisfy a strong discrete maximum principle and if the overlap is sufficiently large, we show that the global discretization converges in optimal order corresponding to the largest truncation errors of the local interpolation maps and discretizations. Our discretization scheme and the corresponding theory allows any combination of lower order and higher order finite difference schemes in different subdomains. We describe also how the resulting linear system can be solved iteratively by a parallel Schwarz alternating method or a Schwarz preconditioned Krylov subspace iterative method. Several numerical results are included to support the theory.

Analysis of block matrix preconditioners for elliptic optimal control problems

In this paper, we describe and analyse several block matrix iterative algorithms for solving a saddle point linear system arising from the discretization of a linear-quadratic elliptic control problem with Neumann boundary conditions. To ensure that the problem is well posed, a regularization term with a parameter α is included. The first algorithm reduces the saddle point system to a symmetric positive definite Schur complement system for the control variable and employs conjugate gradient (CG) acceleration, however, double iteration is required (except in special cases). A preconditioner yielding a rate of convergence independent of the mesh size h is described for Ω ⊂ R2 or R3, and a preconditioner independent of h and α when Ω ⊂ R2. Next, two algorithms avoiding double iteration are described using an augmented Lagrangian formulation. One of these algorithms solves the augmented saddle point system employing MINRES acceleration, while the other solves a symmetric positive definite reformulation of the augmented saddle point system employing CG acceleration. For both algorithms, a symmetric positive definite preconditioner is described yielding a rate of convergence independent of h. In addition to the above algorithms, two heuristic algorithms are described, one a projected CG algorithm, and the other an indefinite block matrix preconditioner employing GMRES acceleration. Rigorous convergence results, however, are not known for the heuristic algorithms. Copyright

Maximum norm stability of difference schemes for parabolic equations on overset nonmatching space-time grids

In this paper, theoretical results are described on the maximum norm stability and accuracy of finite difference discretizations of parabolic equations on overset nonmatching space-time grids. We consider parabolic equations containing a linear reaction term on a space-time domain Ω × [0, T] which is decomposed into an overlapping collection of cylindrical subregions of the form Ωl*×[0, T], for l = 1,..., p. Each of the space-time domains Ωl*[0, T] are assumed to be independently grided (in parallel) according to the local geometry and space-time regularity of the solution, yielding space-time grids with mesh parameters hl and τl. In particular, the different space-time grids need not match on the regions of overlap, and the time steps τl can differ from one grid to the next. We discretize the parabolic equation on each local grid by employing an explicit or implicit θ-scheme in time and a finite difference scheme in space satisfying a discrete maximum principle. The local discretizations are coupled together, without the use of Lagrange multipliers, by requiring the boundary values on each space-time grid to match a suitable interpolation of the solution on adjacent grids. The resulting global discretization yields a large system of coupled equations which can be solved by a parallel Schwarz iterative procedure requiring some communication between adjacent subregions. Our analysis employs a contraction mapping argument.Applications of the results are briefly indicated for reaction-diffusion equations with contractive terms and heterogeneous hyperbolic-parabolic approximations of parabolic equations.

Block iterative algorithms for the solution of parabolic optimal control problems

In this paper, we describe block matrix algorithms for the iterative solution of large scale linear-quadratic optimal control problems arising from the control of parabolic partial differential equations over a finite control horizon. After spatial discretization, by finite element or finite difference methods, the original problem reduces to an optimal control problem for n coupled ordinary differential equations, where n can be quite large. As a result, its solution by conventional control algorithms can be prohibitively expensive in terms of computational cost and memory requirements. We describe two iterative algorithms. The first algorithm employs a CG method to solve a symmetric positive definite reduced linear system for the unknown control variable. A preconditioner is described, which we prove has a rate of convergence independent of the space and time discretization parameters, however, double iteration is required. The second algorithm is designed to avoid double iteration by introducing an auxiliary variable. It yields a symmetric indefinite system, and for this system a positive definite block preconditioner is described. We prove that the resulting rate of convergence is independent of the space and time discretization parameters, when MINRES acceleration is used. Numerical results are presented for test problems.

Efficient variants of the vertex space domain decomposition algorithm

Several variants of the vertex space algorithm of Smith for two-dimensional elliptic problems are described. The vertex space algorithm is a domain decomposition method based on nonoverlapping subregions, in which the reduced Schur complement system on the interface is solved using a generalized block Jacobi-type preconditioner, with the blocks corresponding to the vertex space, edges, and a coarse grid. Two kinds of approximations are considered for the edge and vertex space subblocks, one based on Fourier approximation, and another based on an algebraic probing technique in which sparse approximations to these subblocks are computed. Our motivation is to improve the efficiency of the algorithm without sacrificing the optimal convergence rate. Numerical and theoretical results on the performance of these algorithms, including variants of an algorithm of Bramble, Pasciak, and Schatz are presented.

Block Diagonal Parareal Preconditioner for Parabolic Optimal Control Problems

We describe a block matrix iterative algorithm for solving a linearquadratic parabolic optimal control problem (OCP) on a finite time interval. We derive a reduced symmetric indefinite linear system involving the control variables and auxiliary variables, and solve it using a preconditioned MINRES iteration, with a symmetric positive definite block diagonal preconditioner based on the parareal algorithm. Theoretical and numerical results show that the preconditioned algorithm converges at a rate independent of the mesh size h, and has parallel scalability.

A Robust Preconditioner for the Hessian System in Elliptic Optimal Control Problems

We consider an elliptic optimal control problem in two dimensions, in which the control variable corresponds to the Neumann data on a boundary segment, and where the performance functional is regularized to ensure that the problem is well posed. A finite element discretization of this control problem yields a saddle point linear system, which can be reduced to a symmetric positive definite Hessian system for determining the control variables. We formulate a robust preconditioner for this reduced Hessian system, as a matrix product involving the discrete Neumann to Dirichlet map and a mass matrix, and show that it yields a condition number bound which is uniform with respect to the mesh size and regularization parameters. On a uniform grid, this preconditioner can be implemented using a fast sine transform. Numerical tests verify the theoretical bounds.