Browsing by Author "Elman, Howard"
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Item Efficient Iterative Algorithms for the Stochastic Finite Element Method with Application to Acoustic Scattering(2002-12-19) Elman, Howard; Ernst, Oliver G.; O'Leary, Dianne P.; Stewart, MichaelIn this study, we describe the algebraic computations required to implement the stochastic finite element method for solving problems in which uncertainty is restricted to right hand side data coming from forcing functions or boundary conditions. We show that the solution can be represented in a compact outer product form which leads to efficiencies in both work and storage, and we demonstrate that block iterative methods for algebraic systems with multiple right hand sides can be used to advantage to compute this solution. We also show how to generate a variety of statistical quantities from the computed solution. Finally, we examine the behavior of these statistical quantities in one setting derived from a model of acoustic scattering. UMIACS-TR-2002-102Item Efficient Preconditioning of the Linearized Navier-Stokes Equations}(1999-10-16) Silvester, David; Elman, Howard; Kay, David; Wathen, AndrewWe outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of Navier-Stokes equations. We describe a very general strategy for preconditioning that has two basic building blocks; a multigrid V-cycle for the scalar convection-diffusion operator, and a multigrid V-cycle for a pressure Poisson operator. We present numerical experiments illustrating that a simple implementation of our approach leads to an effective and robust solver strategy in that the convergence rate is independent of the grid and the time-step, and only deteriorates very slowly as the Reynolds number is increased. (Also cross-referenced as UMIACS-TR-99-66)Item Fast Iterative Solver for Convection-Diffusion Systems with Spectral Elements(2009-03-04) Lott, P. Aaron; Elman, HowardWe introduce a solver and preconditioning technique based on Domain Decomposition and the Fast Diagonalization Method that can be applied to tensor product based discretizations of the steady convection-diffusion equation. The method is based on iterative substructuring where fast diagonalization is used to efficiently eliminate the interior degrees of freedom and subsidiary subdomain solves. We demonstrate the effectiveness of this method in numerical simulations using a spectral element discretization.Item H(div) Preconditioning for a Mixed Finite Element Formulation of the Stochastic Diffusion Problem(2008-06-27) Elman, Howard; Furnival, Darran G.; Powell, Catherine E.We study H(div) preconditioning for the saddle-point systems that arise in a stochastic Galerkin mixed formulation of the steady-state diffusion problem with random data. The key ingredient is a multigrid V-cycle for a weighted, stochastic H(div) operator, acting on a certain tensor product space of random fields with finite variance. We build on a multigrid algorithm described by Arnold, Falk, and Winther for the deterministic problem by varying the spatial discretization from grid to grid whilst keeping the stochastic discretization fixed. We extend the deterministic analysis to accommodate the modified H(div) operator and establish spectral equivalence bounds with a new multigrid V-cycle operator that are independent of the spatial and stochastic discretization parameters. We implement multigrid within a block-diagonal preconditioner for the full stochastic saddle-point problem, derive eigenvalue bounds for the preconditioned system matrices and investigate the impact of all the discretization parameters on the convergence rate of preconditioned MINRES.Item Least Squares Preconditioners for Stabilized Discretizations of the Navier-Stokes Equations(2006-04-20T16:09:59Z) Elman, Howard; Howle, Victoria E.; Shadid, John; Silvester, David; Tuminaro, RayThis paper introduces two stabilization schemes for the Least Squares Commutator (LSC) preconditioner developed by Elman, Howle, Shadid, Shuttleworth and Tuminaro [SIAM J. Sci. Comput., 27, 2006, pp. 1651--1668] for the incompressible Navier-Stokes equations. This preconditioning methodology is one of several choices that are effective for Navier-Stokes equations, and it has the advantage of being defined from strictly algebraic considerations. It has previously been limited in its applicability to div-stable discretizations of the Navier-Stokes equations. This paper shows how to extend the same methodology to stabilized low-order mixed finite element approximation methods.Item A Parallel Block Multi-level Preconditioner for the 3D Incompressible Navier--Stokes Equations(2002-10-25) Elman, Howard; Howle, V. E.; Shadid, John; Tuminaro, RayThe development of robust and efficient algorithms for both steady-state simulations and fully-implicit time integration of the Navier--Stokes equations is an active research topic. To be effective, the linear subproblems generated by these methods require solution techniques that exhibit robust and rapid convergence. In particular, they should be insensitive to parameters in the problem such as mesh size, time step, and Reynolds number. In this context, we explore a parallel preconditioner based on a block factorization of the coefficient matrix generated in an Oseen nonlinear iteration for the primitive variable formulation of the system. The key to this preconditioner is the approximation of a certain Schur complement operator by a technique first proposed by Kay, Loghin, and Wathen [25] and Silvester, Elman, Kay, and Wathen [45]. The resulting operator entails subsidiary computations (solutions of pressure Poisson and convection--diffusion subproblems) that are similar to those required for decoupled solution methods; however, in this case these solutions are applied as preconditioners to the coupled Oseen system. One important aspect of this approach is that the convection--diffusion and Poisson subproblems are significantly easier to solve than the entire coupled system, and a solver can be built using tools developed for the subproblems. In this paper, we apply smoothed aggregation algebraic multigrid to both subproblems. Previous work has focused on demonstrating the optimality of these preconditioners with respect to mesh size on serial, two-dimensional, steady-state computations employing geometric multi-grid methods; we focus on extending these methods to large-scale, parallel, three-dimensional, transient and steady-state simulations employing algebraic multigrid (AMG) methods. Our results display nearly optimal convergence rates for steady-state solutions as well as for transient solutions over a wide range of CFL numbers on the two-dimensional and three-dimensional lid-driven cavity problem. Also UMIACS-TR-2002-95Item Solving the Stochastic Steady-State Diffusion Problem using(2006-03-30T17:48:39Z) Elman, Howard; Furnival, DarranWe study multigrid for solving the stochastic steady-state diffusion problem. We operate under the mild assumption that the diffusion coefficient takes the form of a finite Karhunen-Loeve expansion. The problem is discretized using a finite element methodology using the polynomial chaos method to discretize the stochastic part of the problem. We apply a multigrid algorithm to the stochastic problem in which the spatial discretization is varied from grid to grid while the stochastic discretization is held constant. We then show, theoretically and experimentally, that the convergence rate is independent of the spatial discretization, as in the deterministic case.Item Stagnation of GMRES(2001-11-12) Zavorin, Ilya; O'Leary, Dianne P.; Elman, HowardWe study problems for which the iterative method \gmr for solving linear systems of equations makes no progress in its initial iterations. Our tool for analysis is a nonlinear system of equations, the stagnation system, that characterizes this behavior. For problems of dimension 2 we can solve this system explicitly, determining that every choice of eigenvalues leads to a stagnating problem for eigenvector matrices that are sufficiently poorly conditioned. We partially extend this result to higher dimensions for a class of eigenvector matrices called extreme. We give necessary and sufficient conditions for stagnation of systems involving unitary matrices, and show that if a normal matrix stagnates then so does an entire family of nonnormal matrices with the same eigenvalues. Finally, we show that there are real matrices for which stagnation occurs for certain complex right-hand sides but not for real ones. (Also UMIACS-TR-2001-74)