Iterative Solution Methods for Reduced-Order Models of Parameterized Partial Differential Equations

This dissertation considers efficient computational algorithms for solving parameterized discrete partial differential equations (PDEs) using techniques of reduced-order modeling. Parameterized equations of this type arise in numerous mathematical models. In some settings, e.g. sensitivity analysis, design optimization, and uncertainty quantification, it is necessary to compute discrete solutions of the PDEs at many parameter values. Accuracy considerations often lead to algebraic systems with many unknowns whose solution via traditional methods can be expensive. Reduced-order models use a reduced space to approximate the parameterized PDE, where the reduced space is of a significantly smaller dimension than that of the discrete PDE. Solving an approximation of the problem on the reduced space leads to reduction in cost, often with little loss of accuracy. In the reduced basis method, an offline step finds an approximation of the solution space and an online step utilizes this approximation to solve a smaller reduced problem, which provides an accurate estimate of the solution. Traditionally, the reduced problem is solved using direct methods. However, the size of the reduced system needed to produce solutions of a given accuracy depends on the characteristics of the problem, and it may happen that the size is significantly smaller than that of the original discrete problem but large enough to make direct solution costly. In this scenario, it is more effective to use iterative methods to solve the reduced problem. To demonstrate this we construct preconditioners for the reduced-order models or construct well-conditioned reduced-order models. We demonstrate that by using iterative methods, reduced-order models of larger dimension can be effective. There are several reasons that iterative methods are well suited to reduced- order modeling. In particular, we take advantage of the similarity of the realizations of parameterized systems, either by reusing preconditioners or by recycling Krylov vectors. These two approaches are shown to be effective when the underlying PDE is linear. For nonlinear problems, we utilize the discrete empirical interpolation method (DEIM) to cheaply evaluate the nonlinear components of the reduced model. The method identifies points in the PDE discretization necessary for representing the nonlinear component of the reduced model accurately. This approach incurs online computational costs that are independent of the spatial dimension of the discretized PDE. When this method is used to assemble the reduced model cheaply, iterative methods are shown to further improve efficiency in the online step. Finally, when the traditional offline/online approach is ineffective for a given problem, reduced-order models can be used to accelerate the solution of the full model. We follow the solution model of Krylov subspace recycling methods for sequences of linear systems where the coefficient matrices vary. A Krylov subspace recycling method contains a reduced-order model and an iterative method that searches the space orthogonal to the reduced space. We once again use iterative solution techniques for the solution of the reduced models that arise in this context. In this case, the iterative methods converge quickly when the reduced basis is constructed to be naturally well conditioned

Solving the steady state diffusion equation with uncertainty

Abstract The goal of this project is to efficiently solve a steady state diffusion equation with a random coefficient. Although, such equations can be solved using Monte-Carlo methods, the lengthy computation time can be constraining. Using a Karhunen-Loéve expansion allows the random coefficient to be approximated with a finite sum of random variables. This expansion combined with a Galerkin method or stochastic collocation method reduces computation time

Preconditioning Techniques for Reduced Basis Methods for Parameterized Partial Differential Equations

The reduced basis methodology is an efficient approach to solve parameterized discrete partial differential equations when the solution is needed at many parameter values. An offline step approximates the solution space and an online step utilizes this approximation, the reduced basis, to solve a smaller reduced problem, which provides an accurate estimate of the solution. Traditionally, the reduced problem is solved using direct methods. However, the size of the reduced system needed to produce solutions of a given accuracy depends on the characteristics of the problem, and it may happen that the size is significantly smaller than that of the original discrete problem but large enough to make direct solution costly. In this scenario, it may be more effective to use iterative methods to solve the reduced problem. We construct preconditioners for reduced iterative methods which are derived from preconditioners for the full problem. This approach permits reduced basis methods to be practical for larger bases than direct methods allow. We illustrate the effectiveness of iterative methods for solving reduced problems by considering two examples, the steady-state diffusion and convection-diffusion-reaction equations

Solving the steady state diffusion equation with uncertainty

Abstract Goal: to efficiently solve a steady state diffusion equation with a random coefficient. Monte-Carlo methods are time intensive. Using principal components analysis (also known as the Karhunen-Loéve expansion) allows the random coefficient to be approximated with a finite sum of random variables. This expansion combined with a stochastic finite element method should reduce computation time

Properties and preservers of the pseudospectrum

AbstractThe interplay between the algebraic and analytic properties of a matrix and the geometric properties of its pseudospectrum is investigated. It is shown that one can characterize Hermitian matrices, positive semi-definite matrices, orthogonal projections, unitary matrices, etc. in terms of the pseudospectrum. Also, characterizations are given to maps on matrices leaving invariant the pseudospectrum of the sum, difference, or product of matrix pairs. It is shown that such a map is always a unitary similarity transform followed by some simple operations such as adding a constant matrix, taking the matrix transpose, or multiplying by a scalar in {1,-1}

Preservers of eigenvalue inclusion sets of matrix products

AbstractFor a square matrix A, let S(A) be an eigenvalue inclusion set such as the Gershgorin region, the union of Cassini ovals, and the Ostrowski’s set. Characterization is obtained for maps Φ on n×n matrices satisfying S(Φ(A)Φ(B))=S(AB) for all matrices A and B