Numerical methods for parameterized linear systems
Time: Tue 2024-05-07 14.00
Location: F3 (Flodis), Lindstedtsvägen 26 & 28, Stockholm
Language: English
Subject area: Applied and Computational Mathematics, Numerical Analysis
Doctoral student: Siobhán Correnty , Numerisk analys, NA, SeRC - Swedish e-Science Research Centre
Opponent: Prof. Andrew J. Wathen, University of Oxford
Supervisor: Elias Jarlebring, Numerisk analys, NA, SeRC - Swedish e-Science Research Centre; Johan Karlsson, Optimeringslära och systemteori, Strategiskt centrum för industriell och tillämpad matematik, CIAM; Kirk M. Soodhalter, Trinity College Dublin
QC 2024-04-08
Abstract
Solving linear systems of equations is a fundamental problem in engineering. Moreover, applications involving the solution to linear systems arise in the social sciences, business, and economics. Specifically, the research conducted in this dissertation explores solutions to linear systems where the system matrix depends nonlinearly on a parameter. The parameter can be a scalar or a vector, and a change in the parameter results in a change in the solution. Such a setting arises in the study of partial differential equations and time-delay systems, and we are interested in obtaining solutions corresponding to many values of the parameter simultaneously. The methods developed in this thesis can also be used to solve parameter estimation problems. Furthermore, software has been developed and is available online.
This thesis consists of four papers and presents both algorithms and theoretical analysis. In Paper A, a linearization based on an infinite Taylor series expansion is considered. Specifically, the linearized system is a shifted parameterized system, and the parameter is a scalar. The GMRES method is used to solve the systems corresponding to many values of the parameter, and only one Krylov subspace basis matrix is required. Convergence analysis is based on solutions to a nonlinear eigenvalue problem and the magnitude of the parameter. Notably, the algorithm is carried out in a finite number of computations.
The approach in Paper B is based on a preconditioned linearized system solved using the inexact GMRES method. In this setting, the linearization incorporates all terms in an infinite Taylor series expansion, and the preconditioner is applied approximately using iterative methods. Solutions corresponding to many values of the scalar parameter are generated from one subspace, and this is done in a finite number of linear algebra operations. Theoretical analysis, based on the error in the application of the preconditioner and the magnitude of the parameter, leads to a bound on the residual.
Paper C proposes a short recurrence Krylov subspace method for solving linear systems that depend on a scalar parameter. In particular, a Chebyshev approximation is used to construct a linearization, and the linearized system is solved in a Bi-CG setting. Additionally, shift-and-invert preconditioning leads to fast convergence of the Krylov method for many different values of the parameter. An inexact variant of the method is also derived and analyzed.
In Paper D, a reduced order model is constructed from snapshots to solve parameterized linear systems. Specifically, the parameter is a vector of dimension 2, and the sampling is performed on a sparse grid using the method proposed in Paper C. A tensor decomposition is utilized to build the model. Approaches of this kind are not always successful, and it is not known a priori if a decomposition will converge on a given set of snapshots. This work offers a novel way to generate a new set of snapshots in the same parameter space, to be used if the decomposition does not converge, with little extra computation.