Reduced basis approximation and error estimation of parametrized partial differential equations

This thesis focuses on reduced basis (RB) methods for parametrized partial differential equations. Specifically, an RB method featuring exact error bounds for linear coercive elliptic problems is presented, and a neural network RB approach is demonstrated for time-dependent problems which lack parametric separability. An RB method based on a least-squares finite element method framework (LSFEM-RB) is introduced, which features a rigorous bound on the error as measured with respect to an exact solution. This is in contrast to typical RB error bounds, which measure the error with respect to a finite-dimensional approximation. The first-order formulation of the LSFEM-RB is shown to be a key ingredient of the method. Several numerical examples demonstrate the validity of the method. A lower bound of the exact coercivity constant is required to guarantee the rigor of the LSFEM-RB error estimate, but standard algorithms which construct these bounds are only applicable to the coercivity of the discretized problem. The consequences of using bounds of the discrete coercivity constant is investigated by establishing numerical convergence rates with respect to the continuous problem. The coercivity constant is characterized as a spectral value of a self-adjoint linear operator, and a relationship is established with the eigenvalues of a compact operator for a variety of differential equations. The discrete coercivity constant is shown to converge at twice the rate of the underlying finite-dimensional approximation, and is verified with numerical evidence. A time-stepping RB method is proposed for time-dependent problems that are not parametrically separable. This method learns the dynamics of the reduced-order expansion coefficients by training a residual neural network. This results in an online stage with a computational cost independent of the size of the underlying problem. The method is verified through several example parabolic equations.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Computer Science

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Sentz, Peter

Contributors dc:contributor

• Olson, Luke
• Kloeckner, Andreas
• Fischer, Paul
• Cyr, Eric

• reduced basis
• partial differential equations
• finite elements

Statement dc:rights

• Copyright 2022 Peter Sentz

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/116208

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F116208