Reduced basis approximation and error estimation of parametrized partial differential equations

Sentz, Peter

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

Description

Title

Reduced basis approximation and error estimation of parametrized partial differential equations

Author(s)

Sentz, Peter

Issue Date

2022-07-12

Director of Research (if dissertation) or Advisor (if thesis)

Olson, Luke

Doctoral Committee Chair(s)

Olson, Luke

Committee Member(s)

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

Department of Study

Computer Science

Discipline

Computer Science

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• reduced basis
• partial differential equations
• finite elements

Abstract

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.

Graduation Semester

2022-08

Type of Resource

Thesis

Copyright and License Information

Copyright 2022 Peter Sentz

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