Publication Date

Summer 8-2022

Abstract

This thesis derives two Uncertainty Quantification (UQ) methods for differential equations that depend on random parameters: (\textbf{i}) error bounds for a computed cumulative distribution function (\textbf{ii}) a multi-level Monte Carlo (MLMC) algorithm with adaptively refined meshes and accurately computed stopping-criteria. Both UQ approaches utilize adjoint-based \textit{a posteriori} error analysis in order to accurately estimate the error in samples of numerically approximated quantities of interest. The adaptive MLMC algorithm developed in this thesis relies on the adjoint-based error analysis to adaptively create meshes and accurately monitor a stopping criteria. This is in contrast to classical MLMC algorithms which employ either a hierarchy of uniform meshes or adaptively refined meshes based on Richardson extrapolation. Moreover, they also use a stopping criteria that relies on assumptions on the convergence rate of the MLMC levels. This thesis overcomes these drawbacks of the classical algorithms.

The analysis and UQ methods developed in this these are applied to several types of differential equations and quantities of interest. Classical \emph{a posteriori} error analysis provides a formulation of the error in a Quantity of Interest (QoI) which is represented as a bounded linear functional of the solution to a differential equation. This thesis derives error estimates for a QoI that describes the \textit{time at which} a linear functional of the solution achieves a threshold value. This QoI is referred to as ``non-standard'' since it cannot be represented as a linear functional of the solution. The classical analysis does not directly apply to non-standard QoIs.

The adjoint-based error analysis for the different QoIs not only develops accurate error estimates, they also provide a decomposition of the error into contributions from different regions of the domain. The decompositions are utilized to adaptively create meshes when adding new levels in the MLMC algorithm. Two adaptive mesh creation methods are described that can be used to build the MLMC estimator. Many numerical experiments demonstrate the accuracy of the error estimates and the advantages of using adaptive mesh creation in the MLMC algorithm.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Jehanzeb Chaudhary

Second Committee Member

Simon Tavener

Third Committee Member

Jacob Schroder

Fourth Committee Member

Stephen Lau

Language

English

Keywords

Uncertatinty Quantification, Differential Equations, a Posteriori, Error Estimation

Document Type

Dissertation

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