Yang Liu successfully defended his PhD Thesis

About

Abstract: 

This thesis explores numerical methods for uncertainty quantification in systems governed by random partial differential equations (PDEs) and advances in randomized quasi-Monte Carlo (RQMC) techniques.

In the first part, we present an adaptive multilevel Monte Carlo (AMLMC) algorithm to efficiently approximate expected quantities of interest (QoIs). The AMLMC framework integrates isoparametric finite element approximations with dual-weighted residual error representations, employing non-uniform adaptive meshes to handle geometric singularities. This method excels with lognormal diffusivity coefficients, addressing their variability while achieving improved computational complexity over standard multilevel Monte Carlo (MLMC). We also extend AMLMC to a quasi-Monte Carlo (QMC) approach, leveraging its faster convergence and incorporating importance sampling (IS) and control variates for enhanced efficiency.

The second part investigates theoretical advancements in RQMC methods, emphasizing nonasymptotic convergence for linear elliptic PDEs with lognormal coefficients and variance analysis of RQMC estimators. The nonasymptotic analysis establishes error bounds for RQMC in the pre-asymptotic regime, while spectral analysis determines the convergence rate for both smooth and singular integrands.

Advisor: 

Prof. Raúl Tempone

Committee Members:
 

Prof. Daniele Boffi, Prof. David Bolin, Prof. Diogo A. Gomes, and Prof. Fabio Nobile