A phd student of our group Yang Liu participated in the confernce ECCOMAS-YIC23 and had a presentation, on Nonasymptotic QMC convergence analysis for elliptic PDE with lognormal coefficients. The Conference held at Porto, Portugal between June 19-21, 2023.
Abstract:
This study analyzes the nonasymptotic convergence behavior of the quasi- Monte Carlo (QMC) method with applications to linear elliptic partial differen- tial equations (PDEs) with lognormal coefficients. Following the error analysis in (Owen, 2006), we derive a non-asymptotic convergence estimate, which de- pends on the specific integrands, the finite dimensions and the finite samples. We discussed the effect of random variable (random field) variance as well as dimensions. We then applied the quasi-Monte Carlo with importance sampling for approximating deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity co- efficient in bounded domains of Rd, where the random coefficient is modeled as stationary Gaussian random field parameterized by both the trigonometric and wavelets-type basis. We proposed two types of importance sampling distribu- tions, analyzed their effect on the QMC convergence rate, and discussed their limitations.
