Error Estimates Of Wiener Chaos Methods For Some Stochastic Partial Differential Equations

The polynomial chaos method, one of the most popular stochastic uncertainty quantifica-tion methods, has attracted more attention in the last two decades. Numerical solutions forimportant stochastic models, especially stochastic (partial) di?erential equations, involvemany e?orts.Chapter 2 provides more rigorous error estimates of Wiener chaos methods for dif-fusion equations than existing analysis with purely spatial multiplicative white noise. Theerror estimates shows delicately how the total error behaves with the truncation param-eter in physical space and random space. The estimate of truncation in random space isimproved.Long-term integration is addressed in Chapters 3 and 4 with Wiener chaos methodconcerning white noise SPDEs and SDEs with single random variable. Chapter 3 considersWiener chaos methods for a passive scalar equation in Gaussian field. Spectral separatingscheme breaks the curse of dimensionality in random space and also allows long-termintegration for linear problems with error growing linearly with time. The technique canbe extended to some nonlinear problems linearizing nonlinear terms properly.Chapter 4 discusses a simple SODE problem with only one uniform random variableas the coe?cient. The error analysis suggests that the Wiener chaos method fails aftercertain time and hence longer-term integration would require more modes or nodes inrandom space. The di?culty of the problem itself is also viewed from multi-scale andsingular perturbation point of view.In Chapter 5, two stochastic advection models are compared with di?erent velocities,white noise and second-order process velocities. The comparison shows that solutions tothese two models are distinct and not close to each other even intuition predicts so.Chapter 6 consider a model reduction technique in high dimensional problems,ANOVA, with polynomial (Wiener chaos) interpolation. Error estimates are presentedwith the weight argument borrowed from quasi-Monte-Carlo error theory. This o?ers a di?erent prospect for revising this reduction technique.Some possible future research topics are discussed in the end. All the proposedmethods are accompanied by their theoretical analysis and tested on model problems.