An Adaptive Wavelet Stochastic Collocation Method for Irregular Solutions of Partial Differential Equations with Random Input Data

Abstract

A novel multi-dimensional multi-resolution adaptive wavelet stochastic collocation method (AWSCM) for solving partial differential equations with random input data is proposed. The uncertainty in the input data is assumed to depend on a finite number of random variables. In case the dimension of this stochastic domain becomes moderately large, we show that utilizing a hierarchical sparse-grid AWSCM (sg-AWSCM) not only combats the curse of dimensionality but, in contrast to the standard sg-SCMs built from global Lagrange-type interpolating polynomials, maintains fast convergence without requiring sufficiently regular stochastic solutions. Instead, our non-intrusive approach extends the sparse-grid adaptive linear stochastic collocation method (sg-ALSCM) by employing a compactly supported wavelet approximation, with the desirable multi-scale stability of the hierarchical coefficients guaranteed as a result of the wavelet basis having the Riesz property. This property provides an additional lower bound estimate for the wavelet coefficients that are used to guide the adaptive grid refinement, resulting in the sg-AWSCM requiring a significantly reduced number of deterministic simulations for both smooth and irregular stochastic solutions. Second-generation wavelets constructed from a lifting scheme allows us to preserve the framework of the multi-resolution analysis, compact support, as well as the necessary interpolatory and Riesz property of the hierarchical basis. Several numerical examples are given to demonstrate the improved convergence of our numerical scheme and show the increased efficiency when compared to the sg-ALSCM method.