An Adaptive Wavelet Stochastic Collocation Method for Irregular Solutions of PDEs with Random Input Data
Accurate predictive simulations of complex real world applications require numerical approximations to first, oppose the curse of dimensionality and second, converge quickly in the presence of steep gradients, sharp transitions, bifurcations or finite discontinuities in high-dimensional random parameter spaces. In this talk we present a novel multi-dimensional multi-resolution sparse grid adaptive wavelet stochastic collocation method (AWSCM), that utilizes hierarchical multi-scale piecewise Riesz basis functions constructed from interpolating wavelets.
The basis for our non-intrusive method forms a stable multi-scale splitting and thus, optimal adaptation is achieved. More importantly, when 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. Error estimates and numerical examples will used to compare the efficiency of the method with several other well-known techniques.