TITLE: Adaptive Hierarchical Stochastic Collocation Methods for High-Dimensional Approximation, Discontinuity Detection and Parameter Estimation

SPEAKER: Guannan Zhang

ABSTRACT:

Abstract: We will discuss an adaptive hierarchical stochastic collocation (AHSC) framework that addresses several challenges arising in uncertainty quantification (UQ) including: quantification of high-dimensional quantities of interest; high-dimensional discontinuity detection; and reducing computational complexity of parameter estimation. For high-dimensional approximation, we extended the conventional AHSC method by incorporating the wavelet basis into the sparse-grid framework. Second-generation wavelets are used constructed from a lifting scheme which allow 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. For high-dimensional discontinuity detection, we developed a hyper-spherical stochastic collocation method for identifying jump discontinuities by incorporating a hyper-spherical coordinate system (HSCS) into the sparse-grid approximation framework. An approximate discontinuity surface is constructed directly in the hyper-spherical system with a greatly reduced number of sparse grid points compared to existing methods. For parameter estimation problems with computationally expense physical simulations, we incorporate the AHSC approach with high-order hierarchical basis into Bayesian inference framework to reduce computational complexity in MCMC sampling. A surrogate of the posterior probability density function (PPDF) is constructed using the AHSC methods. High-order local hierarchical polynomials (e.g. quadratic, cubic basis) are used to further improve the accuracy and cost of the surrogate. Moreover, to efficiently approximate PPDFs with multiple significant modes, we also incorporate optimization into our new approaches.

Bio: Dr. Guannan Zhang is the Householder fellow in the Computer Science and Mathematics Division at Oak Ridge National Laboratory since August 2012. Previously, he enrolled in the doctoral program in computational science at the Florida State University in 2009 and obtained his Ph.D. degree in the summer of 2012. Zhang is currently a member of the Applied Mathematics Group (AMG) at ORNL. His research is focused on mathematical foundations of uncertainty quantification (UQ), including numerical methods for stochastic partial differential equations, high-dimensional stochastic approximation and discontinuity detection, Data assimilation, stochastic optimization and related applications.