Multi-element probabilistic collocation in high dimensions: Applications to systems biology and physical systems

In this work we develop and apply numerical methods for quantifying parametric uncertainty in mathematical models. In Part I, we introduce the Multi-Element Probabilistic Collocation Method (ME-PCM) and investigate its convergence properties. We prove that as the parameter space mesh is refined, the convergence rate of the solution depends on the quadrature rule of each element only through its degree of exactness. In addition, the L2 error of the tensor product ME-PCM interpolant is examined and an adaptivity algorithm is provided. We test the ME-PCM on Navier-Stokes examples and a stochastic diffusion problem with various random inputs with up to 50 dimensions. The computational cost of the ME-PCM is found to be favorable when compared to the cost of other methods including stochastic Galerkin (ME-gPC), probabilistic collocation (PCM), Monte Carlo (MC) and low discrepancy sequence methods (e.g. quasi-Monte Carlo).;In Part II we apply stochastic spectral methods to various applications from systems biology and physical systems. In particular, we demonstrate how the ME-PCM may be used to understand parametric sensitivity in cellular signaling networks. We focus specifically two models: a regulatory network model of cell apoptosis and a model of morphogen patterning in the Drosophila embryo. We also apply stochastic Galerkin and collocation methods to quantify the effect of uncertain forces and material properties on three-dimensional riser sections undergoing elastic deformations. Lastly, we apply the MEPCM-A to aid in modeling subsurface contaminant transport through heterogeneous media at the U.S. Department of Energy's Hanford Site in southeastern Washington state.;We also introduce an extension of the ME-PCM wherein high-dimensional problems are split with an ANOVA-type decomposition into a series of lower-dimensional problems, and the ME-PCM is applied to each subproblem. We perform numerical studies to analyze the efficiency of this method in comparison to other existing methods. We also test the MEPCM-A method for integration of discontinuous functions in 100 to 500 dimensions using the GENZ testing package for high dimensional integration.