Realistic analysis and design of multi-disciplinary engineering systems requires not only a fine understanding and modeling of the underlying physics and their interactions but also recognition of intrinsic uncertainties and their influences on the quantities of interest. Uncertainty Quantification (UQ) is an emerging discipline that attempts to address the latter issue: It aims at a meaningful characterization of uncertainties from the available measurements, as well as efficient propagation of these uncertainties through the governing equations for a quantitative validation of model predictions.

This talk first provides a brief introduction to uncertainty propagation using spectral methods, specifically Polynomial Chaos (PC) expansions, along with numerical challenges associated with these methods. Following that, recent developments on sparse PC approximation via compressive sampling, specifically l1-minimization, will be introduced as a means to tackle these difficulties. The rest of the talk focuses on the convergence analysis, weighted approximation, and random sampling of PC expansions for successful solution recovery via l1-minimization. In particular, it will be shown that bounding a probabilistic parameter, referred to as coherence, yields a bound on the number of samples necessary to identify coefficients in a sparse PC expansion. Utilizing properties of orthogonal polynomials, bounds on the coherence parameter will be provided for polynomials of Hermite and Legendre type under their respective natural sampling distribution. These bounds reveal that sampling the solution of interest from the probability distribution of the random inputs may be sub-optimal. In both polynomial bases importance sampling distributions will be identified, which yield bounds on the coherence with weaker dependence on the order or dimension of the approximation. Numerical examples will be provided to illustrate the convergence as well as the effect of different sampling strategies.

This is a joint work with Jerrad Hampton and Ji Peng from CU Boulder.