Sparse multiresolution stochastic approximation for uncertainty quantification

Most physical systems are inevitably affected by uncertainties due to natural variabili-ties or incomplete knowledge about their governing laws. To achieve predictive computer simulations of such systems, a major task is, therefore, to study the impact of these uncertainties on response quantities of interest. Within the probabilistic framework, uncertainties may be represented in the form of random variables/processes. Several computational strategies may then be applied to propagate these uncertainties in order to obtain the statistics of quantities of interest. Monte Carlo (MC) sampling methods have been widely used for this purpose. Besides their generally slow convergence, they offer a blend of simplicity, robustness, and efficiency for high-dimensional problems. Recently, there has been an increasing interest in developing more efficient alternative numerical methods. Most notably, stochastic approximation schemes based on Wiener-Askey polynomial chaos expansions (Ghanem & Spanos 2003; Xiu & Karniadakis 2002) or interpolations on sparse grids (Xiu & Hesthaven 2006) have been applied successfully to a variety of problems with random inputs. For sufficiently smooth solutions, these methods achieve as high as exponential convergence rate in the mean-squares sense. For non-smooth responses such as those exhibiting sharp gradients or discontinuities, however, these methods may result in poor approximations. To address this shortcoming, various methodologies including adaptive multiresolution (Le Maıtre et al. 2004), multi-element (generalized) polynomial chaos (Wan & Karni-adakis 2005), adaptive sparse grid collocation (Ma & Zabaras 2009), rational approximation (Chantrasmi et al. 2009), and simplex stochastic collocation (Witteveen & Iaccarino 2012), have been suggested in recent years. In this work, we present a multiresolution approach based on sparse multiwavelet expansions for uncertainty propagation. Unlike the work in Le Maıtre et al. (2004) where the multiwavelet coefficients are computed via Galerkin projections (typically requiring modifications of deterministic physics solvers), we propose a Compressive Sampling (CS) strategy in which physics solvers are treated as " black boxes ". CS is a new direction in signal processing that enables (up to) exact reconstruction of signals admitting sparse representations, in some suitable basis, using samples obtained at a sub-Nyquist rate (Candes & Tao 2005; Bruckstein et al. 2009). In Doostan & Owhadi (2011), the use of CS in the approximation of sparse polynomial chaos solutions to stochastic PDEs was first introduced. In the present study, we demonstrate the efficiency of CS in recovering stochastic functions having sparse expansions in multiwavelet bases. At the core of the proposed CS framework are the application of …