A Bayesian mixed shrinkage prior procedure for spatial-stochastic basis selection and evaluation of gPC expansions: Applications to elliptic SPDEs

We propose a new fully Bayesian method to efficiently obtain the spectral representation of a spatial random field, which can conduct spatial-stochastic basis selection and evaluation of generalized Polynomial Chaos (gPC) expansions when the number of the available basis functions is significantly larger than the size of the training data-set. We develop a fully Bayesian stochastic procedure, called mixed shrinkage prior (MSP), which performs both basis selection and coefficient evaluation simultaneously. MSP involves assigning a prior probability to the gPC structure and assigning conjugate priors to the expansion coefficients that can be thought of as mixtures of Ridge-LASSO shrinkage priors, in augmented form. The method offers a number of advantages over existing compressive sensing methods in gPC literature, such that it recovers possible sparse structures in both stochastic and spatial domains while the resulting expansion can be re-used directly to economically obtain results at any spatial input values. Yet, it inherits all the advantages of Bayesian model uncertainty methods, e.g. accounts for uncertainty about basis significance and provides interval estimation through posterior distributions. A unique highlight of the MSP procedure is that it can address heterogeneous sparsity in the spatial domain for different random dimensions. Furthermore, it yields a compromise between Ridge and LASSO regressions, and hence combines a weak (l2-norm) and strong (l1-norm) shrinkage, in an adaptive, data-driven manner. We demonstrate the good performance of the proposed method, and compare it against other existing compressive sensing ones on elliptic stochastic partial differential equations.