Data-driven compressive sensing and applications in uncertainty quantification

In this paper, we propose a data-driven compressive sensing method for the effective and sparse representation of solutions to a stochastic system, in which a problem-dependent basis is constructed and used to exploit the sparse patterns. Given what appears to be a highly incomplete set of sample observations, the essential ingredients of the proposed method involve (i) estimating the covariance function from low-fidelity simulations, (ii) extracting the most dominant energetic modes from the Karhunen-Loève analysis, and (iii) solving the basis pursuit problem associated with data-driven functions for high-fidelity models. Compared with other conventional compressive sensing methods that use a problem-independent basis, our approach can significantly increase the sparsity of expansion coefficients in many situations. When applied to uncertainty quantification problems such as partial differential equations with random input data, numerical experiments are carried out to show the effectiveness of our data-driven method in recovering the sparse solution globally over the physical domain.