We consider sparsity-based techniques for the approximation of high-dimensional functions from random pointwise evaluations. To date, almost all the works published in this field contain some a priori assumptions about the error corrupting the samples that are hard to verify in practice. In this paper, we instead focus on the scenario where the error is unknown. We study the performance of four sparsity-promoting optimization problems: weighted quadratically-constrained basis pursuit, weighted LASSO, weighted square-root LASSO, and weighted LAD-LASSO. From the theoretical perspective, we prove uniform recovery guarantees for these decoders, deriving recipes for the optimal choice of the respective tuning parameters. On the numerical side, we compare them in the pure function approximation case and in applications to uncertainty quantification of ODEs and PDEs with random inputs. Our main conclusion is that the lesser-known square-root LASSO is better suited for high-dimensional approximation than the other procedures in the case of bounded noise, since it avoids (both theoretically and numerically) the need for parameter tuning.

The recovery of approximately sparse or compressible coefficients in a polynomial chaos expansion is a common goal in many modern parametric uncertainty quantification (UQ) problems. However, relatively little effort in UQ has been directed toward theoretical and computational strategies for addressing the sparse corruptions problem, where a small number of measurements are highly corrupted. Such a situation has become pertinent today since modern computational frameworks are sufficiently complex with many interdependent components that may introduce hardware and software failures, some of which can be difficult to detect and result in a highly polluted simulation result.In this paper we present a novel compressive sampling-based theoretical analysis for a regularized 1 minimization algorithm that aims to recover sparse expansion coefficients in the presence of measurement corruptions. Our recovery results are uniform (the theoretical guarantees hold for all compressible signals and compressible corruptions vectors), and prescribe algorithmic regularization parameters in terms of a user-defined a priori estimate on the ratio of measurements that are believed to be corrupted. We also propose an iteratively reweighted optimization algorithm that automatically refines the value of the regularization parameter, and empirically produces superior results. Our numerical results test our framework on several medium-to-high dimensional examples of solutions to parameterized differential equations, and demonstrate the effectiveness of our approach.

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