# Compressed sensing with sparse corruptions: Fault-tolerant sparse collocation approximations

## TL;DR

This paper develops a robust compressive sensing method to recover sparse coefficients in Polynomial Chaos Expansions despite measurement corruptions, addressing hardware/software failures in complex computational frameworks.

## Contribution

It introduces a novel theoretical analysis and an iterative reweighted algorithm for fault-tolerant sparse recovery in polynomial chaos expansions.

## Key findings

- The method effectively recovers sparse coefficients with corrupted measurements.
- The iterative algorithm improves regularization parameter selection.
- Numerical tests demonstrate robustness in high-dimensional differential equation solutions.

## Abstract

The recovery of approximately sparse or compressible coefficients in a Polynomial Chaos Expansion is a common goal in modern parametric uncertainty quantification (UQ). 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 $\ell^1$ minimization algorithm that aims to recover sparse expansion coefficients in the presence of measurement corruptions. Our recovery results are uniform, 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.