Sparse representations and compressive sampling approaches in engineering mechanics: A review of theoretical concepts and diverse applications

Abstract

A review of theoretical concepts and diverse applications of sparse representations and compressive sampling (CS) approaches in engineering mechanics problems is provided from a broad perspective. First, following a presentation of well-established CS concepts and optimization algorithms, attention is directed to currently emerging tools and techniques for enhancing solution sparsity and for exploiting additional information in the data. These include alternative to ℓ1-norm minimization formulations and iterative re-weighting solution schemes, Bayesian approaches, as well as structured sparsity and dictionary learning strategies. Next, CS-based research work of relevance to engineering mechanics problems is categorized and discussed under three distinct application areas: a) inverse problems in structural health monitoring, b) uncertainty modeling and simulation, and c) computationally efficient uncertainty propagation. Notably, the vast majority of problems in all three areas share the challenge of “incomplete data”, addressed by the versatile CS framework. In this regard, incomplete data may manifest themselves in various different forms and can correspond to missing or compressed data, or even refer generally to insufficiently few function evaluations. The primary objective of this review paper relates to identifying and presenting significant contributions in each of the above three application areas in engineering mechanics, with the goal of expediting additional research and development efforts. To this aim, an extensive list of 248 references is provided, composed almost exclusively of books and archival papers, which can be readily available to a potential reader.