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 l 1 -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.

中文翻译：

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工程力学中的稀疏表示和压缩采样方法：理论概念和不同应用的回顾

摘要 从广泛的角度回顾了工程力学问题中稀疏表示和压缩采样 (CS) 方法的理论概念和各种应用。首先，在介绍完善的 CS 概念和优化算法之后，关注当前用于增强解决方案稀疏性和利用数据中的附加信息的新兴工具和技术。这些包括替代 l 1 -范数最小化公式和迭代重新加权解决方案、贝叶斯方法以及结构化稀疏和字典学习策略。接下来，与工程力学问题相关的基于 CS 的研究工作在三个不同的应用领域进行分类和讨论：a) 结构健康监测中的反问题，b) 不确定性建模和模拟，以及 c) 计算效率高的不确定性传播。值得注意的是，这三个领域的绝大多数问题都面临着“不完整数据”的挑战，由通用 CS 框架解决。在这方面，不完整的数据可能以各种不同的形式表现出来，并且可能对应于缺失或压缩的数据，甚至通常指的是功能评估不足。这篇评论论文的主要目标是确定和展示在上述三个工程力学应用领域中的每一个领域的重大贡献，目的是加快额外的研究和开发工作。为此，提供了 248 篇参考文献的广泛列表，几乎完全由书籍和档案文件组成，