On Data-Driven Sparse Sensing and Linear Estimation of Fluid Flows

The reconstruction of fine-scale information from sparse data measured at irregular locations is often needed in many diverse applications, including numerous instances of practical fluid dynamics observed in natural environments. This need is driven by tasks such as data assimilation or the recovery of fine-scale knowledge including models from limited data. Sparse reconstruction is inherently badly represented when formulated as a linear estimation problem. Therefore, the most successful linear estimation approaches are better represented by recovering the full state on an encoded low-dimensional basis that effectively spans the data. Commonly used low-dimensional spaces include those characterized by orthogonal Fourier and data-driven proper orthogonal decomposition (POD) modes. This article deals with the use of linear estimation methods when one encounters a non-orthogonal basis. As a representative thought example, we focus on linear estimation using a basis from shallow extreme learning machine (ELM) autoencoder networks that are easy to learn but non-orthogonal and which certainly do not parsimoniously represent the data, thus requiring numerous sensors for effective reconstruction. In this paper, we present an efficient and robust framework for sparse data-driven sensor placement and the consequent recovery of the higher-resolution field of basis vectors. The performance improvements are illustrated through examples of fluid flows with varying complexity and benchmarked against well-known POD-based sparse recovery methods.