Abstract
Sparse linear estimation of fluid flows using data-driven proper orthogonal decomposition (POD) basis is systematically explored in this work. Fluid flows are manifestations of nonlinear multiscale partial differential equations (PDE) dynamical systems with inherent scale separation that impact the system dimensionality. Given that sparse reconstruction is inherently an ill-posed problem, the most successful approaches require the knowledge of the underlying low-dimensional space spanning the manifold in which the system resides. In this paper, we adopt an approach that learns basis from singular value decomposition (SVD) of training data to recover sparse information. This results in a set of four design parameters for sparse recovery, namely, the choice of basis, system dimension required for sufficiently accurate reconstruction, sensor budget and their placement. The choice of design parameters implicitly determines the choice of algorithm as either l 2 minimization reconstruction or sparsity promoting l 1 minimization reconstruction. In this work, we systematically explore the implications of these design parameters on reconstruction accuracy so that practical recommendations can be identified. We observe that greedy-smart sensor placement, particularly interpolation points from the discrete empirical interpolation method (DEIM), provide the best balance of computational complexity and accurate reconstruction.
Highlights
Multiscale fluid flow phenomena in engineering and geophysical settings are invariably data-sparse, i.e., there are more scales to resolve than there are sensors
We systematically assess sparse reconstruction (SR) of fluid flows based on linear estimation principles using a chosen set of basis vectors
We characterize the accuracy of proper orthogonal decomposition (POD)-based linear estimation of the full state using system normalized error metrics and basis/sensor budget to shed light on the interplay between design parameters
Summary
Multiscale fluid flow phenomena in engineering and geophysical settings are invariably data-sparse, i.e., there are more scales to resolve than there are sensors. A major goal is to recover more information about the dynamical system through reconstruction of the higher dimensional state To expand on this view, in many practical fluid flow applications, accurate simulations may not be feasible for a multitude of reasons, including lack of accurate models, unknown governing equations or extremely complex boundary conditions. In such situations, measurement data represent the absolute truth and are often acquired from very few probes, and offering limited scope for analysis. With growth in computing power and generation of big data, Fluids 2019, 4, 109; doi:10.3390/fluids4020109 www.mdpi.com/journal/fluids
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