Low-cost singular value decomposition with optimal sensor placement

This paper presents a novel mathematical methodology capable of reconstructing complex data, using a modal decomposition-based data assimilation approach, with great precision and very low computational cost. The method presented in this work consists of a new variant of the singular value decomposition (SVD) algorithm, which has been named low-cost SVD (lcSVD) (The website of the software is available at https://modelflows.github.io/modelflowsapp/.). The lcSVD algorithm has been applied to a diverse set of test cases, consisting of two- and three-dimensional, numerical and experimental, laminar and turbulent fluid dynamics data, which have been selected based on their high complexity to prove the high reconstruction accuracy, robustness, and computational resource optimization of this method. The method is applied to under-resolved data, which can be equidistant in space, or reduced to the most relevant data points using the optimal sensor variant of this methodology named OS-lcSVD. A comparison between SVD, lcSVD, and OS-lcSVD is presented in this work, showing that lcSVD is capable of reconstructing under-resolved data 630 times faster than SVD, saving 37% of memory. This novel algorithm is proposed as an alternative to current parallelization strategies, but it can also be combined with these to reduce memory consumption and computational cost even further, allowing for working with big data and industrial datasets.