Exploiting Sparsity and Equation-Free Architectures in Complex Systems

December 10, 2014

Abstract: 

We argue that data-driven dimensionality reduction methods integrate naturally with sparse sensing in the context of complex systems. This framework works equally well with a physical model or in an equation-free context, where data is available but the governing equations may be unknown. We demonstrate the advantages of combining these methods on three prototypical examples: classification of dynamical regimes, optimal sensor placement, and equation-free dynamic model reduction. These examples motivate the potentially transformative role that state-of-the-art data methods and machine learning can play in the analysis of complex systems.

Background Model reduction methods such as Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) are often grouped together as data-driven methods for analyzing complex dynamical systems. Compressive sensing and sparse approximation have focused primarily on the task of either reconstruction or representation. However, these methods have a significant amount of overlap. In this section, these methods are reviewed to familiarize the reader with the concepts, formulation, and challenges of these methods. The section begins with a review of the singular value decomposition (SVD), which is an essential tool to POD, DMD, and dimensionality reduction in general.

The model reduction methods described in this review rely heavily on the properties of the SVD. The SVD is a matrix factorization that can be used to construct optimal, low-rank approximations

This figure is an illustration of the framework to connect the concepts of sparsity and model reduction in complex systems. The top box describes a typical SVD–based model reduction approach that builds a basis in which a sparse representation exists for the data. The bottom box contains sketches of the three approaches to exploit sparsity and an equation-free architecture in characterizing dynamic regimes of a complex system.

Conclusions Data fluency is becoming increasingly important in modern science. Indeed, as data from simulations and experiments grow in size and complexity, the fourth-paradigm of data-driven discovery proposed by Jim Gray will become essential for making acquisition and analysis more tractable. We show that there are significant opportunities to combine the low-dimensional tailored bases from the SVD with compressive sampling to reduce the burden of data collection, while still providing high-fidelity information about the high-dimensional system.