IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

Model reduction for nonlinear dynamical systems: discrete optimality and time parallelism

  • Oct. 17, 2016
  • 1:15 p.m.
  • LeConte 312


Large-scale models of nonlinear dynamical systems arise in applications ranging from compressible fluid dynamics to structural dynamics. Due to the large computational cost incurred by these models, it is impractical to use them in time-critical scenarios such as control, design, and uncertainty quantification (UQ). Model reduction aims to mitigate this computational burden. To date, reduced-order models (ROMs) for relatively simple models (e.g., linear-time-invariant systems; elliptic, parabolic, and linear hyperbolic PDEs) have been widely adopted, as researchers have developed methods that are accurate, reliable, and certified. In contrast, model reduction for nonlinear dynamical systems lacks these assurances and thus remains in its infancy; the most common method---POD--Galerkin---is often unstable.

This talk will describe several advances that have made nonlinear model reduction viable for a new frontier of problems. However, doing so has required fundamentally new data-driven approaches, as the critical tools leveraged for simpler models (e.g., Gramians, coercivity constants) are no longer available. First, I will introduce the notion of least-squares Petrov–Galerkin (LSPG) projection---and the associated GNAT method---which enables accuracy via discrete optimality: it performs optimal projection after the dynamical system has been discretized in time. Comparative theoretical and numerical studies will highlight the benefits of LSPG projection over Galerkin projection.

Second, I will introduce a new approach for data-driven time parallelism. Because the GNAT model incurs a small computational footprint, parallelizing the computation in the spatial domain quickly saturates; this limits the realizable wall-time speedup for the GNAT ROM. To address this, we introduce a new method for parallelizing the simulation in time. The technique relies on a coarse propagator that ensures rapid convergence by leveraging previously available time-domain data.

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