Time-critical applications for systems governed by dynamical systems—such as control, fast-turnaround design, and uncertainty quantification—often demand the accuracy provided by large-scale computational models, but cannot afford their computational cost. To mitigate this bottleneck, researchers have developed model-reduction techniques that decrease the dimension of the dynamical system while preserving its key features. Such methods are effective when applied to specialized problems such as linear time-invariant systems (e.g., balanced truncation). However, model reduction for nonlinear dynamical systems has been primarily limited to methods based on the proper orthogonal decomposition (POD)–-Galerkin approach, which lacks `discrete optimality' and leads to unstable responses in many cases.
In this talk, I will present the Gauss–Newton with approximated tensors (GNAT) nonlinear model-reduction method. This method is discrete optimal, is equipped with an error bound, and leads to highly accurate responses for practical problems across a wide range of physics. I will also describe the `sample mesh' concept, which enables a practical, distributed, computationally efficient implementation of GNAT in computational-mechanics codes. Finally, I will present results for the method applied to a validated CFD model (with over 17 million unknowns) of a compressible, turbulent flow problem. Results illustrate GNAT’s favorable performance compared with other model-reduction techniques; it achieves speedups exceeding 350 with errors below 1%.
Joint work with Charbel Farhat, Julien Cortial, and David Amsallem.