ST-GNAT and SNS: Model order reduction techniques for time-dependent nonlinear system of equations

Youngsoo Choi, Lawrence Livermore National Laboratory
September 13th, 2018 at 11:00AM–12:00PM in Evans 732

Several reduced order models have been successfully developed for nonlinear dynamical systems. To achieve a considerable speedup, a hyper-reduction step is needed to reduce the computational complexity due to nonlinear terms. A new space–time reduced order model, the ST-GNAT method, for nonlinear dynamical systems will be introduced as well as the traditional methods, such as the DEIM and GNAT methods. The ST-GNAT method applies a space–time least-squares Petrov–Galerkin projection and space–time gappy POD approach to reduce both the dimensionality and complexity of the system. An attractive error bound associated with the ST-GNAT method and several compelling numerical results will be shown. One drawback of the ST-GNAT method is the computationally expensive offline phase where solution and nonlinear term bases as well as corresponding sample elements are constructed. To reduce the offline cost, the SNS method is developed. In contrast to the traditional hyper-reduction techniques where collection of nonlinear term snapshots is required, the SNS method completely avoids the use of the nonlinear term snapshots. Instead, it uses the solution snapshots that are used for building a solution basis. Furthermore, it avoids an extra data compression of nonlinear term snapshots. As a result, the SNS method provides a more efficient offline strategy than the traditional model order reduction techniques, such as the DEIM, GNAT, and ST-GNAT methods. Numerical results support that the accuracy of the solution from the SNS method is comparable to the traditional methods.