Model Order Reduction for Sine-Gordon Equation Using POD and DEIM

Norapon Sukuntee, Saifon Chaturantabut

Abstract

This work applies model reduction techniques for efficiently approximating the solution of the sin-Gordon equation. The proper orthogonal decomposition (POD) is employed to construct a low dimensional basis that can accurately capture the dynamics of the solution space. This POD basis is used with the Galerkin projection to obtain a reduced-order model that is much smaller than the original discretized system. However, the effective dimension reduction of the POD-Galerkin approach is limited to the linearity of the system. The discrete empirical interpolation method (DEIM) is then applied to further reduce the computational complexity of the nonlinear term. This work investigates the effect of using different amount of snapshots from coarse discretization in constructing the POD basis, as well as demonstrates the applicability of the POD-DEIM approach on predicting solution of the parametrized sine-Gordon equation. The POD-DEIM solutions are shown to be accurate for both numerical tests and can be solved with much less computational time and memory storage when compared to the high-dimensional discretized sine-Gordon equation.