Reduced order optimal control of the convective FitzHugh–Nagumo equations

Abstract

We compare three different model order reduction techniques with Galerkin projection: the proper orthogonal decomposition (POD), POD-DEIM (discrete empirical interpolation) and POD-DMD (dynamic mode decomposition) for solving optimal control problems governed by the convective FitzHugh-Nagumo (FHN) equation. The convective FHN equation consists of the semilinear activator and the linear inhibitor equation, modelling blood coagulation in moving excitable media. The POD and POD-DEIM reduced optimal control problems are nonconvex due to the nonlinear activator equation. DMD is an equation-free, datadriven method which extracts dynamically relevant information content without explicitly knowing the dynamical operator. We use DMD as an alternative method to DEIM in order to approximate the nonlinear term in the convective FHN equation. Applying the POD-DMD Galerkin projection gives rise to a linear system of equations for the activator, and the optimal control problem becomes convex. We compare the accuracy and CPU times of three reduced order methods(ROM) with respect to the full order discontinuous Galerkin finite element solutions for convection dominated wave type solutions with terminal controls. Numerical results show that POD is the most accurate whereas POD-DMD is the fastest.

Multilinear POD-DEIM model reduction for 2D and 3D semilinear systems of differential equations

“…As a result, model order reduction techniques have been applied to dramatically reduce the dimension and the complexity of the resulting system of ODEs, see e.g., [9,53,26,44,25,27,30]. In particular, the common approach is to form a lexicographic ordering of the spatial nodes, unrolling the arrays (i.e., matrices or tensors when d = 2 and d = 3 respectively) of nodal values into long vectors in R N , i.e., the unknown vectors u i (t) ∈ R N .…”