Reduced order optimal control of the convective FitzHugh-Nagumo equations. (English) Zbl 1448.49038
Summary: In this paper, we compare three model order reduction methods: the proper orthogonal decomposition (POD), discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD) for the optimal control of the convective FitzHugh-Nagumo (FHN) equations. The convective FHN equations consist of the semi-linear activator and the linear inhibitor equations, modeling blood coagulation in moving excitable media. The semilinear activator equation leads to a non-convex optimal control problem (OCP). The most commonly used method in reduced optimal control is POD. We use DEIM and DMD to approximate efficiently the nonlinear terms in reduced order models. We compare the accuracy and computational times of three reduced-order optimal control solutions with the full order discontinuous Galerkin finite element solution of the convection dominated FHN equations with terminal controls. Numerical results show that POD is the most accurate whereas POD-DMD is the fastest.
MSC:
| 49M41 | PDE constrained optimization (numerical aspects) |
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 49M05 | Numerical methods based on necessary conditions |
| 49M25 | Discrete approximations in optimal control |
| 65K10 | Numerical optimization and variational techniques |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
FitzHugh-Nagumo equation; optimal control; discontinuous Galerkin method; proper orthogonal decomposition; discrete empirical interpolation; dynamic mode decompositionSoftware:
CG_DESCENTReferences:
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