Higher order optimization and adaptive numerical solution for optimal control of monodomain equations in cardiac electrophysiology. (English) Zbl 1201.92026

Nagaiah, Chamakuri; Kunisch, Karl

Appl. Numer. Math. 61, No. 1, 53-65 (2011).

Summary: Adaptive and high resolution numerical discretization techniques are demonstrated for solving optimal control of the monodomain equations in cardiac electrophysiology. A monodomain model, which is a well established model for describing the wave propagation of the action potential in the cardiac tissue, will be employed for the numerical experiments. The optimal control problem is considered as a PDE constrained optimization problem. We present an optimal control formulation for the monodomain equations with an extra-cellular current as the control variable which must be determined in such a way that excitations of the transmembrane voltage are damped in an optimal manner.
The focus of this work is on the development and implementation of an efficient numerical technique to solve an optimal control problem related to a reaction-diffusions system arising in cardiac electrophysiology. Specifically a Newton-type method for the monodomain model is developed. The numerical treatment is enhanced by using a second order time stepping method and adaptive grid refinement techniques. The numerical results clearly show that super-linear convergence is achieved in practice.

MSC:

92C30	Physiology (general)
92C05	Biophysics
49N90	Applications of optimal control and differential games
49J30	Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
35K57	Reaction-diffusion equations
65K10	Numerical optimization and variational techniques

Keywords:

reaction-diffusion equations; monodomain model; PDE constraint optimization; adaptive FEM; Newton’s optimization algorithm

Software:

DUNE

References:

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