Higher order optimization and adaptive numerical solution for optimal control of monodomain equations in cardiac electrophysiology

Authors:
Chamakuri Nagaiah;Karl Kunisch
Affiliations:
Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstr. 36, Graz, A-8010, Austria;Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstr. 36, Graz, A-8010, Austria
Venue:
Applied Numerical Mathematics
Year:
2011

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Abstract

In this work 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.