Numerical Analysis of a Finite Element Method for an Optimal Control of Bidomain-bath Model

Abstract : This work is concerned with the study of the convergence analysis for an optimal control of bidomain-bath model by using the finite element scheme. The bidomain-bath model equations describe the cardiac bioelectric activity at the tissue and bath volumes where the control acts at the boundary of the tissue domain. We establish the existence of the finite element scheme, and convergence of the unique weak solution of the direct bidomain-bath model. The convergence proof is based on deriving a series of a priori estimates and using a general L 2-compactness criterion. Moreover, the well-posedness of the adjoint problem and the first order necessary optimality conditions are shown. Comparing to the direct problem, the convergence proof of the adjoint problem is based on using a general L 1-compactness criterion. The numerical tests are demonstrated which achieve the successful cardiac defibrillation by utilizing less total current. Finally, the robustness of the Newton optimization algorithm is presented for different finer mesh geometries. 1. Introduction. The electrical behavior of the cardiac tissue surrounded by a nonconductive bath is described by a coupled partial and ordinary differential equations which are so called bidomain model equations [17, 22, 24]. The bidomain model equations consist of two parabolic partial differential equations (PDEs) which describe the dynamics of the intra and the extracellular potentials. The PDEs coupled with an ordinary differential equations which model the ionic currents associated with the reaction terms. Furthermore , an additional Poisson problem has to be solved when the cardiac tissue is immersed in a conductive fluid, e.g. tissue bath in an experimental context or a surrounding torso to model in vivo scenarios.
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Journal of Differential Equations, Elsevier, 2017, 263 (5), pp.2419-2456
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Mostafa Bendahmane, Nagaiah Chamakuri. Numerical Analysis of a Finite Element Method for an Optimal Control of Bidomain-bath Model. Journal of Differential Equations, Elsevier, 2017, 263 (5), pp.2419-2456. 〈hal-01259773〉

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