Stability analysis of the POD reduced order method for solving the bidomain model in cardiac electrophysiology. (English) Zbl 1332.92013
Summary: In this paper we show the numerical stability of the proper orthogonal decomposition (POD) reduced order method used in cardiac electrophysiology applications. The difficulty of proving the stability comes from the fact that we are interested in the bidomain model, which is a system of degenerate parabolic equations coupled to a system of ODEs representing the cell membrane electrical activity. The proof of the stability of this method is based on a priori estimates controlling the gap between the reduced order solution and the Galerkin finite element one. We present some numerical simulations confirming the theoretical results. We also combine the POD method with a time splitting scheme allowing a faster solving of the bidomain problem and show numerical results. Finally, we conduct numerical simulation in 2D illustrating the stability of the POD method in its sensitivity to the ionic model parameters. We also perform 3D simulation using a massively parallel code. We show the computational gain using the POD reduced order model. We also show that this method has a better scalability than the full finite element method.
MSC:
| 92C30 | Physiology (general) |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35B45 | A priori estimates in context of PDEs |
| 35K57 | Reaction-diffusion equations |
| 35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 49M27 | Decomposition methods |
Keywords:
bidomain equation; reduced order method; proper orthogonal decomposition; stability analysis; a priori estimates; Mitchell-Schaeffer modelSoftware:
ChasteReferences:
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