Abstract
We apply POD-DEIM model order reduction to a 0D/1D model used to simulate the propagation of action potentials through the myocardium or along skeletal muscle fibers. This corresponding system of ODEs (reaction) and PDEs (diffusion) is called the monodomain equation. 0D sets of ODEs describing the ionic currents flowing across the cell membrane are coupled along muscle fibers through a 1D diffusion process for the transmembrane potential. Due to the strong coupling of the transmembrane potential and other state variables describing the behavior of the membrane, a total reduction strategy including all degrees of freedom turns out to be more efficient than a reduction of only the transmembrane potential. The total reduction approach is four orders of magnitude more accurate than partial reduction and shows a faster convergence in the number of POD modes with respect to the mesh refinement. A speedup of 2.7 is achieved for a 1D mesh with 320 nodes. Considering the DEIM approximation in combination with the total reduction, the nonlinear functions corresponding to the ionic state variables are also approximated in addition to the nonlinear ionic current in the monodomain equation. We observe that the same number of DEIM interpolation points as the number of POD modes is the optimal choice regarding stability, accuracy and runtime for the current POD-DEIM approach.
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Snapshots may be subsequent time steps or only a selection of time steps. In general, snapshots may be based on parameters independent of time. Therefore, we do not use parenthesis for the superscripts denoting the snapshots as we do for time step data.
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This research was funded by the Baden-Württemberg Stiftung as part of the DiHu project of the High Performance Computing II program and the Cluster of Excellence for Simulation Technology (EXC 310/1).
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Emamy, N., Litty, P., Klotz, T., Mehl, M., Röhrle, O. (2020). POD-DEIM Model Order Reduction for the Monodomain Reaction-Diffusion Sub-Model of the Neuro-Muscular System. In: Fehr, J., Haasdonk, B. (eds) IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018. IUTAM Bookseries, vol 36. Springer, Cham. https://doi.org/10.1007/978-3-030-21013-7_13
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