Zusammenfassung
We apply the POD-DEIM model order reduction to the propagation of the
transmembrane potential along \$1\$D muscle fibers. This propagation is
represented using the monodomain partial differential equation. The monodomain
equation, which is a reaction-diffusion equation, is coupled through its
reaction term with a set of ordinary differential equations, which provide the
ionic current across the cell membrane. Due to the strong coupling of the
transmembrane potential and ionic state variables, we reduce them all together
proposing a total reduction strategy. We compare the current strategy with the
conventional strategy of reducing the transmembrane potential. Considering the
current approach, the discrete system matrix is slightly modified to adjust for
the size. However, size of the precomputed reduced system matrix remains the
same, which means the same computational cost. The current approach appears to
be four orders of magnitude more accurate considering the equivalent number of
modes on the same grid in comparison to the conventional approach. Moreover, it
shows a faster convergence in the number of POD modes with respect to the grid
refinement. Using the DEIM approximation of nonlinear functions in combination
with the total reduction, the nonlinear functions corresponding to the ionic
state variables are also approximated besides the nonlinear ionic current in
the monodomain equation. For the current POD-DEIM approach, it appears that the
same number of DEIM interpolation points as the number of POD modes is the
optimal choice regarding stability, accuracy and runtime.
Nutzer
