Numerical study of POD-Galerkin-DEIM reduced order modeling of cardiac monodomain formulation

The three-dimensional cardiac monodomain model with inhomogeneous and anisotropic conductivity characterizes a complicated system that contains spatial and temporal approximation coefficients along with a nonlinear ionic current term. These complexities make its numerical modeling computationally challenging, and therefore, the formation of an efficient computational approximation is important for studying cardiac propagation. In this paper, a reduced order modeling approach has been developed for the simplified cardiac monodomain model, which yields a significant reduction of the full order dynamics of the cardiac tissue, reducing the required computational resources. Additionally, the discrete empirical interpolation technique has been implemented to accurately estimate the nonlinearity of the ionic current of the cardiac monodomain scheme. The proper orthogonal decomposition technique has been utilized, which transforms a given dataset called 'snapshots' to a new coordinate system. The snapshots are computed first from the original system, and they encapsulate all the information observed over both time and parameter variations. Next, the proper orthogonal decomposition provides a reduced order basis for projecting the original solution onto a low-dimensional orthonormal subspace. Finally, a reduced set of unknowns of the forward problem is obtained for which the solution involves significant computational savings compared to that for the original system of unknowns. The efficiency of the model order reduction technique for finite difference solution of cardiac electrophysiology is examined concerning simulation time, error potential, activation time, maximum temporal derivative, and conduction velocity. Numerical results for the monodomain show that its solution time can be reduced by a significant factor, with only 0.474 mV RMS error between the full order and reduced dimensions solution.