[Paper Review] Efficient adaptivity for simulating cardiac electrophysiology with spectral deferred correction methods

Fatemeh Chegini, Thomas Steinke+1 more|arXiv (Cornell University)|Nov 13, 2023

Cardiac electrophysiology and arrhythmias4 citations

TL;DR

This paper proposes algebraic adaptivity in spectral deferred correction (SDC) methods for cardiac electrophysiology simulations, using progressive subset selection of degrees of freedom (dofs) in later SDC sweeps to reduce computational cost. By exploiting the localized nature of solution features and leveraging a posteriori error estimates, the method achieves 2–4× speedups in monodomain and EMI models with negligible overhead compared to mesh-based adaptivity.

ABSTRACT

The locality of solution features in cardiac electrophysiology simulations calls for adaptive methods. Due to the overhead incurred by established mesh refinement and coarsening, however, such approaches failed in accelerating the computations. Here we investigate a different route to spatial adaptivity that is based on nested subset selection for algebraic degrees of freedom in spectral deferred correction methods. This combination of algebraic adaptivity and iterative solvers for higher order collocation time stepping realizes a multirate integration with minimal overhead. This leads to moderate but significant speedups in both monodomain and cell-by-cell models of cardiac excitation, as demonstrated at four numerical examples.

Motivation & Objective

• Address the high computational cost of high-resolution cardiac electrophysiology simulations, particularly in monodomain and EMI models.
• Overcome the inefficiency of traditional mesh-based spatial adaptivity due to high overhead from mesh refinement, coarsening, and matrix assembly.
• Develop a low-overhead, algebraic adaptivity strategy that operates on the linear system level without mesh modifications.
• Enable efficient multirate integration by dynamically restricting dofs to regions of active excitation fronts during SDC iterations.
• Demonstrate that algebraic adaptivity can significantly reduce simulation time while maintaining accuracy and tolerance requirements.

Proposed method

• Apply spectral deferred correction (SDC) methods with implicit-explicit (IMEX) time integration for high-order accuracy in solving the monodomain and EMI models.
• Use a posteriori error estimation based on the linear convergence of SDC to identify and retain only the most significant degrees of freedom (dofs) in later sweeps.
• Implement progressive subset selection: the first SDC sweep uses all dofs, while subsequent sweeps restrict to dofs with significant correction contributions, reducing system size.
• Interleave SDC iterations with adaptive dofs selection to maintain accuracy while minimizing computational effort in regions with low solution activity.
• Utilize a drop tolerance criterion to discard dofs with negligible correction impact, ensuring the method remains efficient and stable.
• Apply the method to both 2D and 3D monodomain and EMI models, comparing adaptive and non-adaptive SDC baselines in terms of wall clock time and accuracy.

Experimental results

Research questions

• RQ1 Can algebraic adaptivity based on SDC correction localization reduce computational cost in cardiac electrophysiology simulations without introducing significant overhead?
• RQ2 How effective is progressive dofs subset selection in later SDC sweeps in reducing system size and simulation time?
• RQ3 What speedup can be achieved with this approach compared to non-adaptive SDC in monodomain and EMI models?
• RQ4 How does the method perform across different problem sizes and excitation front dynamics, particularly in 3D simulations?
• RQ5 To what extent does the method’s efficiency depend on the number of SDC sweeps and the choice of drop tolerance?

Key findings

• The proposed algebraic adaptivity achieves 2.13× speedup in the 2D monodomain model and 3.29× in the 3D monodomain model, with even higher gains in EMI models (3.336× and 4.344× respectively).
• In the 3D monodomain example, the number of active dofs dropped from 646,166 in the first sweep to 425 in the third sweep, indicating strong localization of corrections to the excitation front.
• The method maintains accuracy within the specified tolerance (TOL = 1e-4) while reducing the number of dofs by up to 99.9% in later SDC sweeps.
• The overhead from error estimation and dofs selection is negligible, allowing direct translation of reduced system size into simulation time savings.
• Speedup increases with larger domains and smaller excitation front fractions, as the spatial support of significant corrections becomes more localized.
• The method is most effective when the number of SDC sweeps is moderate (3–10), and future improvements could include higher-order collocation or heuristic first-sweep dofs selection.