SIAM Journal on Numerical Analysis
Computational Differential Equations
Computational Differential Equations
Multi-Adaptive Galerkin Methods for ODEs I
SIAM Journal on Scientific Computing
Multi-Adaptive Galerkin Methods for ODEs II: implementation and Applications
SIAM Journal on Scientific Computing
Proceedings of the 2006 ACM SIGPLAN symposium on Partial evaluation and semantics-based program manipulation
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Simulating the human heart is a challenging computational problem, with simulations being very time-consuming, to the extent that some can take days to compute even on high performance computing resources. There is considerable interest in computational optimization techniques, with a view to making whole-heart simulations tractable. A common example is the use of lookup tables to approximate computationally intensive expressions in cardiac electrophysiology cell models, which are given mathematically as systems of ordinary differential equations. The approximation error introduced through the use of lookup tables may not be checked thoroughly, and when models are included within large simulation codes any error could go unnoticed; this may impact the reliability of results obtained from the simulation. We thus present a mathematical analysis of the error, using a posteriori error analysis, to determine whether it will remain within acceptable limits even when accumulated over the course of a simulation. This analysis has been implemented and can be applied automatically to models described in CellML, an XML-based language for mathematical models of biology. We demonstrate the analysis on a sample of models. A key feature of the approach is that it can be used to obtain bounds on quantities of physiological interest computed from simulation results. We also discuss how the error bounds obtained may be used to choose an appropriate granularity for the lookup tables.