Numerical comparison of conservation-based state transformations

Determine whether different choices of the transformation matrix \(\mathbf{T}\) require fewer solver steps for the Wang et al. rapid delayed rectifier potassium-current (\(I_{\mathrm{Kr}}\)) model and other example Markov models.

Background

The paper derives a family of conservation-preserving linear transformations $\mathbf{T}$ that reduce an $N$-state Markov ODE system to an $(N-1)$-state system. Different choices can eliminate different state variables or linear combinations of states while retaining the conserved total occupancy and the nonzero eigenvalues of the original system.

Numerical experiments with the Wang et al. $I_{\mathrm{Kr}}$ model indicate that eliminating any one of its five state variables gives broadly similar computational performance. The paper leaves unresolved whether more general choices of $\mathbf{T}$ can reduce the number of solver steps, either for that model or for other models, particularly in settings with widely differing transition timescales.