Solving linear ODEs using Krylov subspace projection methods
Sebastian Ueckert
Dept Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden
Objectives: Within the field of pharmacometrics, linear ordinary differential equations (ODE) are used for pharmacokinetic models as well as for modeling of continuous time Markov-Models. The solution of these linear ODEs generally involves the numerical calculation of the matrix exponential. The objective of this work was to evaluate the performance of Krylov subspace projection methods for calculating the matrix exponential and compare it to existing methods.
Methods: A new solver for pharmacometric linear ODE models was implemented by combining a method for the calculation of derivatives of the matrix exponential [1] with a Krylov subspace solver from the EXPOKIT package [2] and linking it to NONMEM 7.3 [3]. The solver was evaluated by comparing the FOCE likelihood evaluation time for a three compartment model with linear absorption, 100 subjects and 15 observations per subject, with the evaluations times from the ADVAN5 and ADVAN6 solvers.
Results: For the evaluated example, the Krylov subspace-based solver was 3.6 times faster than the standard solver for linear ODE ADVAN5 (1.2 vs. 4.3 sec). Compared to the general ODE solver ADVAN6, the novel solver performed 12 times faster (1.2 vs. 12 sec).
Conclusions: Krylov subspace projection methods constitute a promising approach to significantly reduce the runtime of models implemented as linear ODE.
References:
[1] Fung, TC. 2004. “Computation of the Matrix Exponential and Its Derivatives by Scaling and Squaring.” International Journal for Numerical Methods in Engineering 59 (10). Wiley Online Library: 1273–86.
[2] Sidje, Roger B. 1998. “Expokit: A Software Package for Computing Matrix Exponentials.” ACM Transactions on Mathematical Software (TOMS) 24 (1). ACM: 130–56.
[3] Beal SL, Boeckmann AJ, Sheiner LB, and Bauer RJ. 1989–2013. NONMEM 7.3.0 Users Guides. ICON Development Solutions, Hanover, MD.