Importance sampling for a robust and efficient multilevel Monte Carlo estimator for stochastic reaction networks

Statistics and Computing, vol. 30, pp. 1665–1689

Abstract

The multilevel Monte Carlo (MLMC) method for continuous-time Markov chains, first introduced by Anderson and Higham (SIAM Multiscal Model Simul 10(1):146–179, 2012), is a highly efficient simulation technique that can be used to estimate various statistical quantities for stochastic reaction networks, in particular for stochastic biological systems. Unfortunately, the robustness and performance of the multilevel method can be affected by the high kurtosis, a phenomenon observed at the deep levels of MLMC, which leads to inaccurate estimates of the sample variance. In this work, we address cases where the high-kurtosis phenomenon is due to catastrophic coupling (characteristic of pure jump processes where coupled consecutive paths are identical in most of the simulations, while differences only appear in a tiny proportion) and introduce a pathwise-dependent importance sampling (IS) technique that improves the robustness and efficiency of the multilevel method. Our theoretical results, along with the conducted numerical experiments, demonstrate that our proposed method significantly reduces the kurtosis of the deep levels of MLMC, and also improves the strong convergence rate from $$\beta =1$$ for the standard case (without IS), to $$\beta =1+\delta $$ , where $$0<\delta <1$$ is a user-selected parameter in our IS algorithm. Due to the complexity theorem of MLMC, and given a pre-selected tolerance, $$\text {TOL}$$ , this results in an improvement of the complexity from $${\mathcal {O}}\left( \text {TOL}^{-2} \log (\text {TOL})^2\right) $$ in the standard case to $${\mathcal {O}}\left( \text {TOL}^{-2}\right) $$ , which is the optimal complexity of the MLMC estimator. We achieve all these improvements with a negligible additional cost since our IS algorithm is only applied a few times across each simulated path.

Authors 3

1. Chiheb Ben Hammouda corresponding

   King Abdullah University of Science and Technology

   Affiliation as printed

   Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia

   KING ABDULLAH UNIVERSITY OF SCIENCE AND TECHNOLOGY

2. Nadhir Ben Rached Aachen Chair of Mathematics for Uncertainty Quantification

   RWTH Aachen University

   Affiliation as printed

   Chair of Mathematics for Uncertainty Quantification, RWTH Aachen University, 52072, Aachen, Germany

   RWTH Aachen University

3. Raul F. Tempone Aachen

   RWTH Aachen University · King Abdullah University of Science and Technology

   Affiliation as printed

   Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification, RWTH Aachen University, 52072, Aachen, Germany

   Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia

   KING ABDULLAH UNIVERSITY OF SCIENCE AND TECHNOLOGY