Importance sampling for a robust and efficient multilevel Monte Carlo estimator for stochastic reaction networks
- PDF / 1,690,566 Bytes
- 25 Pages / 595.276 x 790.866 pts Page_size
- 46 Downloads / 206 Views
Importance sampling for a robust and efficient multilevel Monte Carlo estimator for stochastic reaction networks Chiheb Ben Hammouda1
· Nadhir Ben Rached2 · Raúl Tempone1,3
Received: 19 November 2019 / Accepted: 11 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 highkurtosis 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 β = 1 for the standard case (without IS), to β = 1+δ, where 0 < δ < 1 is a user-selected parameter in our IS algorithm. Due to the complexity theorem of MLMC, −2 log(TOL)2 in the and given a pre-selected tolerance, TOL, this results in an improvement of the complexity from O TOL standard case to O TOL−2 , 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. Keywords Multilevel Monte Carlo · Continuous-time Markov chains · Stochastic reaction networks · Stochastic biological systems · Importance sampling Mathematics Subject Classification 60H35 · 60J27 · 60J75 · 92C40
1 Introduction
B
Chiheb Ben Hammouda [email protected] Nadhir Ben Rached [email protected] Raúl Tempone [email protected]
1
Computer, Electrical and Mathematical Sciences and Engineering Division (CEMSE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia
2
Chair of Mathematics for Uncertainty Quantification, RWTH Aachen University, 52072 Aachen, Germany
3
Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification, RWTH Aachen University, 52072 Aachen, Germany
In this work, we propose a novel importance sampling (IS) algorithm that can be combined with the multilevel Monte Carlo (MLMC) estimator to numerically solve stochastic differential equations (SDEs) driven by Poisson random measures (Li 2007; Çınlar 2011). We focus o
Data Loading...
