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Center for Uncertainty Quantification Multilevel Hybrid Chernoff Tau-Leap

Alvaro Moraes, Rá Ul Tempone, Pedro Vilanova

unpublished

Markovian pure jump processes can model many phenomena, e.g. chemical reactions at molecular level, protein transcription and translation , spread of epidemics diseases in small populations and in wireless communication networks, among many others. In this work [6] we present a novel multilevel algorithm for the Chernoff-based hybrid tau-leap algorithm. This variance reduction technique allows us to: (a) control the global exit probability of any simulated trajectory, (b) obtain accurate and
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... ain accurate and computable estimates for the expected value of any smooth observable of the process with minimal computational work. Statement of the problem Let X be a Pure Jump Process X = (X 1 ,. .. , X d) : [0, T ] × Ω → Z d + described by • Finite number of possible reactions, ν j ∈ Z d , x ∈ Z d + , x → x + ν j , j=1, ..., J • and propensity (jump intensity) functions, a j : R d → R + such that P X(t + dt) = x + ν j X(t) = x = a j (x)dt + o (dt) , (1) Typically, X k (t) is the population size at time t of the k − th species in the chemical kinetics jargon. Goal: accurately approximate the Quantity of Interest (QoI) E [g(X(T))], for some real observable g : R d → R. The process X can be characterized by X(t) = X(0) + J j=1 ν j Y j t 0 a j (X(s)) ds , where Y j : R + × Ω → Z + are indep. unit-rate Poisson processes [3]. Idea: Multilevel simulation of hybrid paths • Multivel Monte Carlo (MLMC). Use a control variate multilevel approach to reduce the computational work (runtime) of the standard Monte Carlo method. • Coupled Hybrid Paths. The MLMC method requires the generation of coupled hybrid paths. The exact method used in the MLMC hybrid algorithm is the Modified Next Reaction Method [2]. The modified next reaction method • This is an algorithm for simulating exact trajectories of the process X based on the random time change representation. • Uses explicitly the firing times of the involved independent Poisson process. • Advantages: (i) is faster than the SSA since we only need only one uniform r.v. at each step. (ii) We can sample in the cases where the rate functions depend on time and there are delayed reactions. • Finally, it is possible to simulate correlated (tau-leap, pure jump) trajectories, and also nested (tau-leap, tau-leap) trajectories. In [1] this technique is used to develop a uniform step MLMC algorithm. Coupling algorithm Couplig idea [1]. Let Y 1 , Y 2 be two independent unit-rate Poisson processes. Set Z 1 (t) = Y 1 (λ 1 t) + Y 2 (λ 2 t) Z 2 (t) = Y 1 (λ 1 t) where λ 1 , λ 2 ∈ R +. The idea is to use Y 1 to generate simultaneous jumps and Y 2 to model the extra jumps. Then, Z 1 and Z 2 are coupled homogeneous Poisson processes. By construction, Var [Z 1 (t) − Z 2 (t)] = Var [Y 2 (λ 2 t)] = λ 2 t. In the non-homogeneous case, where λ 1 = λ 1 (t), λ 2 = λ 2 (t), we have Z 1 (t) = Y 1 t 0 ˆ λ(s)ds + Y 2 t 0 λ 1 (s)− ˆ λ(s)ds Z 2 (t) = Y 1 t 0 ˆ λ(s)ds + Y 3 t 0 λ 2 (s)− ˆ λ(s)ds wherê λ(t) ≡ min{λ 1 (t), λ 2 (t)}. We observe that Z 1 (t) D = Y t 0 λ 1 (s)ds with Y a unit-rate Poisson process. Couplig of hybrid trajectories. The MLMC estimator requires the generation of [g − g −1 ](ω m). This is a functional of two coupled hybrid paths.

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