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Colgin, Zane
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Middle Tennessee State University
In this work we implement various techniques to improve the multilevel Monte Carlo (MLMC) method as it is applied to a variety of stochastic models. In each case we were able to show a quantifiable computational benefit.
First we explore the use of a parallel antithetic MLMC algorithm to simulate systems of stochastic differential equations (SDEs) with correlated noise. Since Le ́vy area approxima- tion is unnecessary with antithetic MLMC, it is a natural choice for the solution of systems with non-diagonal, non-commutative noise. The Milstein method used with antithetic MLMC provides a computation complexity of O(ε^−2) root-mean-square error. Furthermore, MLMC uses independent sampling, which is naturally suited for parallel algorithms. We display the advantages of this approach with a case study in stochastic pricing models.
Secondly, we analyze the effects of stiffness on the convergence rate to the solution of a system of SDEs. Similarly to their deterministic counterparts, stochastic differential solvers can be unstable when used with a stiff system. When unstable step sizes are taken on the lower levels of MLMC, convergence is not guaranteed. We examine two approaches to remedy this problem: 1) the use of a semi-implicit method with a larger step-size stability region and 2) simply using a more fine discretization as the initial level for the MLMC simulator. We apply this approach to a case study in biochemical reaction networks.
Lastly, we improve a recently developed MLMC algorithm, which uses an iterative solver for the solution a partial differential equation (PDE) with random input. The innovation of the original algorithm is that each sample utilizes data gathered from all previously computed samples to create a better initial guess for the iterative solver. The drawback of this method is that the computation of a sample is no longer independent in a computational sense. We use a K-dimensional tree to sort the random input initially so that groups of locally distributed samples may be computed in batches at each parallel computing node.
Chemical Langevin Equation, Multilevel Monte Carlo, Stiff Systems, Stochastic Differential Equati, Stochastic PDE, Stochastic Systems