“…The most significant prior research on adaptive timestepping in MLMC has been by Hoel, von Schwerin, Szepessy and Tempone [9] and [10]. In their research, they construct a multilevel adaptive timestepping discretisation in which the timesteps used on level are a subdivision of those used on level −1, which in turn are a subdivision of those on level −2, and so on.…”
Section: In the Particular Case In Which |E[p ]−E[p] | ∝mentioning
This paper shows that it is relatively easy to incorporate adaptive timesteps into multilevel Monte Carlo simulations without violating the telescoping sum on which multilevel Monte Carlo is based. The numerical approach is presented for both SDEs and continuous-time Markov processes. Numerical experiments are given for each, with the full code available for those who are interested in seeing the implementation details.Keywords multilevel Monte Carlo · adaptive timestep · SDE · continuous-time
Markov process
Multilevel Monte Carlo and Adaptive SimulationsMultilevel Monte Carlo methods [4,6,8] are a very simple and general approach to improving the computational efficiency of a wide range of Monte Carlo applications. Given a set of approximation levels = 0, 1, . . . , L giving a sequence of approximations P of a stochastic output P, with the cost and accuracy both increasing as increases, then a trivial telescoping sum givesexpressing the expected value on the finest level as the expected value on the coarsest level of approximation plus a sum of expected corrections.
“…The most significant prior research on adaptive timestepping in MLMC has been by Hoel, von Schwerin, Szepessy and Tempone [9] and [10]. In their research, they construct a multilevel adaptive timestepping discretisation in which the timesteps used on level are a subdivision of those used on level −1, which in turn are a subdivision of those on level −2, and so on.…”
Section: In the Particular Case In Which |E[p ]−E[p] | ∝mentioning
This paper shows that it is relatively easy to incorporate adaptive timesteps into multilevel Monte Carlo simulations without violating the telescoping sum on which multilevel Monte Carlo is based. The numerical approach is presented for both SDEs and continuous-time Markov processes. Numerical experiments are given for each, with the full code available for those who are interested in seeing the implementation details.Keywords multilevel Monte Carlo · adaptive timestep · SDE · continuous-time
Markov process
Multilevel Monte Carlo and Adaptive SimulationsMultilevel Monte Carlo methods [4,6,8] are a very simple and general approach to improving the computational efficiency of a wide range of Monte Carlo applications. Given a set of approximation levels = 0, 1, . . . , L giving a sequence of approximations P of a stochastic output P, with the cost and accuracy both increasing as increases, then a trivial telescoping sum givesexpressing the expected value on the finest level as the expected value on the coarsest level of approximation plus a sum of expected corrections.
“…] is in this setting based on constructing numerical realizations X (t) on stochastic adaptively refined meshes ∆t { } so that the 17) are asymptotically fulfilled, and by determining the number of samples M 0 to ensure that the statistical error…”
Section: Stochastic Time Steppingmentioning
confidence: 99%
“…[26]. The idea of extending the MLMC method [11] to hierarchies of adaptively refined, non uniform time discretizations that are generated by the adaptive algorithm introduced in [26,25,8] was first introduced and tested computationally by the authors in [17].…”
Section: Introductionmentioning
confidence: 99%
“…Later, the work [17] presented a MLMC method using a hierarchy of adaptively refined, non uniform time discretizations that are generated by the adaptive algorithm introduced in [26,25,8], and, as such, it may be considered a generalization of Giles' work [11]. This work improves the algorithms presented in [17] and furthermore, it provides mathematical analysis of these new adaptive MLMC algorithms. In particular, we show that under some assumptions our adaptive MLMC algorithms are asymptotically accurate and essentially have the correct complexity but with improved control of the complexity constant factor in the asymptotic analysis.…”
Abstract. We present an adaptive multilevel Monte Carlo (MLMC) method for weak approximations of solutions to Itô stochastic differential equations (SDE). The work [11] proposed and analyzed a MLMC method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a standard, single level, forward Euler Monte Carlo method from O TOL −3 to O (TOL −1 log(TOL −1 )) 2 for a mean square error of O TOL 2 . Later, the work [17] presented a MLMC method using a hierarchy of adaptively refined, non uniform time discretizations that are generated by the adaptive algorithm introduced in [26,25,8], and, as such, it may be considered a generalization of Giles' work [11]. This work improves the algorithms presented in [17] and furthermore, it provides mathematical analysis of these new adaptive MLMC algorithms. In particular, we show that under some assumptions our adaptive MLMC algorithms are asymptotically accurate and essentially have the correct complexity but with improved control of the complexity constant factor in the asymptotic analysis. Numerical tests include one case with singular drift and one with stopped diffusion, where the complexity of a uniform single level method is O TOL −4 . For both these cases the results confirm the theory, exhibiting savings in the computational cost for achieving the accuracy O (TOL) from O TOL −3 for the adaptive single level algorithm to essentially O TOL −2 log TOL −1 2 for the adaptive MLMC.
“…To mitigate the known slow convergence of Monte Carlo methods, multilevel Monte Carlo methods have been recently developed; see, e.g., [2,3,4,5,6,7,8]. A hierarchy of nested finite element subspaces…”
Abstract. We present a multilevel stochastic collocation method that, as do multilevel Monte Carlo methods, uses a hierarchy of spatial approximations to reduce the overall computational complexity when solving partial differential equations with random inputs. For approximation in parameter space, a hierarchy of multi-dimensional interpolants of increasing fidelity are used. Rigorous convergence and computational cost estimates for the new multilevel stochastic collocation method are derived and used to demonstrate its advantages compared to standard single-level stochastic collocation approximations as well as multilevel Monte Carlo methods.
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