Non-nested Adaptive Timesteps in Multilevel Monte Carlo Computations

Abstract

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.

Adaptive Euler–Maruyama method for SDEs with nonglobally Lipschitz drift

“…Note that if another timestep function h δ (x) is smaller than h(x), then h δ (x) also satisfies the Assumption 2. Note also that the form of (7), which is motivated by the requirements of the proof of the next theorem, is very similar to (5). Indeed, if ( 7) is satisfied then ( 5) is also true for the same values of α and β .…”

“…For the ergodic SDEs, by setting a suitable condition for h, we can show that, instead of an exponential bound, the numerical solution has a uniform bound with respect to T for both moments and the strong error. Then, multi-level Monte Carlo (MLMC) methodology [5,6] is employed and non-nested timestepping is used to construct an adaptive MLMC [7]. Following the idea of Glynn and Rhee [8] to estimate the invariant measure of some Markov chains, we introduce an adaptive MLMC algorithm for the infinite time interval, in which each level has a different time interval length T , to achieve a better computational performance.…”

Multilevel Monte Carlo and improved timestepping methods in atmospheric dispersion modelling

“…Various methods for adaptive timestepping in the MLMC method have been proposed in the literature [25,36,37,38]. Instead of using a fixed timestep h for all timesteps, in an adaptive algorithm the timestep size is adjusted at every time t n .…”

“…When using adaptivity within MLMC, the coarse and fine timesteps are not necessarily nested. However, it is easy to adapt the MLMC algorithm to account for this [25]. For this, the time interval [0, T ] is divided into a number of intervals [τ j , τ j+1 ] such that τ 0 = 0, τ N = T and each of the points τ j is either a fine or a coarse time point (see Fig.…”

“…In this work, we focus on stability issues and therefore employ a simple heuristic based on the local velocity decorrelation time. The implementation of the adaptive MLMC algorithm that we use is described in [25]. We find that adaptive timestepping can further reduce the cost for the symplectic Euler integrator.…”