Stabilized multilevel Monte Carlo method for stiff stochastic differential equations

Abstract

A multilevel Monte Carlo (MLMC) method for mean square stable stochastic differential equations with multiple scales is proposed. For such problems, that we call stiff, the performance of MLMC methods based on classical explicit methods deteriorates because of the time step restriction to resolve the fastest scales that prevents to exploit all the levels of the MLMC approach. We show that by switching to explicit stabilized stochastic methods and balancing the stabilization procedure simultaneously with the hierarchical sampling strategy of MLMC methods, the computational cost for stiff systems is significantly reduced, while keeping the computational algorithm fully explicit and easy to implement. Numerical experiments on linear and nonlinear stochastic differential equations and on a stochastic partial differential equation illustrate the performance of the stabilized MLMC method and corroborate our theoretical findings.

Improved Stabilized Multilevel Monte Carlo Method for Stiff Stochastic Differential Equations

“…For stiff problems as well as for nonstiff problems with no small noise, this means a significant improvement over the standard MLMC approach (which uses EM) as it is shown in [2].…”

“…In [2] a stabilized multilevel Monte Carlo method is introduced which uses as numerical integrator S-ROCK1 (2), an explicit Runge-Kutta method based on orthogonal Chebyshev polynomials. The stability constraint of this scheme is given by k −ℓ ρ cSR1s 2 ℓ ≤ 1, where s l is the number of stages at level ℓ and c SR1 a positive constant.…”

“…Complexity. The computational cost of E and E, the estimator of the stabilized MLMC of [2], can both be estimated as…”

“…A popular approach for such problems is the use of classical and improved Monte Carlo (MC) techniques, in particular the multilevel Monte Carlo (MLMC) method using Euler-Maruyama (EM) [10] as numerical integrator [7]. An explicit stabilized multilevel Monte Carlo method has been proved to be useful and efficient for stiff problems in a mean square sense and specially attractive for problems of large dimensions (as it avoids solving nonlinear systems typically arising with implicit methods) [2]. Here we present an improved version of the stabilized MLMC method by using a higher weak order scheme on the finest time grid.…”

Weak Second Order Multirevolution Composition Methods for Highly Oscillatory Stochastic Differential Equations with Additive or Multiplicative Noise

“…We now formulate the fully-discrete multi-revolution stochastic methods for the class of problems (1). It involves a micro stepsize h and a macro stepsize H where H ≫ ε is allowed.…”