Lee Ricketson

L. F. Ricketson

We generalize the multilevel Monte Carlo (MLMC) method of Giles to the simulation of systems of particles that interact via a mean field. When the number of particles is large, these systems are described by a McKean-Vlasov process - a stochastic differential equation (SDE) whose coefficients depend on expectations of the solution as well as pathwise data. In contrast to standard MLMC, the new method uses mean field estimates at coarse levels to inform the fine level computations. Using techniques from the theory of propagation of chaos, we prove convergence and complexity results for the algorithm in a special case. We find that the new method achieves $L^1$ error of size $\varepsilon$ with $O(\varepsilon^{-2} (\log \varepsilon)^5)$ complexity, in contrast to the $O(\varepsilon^{-3})$ complexity of standard methods. We also prove a variance scaling result that strongly suggests similar performance of the algorithm in a more general context. We present numerical examples from applications and observe the expected behavior in each case.

L. F. Ricketson, A. J. Cerfon, M. Rachh et al.

We present a numerical scheme that can be combined with any fixed boundary finite element based Poisson or Grad-Shafranov solver to compute the first and second partial derivatives of the solution to these equations with the same order of convergence as the solution itself. At the heart of our scheme is an efficient and accurate computation of the Dirichlet to Neumann map through the evaluation of a singular volume integral and the solution to a Fredholm integral equation of the second kind. Our numerical method is particularly useful for magnetic confinement fusion simulations, since it allows the evaluation of quantities such as the magnetic field, the parallel current density and the magnetic curvature with much higher accuracy than has been previously feasible on the affordable coarse grids that are usually implemented.