Multilevel Monte Carlo simulation of Coulomb collisions

Abstract

We present a new, for plasma physics, highly efficient multilevel Monte Carlo numerical method for simulating Coulomb collisions. The method separates and optimally minimizes the finite-timestep and finite-sampling errors inherent in the Langevin representation of the Landau-Fokker-Planck equation. It does so by combining multiple solutions to the underlying equations with varying numbers of timesteps. For a desired level of accuracy ε, the computational cost of the method is O(ε −2 ) or O(ε −2 (ln ε) 2 ), depending on the underlying discretization, Milstein or Euler-Maruyama respectively. This is to be contrasted with a cost of O(ε −3 ) for direct simulation Monte Carlo or binary collision methods. We successfully demonstrate the method with a classic beam diffusion test case in 2D, making use of the Lévy area approximation for the correlated Milstein cross terms, and generating a computational saving of a factor of 100 for ε = 10 −5 . We discuss the importance of the method for problems in which collisions constitute the computational rate limiting step, and its limitations.

Anisotropic Radio-wave Scattering and the Interpretation of Solar Radio Emission Observations

“…where ξ(t) is a Gaussian white noise with the properties ξ(t) = 0 and ξ i (0) ξ j (t) = δ ij δ(t), where ... denotes an ensemble average, δ(t) is the Dirac delta function, and the k-dependent deterministic vectors ∂D ij /∂k j and B ij correspond to the diffusion tensor D ij . These are analogous to the equations describing binary collisions in a plasma (see, e.g., Ivanov & Shvets 1978;Shvets 1979;Rosin et al 2014). Equation (19) is similar to the equation by Arzner & Magun (1999); it is the definition of the stochastic integral in Itô's sense, adopted in the theory of random processes.…”

Multilevel Monte Carlo methods

“…Another is uncertainty quantification in engineering and science, which has led to a new SIAM journal and associated annual conferences. Stochastic modelling is also important in diverse areas such as biochemical reactions (Anderson and Higham 2012) and plasma physics (Rosin et al 2014).…”

“…Haji-Ali (2012) used it in a mean field model for the motion of crowds, in which each person is modelled as a independent agent subject to random forcing and an additional force due to the collective influence of the crowd. This same approach is also relevant to mean field problems which arise in plasma physics (Rosin et al 2014). Bujok et al (2013) used multilevel nested simulation for a financial credit derivative application.…”

Multilevel and Multi-index Monte Carlo methods for the McKean–Vlasov equation

“…We also consider another estimator that is more correlated with φ N P (ω ( ,m) 1:P ). The "antithetic" estimator was first independently introduced in Haji-Ali (2012, Chapter 5) and Bujok et al (2013) and subsequently used in other works on particle systems (Rosin et al 2014) and nested simulations (Giles 2015). In this work, we call this estimator a "partitioning" estimator to clearly distinguish it from the antithetic estimator in Giles and Szpruch (2014).…”

“…3.2 for the case of stochastic particle systems. The application of MLMC to particle systems has been investigated in many works (Bujok et al 2013;Haji-Ali 2012;Rosin et al 2014). The same concepts have also been applied to nested expectations (Giles 2015).…”