# A Multilevel Monte Carlo Asymptotic-Preserving Particle Method for Kinetic Equations in the Diffusion Limit

Abstract: We propose a multilevel Monte Carlo method for a particle-based asymptotic-preserving scheme for kinetic equations. Kinetic equations model transport and collision of particles in a position-velocity phase-space. With a diffusive scaling, the kinetic equation converges to an advection-diffusion equation in the limit of zero mean free path. Classical particle-based techniques suffer from a strict time-step restriction to maintain stability in this limit. Asymptotic-preserving schemes provide a solution to this time step restriction, but introduce a first-order error in the time step size. We demonstrate how the multilevel Monte Carlo method can be used as a bias reduction technique to perform accurate simulations in the diffusive regime, while leveraging the reduced simulation cost given by the asymptotic-preserving scheme. We describe how to achieve the necessary correlation between simulation paths at different levels and demonstrate the potential of the approach via numerical experiments.

## Summary (2 min read)

### 1 Introduction

- Kinetic equations, modeling particle behavior in a position-velocity phase space, occur in many domains.
- Deterministic methods become prohibitively expensive for higher dimensional applications.
- In the diffusive scaling, there are only of two works [9, 16] so far, to the best of their knowledge.
- In Section 3, the authors cover the multilevel Monte Carlo method that is the core contribution of this paper.

### 2.1 Model equation in the diffusive limit

- (6) Page:3 job:MLMCforAP macro:svmult.cls date/time:15-Jan-2019/18:31 Equation (6) is also known as the Goldstein-Taylor model, and can be solved using a particle scheme.
- Equation (6) is then solved via operator splitting as 1.
- Transport step. (9) To maintain stability, this approximation requires a time step restriction ∆ t = O(ε2), leading to unacceptably high computational costs.

### 2.2 Asymptotic-preserving Monte Carlo scheme

- Discretizing this equation, using operator splitting as above, again leads to a Monte Carlo scheme.
- For each particle Xp and for each time step n, one time step now consists of a transport-diffusion and a collision step: 1. Transport-diffusion step. (13).
- For more details, the authors refer the reader to [16].

### 3.1 Method and notation

- That is a choice the authors make for notational convenience, and is not essential for the method they present.
- On the one hand, a small time step reduces the bias of the simulation of each sampled trajectory, and thus of the estimated quantity of interest.
- The key idea behind the Multilevel Monte Carlo method [17] is to generate a sequence of estimates with varying discretization accuracy and a varying number of realizations.
- The estimators (17) estimate the bias induced by sampling with a simulation time step size ∆ t`−1 by comparing the sample results with a simulation using a time step size ∆ t`.

### 3.2.1 Coupled trajectories and notation

- The differences in (17) will only have low variance if the simulated paths Xn,m∆ t`,p and Xn∆ t`−1,p are correlated.
- In each time step using the asymptotic-preserving particle scheme (11)–(13), there are two sources of stochastic behavior.
- Particle trajectories can be coupled by separately correlating the random numbers used for each individual particle in the transport-diffusion and collision phase of each time step.
- The authors present the complete algorithm in Section 3.2.4.

### 3.2.2 Coupling the transport-diffusion phase

- The authors observe that the paths have similar behavior, i.e., if the fine simulation tends towards negative values, so does the coarse simulation and vice versa.
- Still, there is an observable difference between them.
- This is due to the bias caused by the paths having different diffusion coefficients.

### 3.2.3 Coupling the collision phase

- While correlating the Brownian paths is relatively straightforward, the coupling of the velocities in the collision phase is more involved.
- Afterwards, when the authors decide to perform a collision both at level ` and `−1, they will correlate the new velocities generated in both simulations.
- The maximum of a set of uniformly distributed random number is not uniformly distributed.
- This approach to selecting the sign of V n+1p,∆ t`−1 means that the velocities going into the next time step will have the same sign.
- This is one source of the bias that the authors want to estimate using the multilevel Monte Carlo method.

### 4 Experimental Results

- The authors will now demonstrate the viability of the suggested approach through some numerical experiments.
- The authors will simulate the model given by (10), using the multilevel Monte Carlo method to estimate a selected quantity of interest, which is the expected Page:10 job:.
- The ensemble of particles is initialized at the origin with equal probability of having a left and right velocity.
- When discussing results the authors will replace the full expression for a sample of the quantity of interest, based on an arbitrary particle p, F(XN,0∆ t`,p), with the symbol F̀ to simplify notation.

### 4.1 Model correlation behavior

- The authors fix the number of samples per difference estimator at 100 000.
- This matches the weak convergence order of the Euler-Maruyama scheme, used to simulate the model (11)–(13), as well as the expected behavior from the time step dependent bias in the asymptotic-preserving model.
- This means that the existing general theory for multilevel Monte Carlo methods [18] concerning, e.g. samples per level, convergence criteria and conditions for adding levels, can be applied in this regime.
- This explains the decrease in the means of the differences and in the variance of the differences for both small and large ∆ t which is clearly visible in Figure 6.

### 5 Conclusion

- The authors have derived a new multilevel scheme for asymptotic-preserving particle schemes of the form given in (10).
- The authors have demonstrated that this scheme has interesting convergence behavior as the time step is refined, which is apparent in the expected value and variance of sampled differences of the quantity of interest.
- The authors have proposed a strategy for selecting the coarsest level in the multilevel Monte Carlo method in this context, and shown that this simulation approach gives a significant speedup over a classical Monte Carlo simulation.
- More complicated models including, for example, absorption terms can also be studied.

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