arXiv subjects

Giovanni Samaey

Publications and source records attributed to Giovanni Samaey.

At least 55 records · Page 3Linked to original sources

arXiv2021

Spatially Adaptive Projective Integration Schemes For Stiff Hyperbolic Balance Laws With Spectral Gaps

Stiff hyperbolic balance laws exhibit large spectral gaps, especially if the relaxation term significantly varies in space. Using examples from rarefied gases and the general form of the underlying balance law model, we perform a detailed spectral analysis of the semi-discrete model that reveals the spectral gaps. Based on that, we show the inefficiency of standard time integration schemes expressed by a severe restriction of the CFL number. We then develop the first spatially adaptive projective integration schemes to overcome the prohibitive time step constraints of standard time integration schemes. The new schemes use different time integration methods in different parts of the computational domain, determined by the spatially varying value of the relaxation time. We use our analytical results to derive accurate stability bounds for the involved parameters and show that the severe time step constraint can be overcome. The new adaptive schemes show good accuracy in a numerical test case and can obtain a large speedup with respect to standard schemes.

math.NA↗

arXiv2021

A multilevel Monte Carlo method for asymptotic-preserving particle schemes in the diffusive limit

Kinetic equations model distributions of particles in position-velocity phase space. Often, one is interested in studying the long-time behavior of particles in high-collisional regimes in which an approximate (advection)-diffusion model holds. In this paper we consider the diffusive scaling. Classical particle-based techniques suffer from a strict time-step restriction in this limit, to maintain stability. Asymptotic-preserving schemes avoid this problem, but introduce an additional time discretization error, possibly resulting in an unacceptably large bias for larger time steps. Here, we present and analyze a multilevel Monte Carlo scheme that reduces this bias by combining estimates using a hierarchy of different time step sizes. We demonstrate how to correlate trajectories from this scheme, using different time steps. We also present a strategy for selecting the levels in the multilevel scheme. Our approach significantly reduces the computation required to perform accurate simulations of the considered kinetic equations, compared to classical Monte Carlo approaches.

math.NA↗

arXiv2021

Projective Integration Schemes for Hyperbolic Moment Equations

In this paper, we apply projective integration methods to hyperbolic moment models of the Boltzmann equation and the BGK equation, and investigate the numerical properties of the resulting scheme. Projective integration is an explicit, asymptotic-preserving scheme that is tailored to problems with large spectral gaps between slow and (one or many) fast eigenvalue clusters of the model. The spectral analysis of a linearized moment model clearly shows spectral gaps and reveals the multi-scale nature of the model for which projective integration is a matching choice. The combination of the non-intrusive projective integration method with moment models allows for accurate, but efficient simulations with significant speedup, as demonstrated using several 1D and 2D test cases with different collision terms, collision frequencies and relaxation times.

physics.comp-ph↗

arXiv2020

A comparison of source term estimators in coupled finite-volume/Monte-Carlo methods with applications to plasma edge simulations in nuclear fusion

In many applications, such as plasma edge simulation of a nuclear fusion reactor, a coupled PDE/kinetic description is required, which is usually solved with a coupled finite-volume/Monte-Carlo method. Different procedures have been proposed to estimate the source terms in the finite volume part that appear from the Monte Carlo part of the simulation. In this paper, we present a systematic comparison of the variance and computational cost of a coherent set of such estimation procedures. We compare the different estimation procedures for mass in a simplified forward-backward scattering model problem, where an analytical comparison is possible, and for mass and momentum in a model problem with realistic scattering. Our results reveal a non-trivial dependence of the optimal choice of estimator on the model parameters and show that different estimation procedures prevail for different quantities of interest.

math.NA↗

arXiv2020

Kinetic-diffusion asymptotic-preserving Monte Carlo algorithm for Boltzmann-BGK in the diffusive scaling

We develop a novel Monte Carlo strategy for the simulation of the Boltzmann-BGK model with both low-collisional and high-collisional regimes present. The presented solution to maintain accuracy in low-collisional regimes and remove exploding simulation costs in high-collisional regimes uses hybridized particles that exhibit both kinetic behaviour and diffusive behaviour depending on the local collisionality. In this work, we develop such a method that maintains the correct mean, variance, and correlation of the positional increments over multiple time steps of fixed step size for all values of the collisionality, under the condition of spatial homogeneity during the time step. In the low-collisional regime, the method reverts to the standard velocity-jump process. In the high-collisional regime, the method collapses to a standard random walk process. We analyze the error of the presented scheme in the low-collisional regime for which we obtain the order of convergence in the time step size. We furthermore provide an analysis in the high-collisional regime that demonstrates the asymptotic-preserving property.

math.NA↗

arXiv2020

Multilevel Asymptotic-Preserving Monte Carlo for Particle Simulations

We develop a novel multilevel asymptotic-preserving Monte Carlo method, called Multilevel Kinetic-Diffusion Monte Carlo (ML-KDMC), for simulating the kinetic Boltzmann transport equation with a Bhatnagar-Gross-Krook (BGK) collision operator. This equation occurs, for instance, in mathematical models of the neutral particles in the plasma edge of nuclear fusion reactors. In this context, the Kinetic-Diffusion Monte Carlo method is known to maintain accuracy both in the low-collisional and the high-collisional limit, without an exploding simulation cost in the latter. We show that, by situating this method within a Multilevel Monte Carlo (MLMC) framework, using a hierarchy of larger time step sizes, the simulation cost is reduced even further. The different levels in our ML-KDMC method are connected via a new and improved recipe for correlating particle trajectories with different time step sizes. Furthermore, a new and more general level selection strategy is presented. We illustrate the efficiency of our ML-KDMC method by applying it to a one-dimensional test case with nonhomogeneous and anisotropic plasma background. Our method yields significant speedups compared to the single-level KDMC scheme, both in the low and high collisional regime. In the high-collisional case, our ML-KDMC outperforms the single-level KDMC method by several orders of magnitude.

math.NA↗

arXiv2020

Linking Machine Learning with Multiscale Numerics: Data-Driven Discovery of Homogenized Equations

The data-driven discovery of partial differential equations (PDEs) consistent with spatiotemporal data is experiencing a rebirth in machine learning research. Training deep neural networks to learn such data-driven partial differential operators requires extensive spatiotemporal data. For learning coarse-scale PDEs from computational fine-scale simulation data, the training data collection process can be prohibitively expensive. We propose to transformatively facilitate this training data collection process by linking machine learning (here, neural networks) with modern multiscale scientific computation (here, equation-free numerics). These equation-free techniques operate over sparse collections of small, appropriately coupled, space-time subdomains ("patches"), parsimoniously producing the required macro-scale training data. Our illustrative example involves the discovery of effective homogenized equations in one and two dimensions, for problems with fine-scale material property variations. The approach holds promise towards making the discovery of accurate, macro-scale effective materials PDE models possible by efficiently summarizing the physics embodied in "the best" fine-scale simulation models available.

math.NA↗

arXiv2020

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

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.

math.NA↗

arXiv2020

Study of micro-macro acceleration schemes for linear slow-fast stochastic differential equations with additive noise

Computational multi-scale methods capitalize on a large time-scale separation to efficiently simulate slow dynamics over long time intervals. For stochastic systems, one often aims at resolving the statistics of the slowest dynamics. This paper looks at the efficiency of a micro-macro acceleration method that couples short bursts of stochastic path simulation with extrapolation of spatial averages forward in time. To have explicit derivations, we elicit an amenable linear test equation containing multiple time scales. We make derivations and perform numerical experiments in the Gaussian setting, where only the evolution of mean and variance matters. The analysis shows that, for this test model, the stability threshold on the extrapolation step is largely independent of the time-scale separation. In consequence, the micro-macro acceleration method increases the admissible time steps far beyond those for which a direct time discretization becomes unstable.

math.NA↗

arXiv2020

A micro-macro Markov chain Monte Carlo method for molecular dynamics using reaction coordinate proposals I: direct reconstruction

We introduce a new micro-macro Markov chain Monte Carlo method (mM-MCMC) to sample invariant distributions of molecular dynamics systems that exhibit a time-scale separation between the microscopic (fast) dynamics, and the macroscopic (slow) dynamics of some low-dimensional set of reaction coordinates. The algorithm enhances exploration of the state space in the presence of metastability by allowing larger proposal moves at the macroscopic level, on which a conditional accept-reject procedure is applied. Only when the macroscopic proposal is accepted, the full microscopic state is reconstructed from the newly sampled reaction coordinate value and is subjected to a second accept/reject procedure. The computational gain stems from the fact that most proposals are rejected at the macroscopic level, at low computational cost, while microscopic states, once reconstructed, are almost always accepted. We analytically show convergence and discuss the rate of convergence of the proposed algorithm, and numerically illustrate its efficiency on a number of standard molecular test cases. We also investigate the effect of the choice of different numerical parameters on the efficiency of the resulting mM-MCMC method.

math.NA↗

arXiv2019

Parareal computation of stochastic differential equations with time-scale separation: a numerical study

The parareal algorithm is known to allow for a significant reduction in wall clock time for accurate numerical solutions by parallelising across the time dimension. We present and test a micro-macro version of parareal, in which the fine propagator is based on a (high-dimensional, slow-fast) stochastic microscopic model, and the coarse propagator is based on a low-dimensional approximate effective dynamics at slow time scales. At the microscopic level, we use an ensemble of Monte Carlo particles, whereas the approximate coarse propagator uses the (deterministic) Fokker-Planck equation for the slow degrees of freedom. The required coupling between microscopic and macroscopic representations of the system introduces several design options, specifically on how to generate a microscopic probability distribution consistent with a required macroscopic probability distribution and how to perform the coarse-level updating of the macroscopic probability distribution in a meaningful manner. We numerically study how these design options affect the efficiency of the algorithm in a number of situations. The choice of the coarse-level updating operator strongly impacts the result, with a superior performance if addition and subtraction of the quantile function (inverse cumulative distribution) is used. How microscopic states are generated has a less pronounced impact, provided a suitable prior microscopic state is used.

math.NA↗

arXiv2019

Efficiency of a micro-macro acceleration method for scale-separated stochastic differential equations

We discuss through multiple numerical examples the accuracy and efficiency of a micro-macro acceleration method for stiff stochastic differential equations (SDEs) with a time-scale separation between the fast microscopic dynamics and the evolution of some slow macroscopic state variables. The algorithm interleaves a short simulation of the stiff SDE with extrapolation of the macroscopic state variables over a longer time interval. After extrapolation, we obtain the reconstructed microscopic state via a matching procedure: we compute the probability distribution that is consistent with the extrapolated state variables, while minimally altering the microscopic distribution that was available just before the extrapolation. In this work, we numerically study the accuracy and efficiency of micro-macro acceleration as a function of the extrapolation time step and as a function of the chosen macroscopic state variables. Additionally, we compare the effect of different hierarchies of macroscopic state variables. We illustrate that the method can take significantly larger time steps than the inner microscopic integrator, while simultaneously being more accurate than approximate macroscopic models.

math.NA↗

arXiv2019

Convergence and stability of a micro-macro acceleration method:linear slow-fast stochastic differential equations with additive noise

We analyse the convergence and stability of a micro-macro acceleration algorithm for Monte Carlo simulations of stiff stochastic differential equations with a time-scale separation between the fast evolution of the individual stochastic realizations and some slow macroscopic state variables of the process. The micro-macro acceleration method performs a short simulation of a large ensemble of individual fast paths, before extrapolating the macroscopic state variables of interest over a larger time step. After extrapolation, the method constructs a new probability distribution that is consistent with the extrapolated macroscopic state variables, while minimizing Kullback-Leibler divergence with respect to the distribution available at the end of the Monte Carlo simulation. In the current work, we study the convergence and stability of this method on linear stochastic differential equations with additive noise, when only extrapolating the mean of the slow component. For this case, we prove convergence to the microscopic dynamics when the initial distribution is Gaussian and present a stability result for non-Gaussian initial laws.

math.NA↗

arXiv2018

Analysis of a micro-macro acceleration method with minimum relative entropy moment matching

We analyse convergence of a micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations with time-scale separation between the (fast) evolution of individual trajectories and the (slow) evolution of the macroscopic function of interest. We consider a class of methods, presented in [Debrabant, K., Samaey, G., Zieliński, P. A micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations. SINUM, 55 (2017) no. 6, 2745-2786], that performs short bursts of path simulations, combined with the extrapolation of a few macroscopic state variables forward in time. After extrapolation, a new microscopic state is then constructed, consistent with the extrapolated variable and minimising the perturbation caused by the extrapolation. In the present paper, we study a specific method in which this perturbation is minimised in a relative entropy sense. We discuss why relative entropy is a useful metric, both from a theoretical and practical point of view, and rigorously study local errors and numerical stability of the resulting method as a function of the extrapolation time step and the number of macroscopic state variables. Using these results, we discuss convergence to the full microscopic dynamics, in the limit when the extrapolation time step tends to zero and the number of macroscopic state variables tends to infinity.

math.NA↗

arXiv2017

Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations

We present high-order, fully explicit projective integration schemes for nonlinear collisional kinetic equations such as the BGK and Boltzmann equation. The methods first take a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution. Then, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. The procedure can be recursively repeated on a hierarchy of projective levels to construct telescopic projective integration methods. Based on the spectrum of the linearized collision operator, we deduce that the computational cost of the method is essentially independent of the stiffness of the problem: with an appropriate choice of inner step size, the time step restriction on the outer time step, as well as the number of inner time steps, is independent of the stiffness of the (collisional) source term. In some cases, the number of levels in the telescopic hierarchy depends logarithmically on the stiffness. We illustrate the method with numerical results in one and two spatial dimensions.

math.AP↗

arXiv2017

A micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations

We present and analyse a micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations with separation between the (fast) time-scale of individual trajectories and the (slow) time-scale of the macroscopic function of interest. The algorithm combines short bursts of path simulations with extrapolation of a number of macroscopic state variables forward in time. The new microscopic state, consistent with the extrapolated variables, is obtained by a matching operator that minimises the perturbation caused by the extrapolation. We provide a proof of the convergence of this method, in the absence of statistical error, and we analyse various strategies for matching, as an operator on probability measures. Finally, we present numerical experiments that illustrate the effects of the different approximations on the resulting error in macroscopic predictions.

math.NA↗

arXiv2017

Projective integration for nonlinear BGK kinetic equations

We present a high-order, fully explicit, asymptotic-preserving projective integration scheme for the nonlinear BGK equation. The method first takes a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution. Then, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. Based on the spectrum of the linearized BGK operator, we deduce that, with an appropriate choice of inner step size, the time step restriction on the outer time step as well as the number of inner time steps is independent of the stiffness of the BGK source term. We illustrate the method with numerical results in one and two spatial dimensions.

math.NA↗

arXiv2016

Variance-reduced multiscale simulation of slow-fast stochastic differential equations

We study a variance reduction strategy based on control variables for simulating the averaged macroscopic behavior of a stochastic slow-fast system. We assume that this averaged behavior can be written in terms of a few slow degrees of freedom, and that the fast dynamics is ergodic for every fixed value of the slow variable. The time derivative for the averaged dynamics can then be approximated by a Markov chain Monte Carlo method. The variance-reduced scheme that is introduced here uses the previous time instant as a control variable. We analyze the variance and bias of the proposed estimator and illustrate its performance when applied to a linear and nonlinear model problem.

math.NA↗