arXiv subjects

Vince Maes

Publications and source records attributed to Vince Maes.

6 recordsLinked to original sources

arXiv2025

A noise-robust Monte Carlo method for electric field calculations in EMC3

EMC3 is a state-of-the-art 3D Monte Carlo code for plasma edge transport in stellarator configurations, but it does not yet treat the $E \times B$ drift self-consistently. Computing the drift requires the electric field $E = -\nabla \varphi$, and hence an accurate gradient of the electric potential $\varphi$. Because the plasma fields produced by EMC3 are inherently noisy, the finite difference approximation used previously amplifies this noise, increasingly so as the grid is refined. We extend the Monte Carlo Gradient Approximation method, originally developed for 1D Fokker--Planck equations, to a 2D setting and apply it to the electric potential. For an isotropic diffusion coefficient, we derive a PDE governing the evolution of the electric field itself, which allows $E$ to be approximated directly by a Monte Carlo simulation, avoiding finite differences altogether. A numerical experiment based on manufactured solutions demonstrates the accuracy of the method and shows that the variance grows substantially more slowly under grid refinement than for finite differences. The resulting formulation contains a source term involving the second poloidal derivative of the potential, which we neglect. This is admissible only when the radial scale of the plasma edge is thin compared to the poloidal variation scale, an ordering expected to fail near X-points, during detachment, and in island divertors such as W7-X. The present work should therefore be regarded as a proof of concept rather than a general electric field solver for EMC3.

math.NA↗

arXiv2025

Fluid boundary conditions in kinetic-diffusion Monte Carlo

The Kinetic-Diffusion Monte Carlo (KDMC) method is a powerful tool for simulating neutral particles in fusion reactors. It is a hybrid fluid-kinetic method that is significantly faster than pure kinetic methods at the cost of a small bias due to fluid approximations. Unfortunately, when simulating particles close to a boundary, it needs to switch to a purely kinetic method, which is significantly slower. In this paper, we will extend the method so that it can accurately take boundary conditions into account without switching to a purely kinetic method. Experiments show that this extension can lead to a speedup of up to 500 times compared to a KDMC method that switches to a purely kinetic method, while not sacrificing too much accuracy.

math.NA↗

arXiv2024

Kinetic-Diffusion-Rotation Algorithm for Dose Estimation in Electron Beam Therapy

Monte Carlo methods are state-of-the-art when it comes to dosimetric computations in radiotherapy. However, the execution time of these methods suffers in high-collisional regimes. We address this problem by introducing a kinetic-diffusion particle tracing scheme. This algorithm, first proposed in the context of neutral transport in fusion energy, relies on explicit simulation of the kinetic motion in low-collisional regimes and dynamically switches to motion based on a random walk in high-collisional regimes. The random walk motion maintains the first two moments (mean and variance) of the kinetic motion. We derive an analytic formula for the mean kinetic motion and discuss the addition of a multiple scattering distribution to the algorithm. In contrast to neutral transport, the radiation transfer setting does not readily admit to an analytical expression for the variance of the kinetic motion, and we therefore resort to the use of a lookup table. We test the algorithm for dosimetric computations in radiation therapy on a 2D CT scan of a lung patient. Using a simple particle model, our Python implementation of the algorithm is nearly 33 times faster than an equivalent kinetic simulation at the cost of a small modeling error.

q-bio.QM↗

arXiv2024

Cell Escape Probabilities for Markov Processes on a Grid

Kinetic equations describe physical processes in a high-dimensional phase space and are often simulated using Markov process-based Monte Carlo routines. The quantities of interest are typically defined on the lower-dimensional position space and estimated on a grid (histogram). In several applications, such as the construction of diffusion Monte Carlo-like techniques and variance prediction for particle tracing Monte Carlo methods, the cell escape probabilities, i.e., the probabilities with which particles escape a grid cell during one step of the Markov process, are of interest. In this paper, we derive formulas to calculate the cell escape probabilities for common mesh elements in one, two, and three dimensions. Deterministic calculation of cell escape probabilities in higher dimensions becomes expensive and prone to quadrature errors due to the involved high-dimensional integrals. We therefore also introduce a stochastic Monte Carlo algorithm to calculate the escape probabilities, which is more robust at the cost of a statistical error. The code used to perform the numerical experiments and accompanying GeoGebra tutorials are openly available at https://gitlab.kuleuven.be/numa/public/escape-probabilities-markov-processes.

math-ph↗

arXiv2024

Predicting the statistical error of analog particle tracing Monte Carlo

Large particle systems are often described by high-dimensional (linear) kinetic equations that are simulated using Monte Carlo methods for which the asymptotic convergence rate is independent of the dimensionality. Even though the asymptotic convergence rate is known, predicting the actual value of the statistical error remains a challenging problem. In this paper, we show how the statistical error of an analog particle tracing Monte Carlo method can be calculated (expensive) and predicted a priori (cheap) when estimating quantities of interest (QoI) on a histogram. We consider two types of QoI estimators: point estimators for which each particle provides one independent contribution to the QoI estimates, and analog estimators for which each particle provides multiple correlated contributions to the QoI estimates. The developed statistical error predictors can be applied to other QoI estimators and nonanalog simulation routines as well. The error analysis is based on interpreting the number of particle visits to a histogram bin as the result of a (correlated) binomial experiment. The resulting expressions can be used to optimize (non)analog particle tracing Monte Carlo methods and hybrid simulation methods involving a Monte Carlo component, as well as to select an optimal particle tracing Monte Carlo method from several available options. Additionally, the cheap statistical error predictors can be used to determine a priori the number of particles N that is needed to reach a desired accuracy. We illustrate the theory using a linear kinetic equation describing neutral particles in the plasma edge of a fusion device and show numerical results. The code used to perform the numerical experiments is openly available.

physics.comp-ph↗

arXiv2023

Hilbert expansion based fluid models for kinetic equations describing neutral particles in the plasma edge of a fusion device

Neutral particles in the plasma edge of fusion devices based on magnetic confinement are described by a transient kinetic equation incorporating ionization, recombination, and charge-exchange collisions. In charge-exchange dominated regimes, the neutral particle velocity distribution approaches the drifting Maxwellian defined by the mean velocity and temperature of the plasma. This enables model order reduction from the kinetic equation to approximate fluid models. We derive transient fluid models consistent with the kinetic equation by exploring a splitting based approach. We split the kinetic equation in sources and sinks on the one hand, and transport combined with charge-exchange on the other hand. Combining transport with charge-exchange collisions allows for deriving Hilbert expansion based fluid models. The retrieved fluid models depend on the assumed importance (scaling) of the different terms in the split equation describing transport and charge-exchange. We explore two scalings: the hydrodynamic scaling and the diffusive scaling. The performance of the fluid models with respect to a discrete velocity model and a Monte Carlo reference solver is assessed in numerical experiments. The code used to perform the numerical experiments is openly available.

physics.plasm-ph↗