Full-horizon magnetic control of the Vlasov--Poisson system for plasma confinement

Giacomo Albi, Giacomo Dimarco, Federica Ferrarese, Lorenzo Pareschi (preprint arXiv:2610.26110)

We study the full-horizon control of a two-dimensional Vlasov--Poisson system through an externally imposed magnetic field. The control is represented by time-dependent coefficients associated with smooth, spatially localized basis functions, and is designed to steer selected position and velocity moments of the distribution function while limiting particle dispersion and the control effort. Starting from the kinetic constrained optimization problem, we derive the continuous first-order optimality conditions, including a backward Vlasov equation, an elliptic adjoint equation, and the variational inequality associated with bounded controls. We then project this system onto a particle--grid representation and identify the resulting particle adjoints and control gradient.

An IMEX multi-level domain-decomposition method for the Boltzmann equation on unstructured meshes

Domenico Caparello, Lorenzo Pareschi, Thomas Rey (preprint arXiv:2610.21740)

We introduce a dynamically adaptive multi-level method for the multiscale Boltzmann equation on unstructured meshes, coupling the compressible Euler, Navier-Stokes, and full Boltzmann descriptions within a single asymptotic-preserving IMEX framework. Unlike standard domain-decomposition approaches, where independent solvers exchange data only through interface conditions, the three models form a unified multiregime scheme. They share the same globally stiffly accurate IMEX Runge-Kutta discretization and are synchronized at every internal stage through kinetic-to-fluid moment projections and fluid-to-kinetic lifting. This stage-wise coupling maintains temporal consistency across model interfaces, while the asymptotic-preserving structure ensures compatibility with the fluid limit as the Knudsen number becomes small. The active description is selected in each cell through moment-realizability indicators derived from successive Chapman-Enskog approximations.

A relaxation system for the low-Mach limit of kinetic equations: Stability and higher order asymptotic preserving scheme

Giacomo Dimarco, Axel Klar, Theresa Köfler, Lorenzo Pareschi, Sudarshan Tiwari, Yizhou Zhou (preprint arXiv:2610.00811)

This work introduces a hyperbolic relaxation system designed for the simulation of kinetic equations in the low-Mach number limit. The methodology is built upon a micro-macro decomposition of the scaled BGK model, which reformulates the kinetic distribution function into a coupled system consisting of a macroscopic equilibrium part and a microscopic non-equilibrium remainder. By projecting the microscopic deviations onto a set of orthogonal polynomials, we derive a closed moment relaxation system. The resulting system is a version of Grad's 13 moment system with a linear hyperbolic part and a relaxation adapted to the incompressible limit. We prove the model's structural stability in the incompressible Navier-Stokes limit. Moreover, we develop a high order Asymptotic-Preserving (AP) numerical framework using Implicit-Explicit (IMEX) Runge Kutta schemes for temporal accuracy and finite difference WENO reconstructions as well as central difference approximations for high order spatial resolution.

A fast spectral particle method for the Landau equation

 

Giacomo Borghi, Lorenzo Pareschi (preprint arXiv:2609.23086)

We propose a fast deterministic particle method for the spatially homogeneous Landau equation. The method combines a particle representation of the solution with a Fourier approximation of the nonlinear collision flux. Nonuniform fast Fourier transforms are used to reconstruct the density from the particles and to evaluate the flux and density at the particle locations, while the convolutional structure of the flux enables its efficient computation by FFTs. For a fixed transform tolerance, the cost per time step is O(N+M^d logM), where N is the number of particles and M the number of Fourier modes per velocity dimension. We establish a consistency estimate for the reconstructed velocity field on regions where the reference density is bounded away from zero. The estimate separates the spectral truncation error from the particle-density reconstruction error, showing how the latter can dominate for sufficiently smooth densities.

Control variates with neural surrogates for uncertainty quantification in kinetic equations

Wei Chen, Giacomo Dimarco, Lorenzo Pareschi (preprint arXiv:2608.01360)

Efficient uncertainty quantification for kinetic equations with random inputs is challenging because it requires repeated simulations of high-dimensional models, such as the Boltzmann, Landau, and related collisional equations, whose computational cost can quickly become prohibitive. Multifidelity control variates address this difficulty by coupling a small number of high-fidelity simulations with many evaluations of lower-complexity reduced models. In this work, we analyze the case in which the reduced model is replaced by a neural surrogate rather than evaluated through a classical numerical scheme.