arXiv subjects

Lukas Einkemmer

Publications and source records attributed to Lukas Einkemmer.

At least 19 recordsLinked to original sources

arXiv2026

Domain decomposition dynamical low-rank for multi-dimensional radiative transfer equations

In this paper, we propose a domain decomposition dynamical low-rank method to solve high-dimensional radiative transfer problems and similar kinetic equations. The algorithm uses a separate low-rank approximation on each spatial subdomain, which means that, for a given accuracy, we can often use a smaller overall rank compared to classic dynamical low-rank methods. In particular, we can solve problems with point sources efficiently, that for classic algorithms require almost full rank. Our algorithm only transfers boundary data between subdomains and is thus very attractive for distributed memory parallelization, where classic dynamical low-rank algorithms suffer from global data dependency. We demonstrate the efficiency of our algorithm by a number of challenging test examples that have both very optical thin and thick regions.

math.NA↗

arXiv2026

Global Optimization of Beam Profiles for Suppressing the Bump-On-Tail Instability

A particle beam with constant intensity that propagates into a plasma triggers a bump-on-tail instability. It has been demonstrated that beam modulation can be used to significantly reduce this instability. However, finding the optimal beam profiles that accomplish this is a complex optimization problem that suffers from an extremely rugged optimization landscape and a relatively high-dimensional parameter space. In this paper, we evaluate a number of optimization algorithms and find that population based methods (in particular, CMA-ES and Differential Evolution) perform best. In particular, we investigate beam profiles where the beam is either completely turned ON or OFF. This configuration is easier to accomplish experimentally, but leads to a higher-dimensional and thus more challenging optimization problem. We obtain ON/OFF beam profiles that reduce the instability by more than 80% and thus perform significantly better than what has been reported in the literature.

math.NA↗

arXiv2026

Low-rank SI-DSA and GMRES-DSA for Radiative Transfer Equation with Adaptive Accuracy Control

Recently, efficient sweep-based low-rank iterative solvers have been developed to reduce the computational costs of steady-state solves and implicit time integration of the radiative transfer equation (RTE). In particular, angular sampling based on the discrete empirical interpolation method (DEIM) or on error-indicated randomized greedy selection enables efficient, non-intrusive reuse of existing transport sweep implementations. However, randomized-greedy low-rank source iteration with diffusion synthetic acceleration (SI--DSA) enforces uniformly tight accuracy tolerances across all source iterations, thereby oversolving the early iterations, and it does not explicitly reuse angular bases from preceding iterations. More critically, like its full-rank counterpart, this approach may lose effectiveness in multiscale problems with strong material discontinuities. We address these limitations through two complementary developments. First, we introduce adaptive accuracy control into randomized-greedy low-rank SI--DSA, reducing intermediate ranks and computational cost while maintaining the accuracy required for convergence. A two-stage hybrid sampling strategy can provide further acceleration by switching to DEIM-based sampling near convergence, once the preceding angular basis becomes sufficiently representative. Second, we develop an inexact flexible GMRES--DSA solver for the integral scalar-flux reformulation, in which sweep-based low-rank approximations are confined to the construction of the right-hand side and the matrix--vector products. The scalar-flux Krylov basis is not compressed, thereby preserving Arnoldi orthogonality in exact arithmetic and allowing established inexact-GMRES analysis to guide adaptive accuracy control of low-rank operations. We present numerical results on benchmark problems that demonstrate the effectiveness of the proposed methods.

math.NA↗

arXiv2026

An adaptive dynamical low-rank optimizer for solving kinetic parameter identification problems

The numerical solution of parameter identification problems for kinetic equations can exhibit high computational and memory costs. In this paper, we propose a dynamical low-rank scheme for the reconstruction of the scattering parameter in the radiative transfer equation from a number of macroscopic time-independent measurements. We first work through the PDE constrained optimization procedure in a continuous setting and derive the adjoint equations using a Lagrangian reformulation. For the scattering coefficient, a periodic B-spline approximation is introduced and a gradient descent step for updating its coefficients is formulated. After the discretization, a dynamical low-rank approximation (DLRA) is applied. We make use of the rank-adaptive basis update & Galerkin integrator and a line search approach for the adaptive refinement of the gradient descent step size and the DLRA tolerance. We show that the proposed scheme significantly reduces both memory and computational cost. Numerical results computed with different initial conditions validate the accuracy and efficiency of the proposed DLRA scheme compared to solutions computed with a full solver.

math.NA↗

arXiv2026

A Memory-Efficient Adjoint State Optimization Method Based on Time-Reversible Dynamical Low-Rank Approximation

The primary challenge of conducting PDE-constrained optimization for high-dimensional problems, such as kinetic equations, is the often prohibitive memory cost. Computing gradients using the adjoint state method would require the storage of the entire time history of the forward solution. For such problems, where the memory cost for storing a single instance of the forward solution can already be a limiting factor, this is clearly not feasible. In this paper, we propose a memory-efficient adjoint state method that compresses the forward and adjoint solution with a dynamical low-rank approximation (a model order reduction technique) and bypasses the need to store the entire forward solution by employing a time-reversible low-rank integrator. The dynamical low-rank approach introduces a number of challenges: reversibility can fail in the rank-deficient case and the low-rank trajectories can show chaotic behavior. In particular, the latter has a number of important consequences for the optimization problem. We address those challenges and show that our method can drastically reduce the memory requirement for gradient-based optimization of kinetic equations. In particular, we consider two examples from kinetic plasma physics: optimizing beam profiles to suppress a bump-on-tail instability and shaping a particle beam using external electric fields.

math.NA↗

arXiv2026

Kinetic Optimization of Magnetic Mirror Confinement: Beyond Classical Loss-Cone Theory

Magnetic mirrors are among the conceptually simplest plasma confinement configurations and remain promising candidates for thermonuclear fusion. Their design requires shaping an externally applied magnetic field to confine plasma within an open-ended cylindrical device. In contrast to toroidally closed devices such as tokamaks and stellarators, confinement in magnetic mirrors depends intrinsically on kinetic mechanisms, particularly velocity-space trapping and particle loss through the open ends. We formulate magnetic mirror design as a PDE-constrained optimization problem governed by a reduced multispecies drift-kinetic-Poisson model. The resulting optimization reveals two physical effects not captured by the classical loss-cone argument. First, the self-consistent electric field generated through Poisson coupling acts as a secondary confinement barrier and substantially alters particle retention in the nonlinear regime. Second, the optimized magnetic-field configuration depends qualitatively on the underlying kinetic model: an electron-only model favors an unconventional centrally peaked field, whereas the fully coupled electron-ion model recovers the classical boundary-peaked mirror configuration. These results demonstrate that optimal magnetic mirror design cannot be determined solely from loss-cone considerations, but must account for the self-consistent nonlinear kinetic dynamics of the plasma.

physics.plasm-ph↗

arXiv2026

A stable multiplicative dynamical low-rank discretization for the linear Boltzmann-BGK equation

The numerical method of dynamical low-rank approximation (DLRA) has recently been applied to various kinetic equations showing a significant reduction of the computational effort. In this paper, we apply this concept to the linear Boltzmann-Bhatnagar-Gross-Krook (Boltzmann-BGK) equation which due its high dimensionality is challenging to solve. Inspired by the special structure of the non-linear Boltzmann-BGK problem, we consider a multiplicative splitting of the distribution function. We propose a rank-adaptive DLRA scheme making use of the basis update & Galerkin integrator and combine it with an additional basis augmentation to ensure numerical stability, for which an analytical proof is given and a classical hyperbolic Courant-Friedrichs-Lewy (CFL) condition is derived. This allows for a further acceleration of computational times and a better understanding of the underlying problem in finding a suitable discretization of the system. Numerical results of a series of different test examples confirm the accuracy and efficiency of the proposed method compared to the numerical solution of the full system.

math.NA↗

arXiv2026

An energy stable and conservative multiplicative dynamical low-rank discretization for the Su-Olson problem

Computing numerical solutions of the thermal radiative transfer equations on a finely resolved grid can be costly due to high computational and memory requirements. A numerical reduced order method that has recently been applied to a wide variety of kinetic partial differential equations is the concept of dynamical low-rank approximation (DLRA). In this paper, we consider the thermal radiative transfer equations with Su-Olson closure, leading to a linearized kinetic model. For the conducted theoretical and practical considerations we use a multiplicative splitting of the distribution function that poses additional challenges in finding an energy stable discretization and deriving a hyperbolic Courant-Friedrichs-Lewy (CFL) condition. We propose such an energy stable DLRA scheme that makes use of the augmented basis update & Galerkin integrator. This integrator allows for additional basis augmentations, enabling us to give a mathematically rigorous proof of energy stability and local mass conservation. Numerical examples confirm the derived properties and show the computational advantages of the DLRA scheme compared to a numerical solution of the full system of equations.

math.NA↗

arXiv2026

Efficient SN-like and PN-like Dynamic Low Rank methods for Thermal Radiative Transfer

Dynamic Low Rank (DLR) methods are a promising way to reduce the computational cost and memory footprint of the high-dimensional thermal radiative transfer (TRT) equations. The TRT equations are a system of nonlinear PDEs that model the energy exhchange between the material temperature and the radiation energy density; due to their high dimensionality, solving the TRT equations is often bottleneck in multi-physics simulations. DLR methods represent the solution in terms of time-evolving SVD-like factors of angle and space. Although previous work has explored DLR methods for TRT, most of the methods have limitations that make them impractical for realistic scenarios and uncompetitive with current non-DLR production codes. Here we develop new PN-like and SN-like Dynamic Low Rank (DLR) methods for TRT. In the SN-like DLR method, we use the time-evolving angular basis functions to select time-evolving angles; this DLR formulation enables us to use the highly optimized SN transport sweep as our main computational kernel, and results in a practical way of leveraging low-rank methods in production TRT codes. In contrast, our PN-like DLR method uses an even-parity formulation and results in positive-definite linear systems to solve for each time step. We demonstrate the methods on several challenging, highly heterogenous problems in two spatial dimensions $(4$D) that these DLR schemes can give significant reduction in angular artifacts (``ray effects'') with the same cost as gold-standard SN methods.

math.NA↗

arXiv2025

An initial-boundary corrected splitting method for diffusion-reaction problems

Strang splitting is a widely used second-order method for solving diffusion-reaction problems. However, its convergence order is often reduced to order $1$ for Dirichlet boundary conditions and to order $1.5$ for Neumann and Robin boundary conditions, leading to lower accuracy and reduced efficiency. In this paper, we consider a new splitting approach, called an initial-boundary corrected splitting, which avoids order reduction while improving computational efficiency for a wider range of applications. In contrast to the corrections proposed in the literature, it does not require the computation of correction terms that depend on the boundary conditions and boundary data. Through rigorous analytical convergence analysis and numerical experiments, we demonstrate the improved accuracy and performance of the proposed method.

math.NA↗

arXiv2025

A simple predictor-corrector scheme without order reduction for advection-diffusion-reaction problems

Treating diffusion and advection/reaction separately is an effective strategy for solving semilinear advection-diffusion-reaction equations. However, such an approach is prone to suffer from order reduction, especially in the presence of inhomogeneous Dirichlet boundary conditions. In this paper, we extend an approach of Einkemmer and Ostermann [SIAM J. Sci. Comput. 37, A1577-A1592, 2015] to advection-diffusion-reaction problems, where the advection and reaction terms depend nonlinearly on both the solution and its gradient. Starting from a modified splitting method, we construct a predictor-corrector scheme that avoids order reduction and significantly improves accuracy. The predictor only requires the solution of a linear diffusion equation, while the corrector is simply an explicit Euler step of an advection-reaction equation. Under appropriate regularity assumptions on the exact solution, we rigorously establish second-order convergence for this scheme. Numerical experiments are presented to confirm the theoretical results.

math.NA↗

arXiv2025

Control of kinetic plasma instabilities by laser fields

We study the possibility of controlling kinetic plasma instabilities by using lasers to apply external electromagnetic fields. We derive the dispersion relation for the corresponding mathematical description, a reduced Vlasov--Maxwell system, by extending the well-known Penrose condition. It is observed that, under very mild assumptions, the dispersion relation decouples into two parts. The first part is identical to the classic Penrose condition for the Vlasov--Poisson system, while the second part describes the influence of the laser on the transverse dynamics (e.g. a Weibel instability). In particular, this means that the longitudinal dynamics (e.g. a two-stream instability) can not be stabilized in this manner as far as linear theory is concerned. We show, however, that nonlinear effects can be used to couple the two parts and achieve effective control. This is done by determining the control parameters (i.e. the form of the external electric and magnetic fields) by solving a PDE-constrained optimization problem.

physics.plasm-ph↗

arXiv2025

A low-rank, high-order implicit-explicit integrator for three-dimensional convection-diffusion equations

This paper presents a rank-adaptive implicit-explicit integrator for the tensor approximation of three-dimensional convection-diffusion equations. In particular, the recently developed Reduced Augmentation Implicit Low-rank (RAIL) integrator is extended from the two-dimensional matrix case to the three-dimensional tensor case. The solutions are approximated using a Tucker tensor decomposition. The RAIL integrator first discretizes the partial differential equation fully in space and time using traditional methods. Here, spectral methods are considered for spatial discretizations, and implicit-explicit Runge-Kutta (IMEX RK) methods are used for time discretization. At each RK stage: the bases computed at the previous stages are augmented and reduced to construct projection subspaces. After updating the bases in a dimension-by-dimension manner, a Galerkin projection is performed to update the coefficients stored in the core tensor. As such, the algorithm balances high-order accuracy from spanning as many bases as possible from previous stages, with efficiency from leveraging low-rank structures in the solution. A post-processing step follows to maintain a low-rank solution while conserving mass, momentum, and energy. We validate the proposed method on a number of convection-diffusion problems, including a Fokker-Planck model, and a 3d viscous Burgers' equation.

math.NA↗

arXiv2025

Interpolatory dynamical low-rank approximation for the 3+3d Boltzmann-BGK equation

We introduce two novel interpolatory dynamical low-rank (DLR) approximation methods for the efficient time integration of the Boltzmann-BGK equation. Both methods overcome limitations of classic DLR schemes based on orthogonal projections for nonlinear equations. In particular, we demonstrate that the proposed methods can efficiently compute solutions to the full Boltzmann-BGK equation without restricting to e.g. weakly compressible or isothermal flow. The first method we propose directly applies the recently developed interpolatory projector-splitting scheme on low-rank matrix manifolds. The second method is a variant of the rank-adaptive basis update and Galerkin scheme, where the Galerkin step is replaced by a collocation step, resulting in a new scheme we call basis update and collocate (BUC). Numerical experiments in both fluid and kinetic regimes demonstrate the performance of the proposed methods. In particular we demonstrate that the methods can be used to efficiently compute low-rank solutions in the six-dimensional (three spatial and three velocity dimensions) setting on a standard laptop.

math.NA↗

arXiv2025

A review of low-rank methods for time-dependent kinetic simulations

Time-dependent kinetic models are ubiquitous in computational science and engineering. The underlying integro-differential equations in these models are high-dimensional, comprised of a six--dimensional phase space, making simulations of such phenomena extremely expensive. In this article we demonstrate that in many situations, the solution to kinetics problems lives on a low dimensional manifold that can be described by a low-rank matrix or tensor approximation. We then review the recent development of so-called low-rank methods that evolve the solution on this manifold. The two classes of methods we review are the dynamical low-rank (DLR) method, which derives differential equations for the low-rank factors, and a Step-and-Truncate (SAT) approach, which projects the solution onto the low-rank representation after each time step. Thorough discussions of time integrators, tensor decompositions, and method properties such as structure preservation and computational efficiency are included. We further show examples of low-rank methods as applied to particle transport and plasma dynamics.

math.NA↗

arXiv2025

Construction of high-order conservative basis-update and Galerkin dynamical low-rank integrators

Numerical simulations of kinetic problems can become prohibitively expensive due to their large memory requirements and computational costs. A method that has proven to successfully reduce these costs is the dynamical low-rank approximation (DLRA). A major accomplishment in the field of DLRA has been the derivation of robust time integrators that are not limited by the stiffness of the DLRA evolution equations. One key question is whether such robust time integrators can be made locally conservative, i.e., can they preserve the invariants and associated conservation laws of the original problem? In this work, we propose extensions to commonly used basis-update \& Galerkin (BUG) integrators that preserve invariants of the solution as well as the associated conservation laws with little or no additional computational cost. This approach requires only minor modifications of existing implementations. The properties of these integrators are investigated by performing numerical simulations in radiation transport and plasma physics.

math.NA↗

arXiv2025

A Galerkin Alternating Projection Method for Kinetic Equations in the Diffusive Limit

The numerical approximation of high-dimensional evolution equations poses significant computational challenges, particularly in kinetic theory and radiative transfer. In this work, we introduce the Galerkin Alternating Projection (GAP) scheme, a novel integrator derived within the Dynamical Low-Rank Approximation (DLRA) framework. We perform a rigorous error analysis, establishing local and global accuracy using standard ODE techniques. Furthermore, we prove that GAP possesses the Asymptotic-Preserving (AP) property when applied to the Radiative Transfer Equation (RTE), ensuring consistent behavior across both kinetic and diffusive regimes. In the diffusive regime, the K-step of the GAP integrator directly becomes the limit equation. In particular, this means that we can easily obtain schemes that even in the diffusive regime are free of a CFL condition, do not require well prepared initial data, and can have arbitrary order in the diffusive limit (in contrast to the semi-implicit and implicit schemes available in the literature). Numerical experiments support the theoretical findings and demonstrate the robustness and efficiency of the proposed method.

math.NA↗

arXiv2025

On the convergence of split exponential integrators for semilinear parabolic problems

Splitting the exponential-like $φ$ functions, which typically appear in exponential integrators, is attractive in many situations since it can dramatically reduce the computational cost of the procedure. However, depending on the employed splitting, this can result in order reduction. The aim of this paper is to analyze different such split approximations. We perform the analysis for semilinear problems in the abstract framework of commuting semigroups and derive error bounds that depend, in particular, on whether the vector (to which the $φ$ functions are applied) satisfies appropriate boundary conditions. We then present the convergence analysis for two split versions of a second-order exponential Runge--Kutta integrator in the context of analytic semigroups, and show that one suffers from order reduction while the other does not. Numerical results for semidiscretized parabolic PDEs confirm the theoretical findings.

math.NA↗