arXiv subjects

Lukas Einkemmer

Publications and source records attributed to Lukas Einkemmer.

At least 37 records · Page 2Linked to original sources

arXiv2025

Control of Instability in a Vlasov-Poisson System Through an External Electric Field

Plasma instabilities are a major concern in plasma science, for applications ranging from particle accelerators to nuclear fusion reactors. In this work, we consider the possibility of controlling such instabilities by adding an external electric field to the Vlasov--Poisson equations. Our approach to determining the external electric field is based on conducting a linear analysis of the resulting equations. We show that it is possible to select external electric fields that completely suppress the plasma instabilities present in the system when the equilibrium distribution and the perturbation are known. In fact, the proposed strategy returns the plasma to its equilibrium with a rate that is faster than exponential in time. We further perform numerical simulations of the nonlinear two-stream and bump-on-tail instabilities to verify our theory and to compare the different strategies that we propose in this work.

physics.plasm-ph↗

arXiv2025

Asymptotic-Preserving Dynamical Low-Rank Method for the Stiff Nonlinear Boltzmann Equation

In kinetic theory, numerically solving the full Boltzmann equation is extremely expensive. This is because the Boltzmann collision operator involves a high-dimensional, nonlinear integral that must be evaluated at each spatial grid point and every time step. The challenge becomes even more pronounced in the fluid (strong collisionality) regime, where the collision operator exhibits strong stiffness, causing explicit time integrators to impose severe stability restrictions. In this paper, we propose addressing this problem through a dynamical low-rank (DLR) approximation. The resulting algorithm requires evaluating the Boltzmann collision operator only $r^2$ times, where $r$, the rank of the approximation, is much smaller than the number of spatial grid points. We propose a novel DLR integrator, called the XL integrator, which reduces the number of steps compared to the available alternatives (such as the projector splitting or basis update & Galerkin (BUG) integrator). For a class of problems including the Boltzmann collision operator which enjoys a separation property between physical and velocity space, we further propose a specialized version of the XL integrator, called the sXL integrator. This version requires solving only one differential equation to update the low-rank factors. Furthermore, the proposed low-rank schemes are asymptotic-preserving, meaning they can capture the asymptotic fluid limit in the case of strong collisionality. Our numerical experiments demonstrate the efficiency and accuracy of the proposed methods across a wide range of regimes, from non-stiff (kinetic) to stiff (fluid).

math.NA↗

arXiv2025

An initial-corrected splitting approach for convection-diffusion-reaction problems

Splitting methods constitute a well-established class of numerical schemes for solving convection-diffusion-reaction problems. They have been shown to be effective in solving problems with periodic boundary conditions. However, in the case of Dirichlet boundary conditions, order reduction has been observed even with homogeneous boundary conditions. In this paper, we propose a novel splitting approach, the so-called `initial-corrected splitting method', which succeeds in overcoming order reduction. A convergence analysis is performed to demonstrate second-order convergence of this modified Strang splitting method. Furthermore, we conduct numerical experiments to illustrate the performance of the newly developed splitting approach.

math.NA↗

arXiv2025

Automatic partitioning for the low-rank integration of stochastic Boolean reaction networks

Boolean reaction networks are an important tool in biochemistry for studying mechanisms in the biological cell. However, the stochastic formulation of such networks requires the solution of a master equation which inherently suffers from the curse of dimensionality. In the past, the dynamical low-rank (DLR) approximation has been repeatedly used to solve high-dimensional reaction networks by separating the network into smaller partitions. However, the partitioning of these networks was so far only done by hand. In this paper, we present a heuristic, automatic partitioning scheme based on two ingredients: the Kernighan-Lin algorithm and information entropy. Our approach is computationally inexpensive and can be easily incorporated as a preprocessing step into the existing simulation workflow. We test our scheme by partitioning Boolean reaction networks on a single level and also in a hierarchical fashion with tree tensor networks. The resulting accuracy of the scheme is superior to both partitionings chosen by human experts and those found by simply minimizing the number of reaction pathways between partitions.

math.NA↗

arXiv2024

Stabilization of beam heated plasmas by beam modulation

A constant intensity beam that propagates into a stationary plasma results in a bump-on-tail feature in velocity space. This results in an instability that transfers kinetic energy from the plasma to the electric field. We show that there are intensity profiles for the beam (found by numerical optimization) that can largely suppress this instability and drive the system into a state that, after the beam has been switched off, remains stable over long times. The modulated beam intensity requires no feedback, i.e. no knowledge of the physical system during time evolution is required, and the frequency of the modulation scales approximately inversely with system size, which is particularly favorable for large plasma systems. We also show that the results obtained are robust in the sense that perturbations, e.g. deviation from the optimized beam profiles, can be tolerated without losing the ability to suppress the instability.

physics.plasm-ph↗

arXiv2024

LeXInt: GPU-accelerated Exponential Integrators package

We present an open-source CUDA-based package that consists of a compilation of exponential integrators where the action of the matrix exponential or the $φ_l$ functions on a vector is approximated using the method of polynomial interpolation at Leja points. Using a couple of test examples on an NVIDIA A100 GPU, we show that one can achieve significant speedups using CUDA over the corresponding CPU code. LeXInt, written in a modular format, facilitates easy integration into any existing software package, and can be used for temporal integration of any differential equation.

math.NA↗

arXiv2024

Should exponential integrators be used for advection-dominated problems?

In this paper, we consider the application of exponential integrators to problems that are advection dominated, either on the entire or on a subset of the domain. In this context, we compare Leja and Krylov based methods to compute the action of exponential and related matrix functions. We set up a performance model by counting the different operations needed to implement the considered algorithms. This model assumes that the evaluation of the right-hand side is memory bound and allows us to evaluate performance in a hardware independent way. We find that exponential integrators perform comparably to explicit Runge-Kutta schemes for problems that are advection dominated in the entire domain. Moreover, they are able to outperform explicit methods in situations where small parts of the domain are diffusion dominated. We generally observe that Leja based methods outperform Krylov iterations in the problems considered. This is in particular true if computing inner products is expensive.

math.NA↗

arXiv2024

Reduced Augmentation Implicit Low-rank (RAIL) integrators for advection-diffusion and Fokker-Planck models

This paper introduces a novel computational approach termed the Reduced Augmentation Implicit Low-rank (RAIL) method by investigating two predominant research directions in low-rank solutions to time-dependent partial differential equations (PDEs): dynamical low-rank (DLR), and step and truncation (SAT) tensor methods. The RAIL method, along with the development of the SAT approach, is designed to enhance the efficiency of traditional full-rank implicit solvers from method-of-lines discretizations of time-dependent PDEs, while maintaining accuracy and stability. We consider spectral methods for spatial discretization, and diagonally implicit Runge-Kutta (DIRK) and implicit-explicit (IMEX) RK methods for time discretization. The efficiency gain is achieved by investigating low-rank structures within solutions at each RK stage using a singular value decomposition (SVD). In particular, we develop a reduced augmentation procedure to predict the basis functions to construct projection subspaces. This procedure balances algorithm accuracy and efficiency by incorporating as many bases as possible from previous RK stages and predictions, and by optimizing the basis representation through SVD truncation. As such, one can form implicit schemes for updating basis functions in a dimension-by-dimension manner, similar in spirit to the K-L step in the DLR framework. We also apply a globally mass conservative post-processing step at the end of each RK stage. We validate the RAIL method through numerical simulations of advection-diffusion problems and a Fokker-Planck model, showcasing its ability to efficiently handle time-dependent PDEs while maintaining global mass conservation. Our approach generalizes and bridges the DLR and SAT approaches, offering a comprehensive framework for efficiently and accurately solving time-dependent PDEs with implicit treatment.

math.NA↗

arXiv2024

Kinetic scrape off layer simulations with semi-Lagrangian discontinuous Galerkin schemes

In this paper we propose a semi-Lagrangian discontinuous Galerkin solver for the simulation of the scrape off layer for an electron-ion plasma. We use a time adaptive velocity space to deal with fast particles leaving the computational domain, a block structured mesh to resolve the sharp gradient in the plasma sheath, and limiters to avoid oscillations in the density function. In particular, we propose a limiter that can be computed directly from the information used in the semi-Lagrangian discontinuous Galerkin advection step. This limiter is particularly efficient on graphic processing units (GPUs) and compares favorable with limiters from the literature. We provide numerical results for a set of benchmark problems and compare different limiting strategies.

math.NA↗

arXiv2024

A hierarchical dynamical low-rank algorithm for the stochastic description of large reaction networks

The stochastic description of chemical reaction networks with the kinetic chemical master equation (CME) is important for studying biological cells, but it suffers from the curse of dimensionality: The amount of data to be stored grows exponentially with the number of chemical species and thus exceeds the capacity of common computational devices for realistic problems. Therefore, time-dependent model order reduction techniques such as the dynamical low-rank approximation are desirable. In this paper we propose a dynamical low-rank algorithm for the kinetic CME using binary tree tensor networks. The dimensionality of the problem is reduced in this approach by hierarchically dividing the reaction network into partitions. Only reactions that cross partitions are subject to an approximation error. We demonstrate by two numerical examples (a 5-dimensional lambda phage model and a 20-dimensional reaction cascade) that the proposed method drastically reduces memory consumption and shows improved computational performance and better accuracy compared to a Monte Carlo method.

math.NA↗

arXiv2024

A robust second-order low-rank BUG integrator based on the midpoint rule

Dynamical low-rank approximation has become a valuable tool to perform an on-the-fly model order reduction for prohibitively large matrix differential equations. A core ingredient is the construction of integrators that are robust to the presence of small singular values and the resulting large time derivatives of the orthogonal factors in the low-rank matrix representation. Recently, the robust basis-update & Galerkin (BUG) class of integrators has been introduced. These methods require no steps that evolve the solution backward in time, often have favourable structure-preserving properties, and allow for parallel time-updates of the low-rank factors. The BUG framework is flexible enough to allow for adaptations to these and further requirements. However, the BUG methods presented so far have only first-order robust error bounds. This work proposes a second-order BUG integrator for dynamical low-rank approximation based on the midpoint rule. The integrator first performs a half-step with a first-order BUG integrator, followed by a Galerkin update with a suitably augmented basis. We prove a robust second-order error bound which in addition shows an improved dependence on the normal component of the vector field. These rigorous results are illustrated and complemented by a number of numerical experiments.

math.NA↗

arXiv2023

Energy stable and conservative dynamical low-rank approximation for the Su-Olson problem

Computational methods for thermal radiative transfer problems exhibit high computational costs and a prohibitive memory footprint when the spatial and directional domains are finely resolved. A strategy to reduce such computational costs is dynamical low-rank approximation (DLRA), which represents and evolves the solution on a low-rank manifold, thereby significantly decreasing computational and memory requirements. Efficient discretizations for the DLRA evolution equations need to be carefully constructed to guarantee stability while enabling mass conservation. In this work, we focus on the Su-Olson closure leading to a linearized internal energy model and derive a stable discretization through an implicit coupling of internal energy and particle density. Moreover, we propose a rank-adaptive strategy to preserve local mass conservation. Numerical results are presented which showcase the accuracy and efficiency of the proposed low-rank method compared to the solution of the full system.

math.NA↗

arXiv2023

Accelerating exponential integrators to efficiently solve semilinear advection-diffusion-reaction equations

In this paper we consider an approach to improve the performance of exponential Runge--Kutta integrators and Lawson schemes} in cases where the solution of a related, but usually much simpler, problem can be computed efficiently. While for implicit methods such an approach is common (e.g. by using preconditioners), for exponential integrators this has proven more challenging. Here we propose to extract a constant coefficient differential operator from the semilinear advection-diffusion-reaction equation for which, in many situations, efficient methods are known to compute the required matrix functions. Both a linear stability analysis and {\color{black} extensive} numerical experiments show that the resulting schemes can be unconditionally stable. In fact, we find that exponential integrators of Runge--Kutta type and Lawson schemes can have better stability properties than similarly constructed implicit-explicit schemes. We also derive two new Lawson type integrators that further improve on these stability properties. The overall effectiveness of the approach is highlighted by a number of performance comparisons on examples in two and three space dimensions.

math.NA↗

arXiv2023

A low-rank complexity reduction algorithm for the high-dimensional kinetic chemical master equation

It is increasingly realized that taking stochastic effects into account is important in order to study biological cells. However, the corresponding mathematical formulation, the chemical master equation (CME), suffers from the curse of dimensionality and thus solving it directly is not feasible for most realistic problems. In this paper we propose a dynamical low-rank algorithm for the CME that reduces the dimensionality of the problem by dividing the reaction network into partitions. Only reactions that cross partitions are subject to an approximation error (everything else is computed exactly). This approach, compared to the commonly used stochastic simulation algorithm (SSA, a Monte Carlo method), has the advantage that it is completely noise-free. This is particularly important if one is interested in resolving the tails of the probability distribution. We show that in some cases (e.g. for the lambda phage) the proposed method can drastically reduce memory consumption and run time and provide better accuracy than SSA.

math.NA↗

arXiv2023

Efficient numerical methods for Anisotropic Diffusion of Galactic Cosmic Rays

Anisotropic diffusion is imperative in understanding cosmic ray diffusion across the Galaxy, the heliosphere, and the interplay of cosmic rays with the Galactic magnetic field. This diffusion term contributes to the highly stiff nature of the cosmic ray transport equation. To conduct numerical simulations of time-dependent cosmic ray transport, implicit integrators (namely, Crank-Nicolson (CN)) have been traditionally favoured over the CFL-bound explicit integrators in order to be able to take large step sizes. We propose exponential methods to treat the linear anisotropc diffusion equation in the presence of advection and time-independent and time-dependent sources. These methods allow us to take even larger step sizes that can substantially speed-up the simulations whilst generating highly accurate solutions. In or subsequent work, we will use these exponential solvers in the Picard code to study anisotropic cosmic ray diffusion and we will consider additional physical processes such as continuous momentum losses and reacceleration.

astro-ph.HE↗

arXiv2023

Accelerating the simulation of kinetic shear Alfvén waves with a dynamical low-rank approximation

We propose a dynamical low-rank algorithm for a gyrokinetic model that is used to describe strongly magnetized plasmas. The low-rank approximation is based on a decomposition into variables parallel and perpendicular to the magnetic field, as suggested by the physics of the underlying problem. We show that the resulting scheme exactly recovers the dispersion relation even with rank 1. We then perform a simulation of kinetic shear Alfvén waves and show that using the proposed dynamical low-rank algorithm a drastic reduction (multiple orders of magnitude) in both computational time and memory consumption can be achieved. We also compare the performance of robust first and second-order projector splitting, BUG (also called unconventional), and augmented BUG integrators as well as a FFT-based spectral and Lax--Wendroff discretization.

physics.comp-ph↗

arXiv2023

Suppressing Instability in a Vlasov-Poisson System by an External Electric Field Through Constrained Optimization

Fusion energy offers the potential for the generation of clean, safe, and nearly inexhaustible energy. While notable progress has been made in recent years, significant challenges persist in achieving net energy gain. Improving plasma confinement and stability stands as a crucial task in this regard and requires optimization and control of the plasma system. In this work, we deploy a PDE-constrained optimization formulation that uses a kinetic description for plasma dynamics as the constraint. This is to optimize, over all possible controllable external electric fields, the stability of the plasma dynamics under the condition that the Vlasov--Poisson (VP) equation is satisfied. For computing the functional derivative with respect to the external field in the optimization updates, the adjoint equation is derived. Furthermore, in the discrete setting, where we employ the semi-Lagrangian method as the forward solver, we also explicitly formulate the corresponding adjoint solver and the gradient as the discrete analogy to the adjoint equation and the Frechet derivative. A distinct feature we observed of this constrained optimization is the complex landscape of the objective function and the existence of numerous local minima, largely due to the hyperbolic nature of the VP system. To overcome this issue, we utilize a gradient-accelerated genetic algorithm, leveraging the advantages of the genetic algorithm's exploration feature to cover a broader search of the solution space and the fast local convergence aided by the gradient information. We show that our algorithm obtains good electric fields that are able to maintain a prescribed profile in a beam shaping problem and uses nonlinear effects to suppress plasma instability in a two-stream configuration.

math.NA↗

arXiv2023

A semi-Lagrangian discontinuous Galerkin method for drift-kinetic simulations on GPUs

In this paper, we demonstrate the efficiency of using semi-Lagrangian discontinuous Galerkin methods to solve the drift-kinetic equation using graphic processing units (GPUs). In this setting we propose a second order splitting scheme and a 2d semi-Lagrangian scheme in the poloidal plane. The resulting method is able to conserve mass up to machine precision, allows us to take large time steps due to the absence of a CFL condition and provides local data dependency which is essential to obtain good performance on state-of-the art high-performance computing systems. We report simulations of a drift-kinetic ion temperature gradient (ITG) instability and show that our implementation achieves a performance of up to 600 GB/s on an A100 GPU.

physics.comp-ph↗