arXiv subjects

Philipp Krah

Publications and source records attributed to Philipp Krah.

At least 19 recordsLinked to original sources

arXiv2026

Revisiting kinetic electrostatic electron non-linear (KEEN) waves in the presence of dynamical ions

We revisit the kinetic electrostatic electron nonlinear (KEEN) waves studied by Afeyan et. al in 2014 using a hybrid flow-mapping strategy that combines the characteristic mapping method (CMM) with numerical flow iteration (NuFi). The study extends the classical setup to dynamical ions, comparing their impact on the long-time KEEN dynamics with the static-ion case. To this end, we extend the CMM-NuFI framework with a multi-map strategy, assigning one map to the ion and one to the electron characteristic flow. The resulting problem exhibits a wide separation of spatial and temporal scales, driven by the fine structures generated by the ponderomotive force and by the large ion-to-electron mass ratio, which renders it computationally prohibitive for conventional grid-based methods. Our multi-map CMM-NuFI method efficiently resolves these disparate scales, enabling long-time simulations of KEEN dynamics with fully dynamical ions.

physics.plasm-ph↗

arXiv2026

A Hybrid semi-Lagrangian Flow Mapping Approach for Vlasov Systems: Combining Iterative and Compositional Flow Maps

We propose a hybrid semi-Lagrangian scheme for the Vlasov--Poisson equation that combines the Numerical Flow Iteration (NuFI) method with the Characteristic Mapping Method (CMM). Both approaches exploit the semi-group property of the underlying diffeomorphic flow, enabling the reconstruction of solutions through flow maps that trace characteristics back to their initial positions. NuFI builds this flow map iteratively, preserving symplectic structure and conserving invariants, but its computational cost scales quadratically with time. Its advantage lies in a compact, low-dimensional representation depending only on the electric field. In contrast, CMM achieves low computational costs when remapping by composing the global flow map from explicitly stored submaps. The proposed hybrid method merges these strengths: NuFi is employed for accurate and conservative local time stepping, while CMM efficiently propagates the solution through submap composition. This approach reduces storage requirements, maintains accuracy, and improves structural properties. Numerical experiments demonstrate the effectiveness of the scheme and highlight the trade-offs between memory usage and computational cost. We benchmark against a semi-Lagrangian predictor-corrector scheme used in modern gyrokinetic codes, evaluating accuracy and conservation properties.

math.NA↗

arXiv2025

Intrinsic Dimension Estimating Autoencoder (IDEA) Using CancelOut Layer and a Projected Loss

This paper introduces the Intrinsic Dimension Estimating Autoencoder (IDEA), which identifies the underlying intrinsic dimension of a wide range of datasets whose samples lie on either linear or nonlinear manifolds. Beyond estimating the intrinsic dimension, IDEA is also able to reconstruct the original dataset after projecting it onto the corresponding latent space, which is structured using re-weighted double CancelOut layers. Our key contribution is the introduction of the projected reconstruction loss term, guiding the training of the model by continuously assessing the reconstruction quality under the removal of an additional latent dimension. We first assess the performance of IDEA on a series of theoretical benchmarks to validate its robustness. These experiments allow us to test its reconstruction ability and compare its performance with state-of-the-art intrinsic dimension estimators. The benchmarks show good accuracy and high versatility of our approach. Subsequently, we apply our model to data generated from the numerical solution of a vertically resolved one-dimensional free-surface flow, following a pointwise discretization of the vertical velocity profile in the horizontal direction, vertical direction, and time. IDEA succeeds in estimating the dataset's intrinsic dimension and then reconstructs the original solution by working directly within the projection space identified by the network.

cs.LG↗

arXiv2024

A Characteristic Mapping Method with Source Terms: Applications to Ideal Magnetohydrodynamics

This work introduces a generalized characteristic mapping method designed to handle non-linear advection with source terms. The semi-Lagrangian approach advances the flow map, incorporating the source term via the Duhamel integral. We derive a recursive formula for the time decomposition of the map and the source term integral, enhancing computational efficiency. Benchmark computations are presented for a test case with an exact solution and for two-dimensional ideal incompressible magnetohydrodynamics (MHD). Results demonstrate third-order accuracy in both space and time. The submap decomposition method achieves exceptionally high resolution, as illustrated by zooming into fine-scale current sheets. An error estimate is performed and suggests third order convergence in space and time.

math.NA↗

arXiv2024

Automated transport separation using the neural shifted proper orthogonal decomposition

This paper presents a neural network-based methodology for the decomposition of transport-dominated fields using the shifted proper orthogonal decomposition (sPOD). Classical sPOD methods typically require an a priori knowledge of the transport operators to determine the co-moving fields. However, in many real-life problems, such knowledge is difficult or even impossible to obtain, limiting the applicability and benefits of the sPOD. To address this issue, our approach estimates both the transport and co-moving fields simultaneously using neural networks. This is achieved by training two sub-networks dedicated to learning the transports and the co-moving fields, respectively. Applications to synthetic data and a wildland fire model illustrate the capabilities and efficiency of this neural sPOD approach, demonstrating its ability to separate the different fields effectively.

cs.LG↗

arXiv2024

Synthesizing impurity clustering in the edge plasma of tokamaks using neural networks

This work investigates the behavior of impurities in edge plasma of tokamaks using high-resolution numerical simulations based on Hasegawa--Wakatani equations. Specifically, it focuses on the behavior of inertial particles, which has not been extensively studied in the field of plasma physics. Our simulations utilize one-way coupling of a large number of inertial point particles, which model plasma impurities. We observe that with Stokes number ($St$) which characterizes the inertia of particles being much less than one, such light impurities closely track the fluid flow without pronounced clustering. For intermediate $St$ values, distinct clustering appears, with larger Stokes values, {\it i.e.} heavy impurities even generating more substantial clusters. When $St$ is significantly large, very heavy impurities tend to detach from the flow and maintain their trajectory, resulting in fewer observable clusters and corresponding to random motion. A core component of this work involves machine learning techniques. Applying three different neural networks - Autoencoder, U-Net, and Generative Adversarial Network (GAN) - to synthesize preferential concentration fields of impurities, we use vorticity as input and predict impurity number density fields. GAN outperforms the two others by aligning closely with direct numerical simulation data in terms of probability density functions of the particle distribution and energy spectra. This machine learning technique holds the potential to reduce computational costs by eliminating the need to track millions of particles modeling impurities in simulations.

physics.plasm-ph↗

arXiv2024

A robust shifted proper orthogonal decomposition: Proximal methods for decomposing flows with multiple transports

We present a new methodology for decomposing flows with multiple transports that further extends the shifted proper orthogonal decomposition (sPOD). The sPOD tries to approximate transport-dominated flows by a sum of co-moving data fields. The proposed methods stem from sPOD but optimize the co-moving fields directly and penalize their nuclear norm to promote low rank of the individual data in the decomposition. Furthermore, we add a robustness term to the decomposition that can deal with interpolation error and data noises. Leveraging tools from convex optimization, we derive three proximal algorithms to solve the decomposition problem. We report a numerical comparison with existing methods against synthetic data benchmarks and then show the separation ability of our methods on 1D and 2D incompressible and reactive flows. The resulting methodology is the basis of a new analysis paradigm that results in the same interpretability as the POD for the individual co-moving fields.

math.NA↗

arXiv2023

A Characteristic Mapping Method for Vlasov-Poisson with Extreme Resolution Properties

We propose an efficient semi-Lagrangian characteristic mapping method for solving the one+one-dimensional Vlasov-Poisson equations with high precision on a coarse grid. The flow map is evolved numerically and exponential resolution in linear time is obtained. Global third-order convergence in space and time is shown and conservation properties are assessed. For benchmarking, we consider linear and nonlinear Landau damping and the two-stream instability. We compare the results with a Fourier pseudo-spectral method. The extreme fine-scale resolution features are illustrated showing the method's capabilities to efficiently treat filamentation in fusion plasma simulations.

math.NA↗

arXiv2023

Model Order Reduction for the 1D Boltzmann-BGK Equation: Identifying Intrinsic Variables Using Neural Networks

Kinetic equations are crucial for modeling non-equilibrium phenomena, but their computational complexity is a challenge. This paper presents a data-driven approach using reduced order models (ROM) to efficiently model non-equilibrium flows in kinetic equations by comparing two ROM approaches: Proper Orthogonal Decomposition (POD) and autoencoder neural networks (AE). While AE initially demonstrate higher accuracy, POD's precision improves as more modes are considered. Notably, our work recognizes that the classical POD-MOR approach, although capable of accurately representing the non-linear solution manifold of the kinetic equation, may not provide a parsimonious model of the data due to the inherently non-linear nature of the data manifold. We demonstrate how AEs are used in finding the intrinsic dimension of a system and to allow correlating the intrinsic quantities with macroscopic quantities that have a physical interpretation.

physics.flu-dyn↗

arXiv2023

Parametric model order reduction for a wildland fire model via the shifted POD based deep learning method

Parametric model order reduction techniques often struggle to accurately represent transport-dominated phenomena due to a slowly decaying Kolmogorov n-width. To address this challenge, we propose a non-intrusive, data-driven methodology that combines the shifted proper orthogonal decomposition (POD) with deep learning. Specifically, the shifted POD technique is utilized to derive a high-fidelity, low-dimensional model of the flow, which is subsequently utilized as input to a deep learning framework to forecast the flow dynamics under various temporal and parameter conditions. The efficacy of the proposed approach is demonstrated through the analysis of one- and two-dimensional wildland fire models with varying reaction rates, and its performance is evaluated using multiple error measures. The results indicate that the proposed approach yields highly accurate results within the percent range, while also enabling rapid prediction of system states within seconds.

physics.flu-dyn↗

arXiv2023

Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: A study using POD-Galerkin and dynamical low rank approximation

Geophysical flow simulations using hyperbolic shallow water moment equations require an efficient discretization of a potentially large system of PDEs, the so-called moment system. This calls for tailored model order reduction techniques that allow for efficient and accurate simulations while guaranteeing physical properties like mass conservation. In this paper, we develop the first model reduction for the hyperbolic shallow water moment equations and achieve mass conservation. This is accomplished using a macro-micro decomposition of the model into a macroscopic (conservative) part and a microscopic (non-conservative) part with subsequent model reduction using either POD-Galerkin or dynamical low-rank approximation only on the microscopic (non-conservative) part. Numerical experiments showcase the performance of the new model reduction methods including high accuracy and fast computation times together with guaranteed conservation and consistency properties.

math.NA↗

arXiv2022

Crystalline phases at finite winding densities in a quantum link ladder

Condensed matter physics of gauge theories coupled to fermions can exhibit a rich phase structure, but are nevertheless very difficult to study in Monte Carlo simulations when they are afflicted by a sign problem. As an alternate approach, we use tensor network methods to explore the finite density physics of Abelian gauge theories without dynamical matter. As a concrete example, we consider the U(1) gauge invariant quantum link ladder with spin-1/2 gauge fields in an external electric field which cause the winding electric fluxes to condense in the ground state. We demonstrate how the electric flux tubes arrange themselves in the bulk giving rise to crystalline patterns, whose period can be controlled by tuning the external field. We propose observables to detect the transitions in ground state properties not only in numerical experiments, but also in future cold-atom realizations. A systematic procedure for reaching the thermodynamic limit, as well as extending the studies from ladders to extended geometries is outlined.

hep-lat↗

arXiv2022

Front Transport Reduction for Complex Moving Fronts

This work addresses model order reduction for complex moving fronts, which are transported by advection or through a reaction-diffusion process. Such systems are especially challenging for model order reduction since the transport cannot be captured by linear reduction methods. Moreover, topological changes, such as splitting or merging of fronts pose difficulties for many nonlinear reduction methods and the small non-vanishing support of the underlying partial differential equations dynamics makes most nonlinear hyper-reduction methods infeasible. We propose a new decomposition method together with a hyper-reduction scheme that addresses these shortcomings. The decomposition uses a level-set function to parameterize the transport and a nonlinear activation function that captures the structure of the front. This approach is similar to autoencoder artificial neural networks, but additionally provides insights into the system, which can be used for efficient reduced order models. We make use of this property and are thus able to solve the advection equation with the same complexity as the POD-Galerkin approach while obtaining errors of less than one percent for representative examples. Furthermore, we outline a special hyper-reduction method for more complicated advection-reaction-diffusion systems. The capability of the approach is illustrated by various numerical examples in one and two spatial dimensions, including real-life applications to a two-dimensional Bunsen flame.

math.NA↗

arXiv2021

Supervised Learning for Multi Zone Sound Field Reproduction under Harsh Environmental Conditions

This manuscript presents an approach for multi zone sound field reproduction using supervised learning. Traditional multi zone sound field reproduction methods assume constant speed of sound, neglecting nonlinear effects like wind and temperature stratification. We show how to overcome these restrictions using supervised learning of transfer functions. The quality of the solution is measured by the acoustic contrast and the reproduction error. Our results show that for the chosen setup, even with relatively small wind speeds, the acoustic contrast and reproduction error can be improved by up to 16 dB, when wind is considered in the trained model.

cs.SD↗

arXiv2021

Phases at finite winding number of an Abelian lattice gauge theory

Pure gauge theories are rather different from theories with pure scalar and fermionic matter, especially in terms of the nature of excitations. For example, in scalar and fermionic theories, one can create ultra-local excitations. For a gauge theory, such excitations need to be closed loops that do not violate gauge invariance. In this proceedings, we present a study on the condensation phenomenon associated with the string-like excitations of an Abelian lattice gauge theory. These phenomena are studied through numerical simulations of a $U(1)$ quantum link model in 2+1 dimensions in a ladder geometry using matrix product states. In this proceedings, we show the existence of ground states characterized by the presence of such string-like excitations. These are caused due to the condensation of torelons. We also study the relationship between the properties of the plaquettes in the ground state and the presence of such condensation phenomenon.

hep-lat↗

arXiv2020

Wavelet Adaptive Proper Orthogonal Decomposition for Large Scale Flow Data

The proper orthogonal decomposition (POD) is a powerful classical tool in fluid mechanics used, for instance, for model reduction and extraction of coherent flow features. However, its applicability to high-resolution data, as produced by three-dimensional direct numerical simulations, is limited owing to its computational complexity. Here, we propose a wavelet-based adaptive version of the POD (the wPOD), in order to overcome this limitation. The amount of data to be analyzed is reduced by compressing them using biorthogonal wavelets, yielding a sparse representation while conveniently providing control of the compression error. Numerical analysis shows how the distinct error contributions of wavelet compression and POD truncation can be balanced under certain assumptions, allowing us to efficiently process high-resolution data from three-dimensional simulations of flow problems. Using a synthetic academic test case, we compare our algorithm with the randomized singular value decomposition. Furthermore, we demonstrate the ability of our method analyzing data of a 2D wake flow and a 3D flow generated by a flapping insect computed with direct numerical simulation.

physics.flu-dyn↗

arXiv2019

Model Order Reduction of Combustion Processes with Complex Front Dynamics

In this work we present a data driven method, used to improve mode-based model order reduction of transport fields with sharp fronts. We assume that the original flow field $q(\mathbf{x},t)=f(\phi(\mathbf{x},t))$ can be reconstructed by a front shape function $f$ and a level set function $\phi$. The level set function is used to generate a local coordinate, which parametrizes the distance to the front. In this way, we are able to embed the local 1D description of the front for complex 2D front dynamics with merging or splitting fronts, while seeking a low rank description of $\phi$. Here, the freedom of choosing $\phi$ far away from the front can be used to find a low rank description of $\phi$ which accelerates the convergence of $\Vert q- f(\phi_n)\Vert$, when truncating $\phi$ after the $n$th mode. We demonstrate the ability of this new ansatz for a 2D propagating flame with a moving front.

math.DS↗

arXiv2019

An open and parallel multiresolution framework using block-based adaptive grids

A numerical approach for solving evolutionary partial differential equations in two and three space dimensions on block-based adaptive grids is presented. The numerical discretization is based on high-order, central finite-differences and explicit time integration. Grid refinement and coarsening are triggered by multiresolution analysis, i.e. thresholding of wavelet coefficients, which allow controlling the precision of the adaptive approximation of the solution with respect to uniform grid computations. The implementation of the scheme is fully parallel using MPI with a hybrid data structure. Load balancing relies on space filling curves techniques. Validation tests for 2D advection equations allow to assess the precision and performance of the developed code. Computations of the compressible Navier-Stokes equations for a temporally developing 2D mixing layer illustrate the properties of the code for nonlinear multi-scale problems. The code is open source.

physics.comp-ph↗