arXiv subjects

Thomas Richter

Publications and source records attributed to Thomas Richter.

At least 19 recordsLinked to original sources

arXiv2026

A Discontinuous Galerkin discretization for the sea ice dynamics

Sea ice dynamics plays a crucial role in the Earth's climate system, making it an important component of weather and climate prediction models. Its numerical simulation remains challenging, however, as it exhibits complex mechanical behaviors due to a nonlinear ice rheology and interactions with external physical forcings. In particular, sea ice presents linear kinematic features (LKFs), i.e., narrow bands of intense deformation associated with processes such as lead opening or pressure-ridge formation. Accurately representing these features is necessary as they affect thermodynamics and ocean-atmosphere exchange. Yet their number, localization, and structure are highly sensitive to spatial resolution and to the chosen discretization of the sea ice velocity. In this work, we investigate the spatial discretization of Hibler's viscous plastic sea-ice model using a fully discontinuous Galerkin (DG) representation of all variables, including the velocity field. The ability of DG elements to represent discontinuities while having high order local polynomial approximation makes them well suited for resolving the complex ice deformation features, and insuring robustness towards mesh-induced numerical artefacts. We assess the fully DG method using an established sea ice dynamics benchmark and compare the obtained sea ice deformation with state-of-the-art discretizations. Additionally, we study the theoretical convergence of the modified Elastic-Viscous-Plastic (mEVP) formulation as a pseudo-time iterative solver for the viscous-plastic (VP) momentum equations. We prove the convergence of the underlying continuous pseudo-time dynamical system towards the viscous plastic formal limit, thereby providing a theoretical foundation for the use of mEVP as an iterative solver for Hibler's VP model.

math.NA↗

arXiv2026

JPEG AIC2026: A large-scale dataset for fine-grained assessment of image coding

Recent advances in conventional and learning-based image coding have increased the demand for benchmark datasets that support fine-grained assessment of compressed image quality, particularly for learning-based image compression methods. This paper introduces Assessment of Image Coding 2026 (AIC2026), a large-scale dataset for high-fidelity image compression containing 70 source images selected from 2,787 candidates using semantic clustering, inter-metric disagreement among objective image quality assessment (IQA) methods, and manual inspection and refinement. The dataset covers a wide range of compression artifacts produced by eight conventional and four learning-based codecs across 17 coding configurations. Each source image is encoded using seven codecs. For each source-codec pair, decoded images are provided at 20 perceptually spaced distortion levels, corresponding approximately to 0.2-4.0 just-noticeable difference (JND) units using the ColorVideoVDP (CVVDP) metric for distortion estimation, yielding 9,618 distorted images. This fine-grained sampling enables analysis of rate-distortion behavior and objective metric evaluation for subtle quality differences across a wide range of compression artifacts. We report an extensive objective analysis using 24 conventional and 12 learning-based IQA methods. The results show substantial disagreement among current IQA methods for fine-grained quality differences, particularly for artifacts introduced by learning-based codecs. The complete dataset is publicly available at https://doi.org/10.18419/DARUS-6156.

eess.IV↗

arXiv2026

Neural enrichment finite element method: A hybrid framework for problems with strong oscillations or interface problems

We propose a hybrid method, the Neural Enrichment Finite Element Method (NEFEM), designed for problems involving strong oscillations or interface problems with weak discontinuities. This method is based on the stable generalized finite element method (SGFEM) framework, wherein neural networks (NNs) are introduced as enrichment functions for adaptivity, and the Ritz functional is applied for the training process. This works makes two main contributions. First, the method constructs local subspaces with superior approximation properties, significantly reducing the required number of degrees of freedom (DoFs). Second, minimal \emph{a priori} knowledge is required to define enrichment functions, as the NNs evolve heuristically during training. Furthermore, for smooth problems, we provide a residual-based error estimator and prove both its reliability and efficiency. For interface problems, a theoretical analysis on the optimal convergence of the SGFEM is studied, notably without imposing additional regularity assumptions. These analytic results guide the network architecture design and training strategies. The performance and effectiveness of the proposed method is validated through several numerical experiments.

math.NA↗

arXiv2026

Goal oriented error estimation for adaptive sampling of PINNS

Physics-Informed Neural Networks (PINNs) are mesh-free approaches for the numerical approximation of partial differential equations, where a neural network is trained by minimizing a loss function derived from the governing equations and boundary conditions. The Deep Ritz method can be interpreted as a particular variational form of a PINN, where the loss corresponds to the minimization of an energy functional associated with a symmetric positive definite problem. In this work, we study the approximation of the Laplace equation using both the classical PINN formulation and its variational counterpart, the Deep Ritz method, with the objective of accurately estimating prescribed goal functionals. When standard sampling strategies, such as uniform or loss-based sampling, are employed during training, the convergence of the functional error and the attained minimal functional value can be slow. To address this issue, we introduce a functional-oriented importance sampling strategy that can be applied to both PINNs and the Deep Ritz method. The key ingredient is the construction of a reliable and accurate estimator for the error in a given quantity of interest. This estimator is derived using concepts from the Dual Weighted Residual (DWR) framework and is implemented entirely within the neural network setting. It is then used to adaptively guide the sampling of training points in the computational domain, focusing computational effort on regions that have the strongest influence on the functional value. Numerical experiments demonstrate that the proposed adaptive sampling strategy significantly accelerates the convergence of the functional error and improves the minimization of the target functional during training for both PINN and Deep Ritz formulations.

math.NA↗

arXiv2026

A robust and stable hybrid neural network/finite element method for 2D flows that generalizes to different geometries

The deep neural network multigrid solver (DNN-MG) combines a coarse-grid finite element simulation with a deep neural network that corrects the solution on finer grid levels, thereby improving the computational efficiency. In this work, we discuss various design choices for the DNN-MG method and demonstrate significant improvements in accuracy and generalizability when applied to the solution of the nonstationary Navier-Stokes equations. We investigate the stability of the hybrid simulation and show how the neural networks can be made more robust with the help of replay buffers. After an initial single-step training, we run the hybrid simulation for extended periods and compute new reference solutions of the neural network perturbed state for each step. By retraining on this data, the error caused by the neural network over multiple time-steps due to distributional shift can be effectively reduced without the need for a differentiable numerical solver. Furthermore, we compare multiple neural network architectures, including recurrent neural networks and Transformers, and study their ability to utilize more information from an increased temporal and spatial receptive field. Transformers allow us to make use of information from cells outside the predicted patch even with unstructured meshes while maintaining the locality of our approach. This can further improve the accuracy of DNN-MG without a significant impact on performance.

math.NA↗

arXiv2025

Dual Weighted Residual-driven adaptive mesh refinement to enhance biomechanical simulations

This chapter describes how a posteriori error estimates targeting a user-defined quantity of interest, using the Dual Weighted Residual (DWR) technique, can be easily applied for biomechanical simulations in current engineering practice. The proposed method considers a very general setting that encompasses complex geometries, model non-linearities (hyperelasticity, fluid-structure interaction) and multi-goal oriented techniques. The developments are substantiated with some numerical tests.

math.NA↗

arXiv2025

Revising Second Order Terms in Deep Animation Video Coding

First Order Motion Model is a generative model that animates human heads based on very little motion information derived from keypoints. It is a promising solution for video communication because first it operates at very low bitrate and second its computational complexity is moderate compared to other learning based video codecs. However, it has strong limitations by design. Since it generates facial animations by warping source-images, it fails to recreate videos with strong head movements. This works concentrates on one specific kind of head movements, namely head rotations. We show that replacing the Jacobian transformations in FOMM by a global rotation helps the system to perform better on items with head-rotations while saving 40% to 80% of bitrate on P-frames. Moreover, we apply state-of-the-art normalization techniques to the discriminator to stabilize the adversarial training which is essential for generating visually appealing videos. We evaluate the performance by the learned metics LPIPS and DISTS to show the success our optimizations.

eess.IV↗

arXiv2025

On the Propulsion of a Rigid Body in a Viscous Liquid by Time-Periodic Force with a Zero Average

We perform analytical and numerical analyses of the propulsion of a rigid body in a viscous fluid subjected to a periodic force with zero average over a period. This general formulation specifically addresses the significant case, where propulsion is generated by the oscillation of a mass located in an internal cavity of the body. We provide a rigorous proof of the necessary and sufficient conditions for propulsion at the second order of magnitude of the force. These conditions are implemented and confirmed by numerical tests for bodies without fore-and-aft symmetry, while they are silent for bodies with such symmetry, like round ellipsoids. Consequently, in this case, propulsion can only occur at an order higher than the second. This problem is investigated by numerically integrating the entire set of equations, and the result shows that, in fact, propulsion does occur, thus opening new avenues for further analytical studies.

math.AP↗

arXiv2025

Fine-Grained HDR Image Quality Assessment From Noticeably Distorted to Very High Fidelity

High dynamic range (HDR) and wide color gamut (WCG) technologies significantly improve color reproduction compared to standard dynamic range (SDR) and standard color gamuts, resulting in more accurate, richer, and more immersive images. However, HDR increases data demands, posing challenges for bandwidth efficiency and compression techniques. Advances in compression and display technologies require more precise image quality assessment, particularly in the high-fidelity range where perceptual differences are subtle. To address this gap, we introduce AIC-HDR2025, the first such HDR dataset, comprising 100 test images generated from five HDR sources, each compressed using four codecs at five compression levels. It covers the high-fidelity range, from visible distortions to compression levels below the visually lossless threshold. A subjective study was conducted using the JPEG AIC-3 test methodology, combining plain and boosted triplet comparisons. In total, 34,560 ratings were collected from 151 participants across four fully controlled labs. The results confirm that AIC-3 enables precise HDR quality estimation, with 95\% confidence intervals averaging a width of 0.27 at 1 JND. In addition, several recently proposed objective metrics were evaluated based on their correlation with subjective ratings. The dataset is publicly available.

cs.CV↗

arXiv2025

The JPEG XL Image Coding System: History, Features, Coding Tools, Design Rationale, and Future

JPEG XL is a new image coding system offering state-of-the-art compression performance, lossless JPEG recompression, and advanced features. It aims to replace JPEG, PNG, GIF, and other formats with a single universal codec. This article provides an overview of JPEG XL, including its history, design rationale, coding tools, and future potential. It can be used as a companion document to the standard (ISO/IEC 18181), or as a standalone article to better understand JPEG XL, either at a high level or in considerable technical detail.

cs.MM↗

arXiv2025

A CFL-type Condition and Theoretical Insights for Discrete-Time Sparse Full-Order Model Inference

In this work, we investigate the data-driven inference of a discrete-time dynamical system via a sparse Full-Order Model (sFOM). We first formulate the involved Least Squares (LS) problem and discuss the need for regularization, indicating a connection between the typically employed $l_2$ regularization and the stability of the inferred discrete-time sFOM. We then provide theoretical insights considering the consistency and stability properties of the inferred numerical schemes that form the sFOM and exemplify them via illustrative, 1D test cases of linear diffusion and linear advection. For linear advection, we analytically derive a "sampling CFL" condition, which dictates a bound for the ratio of spatial and temporal discretization steps in the training data that ensures stability of the inferred sFOM. Finally, we investigate the sFOM inference for two nonlinear problems, namely a 2D Burgers' test case and the incompressible flow in an oscillating lid driven cavity, and draw connections between the theoretical findings and the properties of the inferred, nonlinear sFOMs.

math.DS↗

arXiv2025

LoC-LIC: Low Complexity Learned Image Coding Using Hierarchical Feature Transforms

Current learned image compression models typically exhibit high complexity, which demands significant computational resources. To overcome these challenges, we propose an innovative approach that employs hierarchical feature extraction transforms to significantly reduce complexity while preserving bit rate reduction efficiency. Our novel architecture achieves this by using fewer channels for high spatial resolution inputs/feature maps. On the other hand, feature maps with a large number of channels have reduced spatial dimensions, thereby cutting down on computational load without sacrificing performance. This strategy effectively reduces the forward pass complexity from $1256 \, \text{kMAC/Pixel}$ to just $270 \, \text{kMAC/Pixel}$. As a result, the reduced complexity model can open the way for learned image compression models to operate efficiently across various devices and pave the way for the development of new architectures in image compression technology.

eess.IV↗

arXiv2025

Compact Latent Representation for Image Compression (CLRIC)

Current image compression models often require separate models for each quality level, making them resource-intensive in terms of both training and storage. To address these limitations, we propose an innovative approach that utilizes latent variables from pre-existing trained models (such as the Stable Diffusion Variational Autoencoder) for perceptual image compression. Our method eliminates the need for distinct models dedicated to different quality levels. We employ overfitted learnable functions to compress the latent representation from the target model at any desired quality level. These overfitted functions operate in the latent space, ensuring low computational complexity, around $25.5$ MAC/pixel for a forward pass on images with dimensions $(1363 \times 2048)$ pixels. This approach efficiently utilizes resources during both training and decoding. Our method achieves comparable perceptual quality to state-of-the-art learned image compression models while being both model-agnostic and resolution-agnostic. This opens up new possibilities for the development of innovative image compression methods.

eess.IV↗

arXiv2025

Inf-sup condition for Stokes with outflow condition

The inf-sup condition is one of the essential tools in the analysis of the Stokes equations and especially in numerical analysis. In its usual form, the condition states that for every pressure $p\in L^2(\Omega)\setminus \mathbb{R}$, (i.e. with mean value zero) a velocity $u\in H^1_0(\Omega)^d$ can be found, so that $(div\,u,p)=\|p\|^2$ and $\|\nabla u\|\le c \|p\|$ applies, where $c>0$ does not depend on $u$ and $p$. However, if we consider domains that have a Neumann-type outflow condition on part of the boundary $\Gamma_N\subset\partial\Omega$, the inf-sup condition cannot be used in this form, since the pressure here comes from $L^2(\Omega)$ and does not necessarily have zero mean value. In this note, we derive the inf-sup condition for the case of outflow boundaries.

math.AP↗

arXiv2025

Non-intrusive reduced-order modeling for dynamical systems with spatially localized features

This work presents a non-intrusive reduced-order modeling framework for dynamical systems with spatially localized features characterized by slow singular value decay. The proposed approach builds upon two existing methodologies for reduced and full-order non-intrusive modeling, namely Operator Inference (OpInf) and sparse Full-Order Model (sFOM) inference. We decompose the domain into two complementary subdomains that exhibit fast and slow singular value decay. The dynamics of the subdomain exhibiting slow singular value decay are learned with sFOM while the dynamics with intrinsically low dimensionality on the complementary subdomain are learned with OpInf. The resulting, coupled OpInf-sFOM formulation leverages the computational efficiency of OpInf and the high resolution of sFOM, and thus enables fast non-intrusive predictions for conditions beyond those sampled in the training data set. A novel regularization technique with a closed-form solution based on the Gershgorin disk theorem is introduced to promote stable sFOM and OpInf models. We also provide a data-driven indicator for subdomain selection and ensure solution smoothness over the interface via a post-processing interpolation step. We evaluate the efficiency of the approach in terms of offline and online speedup through a quantitative, parametric computational cost analysis. We demonstrate the coupled OpInf-sFOM formulation for two test cases: a one-dimensional Burgers' model for which accurate predictions beyond the span of the training snapshots are presented, and a two-dimensional parametric model for the Pine Island Glacier ice thickness dynamics, for which the OpInf-sFOM model achieves an average prediction error on the order of $1 \%$ with an online speedup factor of approximately $8\times$ compared to the numerical simulation.

math.DS↗

arXiv2024

Application of a Temporal Multiscale Method for Efficient Simulation of Degradation in PEM Water Electrolysis under Dynamic Operation

Hydrogen is vital for sectors like chemicals and others, driven by the need to reduce carbon emissions. Proton Electrolyte Membrane Water Electrolysis (PEMWE) is a key technology for the production of green hydrogen under fluctuating conditions of renewable power sources. However, due to the scarcity of noble metal materials, the stability of the anode catalyst layer under dynamic operating conditions must be better understood. Model-aided investigation approaches are essential due to the back-box nature of the electrochemical system and the high costs of experimental long-term testing. In this work, a temporal multiscale method based on a Heterogeneous technique is applied to reduce the computational effort of simulating long-term degradation, focused on catalyst dissolution. Such an approach characterizes the problem in fast locally periodic processes, influenced by the dynamic operation and slow processes attributed to the gradual degradation of the catalyst layer. A mechanistic model that includes the oxygen evolution reaction, catalyst dissolution and hydrogen permeation from the cathode to the anode side is hypothesized and implemented. The multiscale approach notably reduces computational effort of simulation from hours to mere minutes. This efficiency gain is ascribed to the limited evolution of Slow-Scale variables during each period of time of the Fast-Scale variables. Consequently, simulation of the fast processes is required only until local periodicity is achieved within each Slow-Scale time step. Thus, the developed temporal multiscale approach proves to be highly effective in accelerating parameter estimation and predictive simulation steps, as could be verified through the results of this article. In this way, the method can support systematic model development to describe degradation in PEMWE under dynamic operating conditions.

math.NA↗

arXiv2024

Error analysis of a pressure correction method with explicit time stepping

The pressure-correction method is a well established approach for simulating unsteady, incompressible fluids. It is well-known that implicit discretization of the time derivative in the momentum equation e.g. using a backward differentiation formula with explicit handling of the nonlinear term results in a conditionally stable method. In certain scenarios, employing explicit time integration in the momentum equation can be advantageous, as it avoids the need to solve for a system matrix involving each differential operator. Additionally, we will demonstrate that the fully discrete method can be expressed in the form of simple matrix-vector multiplications allowing for efficient implementation on modern and highly parallel acceleration hardware. Despite being a common practice in various commercial codes, there is currently no available literature on error analysis for this scenario. In this work, we conduct a theoretical analysis of both implicit and two explicit variants of the pressure-correction method in a fully discrete setting. We demonstrate to which extend the presented implicit and explicit methods exhibit conditional stability. Furthermore, we establish a Courant-Friedrichs-Lewy (CFL) type condition for the explicit scheme and show that the explicit variant demonstrate the same asymptotic behavior as the implicit variant when the CFL condition is satisfied.

math.NA↗

arXiv2024

Rotational dynamics of a disk in a thin film of weakly nematic fluid subject to linear friction

Dynamics at low Reynolds numbers experiences recent revival in the fields of biophysics and active matter. While in bulk isotropic fluids it is exhaustively studied, this is less so in anisotropic fluids and in confined situations. Here, we combine the latter two by studying the rotation of a disk-like inclusion in a uniaxially anisotropic, globally oriented, incompressible two-dimensional fluid film. In terms of a perturbative expansion in parameters that quantify anisotropies in viscosity and in additional linear friction with a supporting substrate or other type of confinement, we derive analytical expressions for the resulting hydrodynamic flow and pressure fields as well as for the resistance and mobility coefficients of the rotating disk. It turns out that, in contrast to translational motion, the solutions remain well-behaved also in the absence of the additional linear friction. Comparison with results from finite-element simulations show very good agreement with those from our analytical calculations. Besides applications to describe technological systems, for instance, in the area of microfluidics and thin cells of aligned nematic liquid crystals, our solutions are important for quantitative theoretical approaches to fluid membranes and thin films in general featuring a preferred direction.

cond-mat.soft↗