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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., Comment: 29 pages, 9 figures, acknowledgements are provided

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.

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., Comment: 17 pages, 5 figures

In this work, we present a comparison between color spaces namely YUV, LAB, RGB and their effect on learned image compression. For this we use the structure and color based learned image codec (SLIC) from our prior work, which consists of two branches - one for the luminance component (Y or L) and another for chrominance components (UV or AB). However, for the RGB variant we input all 3 channels in a single branch, similar to most learned image codecs operating in RGB. The models are trained for multiple bitrate configurations in each color space. We report the findings from our experiments by evaluating them on various datasets and compare the results to state-of-the-art image codecs. The YUV model performs better than the LAB variant in terms of MS-SSIM with a Bj{\o}ntegaard delta bitrate (BD-BR) gain of 7.5\% using VTM intra-coding mode as the baseline. Whereas the LAB variant has a better performance than YUV model in terms of CIEDE2000 having a BD-BR gain of 8\%. Overall, the RGB variant of SLIC achieves the best performance with a BD-BR gain of 13.14\% in terms of MS-SSIM and a gain of 17.96\% in CIEDE2000 at the cost of a higher model complexity., Comment: Accepter pre-print version for ICIP 2024

Adaptive finite elements combined with geometric multigrid solvers are one of the most efficient numerical methods for problems such as the instationary Navier-Stokes equations. Yet despite their efficiency, computations remain expensive and the simulation of, for example, complex flow problems can take many hours or days. GPUs provide an interesting avenue to speed up the calculations due to their very large theoretical peak performance. However, the large degree of parallelism and non-standard API make the use of GPUs in scientific computing challenging. In this work, we develop a GPU acceleration for the adaptive finite element library Gascoigne and study its effectiveness for different systems of partial differential equations. Through the systematic formulation of all computations as linear algebra operations, we can employ GPU-accelerated linear algebra libraries, which simplifies the implementation and ensures the maintainability of the code while achieving very efficient GPU utilizations. Our results for a transport-diffusion equation, linear elasticity, and the instationary Navier-Stokes equations show substantial speedups of up to 20X compared to multi-core CPU implementations.

This work reviews goal-oriented a posteriori error control, adaptivity and solver control for finite element approximations to boundary and initial-boundary value problems for stationary and non-stationary partial differential equations, respectively. In particular, coupled field problems with different physics may require simultaneously the accurate evaluation of several quantities of interest, which is achieved with multi-goal oriented error control. Sensitivity measures are obtained by solving an adjoint problem. Error localization is achieved with the help of a partition-of-unity. We also review and extend theoretical results for efficiency and reliability by employing a saturation assumption. The resulting adaptive algorithms allow to balance discretization and non-linear iteration errors, and are demonstrated for four applications: Poisson's problem, non-linear elliptic boundary value problems, stationary incompressible Navier-Stokes equations, and regularized parabolic $p$-Laplace initial-boundary value problems. Therein, different finite element discretizations in two different software libraries are utilized, which are partially accompanied with open-source implementations on GitHub.

To make progress towards the development of a theory on the motion of inclusions in thin structured films and membranes, we here consider as an initial step a circular disk in a two-dimensional, uniaxially anisotropic fluid layer. We assume overdamped dynamics, incompressibility of the fluid, and global alignment of the axis of anisotropy. Motion within this layer is affected by additional linear friction with the environment, for instance, a supporting substrate. We investigate the induced flows in the fluid when the disk is translated parallel or perpendicular to the direction of anisotropy. Moreover, expressions for corresponding mobilities and resistance coefficients of the disk are derived. Our results are obtained within the framework of a perturbative expansion in the parameters that quantify the anisotropy of the fluid. Good agreement is found for moderate anisotropy when compared to associated results from finite-element simulations. At pronounced anisotropy, the induced flow fields are still predicted qualitatively correctly by the perturbative theory, although quantitative deviations arise. We hope to stimulate with our investigations corresponding experimental analyses, for example, concerning fluid flows in anisotropic thin films on uniaxially rubbed supporting substrates., Comment: 14 pages, 5 figures

The cryosphere plays a significant role in Earth's climate system. Therefore, an accurate simulation of sea ice is of great importance to improve climate projections. To enable higher resolution simulations, graphics processing units (GPUs) have become increasingly attractive as they offer higher floating point peak performance and better energy efficiency compared to CPUs. However, making use of this theoretical peak performance, which is based on massive data parallelism, usually requires more care and effort in the implementation. In recent years, a number of frameworks have become available that promise to simplify general purpose GPU programming. In this work, we compare multiple such frameworks, including CUDA, SYCL, Kokkos and PyTorch, for the parallelization of \nextsim, a finite-element based dynamical core for sea ice. We evaluate the different approaches according to their usability and performance., Comment: revision for PASC 2024

We propose the structure and color based learned image codec (SLIC) in which the task of compression is split into that of luminance and chrominance. The deep learning model is built with a novel multi-scale architecture for Y and UV channels in the encoder, where the features from various stages are combined to obtain the latent representation. An autoregressive context model is employed for backward adaptation and a hyperprior block for forward adaptation. Various experiments are carried out to study and analyze the performance of the proposed model, and to compare it with other image codecs. We also illustrate the advantages of our method through the visualization of channel impulse responses, latent channels and various ablation studies. The model achieves Bj{\o}ntegaard delta bitrate gains of 7.5% and 4.66% in terms of MS-SSIM and CIEDE2000 metrics with respect to other state-of-the-art reference codecs., Comment: Accepter paper for Data Compression Conference 2024

We describe and analyze a hybrid finite element/neural network method for predicting solutions of partial differential equations. The methodology is designed for obtaining fine scale fluctuations from neural networks in a local manner. The network is capable of locally correcting a coarse finite element solution towards a fine solution taking the source term and the coarse approximation as input. Key observation is the dependency between quality of predictions and the size of training set which consists of different source terms and corresponding fine & coarse solutions. We provide the a priori error analysis of the method together with the stability analysis of the neural network. The numerical experiments confirm the capability of the network predicting fine finite element solutions. We also illustrate the generalization of the method to problems where test and training domains differ from each other.

We present a priori error estimates for a multirate time-stepping scheme for coupled differential equations. The discretization is based on Galerkin methods in time using two different time meshes for two parts of the problem. We aim at surface coupled multiphysics problems like two-phase flows. Special focus is on the handling of the interface coupling to guarantee a coercive formulation as key to optimal order error estimates. In a sequence of increasing complexity, we begin with the coupling of two ordinary differential equations, coupled heat conduction equation, and finally a coupled Stokes problem. For this we show optimal multi-rate estimates in velocity and a suboptimal result in pressure. The a priori estimates prove that the multirate method decouples the two subproblems exactly. This is the basis for adaptive methods which can choose optimal lattices for the respective subproblems.

We extend and analyze the deep neural network multigrid solver (DNN-MG) for the Navier-Stokes equations in three dimensions. The idea of the method is to augment a finite element simulation on coarse grids with fine scale information obtained using deep neural networks. The neural network operates locally on small patches of grid elements. The local approach proves to be highly efficient, since the network can be kept (relatively) small and since it can be applied in parallel on all grid patches. However, the main advantage of the local approach is the inherent generalizability of the method. Since the network only processes data of small sub-areas, it never ``sees'' the global problem and thus does not learn false biases. We describe the method with a focus on the interplay between the finite element method and deep neural networks. Further, we demonstrate with numerical examples the excellent efficiency of the hybrid approach, which allows us to achieve very high accuracy with a coarse grid and thus reduce the computation time by orders of magnitude., Comment: 40 pages, 24 figures, 11 tables. Submitted to Computer Methods in Applied Mechanics and Engineering

In this work, we propose and computationally investigate a monolithic space-time multirate scheme for coupled problems. The novelty lies in the monolithic formulation of the multirate approach as this requires a careful design of the functional framework, corresponding discretization, and implementation. Our method of choice is a tensor-product Galerkin space-time discretization. The developments are carried out for both prototype interface- and volume coupled problems such as coupled wave-heat-problems and a displacement equation coupled to Darcy flow in a poro-elastic medium. The latter is applied to the well-known Mandel's benchmark and a three-dimensional footing problem. Detailed computational investigations and convergence analyses give evidence that our monolithic multirate framework performs well., Comment: 39 pages, 18 figures, 4 tables

The quasi-random discrete ordinates method (QRDOM) is here proposed for the approximation of transport problems. Its central idea is to explore a quasi Monte Carlo integration within the classical source iteration technique. It preserves the main characteristics of the discrete ordinates method, but it has the advantage of providing mitigated ray effect solutions. The QRDOM is discussed in details for applications to one-group transport problems with isotropic scattering in rectangular domains. The method is tested against benchmark problems for which DOM solutions are known to suffer from the ray effects. The numerical experiments indicate that the QRDOM provides accurate results and it demands less discrete ordinates per source iteration when compared against the classical DOM., Comment: 15 pages, 5 figures

The quasi-random discrete ordinates method (QRDOM) is here proposed for the approximation of transport problems. Its central idea is to explore a quasi Monte Carlo integration within the classical source iteration technique. It preserves the main characteristics of the discrete ordinates method, but it has the advantage of providing mitigated ray effect solutions. The QRDOM is discussed in details for applications to one-group transport problems with isotropic scattering in rectangular domains. The method is tested against benchmark problems for which DOM solutions are known to suffer from the ray effects. The numerical experiments indicate that the QRDOM provides accurate results and it demands less discrete ordinates per source iteration when compared against the classical DOM., Comment: 14 pages, 6 figures

We analyze a hybrid method that enriches coarse grid finite element solutions with fine scale fluctuations obtained from a neural network. The idea stems from the Deep Neural Network Multigrid Solver (DNN-MG), (Margenberg et al., J Comput Phys 460:110983, 2022; A neural network multigrid solver for the Navier-Stokes equations) which embeds a neural network into a multigrid hierarchy by solving coarse grid levels directly and predicting the corrections on fine grid levels locally (e.g. on small patches that consist of several cells) by a neural network. Such local designs are quite appealing, as they allow a very good generalizability. In this work, we formalize the method and describe main components of the a-priori error analysis. Moreover, we numerically investigate how the size of training set affects the solution quality.

Deep learning based image compression has gained a lot of momentum in recent times. To enable a method that is suitable for image compression and subsequently extended to video compression, we propose a novel deep learning model architecture, where the task of image compression is divided into two sub-tasks, learning structural information from luminance channel and color from chrominance channels. The model has two separate branches to process the luminance and chrominance components. The color difference metric CIEDE2000 is employed in the loss function to optimize the model for color fidelity. We demonstrate the benefits of our approach and compare the performance to other codecs. Additionally, the visualization and analysis of latent channel impulse response is performed.

Non-intrusive model reduction is a promising solution to system dynamics prediction, especially in cases where data are collected from experimental campaigns or proprietary software simulations. In this work, we present a method for non-intrusive model reduction applied to Fluid-Structure Interaction (FSI) problems. The approach is based on the a priori known sparsity of the full-order system operators, which is dictated by grid adjacency information. In order to enforce this type of sparsity, we solve a local, regularized least-squares problem for each degree of freedom on a grid, considering only the training data from adjacent degrees of freedom, thus making computation and storage of the inferred full-order operators feasible. After constructing the non-intrusive, sparse full-order model, Proper Orthogonal Decomposition (POD) is used for its projection to a reduced dimension subspace and thus the construction of a reduced-order model (ROM). The methodology is applied to the challenging Hron-Turek benchmark FSI3, for Re = 200. A physics-informed, non-intrusive ROM is constructed to predict the two-way coupled dynamics of a solid with a deformable, slender tail, subject to an incompressible, laminar flow. Results considering the accuracy and predictive capabilities of the inferred reduced models are discussed.

Due to their high resolution, dynamic medical 2D+t and 3D+t volumes from computed tomography (CT) and magnetic resonance tomography (MR) reach a size which makes them very unhandy for teleradiologic applications. A lossless scalable representation offers the advantage of a down-scaled version which can be used for orientation or previewing, while the remaining information for reconstructing the full resolution is transmitted on demand. The wavelet transform offers the desired scalability. A very high quality of the lowpass sub-band is crucial in order to use it as a down-scaled representation. We propose an approach based on compensated wavelet lifting for obtaining a scalable representation of dynamic CT and MR volumes with very high quality. The mesh compensation is feasible to model the displacement in dynamic volumes which is mainly given by expansion and contraction of tissue over time. To achieve this, we propose an optimized estimation of the mesh compensation parameters to optimally fit for dynamic volumes. Within the lifting structure, the inversion of the motion compensation is crucial in the update step. We propose to take this inversion directly into account during the estimation step and can improve the quality of the lowpass sub-band by 0.63 and 0.43 dB on average for our tested dynamic CT and MR volumes at the cost of an increase of the rate by 2.4% and 1.2% on average.

Vortex-induced vibrations (VIV) pose computationally expensive problems of high practical interest to several engineering fields. In this work we develop a non-intrusive, reduced-order modelling methodology for two-dimensional (2D) VIV simulations. We consider an elliptical, non-deformable solid mounted on springs, subject to a laminar, incompressible flow. Approximating the Arbitrary Lagrangian-Eulerian (ALE) incompressible Navier-Stokes (NS) formulation, a discrete-time, quadratic-bilinear data-driven model structure is assigned for the velocity flowfield prediction. The full-order, data-driven model operators have a predefined sparsity pattern, following the adjacency-based sparsity of the discretized ALE-NS operators. Thus, the data-driven operators inference requires solving many low-dimensional least squares problems, isolating the contribution of "nearest neighbours" for each degree of freedom. Numerical aspects such as data centering and regularization are extensively discussed. With this approach, Dirichlet boundary conditions at the inlet and the fluid/solid interface can be enforced on the full-order, non-intrusive level. Consequently, a non-intrusive reduced-order model (ROM) for the velocity flowfield is obtained after linear projection. The resulting ROM is coupled with the first-principle, 2D solid oscillation equations and is simulated using an implicit time integration scheme. The coupled solution is then mapped on the deformed grid through the inverse ALE map. This methodology is showcased for two testcases, at different Reynolds numbers (Re = 90, 180). Numerical results indicate a successful coupling between the data-driven velocity flowfield and the solid oscillation, with prediction errors of less than 3% for both the flowfield and the solid oscillation. A comparative study with respect to the ROM dimension indicates the robustness and potential of the approach.

Factorized in the lifting structure, the wavelet transform can easily be extended by arbitrary compensation methods. Thereby, the transform can be adapted to displacements in the signal without losing the ability of perfect reconstruction. This leads to an improvement of scalability. In temporal direction of dynamic medical 3-D+t volumes from Computed Tomography, displacement is mainly given by expansion and compression of tissue. We show that these smooth movements can be well compensated with a mesh-based method. We compare the properties of triangle and quadrilateral meshes. We also show that with a mesh-based compensation approach coding results are comparable to the common slice wise coding with JPEG 2000 while a scalable representation in temporal direction can be achieved.

Lost image areas with different size and arbitrary shape can occur in many scenarios such as error-prone communication, depth-based image rendering or motion compensated wavelet lifting. The goal of image reconstruction is to restore these lost image areas as close to the original as possible. Frequency selective extrapolation is a block-based method for efficiently reconstructing lost areas in images. So far, the actual shape of the lost area is not considered directly. We propose a centroid adaption to enhance the existing frequency selective extrapolation algorithm that takes the shape of lost areas into account. To enlarge the test set for evaluation we further propose a method to generate arbitrarily shaped lost areas. On our large test set, we obtain an average reconstruction gain of 1.29 dB.

Lossless image coding is a crucial task especially in the medical area, e.g., for volumes from Computed Tomography or Magnetic Resonance Tomography. Besides lossless coding, compensated wavelet lifting offers a scalable representation of such huge volumes. While compensation methods increase the details in the lowpass band, they also vary the characteristics of the wavelet coefficients, so an adaption of the coefficient coder should be considered. We propose a simple invertible extension for JPEG 2000 that can reduce the filesize for lossless coding of the highpass band by 0.8% on average with peak rate saving of 1.1%.

For scalable coding, a high quality of the lowpass band of a wavelet transform is crucial when it is used as a downscaled version of the original signal. However, blur and motion can lead to disturbing artifacts. By incorporating feasible compensation methods directly into the wavelet transform, the quality of the lowpass band can be improved. The displacement in dynamic medical 3-D+t volumes from Computed Tomography is mainly given by expansion and compression of tissue over time and can be modeled well by mesh-based methods. We extend a 2-D mesh-based compensation method to three dimensions to obtain a volume compensation method that can additionally compensate deforming displacements in the third dimension. We show that a 3-D mesh can obtain a higher quality of the lowpass band by 0.28 dB with less than 40% of the model parameters of a comparable 2-D mesh. Results from lossless coding with JPEG 2000 3D and SPECK3D show that the compensated subbands using a 3-D mesh need about 6% less data compared to using a 2-D mesh.

The prediction step is a very important part of hybrid video codecs. In this contribution, a novel spatio-temporal prediction algorithm is introduced. For this, the prediction is carried out in two steps. Firstly, a preliminary temporal prediction is conducted by motion compensation. Afterwards, spatial refinement is carried out for incorporating spatial redundancies from already decoded neighboring blocks. Thereby, the spatial refinement is achieved by applying Non-Local Means de-noising to the union of the motion compensated block and the already decoded blocks. Including the spatial refinement into H.264/AVC, a rate reduction of up to 14 % or respectively a gain of up to 0.7 dB PSNR compared to unrefined motion compensated prediction can be achieved.

Multi-view image acquisition systems with two or more cameras can be rather costly due to the number of high resolution image sensors that are required. Recently, it has been shown that by covering a low resolution sensor with a non-regular sampling mask and by using an efficient algorithm for image reconstruction, a high resolution image can be obtained. In this paper, a stereo image reconstruction setup for multi-view scenarios is proposed. A scene is captured by a pair of non-regular sampling sensors and by incorporating information from the adjacent view, the reconstruction quality can be increased. Compared to a state-of-the-art single-view reconstruction algorithm, this leads to a visually noticeable average gain in PSNR of 0.74 dB., Comment: 4 pages, 5 figures, 1 table, International Conference on Image Processing (ICIP)

Recently, it has been shown that a high resolution image can be obtained without the usage of a high resolution sensor. The main idea has been that a low resolution sensor is covered with a non-regular sampling mask followed by a reconstruction of the incomplete high resolution image captured this way. In this paper, a multi-frame reconstruction approach is proposed where a video is taken by a non-regular sampling sensor and fully reconstructed afterwards. By utilizing the temporal correlation between neighboring frames, the reconstruction quality can be further enhanced. Compared to a state-of-the-art single-frame reconstruction approach, this leads to a visually noticeable gain in PSNR of up to 1.19 dB on average., Comment: 4 pages, 9 figures, 1 table, Visual Communications and Image Processing (VCIP)

Increasing spatial image resolution is an often required, yet challenging task in image acquisition. Recently, it has been shown that it is possible to obtain a high resolution image by covering a low resolution sensor with a non-regular sampling mask. Due to the masking, however, some pixel information in the resulting high resolution image is not available and has to be reconstructed by an efficient image reconstruction algorithm in order to get a fully reconstructed high resolution image. In this paper, the influence of different sampling masks with a reduced randomness of the non-regularity on the image reconstruction process is evaluated. Simulation results show that it is sufficient to use sampling masks that are non-regular only on a smaller scale. These sampling masks lead to a visually noticeable gain in PSNR compared to arbitrary chosen sampling masks which are non-regular over the whole image sensor size. At the same time, they simplify the manufacturing process and allow for efficient storage., Comment: 4 pages, 5 figures, 1 table, Visual Communications and Image Processing (VCIP)

We introduce an unfitted finite element method with Lagrange-multipliers to study an Eulerian time stepping scheme for moving domain problems applied to a model problem where the domain motion is implicit to the problem. We consider a parabolic partial differential equation (PDE) in the bulk domain, and the domain motion is described by an ordinary differential equation (ODE), coupled to the bulk partial differential equation through the transfer of forces at the moving interface. The discretisation is based on an unfitted finite element discretisation on a time-independent mesh. The method-of-lines time discretisation is enabled by an implicit extension of the bulk solution through additional stabilisation, as introduced by Lehrenfeld & Olshanskii (ESAIM: M2AN, 53:585-614, 2019). The analysis of the coupled problem relies on the Lagrange-multiplier formulation, the fact that the Lagrange-multiplier solution is equal to the normal stress at the interface and that the motion of the interface is given through rigid body motion. This paper covers the complete stability analysis of the method and an error estimate in the energy norm, under an assumption on the discrete interface velocity. This includes the dynamic error in the domain motion resulting from the discretised ODE and the forces from the discretised PDE. To the best of our knowledge this is the first error analysis of this type of coupled moving domain problem in a fully Eulerian framework. Numerical examples illustrate the theoretical results.