08 Fakultät Mathematik und Physik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9
Browse
51 results
Search Results
Item Open Access On a stochastic Camassa-Holm type equation with higher order nonlinearities(2020) Rohde, Christian; Tang, HaoThe subject of this paper is a generalized Camassa-Holm equation under random perturbation. We first establish local existence and uniqueness results as well as blow-up criteria for pathwise solutions in the Sobolev spaces Hs with s>3/2. Then we analyze how noise affects the dependence of solutions on initial data. Even though the noise has some already known regularization effects, much less is known concerning the dependence on initial data. As a new concept we introduce the notion of stability of exiting times and construct an example showing that multiplicative noise (in Itô sense) cannot improve the stability of the exiting time, and simultaneously improve the continuity of the dependence on initial data. Finally, we obtain global existence theorems and estimate associated probabilities.Item Open Access Efficient simulation of challenging PDE problems on CPU and GPU clusters(2021) Schirwon, Malte; Göddeke, Dominik (Prof. Dr.)The main contribution of this dissertation is to show how efficient parallelization techniques for numerical simulations of partial differential equations (PDEs) can be developed and which aspects have to be considered in order to obtain the best possible performance. For this purpose, the target platforms range from high-performance workstations to small clusters and up to supercomputers. In particular, we focus on platforms accelerated by graphics cards. We emphasize that the efficient numerical simulation of PDE problems comprises and combines, in novel ways, aspects from numerical analysis, numerical methods (algorithmics, data structures and other areas more related to computer science) and hardware details. Many models in science, engineering and economics are based on systems of PDEs. The choice of modeling techniques, the implementation of numerical solution techniques, as well as the chosen target platform limit the accuracy and the duration of the simulation. Increasing the accuracy and/or reducing the duration of the simulation is usually not possible without efficient software. Based on three application scenarios, we adapt already existing methodologies and algorithms to the target platforms or change the way they are implemented in order to achieve optimal efficiency. As a guiding scheme, we consider the challenging case of unstructured data and schemes. The first application is the wave propagation in optical fibers. We present an MPI-parallel implementation that is particularly suitable for small clusters. %Here, we change the numerical method and the implementation technique to increase efficiency and decrease runtime. The second application scenario is the flow in porous media. Based on both applications, we develop implementation techniques that increase their efficiency. Furthermore, we present an adapted version of a neighborhood algorithm that further increases the efficiency for current graphics cards. The increased efficiency and reduced runtime allows to perform more complex simulations. %For example, higher resolutions can be simulated or more physical parameters can be included. One of theses applications is considered to be the third application, which is seismic wave propagation and waveform inversion. The feasibility of developing efficient implementations for computationally powerful target platforms permits us to consider the inversion of seismic waves in viscoelastic materials. In particular, we present an inversion scheme that also allows us to determine the damping parameters of the viscoelastic material. In addition, regularization methods and a modified solver method are presented, which can be used for a more efficient solution of such problems.Item Open Access Analysis of target data-dependent greedy kernel algorithms : convergence rates for f-, f· P- and f/P-greedy(2022) Wenzel, Tizian; Santin, Gabriele; Haasdonk, BernardData-dependent greedy algorithms in kernel spaces are known to provide fast converging interpolants, while being extremely easy to implement and efficient to run. Despite this experimental evidence, no detailed theory has yet been presented. This situation is unsatisfactory, especially when compared to the case of the data-independent P-greedy algorithm, for which optimal convergence rates are available, despite its performances being usually inferior to the ones of target data-dependent algorithms. In this work, we fill this gap by first defining a new scale of greedy algorithms for interpolation that comprises all the existing ones in a unique analysis, where the degree of dependency of the selection criterion on the functional data is quantified by a real parameter. We then prove new convergence rates where this degree is taken into account, and we show that, possibly up to a logarithmic factor, target data-dependent selection strategies provide faster convergence. In particular, for the first time we obtain convergence rates for target data adaptive interpolation that are faster than the ones given by uniform points, without the need of any special assumption on the target function. These results are made possible by refining an earlier analysis of greedy algorithms in general Hilbert spaces. The rates are confirmed by a number of numerical examples.Item Open Access Resilience and fault tolerance in high-performance computing for numerical weather and climate prediction(2021) Benacchio, Tommaso; Bonaventura, Luca; Altenbernd, Mirco; Cantwell, Chris D.; Düben, Peter D.; Gillard, Mike; Giraud, Luc; Göddeke, Dominik; Raffin, Erwan; Teranishi, Keita; Wedi, NilsProgress in numerical weather and climate prediction accuracy greatly depends on the growth of the available computing power. As the number of cores in top computing facilities pushes into the millions, increased average frequency of hardware and software failures forces users to review their algorithms and systems in order to protect simulations from breakdown. This report surveys hardware, application-level and algorithm-level resilience approaches of particular relevance to time-critical numerical weather and climate prediction systems. A selection of applicable existing strategies is analysed, featuring interpolation-restart and compressed checkpointing for the numerical schemes, in-memory checkpointing, user-level failure mitigation and backup-based methods for the systems. Numerical examples showcase the performance of the techniques in addressing faults, with particular emphasis on iterative solvers for linear systems, a staple of atmospheric fluid flow solvers. The potential impact of these strategies is discussed in relation to current development of numerical weather prediction algorithms and systems towards the exascale. Trade-offs between performance, efficiency and effectiveness of resiliency strategies are analysed and some recommendations outlined for future developments.Item Open Access Permeability estimation of regular porous structures : a benchmark for comparison of methods(2021) Wagner, Arndt; Eggenweiler, Elissa; Weinhardt, Felix; Trivedi, Zubin; Krach, David; Lohrmann, Christoph; Jain, Kartik; Karadimitriou, Nikolaos; Bringedal, Carina; Voland, Paul; Holm, Christian; Class, Holger; Steeb, Holger; Rybak, IrynaThe intrinsic permeability is a crucial parameter to characterise and quantify fluid flow through porous media. However, this parameter is typically uncertain, even if the geometry of the pore structure is available. In this paper, we perform a comparative study of experimental, semi-analytical and numerical methods to calculate the permeability of a regular porous structure. In particular, we use the Kozeny-Carman relation, different homogenisation approaches (3D, 2D, very thin porous media and pseudo 2D/3D), pore-scale simulations (lattice Boltzmann method, Smoothed Particle Hydrodynamics and finite-element method) and pore-scale experiments (microfluidics). A conceptual design of a periodic porous structure with regularly positioned solid cylinders is set up as a benchmark problem and treated with all considered methods. The results are discussed with regard to the individual strengths and limitations of the used methods. The applicable homogenisation approaches as well as all considered pore-scale models prove their ability to predict the permeability of the benchmark problem. The underestimation obtained by the microfluidic experiments is analysed in detail using the lattice Boltzmann method, which makes it possible to quantify the influence of experimental setup restrictions.Item Open Access Deep and greedy kernel methods : algorithms, analysis and applications(2023) Wenzel, Tizian; Haasdonk, Bernard (Prof. Dr.)Item Open Access Coupled simulations and parameter inversion for neural system and electrophysiological muscle models(2024) Homs‐Pons, Carme; Lautenschlager, Robin; Schmid, Laura; Ernst, Jennifer; Göddeke, Dominik; Röhrle, Oliver; Schulte, MiriamThe functioning of the neuromuscular system is an important factor for quality of life. With the aim of restoring neuromuscular function after limb amputation, novel clinical techniques such as the agonist‐antagonist myoneural interface (AMI) are being developed. In this technique, the residual muscles of an agonist‐antagonist pair are (re‐)connected via a tendon in order to restore their mechanical and neural interaction. Due to the complexity of the system, the AMI can substantially profit from in silico analysis, in particular to determine the prestretch of the residual muscles that is applied during the procedure and determines the range of motion of the residual muscle pair. We present our computational approach to facilitate this. We extend a detailed multi‐X model for single muscles to the AMI setup, that is, a two‐muscle‐one‐tendon system. The model considers subcellular processes as well as 3D muscle and tendon mechanics and is prepared for neural process simulation. It is solved on high performance computing systems. We present simulation results that show (i) the performance of our numerical coupling between muscles and tendon and (ii) a qualitatively correct dependence of the range of motion of muscles on their prestretch. Simultaneously, we pursue a Bayesian parameter inference approach to invert for parameters of interest. Our approach is independent of the underlying muscle model and represents a first step toward parameter optimization, for instance, finding the prestretch, to be applied during surgery, that maximizes the resulting range of motion. Since our multi‐X fine‐grained model is computationally expensive, we present inversion results for reduced Hill‐type models. Our numerical results for cases with known ground truth show the convergence and robustness of our approach.Item Open Access A Bayesian approach to parameter reconstruction from surface electromyographic signals(2021) Rörich, Anna; Göddeke, Dominik (Prof. Dr.)Applying a Bayesian approach to infer the electrical conductivity of a body or body part from surface electromyographic (EMG) signals yields a non-invasive and radiation-free imaging technique. Further, measuring the surface EMG signals that stem from voluntary muscle contractions, there is no need to apply external electrical stimuli to the body. The electrical conductivity provides structural information of the corresponding tissue that is used to estimate whether the tissue has isotropic or anisotropic properties and which is the preferred conducting direction, if applicable. Additionally, changes in the magnitude of the electrical conductivity indicate changes in the tissue material. Together, these properties of the electrical conductivity provide medical images of the examined body part. This imaging process results in an inverse and mathematically ill-posed problem. Including a stochastic model of the inevitable measurement error into the mathematical problem description, the whole system is embedded into a probabilistic framework. Thus, instead of estimating the structure of the examined body part, the probability distribution of the parameters describing the tissue structure given surface EMG measurements, the so-called posterior distribution, is estimated. This Bayesian approach to inverse problems not only yields more information about the quantities of interest than classical regularization approaches, but also has a regularizing effect on the ill-posed problem. Indeed, the Bayesian inverse problem of inferring the tissue structure from surface EMG measurements is proven to be well-posed. This yields the convergence of the inversion algorithm and allows establishing error bounds and thus quantifying the uncertainties in the solution of the inverse EMG problem. Numerically, Markov chain Monte Carlo methods are used to explore the posterior distribution. Accelerations of these sampling methods are achieved by deriving a data-sparse representation of the discretized forward model for all conceivable discretizations of the parameters describing the tissue structure. The resulting approach is not only mathematically well-founded, but also faster by orders of magnitude. Finally, the proposed sampling algorithms are applied to several use cases that are related to clinical applications.Item Open Access On some distributional properties of subordinated Gaussian random fields(2022) Merkle, Robin; Barth, AndreaMotivated by the subordinated Brownian motion, we define a new class of (in general discontinuous) random fields on higher-dimensional parameter domains: the subordinated Gaussian random field. We investigate the pointwise marginal distribution of the constructed random fields, derive a Lévy-Khinchin-type formula and semi-explicit formulas for the covariance function. Further, we study the pointwise stochastic regularity and present various numerical examples.Item Open Access Interface conditions for arbitrary flows in Stokes-Darcy systems : derivation, analysis and validation(2022) Eggenweiler, Elissa; Rybak, Iryna (PD Dr.)