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arXiv 2020年

作者： Rörich, Anna Werthmann, Tim A. Göddeke, Dominik Grasedyck, Lars Institute of Applied Analysis and Numerical Simulation University of Stuttgart Allmandring 5b Stuttgart70569 Germany Institut für Geometrie und Praktische Mathematik Rwth Aachen University Templergraben 55 Aachen52056 Germany Stuttgart Center for Simulation Science University of Stuttgart Pfaffenwaldring 5a Stuttgart70569 Germany

The reconstruction of the structure of biological tissue using electromyographic data is a non-invasive imaging method with diverse medical applications. Mathematically, this process is an inverse problem. Furthermore, electromyographic data are highly sensitive to changes in the electrical conductivity that describes the structure of the tissue. Modeling the inevitable measurement error as a stochastic quantity leads to a Bayesian approach. Solving the discretized Bayesian inverse problem means drawing samples from the posterior distribution of parameters, e.g., the conductivity, given measurement data. Using, e.g., a Metropolis-Hastings algorithm for this purpose involves solving the forward problem for different parameter combinations which requires a high computational effort. Low-rank tensor formats can reduce this effort by providing a data-sparse representation of all occurring linear systems of equations simultaneously and allow for their efficient solution. The application of Bayes' theorem proves the well-posedness of the Bayesian inverse problem. The derivation and proof of a low-rank representation of the forward problem allow for the precomputation of all solutions of this problem under certain assumptions, resulting in an efficient and theory-based sampling algorithm. numerical experiments support the theoretical results, but also indicate that a high number of samples is needed to obtain reliable estimates for the parameters. The Metropolis-Hastings sampling algorithm, using the precomputed forward solution in a tensor format, draws this high number of samples and therefore enables solving problems which are infeasible using classical methods. Copyright © 2020, The Authors. All rights reserved.

关键词： Tensors

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A novel class of stabilized greedy kernel approximation algorithms: Convergence, stability & uniform point distribution

arXiv

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arXiv 2019年

作者： Wenzel, Tizian Santin, Gabriele Haasdonk, Bernard Institute for Applied Analysis and Numerical Simulation University of Stuttgart Germany Center for Information and Communication Technology Fondazione Bruno Kessler Italy

Kernel based methods provide a way to reconstruct potentially high-dimensional functions from meshfree samples, i.e., sampling points and corresponding target values. A crucial ingredient for this to be successful is the distribution of the sampling points. Since the computation of an optimal selection of sampling points may be an infeasible task, one promising option is to use greedy methods. Although these methods may be very effective, depending on the specific greedy criterion the chosen points might quickly lead to instabilities in the computation. To circumvent this problem, we introduce and investigate a new class of stabilized greedy kernel algorithms, which can be used to create a scale of new selection strategies. We analyze these algorithms, and in particular we prove convergence results and quantify in a precise way the distribution of the selected points. These results allow to prove, in the case of certain Sobolev kernels, that the algorithms have optimal stability and optimal convergence rates, including for functions outside the native space of the kernel. The results also apply to the case of the usual P -greedy algorithm, significantly improving state-of-the-art results available in the literature. Illustrative experiments are presented that support the theoretical findings. Copyright © 2019, The Authors. All rights reserved.

关键词： Approximation algorithms

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A parabolic relaxation model for the Navier-Stokes-Korteweg equations

arXiv

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arXiv 2019年

作者： Hitz, Timon Keim, Jens Munz, Claus-Dieter Rohde, Christian Institute of Aerodynamics and Gasdynamics University of Stuttgart Stuttgart70569 Germany Institute of Applied Analysis and Numerical Simulation University of Stuttgart Stuttgart70569 Germany

The isothermal Navier-Stokes-Korteweg system is a classical diffuse interface model for compressible two-phase flow. However, the numerical solution faces two major challenges: due to a third-order dispersion contribution in the momentum equations, extended numerical stencils are required for the flux calculation. Furthermore, the equation of state given by a Van-der-Waals law, exhibits non-monotone behaviour in the two-phase state space leading to imaginary eigenvalues of the Jacobian of the first-order fluxes. In this work a lower-order parabolic relaxation model is used to approximate solutions of the classical NSK equations. Whereas an additional parabolic evolution equation for the relaxation variable has to be solved, the system involves no differential operator of higher as second order. The use of a modified pressure function guarantees that the first-order fluxes remain hyperbolic. Altogether, the relaxation system is directly accessible for standard compressible flow solvers. We use the higher-order Discontinuous Galerkin spectral element method as realized in the open source code FLEXI. The relaxation model is validated against solutions of the original NSK model for a variety of 1D and 2D test cases. Three-dimensional simulations of head-on droplet collisions for a range of different collision Weber numbers underline the capability of the approach. Copyright © 2019, The Authors. All rights reserved.

关键词： Navier Stokes equations

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PAC-Bayesian Bounds for Deep Gaussian Processes

arXiv

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arXiv 2019年

作者： Föll, Roman Steinwart, Ingo Institute of Applied Analysis and Numerical Simulation University of Stuttgart Stuttgart70569 Germany Institute of Stochastics and Applications University of Stuttgart Stuttgart70569 Germany

Variational approximation techniques and inference for stochastic models in machine learning has gained much attention the last years. Especially in the case of Gaussian Processes (GP) and their deep versions, Deep Gaussian Processes (DGPs), these viewpoints improved state of the art work. In this paper we introduce Probably Approximately Correct (PAC)-Bayesian risk bounds for DGPs making use of variational approximations. We show that the minimization of PAC-Bayesian generalization risk bounds maximizes the variational lower bounds belonging to the specific DGP model. We generalize the loss function property of the log likelihood loss function in the context of PAC-Bayesian risk bounds to the quadratic-form-Gaussian case. Consistency results are given and an oracle-type inequality gives insights in the convergence between the raw model (predictor without variational approximation) and our variational models (predictor for the variational approximation). Furthermore, we give extensions of our main theorems for specific assumptions and parameter cases. Moreover, we show experimentally the evolution of the consistency results for two Deep Recurrent Gaussian Processes (DRGP) modeling time-series, namely the recurrent Gaussian Process (RGP) and the DRGP with Variational Sparse Spectrum approximation, namely DRGP-(V)SS. Keywords: PAC-Bayesian theory, Deep Gaussian Process models, Variational approximations, Consistency, Recurrent models Copyright © 2019, The Authors. All rights reserved.

关键词： Number theory

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Deep recurrent gaussian process with variational sparse spectrum approximation

arXiv

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arXiv 2019年

作者： Föll, Roman Haasdonk, Bernard Hanselmann, Markus Ulmer, Holger Institute of Applied Analysis and Numerical Simulation University of Stuttgart Stuttgart70569 Germany ETAS GmbH ML Team Stuttgart70469 Germany

Modeling sequential data has become more and more important in practice. Some applications are autonomous driving, virtual sensors and weather forecasting. To model such systems, so called recurrent models are frequently used. In this paper we introduce several new Deep recurrent Gaussian process (DRGP) models based on the Sparse Spectrum Gaussian process (SSGP) and the improved version, called variational Sparse Spectrum Gaussian process (VSSGP). We follow the recurrent structure given by an existing DRGP based on a specific variational sparse Nyström approximation, the recurrent Gaussian process (RGP). Similar to previous work, we also variationally integrate out the input-space and hence can propagate uncertainty through the Gaussian process (GP) layers. Our approach can deal with a larger class of covariance functions than the RGP, because its spectral nature allows variational integration in all stationary cases. Furthermore, we combine the (variational) Sparse Spectrum ((V)SS) approximations with a well known inducing-input regularization framework. We improve over current state of the art methods in prediction accuracy for experimental data-sets used for their evaluation and introduce a new data-set for engine control, named Emission. Copyright © 2019, The Authors. All rights reserved.

关键词： Weather forecasting

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simulation of nonlinear signal propagation in multimode fibers on multi-GPU systems

arXiv

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arXiv 2019年

作者： Brehler, Marius Schirwon, Malte Krummrich, Peter M. Göddeke, Dominik TU Dortmund Chair for High Frequency Technology Dortmund44227 Germany University of Stuttgart Institute for Applied Analysis and Numerical Simulation Stuttgart70569 Germany

Mode-division multiplexing (MDM) is seen as a possible solution to satisfy the rising capacity demands of optical communication networks. To make MDM a success, fibers supporting the propagation of a huge number of modes are of interest. Many of the system aspects occurring during the propagation can be evaluated by using appropriate models. However, fibers are a nonlinear medium and, therefore, numerical simulations are required. For a large number of modes, the simulation of the nonlinear signal propagation leads to new challenges, for example regarding the required memory, which we address with an implementation incorporating multiple GPU-accelerators. Within this paper, we evaluate two different approaches to realize the communication between the GPUs and analyze the performance for simulations involving up to 8 Tesla GPUs. We show results for a MDM transmission system utilizing the extremely large but practically very relevant number of 120 spatial modes as an application example and analyze the impact of the nonlinear effects on the transmitted signals. Copyright © 2019, The Authors. All rights reserved.

关键词： Runge Kutta methods

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Error control for statistical solutions

arXiv

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arXiv 2019年

作者： Giesselmann, Jan Meyer, Fabian Rohde, Christian Department of Mathematics TU Darmstadt Dolivostraße 15 Darmstadt64293 Germany Institute of Applied Analysis and Numerical Simulation University of Stuttgart Pfaffenwaldring 57 Stuttgart70569 Germany

Statistical solutions have recently been introduced as a an alternative solution framework for hyperbolic systems of conservation laws. In this work we derive a novel a posteriori error estimate in the Wasserstein distance between dissipative statistical solutions and numerical approximations, which rely on so-called regularized empirical measures. The error estimator can be split into deterministic parts which correspond to spatio-temporal approximation errors and a stochastic part which reflects the stochastic error. We provide numerical experiments which examine the scaling properties of the residuals and verify their *** Codes Primary: 35L65, 65M15, Secondary: 65M60, 65M700 Copyright © 2019, The Authors. All rights reserved.

关键词： Galerkin methods

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Modeling the Kerr-nonlinearity in mode-division multiplexing fiber transmission systems on GPUs

Modeling the Kerr-nonlinearity in mode-division multiplexing...

引用

Bragg Gratings, Photosensitivity and Poling in Glass Waveguides and Materials, BGPPM 2018

作者： Brehler, Marius Schirwon, Malte Göddeke, Dominik Krummrich, Peter M. TU Dortmund Chair for High Frequency Technology Dortmund44227 Germany University of Stuttgart Institute for Applied Analysis and Numerical Simulation Stuttgart70569 Germany

We discuss the GPU-acceleration of MDM transmission system simulations and how the required memory can be drastically reduced to simulate a high number of modes. Furthermore, we show how to reduce the runtime. © ... 详细信息

ISBN: (纸本)9781943580439

关键词： Program processors

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An a posteriori error analysis based on non-intrusive spectral projections for systems of random conservation laws

arXiv

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arXiv 2019年

We present an a posteriori error analysis for one-dimensional random hyperbolic systems of conservation laws. For the discretization of the random space we consider the Non-Intrusive Spectral Projection method, the spatio-temporal discretization uses the Runge–Kutta Discontinuous Galerkin method. We derive an a posteriori error estimator using smooth reconstructions of the numerical solution, which combined with the relative entropy stability framework yields computable error bounds for the space-stochastic discretization error. Moreover, we show that the estimator admits a splitting into a stochastic and deterministic *** Codes Primary 35L65, 35R60, Secondary 65M15, 65M60, 65M70 Copyright © 2019, The Authors. All rights reserved.

关键词： Galerkin methods

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A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws

arXiv

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arXiv 2019年

In this article we consider one-dimensional random systems of hyperbolic conservation laws. We first establish existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws which involve random initial data and random flux functions. Based on these results we present an a posteriori error analysis for a numerical approximation of the random entropy admissible solution. For the stochastic discretization, we consider a non-intrusive approach, the Stochastic Collocation method. The spatio-temporal discretization relies on the Runge–Kutta Discontinuous Galerkin method. We derive the a posteriori estimator using continuous reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. We conclude with various numerical examples investigating the scaling properties of the residuals and illustrating the efficiency of the proposed adaptive *** Codes Primary 35L65, 35R60, Secondary 65M15, 65M60, 65M70 Copyright © 2019, The Authors. All rights reserved.

关键词： Stochastic systems

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