Jan Glaubitz

Philipp Öffner, Jan Glaubitz, Hendrik Ranocha

In this paper, we consider Burgers' equation with uncertain boundary and initial conditions. The polynomial chaos (PC) approach yields a hyperbolic system of deterministic equations, which can be solved by several numerical methods. Here, we apply the correction procedure via reconstruction (CPR) using summation-by-parts operators. We focus especially on stability, which is proven for CPR methods and the systems arising from the PC approach. Due to the usage of split-forms, the major challenge is to construct entropy stable numerical fluxes. For the first time, such numerical fluxes are constructed for all systems resulting from the PC approach for Burgers' equation. In numerical tests, we verify our results and show also the advantage of the given ansatz using CPR methods. Moreover, one of the simulations, i.e. Burgers' equation equipped with an initial shock, demonstrates quite fascinating observations. The behaviour of the numerical solutions from several methods (finite volume, finite difference, CPR) differ significantly from each other. Through careful investigations, we conclude that the reason for this is the high sensitivity of the system to varying dissipation. Furthermore, it should be stressed that the system is not strictly hyperbolic with genuinely nonlinear or linearly degenerate fields.

Jan Glaubitz, Anne Gelb

This paper investigates the use of $\ell^1$ regularization for solving hyperbolic conservation laws based on high order discontinuous Galerkin (DG) approximations. We first use the polynomial annihilation method to construct a high order edge sensor which enables us to flag troubled elements. The DG approximation is enhanced in these troubled regions by activating $\ell^1$ regularization to promote sparsity in the corresponding jump function of the numerical solution. The resulting $\ell^1$ optimization problem is efficiently implemented using the alternating direction method of multipliers. By enacting $\ell^1$ regularization only in troubled cells, our method remains accurate and efficient, as no additional regularization or expensive iterative procedures are needed in smooth regions. We present results for the inviscid Burgers' equation as well as a nonlinear system of conservation laws using a nodal collocation-type DG method as a solver.

Jan Glaubitz, Philipp Öffner, Thomas Sonar

We adapt the spectral viscosity (SV) formulation implemented as a modal filter to a Spectral Difference Method (SD) solving hyperbolic conservation laws. In the SD Method we use selections of different orthogonal polynomials (APK polynomials). Furthermore we obtain new error bounds for filtered APK extensions of smooth functions. We demonstrate that the modal filter also depends on the chosen polynomial basis in the SD Method. Spectral filtering stabilizes the scheme and leaves weaker oscillations. Hence, the selection of the family of orthogonal polynomials on triangles and their specific modal filter possesses a positive influence on the stability and accuracy of the SD Method. In the second part, we initiate a stability analysis for a linear scalar test case with periodic initial condition to find the best selection of APK polynomials and their specific modal filter. To the best of our knowledge, this work is the first that gives a stability analysis for a scheme with spectral filtering. Finally, we demonstrate the influence of the underlying basis of APK polynomials in a well-known test case.

Jan Glaubitz, Anne Gelb

We present a hierarchical Bayesian learning approach to infer jointly sparse parameter vectors from multiple measurement vectors. Our model uses separate conditionally Gaussian priors for each parameter vector and common gamma-distributed hyper-parameters to enforce joint sparsity. The resulting joint-sparsity-promoting priors are combined with existing Bayesian inference methods to generate a new family of algorithms. Our numerical experiments, which include a multi-coil magnetic resonance imaging application, demonstrate that our new approach consistently outperforms commonly used hierarchical Bayesian methods.

Hendrik Ranocha, Jan Glaubitz, Philipp Öffner et al.

The correction procedure via reconstruction (CPR, also known as flux reconstruction) is a framework of high order semidiscretisations used for the numerical solution of hyperbolic conservation laws. Using a reformulation of these schemes relying on summation-by-parts (SBP) operators and simultaneous approximation terms (SATs), artificial dissipation / spectral viscosity operators are investigated in this first part of a series. Semidiscrete stability results for linear advection and Burgers' equation as model problems are extended to fully discrete stability by an explicit Euler method. As second part of this series, Glaubitz, Ranocha, Öffner, and Sonar (Enhancing stability of correction procedure via reconstruction using summation-by-parts operators II: Modal filtering, 2016) investigate connections to modal filters and their application instead of artificial dissipation.

Jan Glaubitz, Hendrik Ranocha, Philipp Öffner et al.

A recently introduced framework of semidiscretisations for hyperbolic conservation laws known as correction procedure via reconstruction (CPR, also known as flux reconstruction) is considered in the extended setting of summation-by-parts (SBP) operators using simultaneous approximation terms (SATs). This reformulation can yield stable semidiscretisations for linear advection and Burgers' equation as model problems. In order to enhance these properties, modal filters are introduced to this framework. As a second part of a series, the results of Ranocha, Glaubitz, Öffner, and Sonar ("Enhancing stability of correction procedure via reconstruction using summation-by-parts operators I: Artificial dissipation", 2016) concerning artificial dissipation / spectral viscosity are extended, yielding fully discrete stable schemes. Additionally, a new adaptive strategy to compute the filter strength is introduced and different possible applications of modal filters are compared both theoretically and numerically.

Yao Xiao, Jan Glaubitz

Recovering temporal image sequences (videos) based on indirect, noisy, or incomplete data is an essential yet challenging task. We specifically consider the case where each data set is missing vital information, which prevents the accurate recovery of the individual images. Although some recent (variational) methods have demonstrated high-resolution image recovery based on jointly recovering sequential images, there remain robustness issues due to parameter tuning and restrictions on the type of the sequential images. Here, we present a method based on hierarchical Bayesian learning for the joint recovery of sequential images that incorporates prior intra- and inter-image information. Our method restores the missing information in each image by "borrowing" it from the other images. As a result, \emph{all} of the individual reconstructions yield improved accuracy. Our method can be used for various data acquisitions and allows for uncertainty quantification. Some preliminary results indicate its potential use for sequential deblurring and magnetic resonance imaging.