Jan Glaubitz

Jan Glaubitz, Henry Haase, Philipp Öffner et al.

Function-space summation-by-parts (FSBP) operators enable conservative and energy-stable numerical methods for hyperbolic conservation laws based on general, non-polynomial approximation spaces. Recent works show that using generalized Gaussian quadrature significantly reduces the number of grid points required compared to existing constructions that have mostly focused on equidistant grids. In this paper, we compare open and closed FSBP operators constructed with generalized Gaussian quadratures and apply them to numerically solve hyperbolic conservation laws. Furthermore, to support open node distributions, we extend the FSBP framework by introducing function-space exact extrapolation operators and operationalize them in numerical schemes for solving hyperbolic conservation laws. Our numerical experiments include the one-dimensional linear advection, non-viscous Burgers, and compressible Euler equations of gas dynamics. We observe that applying FSBP operators in numerical schemes can improve efficiency and accuracy. Notably, we demonstrate these advantages in more challenging time-dependent settings compared to other recent works on Gaussian FSBP operators.

Jan Glaubitz, Youssef Marzouk

We introduce an approach for efficient Markov chain Monte Carlo (MCMC) sampling for challenging high-dimensional distributions in sparse Bayesian learning (SBL). The core innovation involves using hierarchical prior-normalizing transport maps (TMs), which are deterministic couplings that transform the sparsity-promoting SBL prior into a standard normal one. We analytically derive these prior-normalizing TMs by leveraging the product-like form of SBL priors and Knothe--Rosenblatt (KR) rearrangements. These transform the complex target posterior into a simpler reference distribution equipped with a standard normal prior that can be sampled more efficiently. Specifically, one can leverage the standard normal prior by using more efficient, structure-exploiting samplers. Our numerical experiments on various inverse problems -- including signal deblurring, inverting the non-linear inviscid Burgers equation, and recovering an impulse image -- demonstrate significant performance improvements for standard MCMC techniques.

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.