Research

I am particularly interested in exploring the intersection of numerical methods, conservation laws, and hierarchical Bayesian learning. My current research focuses on developing algorithms and methods that leverage prior knowledge to improve the accuracy, robustness, and efficiency of solving partial differential equations (PDEs). In addition, I am interested in tackling the challenges associated with solving inverse problems from noisy, indirect, and under-sampled data and in quantifying the confidence of computational predictions. My research is motivated by the need for improved solutions in various applications, including computational fluid dynamics, medical and radar imaging, geosciences, and electrical engineering.

Please find more details on my research, including publications, below. See also Google Scholar, Research Gate, or ORCiD.

Hierarchical Bayesian learning

Hierarchical Bayesian learning is a sophisticated statistical modeling approach that extends traditional Bayesian methods by incorporating hierarchical structures in prior distributions. This technique enables the capture of complex relationships between parameters at different levels and promotes information sharing across the model. Particularly useful in handling limited, noisy, or multi-source data, hierarchical Bayesian learning leads to more accurate and robust inference, making it a valuable tool in various fields such as machine learning, signal processing, and computational biology.

My research in hierarchical Bayesian learning focuses on developing robust and efficient approaches to address challenges in signal and image recovery based on indirect, noisy, or incomplete data. I have explored sparse Bayesian learning methods for various data acquisitions and priors, with applications in denoising, deblurring, magnetic resonance imaging, and synthetic aperture radar. My work also includes joint recovery of sequential images (videos) by incorporating prior intra- and inter-image information, enabling improved accuracy and uncertainty quantification for various data acquisitions. Furthermore, I have developed a new family of algorithms for inferring jointly sparse parameter vectors from multiple measurement vectors.

Uncertainty quantification

Uncertainty quantification (UQ) is a field focused on assessing and managing uncertainties in mathematical models, simulations, and data. UQ techniques quantify the confidence of computational predictions by considering various sources of uncertainty. By incorporating UQ, researchers can make better-informed decisions and improve system designs across diverse fields, such as engineering, finance, environmental sciences, and medicine.

My research in uncertainty quantification (UQ) is centered on developing advanced hierarchical Bayesian learning methods to better quantify the confidence of computational predictions in various inverse problems. By incorporating prior knowledge such as sparsity and inter-image correlations, these methods not only improve the accuracy and robustness of solutions but also enable uncertainty quantification in diverse applications like image reconstruction, signal recovery, and temporal image sequences. This focus on UQ allows for more informed decision-making and reliable interpretation of results, benefiting fields such as medical imaging, geophysics, and engineering.

Numerical conservation laws

Hyperbolic conservation laws are a class of partial differential equations (PDEs) that describe various physical phenomena, such as fluid dynamics, traffic flow, and wave propagation. These PDEs often exhibit shocks and discontinuities, which makes their numerical treatment challenging. Numerical methods for hyperbolic conservation laws include finite difference, finite volume, and discontinuous Galerkin methods. These methods focus on accurately capturing shocks and discontinuities while maintaining stability.

My recent research focuses on using non-polynomial approximation spaces and summation-by-parts (SBP) operators to solve conservation laws. Traditional SBP operators assume that the solution is well approximated by polynomials. However, polynomials might not be the best approximation for some problems, and other approximation spaces may be more appropriate.

I have developed a general theory for SBP operators based on various function spaces, demonstrating that established results for polynomial-based SBP operators can be applied to a wider class of methods. This research expands the concept of SBP operators to include function spaces such as trigonometric, exponential, and radial basis functions.

By incorporating radial basis function methods into the general framework of SBP operators, I have addressed stability issues that frequently occur when dealing with time-dependent PDEs and boundary conditions. This work paves the way for energy-stable and high-order accurate numerical methods that can handle a broader range of problems, increasing the accuracy of numerical solutions and providing stability to the methods.

Publications

Books

  1. J. Glaubitz.
    Shock capturing and high-order methods for hyperbolic conservation laws.
    Dissertation, Logos Verlag Berlin, 2020. (DOI: 10.30819/5084)

  2. J. Glaubitz, D. Rademacher, T. Sonar.
    Lernbuch Analysis 1 - Das Wichtigste ausführlich für Bachelor und Lehramt.
    (Teaching book in analysis,) Springer, 2019. (DOI: 10.1007/978-3-658-26937-1)

Preprints

  1. J. Glaubitz, H. Ranocha, A. Winters, M. Schlottke-Lakemper, P. Öffner, G. Gassner.
    Generalized upwind summation-by-parts operators and their application to nodal discontinuous Galerkin methods.
    2024, arXiv:2406.14557 [math.NA]

  2. J. Glaubitz, J. Nordström, P. Öffner.
    An optimization-based construction procedure for function space-based summation-by-parts operators on arbitrary grids.
    2024, arXiv:2405.08770 [math.NA]

  3. M. Le Provost, J. Glaubitz, and Y. Marzouk.
    Preserving linear invariants in ensemble filtering methods.
    2024, arXiv:2404.14328 [stat.CO]

  4. J. Lindbloom, J. Glaubitz, A. Gelb.
    Generalized sparsity-promoting solvers for Bayesian inverse problems: Versatile sparsifying transforms and unknown noise variances.
    2024, arXiv:2402.16623 [math.NA]

Refereed Journal Articles

  1. H. Ranocha, A. Winters, M. Schlottke-Lakemper, P. Öffner, J. Glaubitz, G. Gassner.
    On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation laws.
    J Comput Phys, 520, 112889 (2025). (DOI: 10.1016/j.jcp.2024.113471)

  2. J. Glaubitz, A. Gelb.
    Leveraging joint sparsity in hierarchical Bayesian learning.
    SIAM-ASA J Uncertain Quantif 12(2), (2024). (DOI: 10.1137/23M156255X)

  3. J. Glaubitz, S.-C. Klein, J. Nordström, P. Öffner.
    Summation-by-parts operators for general function spaces: The second derivative.
    J Comput Phys, 504, 112889 (2024). (DOI: 10.1016/j.jcp.2024.112889)

  4. J. Glaubitz, J. Nordström, P. Öffner.
    Energy-stable global radial basis function methods on summation-by-parts form.
    J Sci Comput 98, article 30 (2024). (DOI: 10.1007/s10915-023-02427-8)

  5. J. Glaubitz, S.-C. Klein, J. Nordström, P. Öffner.
    Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction.
    J Comput Phys 491, 112370 (2023). (DOI: 10.1016/j.jcp.2023.112370)

  6. J. Glaubitz.
    Construction and application of provable positive and exact cubature formulas.
    IMA J Numer Anal, 43(3), 1616—1652 (2023). (DOI: 10.1093/imanum/drac017)

  7. Y. Xiao, J. Glaubitz.
    Sequential image recovery using joint hierarchical Bayesian learning.
    J Sci Comput 96, Article 4 (2023). (DOI: 10.1007/s10915-023-02234-1)

  8. J. Glaubitz, J. Nordström, P. Öffner.
    Summation-by-parts operators for general function spaces.
    SIAM J Numer Anal 61(2), 733—754 (2023). (DOI: 10.1137/22M1470141)

  9. J. Glaubitz, A. Gelb, G. Song.
    Generalized sparse Bayesian learning and application to image reconstruction.
    SIAM-ASA J Uncertain Quantif 11(1), 262–284 (2023). (DOI: 10.1137/22M147236X)

  10. J. Glaubitz, J. Reeger.
    Towards stability results for global radial basis function based quadrature formulas.
    BIT Numer Math 63, 6 (2023). (DOI: 10.1007/s10543-023-00956-0)

  11. Y. Xiao, J. Glaubitz, A. Gelb, G. Song.
    Sequential image recovery from noisy and under-sampled Fourier data.
    J Sci Comput 91 (3), 79 (2022). (DOI: 10.1007/s10915-022-01850-7)

  12. J. Glaubitz.
    Stable high-order cubature formulas for experimental data.
    J Comput Phys 447, 110693 (2021). (DOI: 10.1016/j.jcp.2021.110693)

  13. J. Glaubitz, E. Le Meledo, P. Öffner.
    Towards stable radial basis function methods for linear advection problems.
    Comput Math Appl 85, 84–97 (2021). (DOI: 10.1016/j.camwa.2021.01.012)

  14. J. Glaubitz, A. Gelb.
    Stabilizing radial basis function methods for conservation laws using weakly enforced boundary conditions.
    J Sci Comput 87, 40 (2021). (DOI: 10.1007/s10915-021-01453-8)

  15. J. Glaubitz.
    Stable high-order quadrature rules for scattered data and general weight functions.
    SIAM J Numer Anal 58, 2144 (2020). (DOI: 10.1137/19M1257901)

  16. J. Glaubitz, P. Öffner.
    Stable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered points.
    Appl Numer Math 151, 98–118 (2020). (DOI: 10.1016/j.apnum.2019.12.020)

  17. P. Öffner, J. Glaubitz, H. Ranocha.
    Analysis of artificial dissipation of explicit and implicit time-integration methods.
    Int J Numer Anal Model, 17.3, 332–349 (2020). (URL: http://www.math.ualberta.ca/ijnam/Volume- 17-2020/No-3-20/2020-03-03.pdf)

  18. J. Glaubitz.
    Shock capturing by Bernstein polynomials for scalar conservation laws.
    Appl Math Comput 363, 124593 (2019). (DOI: 10.1016/j.amc.2019.124593)

  19. J. Glaubitz, A. Gelb.
    High order edge sensors with l1 regularization for enhanced discontinuous Galerkin methods.
    SIAM J Sci Comput, 41(2), A1304–A1330 (2019). (DOI: 10.1137/18M1195280)

  20. J. Glaubitz, A.C. Nogueira Jr., J.L.S. Almeida, R.F. Cantao, C.A.C. Silva.
    Smooth and compactly supported viscous sub-cell shock capturing for discontinuous Galerkin methods.
    J Sci Comput, 79, 249–272 (2019). (DOI: 10.1007/s10915-018-0850-3)

  21. P. Öffner, J. Glaubitz, H. Ranocha.
    Stability of correction procedure via reconstruction with summation-by- parts operators for Burgers’ equation using a polynomial chaos approach.
    ESAIM: M2AN, 52.6, 2215–2245, (2018). (DOI: 10.1051/m2an/2018072)

  22. H. Ranocha, J. Glaubitz, P. Öffner, T. Sonar.
    Stability of artificial dissipation and modal filtering for flux reconstruction schemes using summation-by-parts operators.
    Appl Numer Math, 128, 1–23 (2018). (DOI: 10.1016/j.apnum.2018.01.019)

  23. J. Glaubitz, P. Öffner, T. Sonar.
    Application of modal filtering to a spectral difference method.
    Math Comput, 87.309, 175–207 (2018). (DOI: 10.1090/mcom/3257)

Refereed Conference Proceedings

  1. J. Glaubitz, A. Gelb.
    Using l1-regularization for shock capturing in discontinuous Galerkin methods.
    ICOSAHOM 2020+1. Springer Nature, Vol. 137, p. 337 (2023). (DOI: 10.1007/978-3-031-20432-6_21)

  2. J. Glaubitz, P. Öffner, H. Ranocha, T. Sonar.
    Artificial viscosity for correction procedure via reconstruction using summation-by-parts operators.
    XVI International Conference on Hyperbolic Problems: Theory, Numerics, Applications. Springer, Cham, 363–375 (2016). (DOI: 10.1007/978-3-319-91548-7_28)

Software

  1. J. Glaubitz, H. Ranocha, A.R. Winters, M. Schlottke-Lakemper, P. Öffner, G.J. Gassner.
    Reproducibility repository for ``Generalized upwind summation-by-parts operators and their application to nodal discontinuous Galerkin methods".
    Zenodo (2024). (DOI: 10.5281/zenodo.11661785)

  2. H. Ranocha, A.R. Winters, M. Schlottke-Lakemper, P. Öffner, J. Glaubitz, G.J. Gassner.
    Reproducibility repository for ``On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation laws".
    Zenodo (2023). (DOI: 10.5281/zenodo.10200102)

  3. J. Glaubitz. jglaubitz/2ndDerivativeFSBP. GitHub (2023). (https://github.com/jglaubitz/2ndDerivativeFSBP)

  4. J. Glaubitz. jglaubitz/LeveragingJointSparsity. GitHub (2023). (https://github.com/jglaubitz/LeveragingJointSparsity)

  5. J. Glaubitz. jglaubitz/FSBP. GitHub (2022). (https://github.com/jglaubitz/FSBP)

  6. J. Glaubitz. jglaubitz/generalizedSBL. GitHub (2022). (https://github.com/jglaubitz/generalizedSBL)

  7. J. Glaubitz. jglaubitz/stableCFs (v2.0). Zenodo (2021). (DOI: 10.5281/zenodo.5392394)

  8. J. Glaubitz. jglaubitz/positive CFs (v1.0). Zenodo (2021). (DOI: 10.5281/zenodo.5164000)

  9. J. Glaubitz. jglaubitz/stability RBF CFs (v1.0). Zenodo (2021). (DOI: 10.5281/zenodo.5086347)

  10. J. Glaubitz. jglaubitz/weakRBF (v1.0). Zenodo (2020). (DOI: 10.5281/zenodo.4310328)