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Research Group Numerical Mathematics (Partial Differential Equations)
PD Dr. Ronny Bergmann

PD Dr. Ronny Bergmann

Phone:
+49 371 531 36098
Fax:
+49 371 531 836098
Office:
Reichenhainer Str. 41, Office 609 (C47.609)
Scholar:
ResearchGate:
GitHub:
twitter:
Secretary:
Anne-Kristin Glanzberg, Office 607, Phone +49 371 531 22500
Postal address:
TU Chemnitz, Faculty of Mathematics, 09107 Chemnitz, Germany
Office Hours:
any time when present
since 04/2018
Deputy Head of the Research Group Numerical Mathematics (Partial Differential Equations) of Prof. Dr. R. Herzog
01/2018
Habilitation in Mathematics at the Technischen Universität Kaiserslautern, habilitation thesis: “Variational Methods in Manifold-valued Image Processing”
09/2013 — 03/2018
PostDoc in the workgroup Image Processing and data analysis led by Prof. Dr. Gabriele Steidl at the Technischen Universität Kaiserslautern
06/2013
PhD at the Institute of Mathematics, Universität zu Lübeck, advisor Prof. Dr. Jürgen Prestin.
10/2009 — 08/2013
Teaching Assistant, PhD student at the Institute of Mathematics, Universität zu Lübeck.
09/2009
master's degree in computer science, Institut für Mathematik, Universität zu Lübeck.
10/2004 — 09/2009
studies of computer science at Universität zu Lübeck
Further Links

Preprints

Bergmann, R., Herrmann, M., Herzog, R., Schmidt, S., Vidal Núñez, J.
Discrete Total Variation of the Normal Vector Field as Shape Prior with Applications in Geometric Inverse Problems
Preprint, arXiv: 1908.07916.
An analogue of the total variation prior for the normal vector field along the boundary of piecewise flat shapes in 3D is introduced. A major class of examples are triangulated surfaces as they occur for instance in finite element computations. The analysis of the functional is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. It is found to agree with the discrete total mean curvature known in discrete differential geometry. A split Bregman iteration is proposed for the solution of discretized shape optimization problems, in which the total variation of the normal appears as a regularizer. Unlike most other priors, such as surface area, the new functional allows for piecewise flat shapes. As two applications, a mesh denoising and a geometric inverse problem of inclusion detection type involving a partial differential equation are considered. Numerical experiments confirm that polyhedral shapes can be identified quite accurately.
Bergmann, R., Herzog, R., Tenbrinck, D., Vidal-Núñez, J.
Fenchel-Duality for Convex Optimization and a Primal Dual Algorithm on Riemannian Manifolds
Preprint, arXiv: 1908.02022.
This paper introduces a new duality theory that generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. This notion of conjugation even yields a more general Fenchel conjugate for the case where the manifold is a vector space. We investigate its properties, e.g., the Fenchel-Young inequality and the characterization of the convex subdifferential using the analogue of the Fenchel-Moreau Theorem. These properties of the Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm, and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions. Numerical results illustrate the performance of the algorithm, which competes with the recently derived Douglas-Rachford algorithm on manifolds of nonpositive curvature. Furthermore we show that our novel algorithm numerically converges on manifolds of positive curvature.
Bergmann, R., Herrmann, M., Herzog, R., Schmidt, S., Vidal Núñez, J.
Total variation of the normal vector field as shape prior
Preprint, arXiv: 1902.07240.
An analogue of the total variation prior for the normal vector field along the boundary of smooth shapes in 3D is introduced. The analysis of the total variation of the normal vector field is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. It is shown that spheres are stationary points when the total variation of the normal is minimized under an area constraint. Shape calculus is used to characterize the relevant derivatives. Since the total variation functional is non-differentiable whenever the boundary contains flat regions, an extension of the split Bregman method to manifold valued functions is proposed.
Bergmann, R. and Herzog, R. (2018).
Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds.
SIAM Journal on Optimization (SIOPT), accepted,
arXiv:1804.06214.
Karush-Kuhn-Tucker (KKT) conditions for equality and inequality constrained optimization problems on smooth manifolds are formulated. Under the Guignard constraint qualification, local minimizers are shown to admit Lagrange multipliers. The linear independence, Mangasarian-Fromovitz, and Abadie constraint qualifications are also formulated, and the chain “LICQ implies MFCQ implies ACQ implies GCQ” is proved. Moreover, classical connections between these constraint qualifications and the set of Lagrange multipliers are established, which parallel the results in Euclidean space. The constrained Riemannian center of mass on the sphere serves as an illustrating numerical example.

Published Papers

Bergmann, R. and Gousenbourger, P.-Y. (2018).
A variational model for data fitting on manifolds by minimizing the acceleration of a Bézier curve
Frontiers in Applied Mathematics and Statistics, 4, 2018.
doi: 10.3389/fams.2018.00059, arXiv: 1807.10090.
We derive a variational model to fit a composite Bézier curve to a set of data points on a Riemannian manifold. The resulting curve is obtained in such a way that its mean squared acceleration is minimal in addition to remaining close the data points. We approximate the acceleration by discretizing the squared second order derivative along the curve. We derive a closed-form, numerically stable and efficient algorithm to compute the gradient of a Bézier curve on manifolds with respect to its control points, expressed as a concatenation of so-called adjoint Jacobi fields. Several examples illustrate the capabilites and validity of this approach both for interpolation and approximation. The examples also illustrate that the approach outperforms previous works tackling this problem.

Prior publications

for a comprehensive list of publications see ronnybergmann.net/publications.html.

Topics for Bachelor- and Master Thesis

  • Linear Programs on Manifolds
  • Optimization on Manifolds in Julia.
  • Differential Equations on Manifolds
If you are interested in writing a thesis within optimization, manifolds, variational methods, please write an e-mail to an ronny,bergmann@....

Current Bachelor's Theses

  • Renée Dornig,
    Regularized Clustering of Manifold-Valued Data.

Supervised Master's Theses

  • Felix Maschke,
    Geodesic Regression on Manifolds with Constraints

Supervised Bachelor's Theses

  • Erik Wünsche,
    L1-TV Regularization of SO(3)-valued Images ,
    together with Dr. Ralf Hielscher.
A list of previously supervised theses is available at ronnybergmann.net/teaching/students.html.
The following Links mostly refer to german pages.

Summer Term 2019

Modellierungsseminar

Winter Term 2018

Computer-orientierte Mathematik
Organizer of the exercises and Tutor
Computerpraktikum
Coordination
Höhere Mathematik III (If, Ph, ET)
Tutor

Earlier semesters

For prior lectures and teching obligations, see ronnybergmann.net/teaching.html.