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Professur Numerische Mathematik (Partielle Differentialgleichungen)
PD Dr. Ronny Bergmann

PD Dr. Ronny Bergmann

Stellvertretender Leiter der Professur Numerische Mathematik (Partielle Differentialgleichungen)
Telefon:
+49 371 531 36098
Fax:
+49 371 531 836098
Büro:
Reichenhainer Str. 41, Zimmer 609 (C47.609)
Scholar:
ResearchGate:
GitHub:
twitter:
Sekretariat:
Anne-Kristin Glanzberg, Zimmer 607, Telefon +49 371 531 22500
Postanschrift:
TU Chemnitz, Fakultät für Mathematik, 09107 Chemnitz
Sprechzeit:
jederzeit bei Anwesenheit
seit 04/2018
Stellvertretender Leiter der Professur Numerische Mathematik (Partielle Differentialgleichungen) von Prof. Dr. R. Herzog
01/2018
Habilitation im Fach Mathematik an der Technischen Universität Kaiserslautern, (kumulative) Habilitationsschrift: “Variational Methods in Manifold-valued Image Processing”
09/2013 — 03/2018
Postdoktorand bei Prof. Dr. Gabriele Steidl in der AG Bildverarbeitung und Datenanalyse an der Technischen Universität Kaiserslautern
06/2013
Promotion am Institut für Mathematik der Universität zu Lübeck bei Prof. Dr. Jürgen Prestin
10/2009 — 08/2013
wissenschaftlicher Mitarbeiter (Doktorand) am Institut für Mathematik der Universität zu Lübeck
09/2009
Diplom-Informatiker (Dipl.-Inf.), Institut für Mathematik, Universität zu Lübeck.
10/2004 — 09/2009
Studium der Informatik an der Universität zu Lübeck
Weitere Links

Preprints

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. (2020)
Discrete Total Variation of the Normal Vector Field as Shape Prior with Applications in Geometric Inverse Problems
accepted for publication, Inverse Problems.
doi: 10.1088/1361-6420/ab6d5c, 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., Herrmann, M., Herzog, R., Schmidt, S., Vidal Núñez, J. (2020)
Total Variation of the Normal Vector Field as Shape Prior
accepted for publication, Inverse Problems.
doi: 10.1088/1361-6420/ab6d5b, 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.

veröffentlichte Artikel

Bergmann, R. and Herzog, R. (2019).
Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds.
SIAM Journal on Optimization (SIOPT, 29(4), pp. 2423–2444.
doi: 10.1137/18M1181602, 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.
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.

vorherige Publikationen

eine vollständige Liste meiner Publikationen ist unter ronnybergmann.net/publications.html verfügbar.

Themen für Abschlussarbeiten

  • Lineare Programme auf Mannigfaltigkeiten
  • Optimierung auf Mannigfaltigkeiten in Julia.
  • Differentialgleichungen auf Mannigfaltigkeiten
Bei Interesse an Themen zur Optimierung, Mannigfaltigkeiten und Variationsmethoden, schreiben Sie mir eine E-Mail an ronny,bergmann@....

Laufende Bachelorarbeiten

  • Tom-Christian Riemer,
    The Riemannian BFGS Method and its Implementation in Julia
  • Jan-Philipp Pfaue,
    Constrained optimization on Riemannian manifolds using geodesic polygonal sets

Betreute Masterarbeiten

  • Felix Maschke,
    Geodätische Regression auf Mannigfaltigkeiten mit Nebenbedingungen

Betreute Bachelorarbeiten

  • Renée Dornig,
    Regularized Clustering of Manifold-Valued Data.
  • Erik Wünsche,
    L1-TV Regularisierung SO(3)-wertiger Bilder ,
    gemeinsam mit Dr. Ralf Hielscher.
Eine Liste vorheriger betreuter Arbeiten findet sich unter ronnybergmann.net/teaching/students.html.

Wintersemester 2020

Computer-orientierte Mathematik
Übungskoordination und -leitung
Computerpraktikum
Gesamtkoordination
Optimierung für Nichtmathematiker
Übungsleiter

Sommersemester 2020

Optimierung auf Mannigfaltigkeiten
Übungsleiter
Höhere Mathematik II (für IW,Ch,SK,CC)
Übungsleiter
Computerpraktikum
Gesamtkoordination

Wintersemester 2019/2020

Modellierungsseminar
Organisator und Koordinator
Optimierung für Nichtmathematiker
Übungsleiter
Höhere Mathematik I (für IW,Ch,SK,CC)
Übungsleiter

Sommersemester 2019

Modellierungsseminar
Organisator und Koordinator

Wintersemester 2018

Computer-orientierte Mathematik
Übungskoordination und -leitung
Computerpraktikum
Gesamtkoordination
Höhere Mathematik III (If, Ph, ET)
Übungsleiter

Frühere Semester

Für frühere Lehrveranstaltungen in Kaiserslautern und Lübeck, siehe ronnybergmann.net/teaching.html.