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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., Herrmann, M., Herzog, R., Schmidt, S., Vidal Nuñez, J.
Total variation of the normal vector field as shape prior with applications in geometric inverse problems
Preprint, arXiv: 1902.07240.
An analogue of the total-variation prior for the normal vector field along the boundary of smooth and non-smooth shapes in 3D is introduced. Its analysis in the smooth case is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. This novel functional is subsequently extended to piecewise flat, triangulated surfaces as they occur for instance in finite element computations. The ensuing discrete functional agrees with the discrete total mean curvature known in discrete differential geometry. A split Bregman iteration is proposed for the solution of shape optimization problems in which the total variation of the normal appears as a regularizer. Both the continuous and discrete settings are detailed. Unlike most other priors such as surface area, the new functional allows for piecewise flat shapes in the discrete setting. As an application, a geometric inverse problem of inclusion detection type is considered. Numerical experiments confirm that polyhedral shapes can be identified quite accurately.
Bergmann, R. and Herzog, R. (2018).
Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds.
Preprint, 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.

veröffentlichte Artikel

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 Abschlussarbeiten

  • 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 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.