

1992  Dr. rer. nat., University of Bonn  1993  1996  Postdoc, University of Freiburg  1996  2001  Professor (C3), University of Bonn  2001  2004  Professor (C4), University of DuisburgEssen  Since 2004  Professor (C4/W3), University of Bonn 


Variational problems and evolution problems arising in computer vision, in geometry processing, and in materials science are the major driving force of my research.
In computer vision I'm interested in the infinite dimensional geometry of shape spaces equipped with a Riemannian metric which is motivated by physical models of viscous dissipation. A central theme is a general variational time discrete Riemannian calculus on different shape spaces, including discrete geodesics, exponential map and parallel transport. Applications are warping of images or shell surfaces, shape extrapolation, and pattern or texture transfer. A comprehensive convergence theory based on convergence, finite element and ODE estimates on Hilbert spaces could be developed. I'm also interested in the close links to the theory of optimal transport.
A major goal is to treat textured images and explore inherent multiple scales in image maps. To this end images are considered as pointwise maps into some patch manifold, describing local, high dimensional texture and structure. Furthermore, spline curves and other low dimensional, smooth submanifolds will be particular interest in time dependent data analysis and in geometry animation.
With respect to materials science, I'm particularly interested in twoscale elastic shape optimization and the formation of optimal branching and folding patterns in elastic materials. The minimization of compliance type cost functionals leads to microstructured shapes and branching patterns arising naturally at material interfaces or at boundary incompatibilities.
My focus is on robust a posteriori error control using functional error estimates for BV functionals, duality techniques and relaxation. The aim is an efficient simulation and optimization of the microscopic patterns, and a better understanding of branchingtype patterns observed in natural elastic structures, as for example bones and thin sheets. The vision is to carry over the twoscale analysis of elastic bulk material to thin elastic plates and shells.


DFG Cluster of Excellence “Hausdorff Center for Mathematics”
Deputy coordinator, 2006 – 2012
DFG Cluster of Excellence “Hausdorff Center for Mathematics”
Member of the Board of Directors, since 2006
DFG project “Discrete Riemannian calculus on shape space”
jointly with KarlTheodor Sturm, in the Collaborative Research Center SFB 1060 “The Mathematics of Emergent Effects”, 2012  2020
DFG project “Numerical optimization of shape microstructures”
jointly with Sergio Conti, in the Collaborative Research Center SFB 1060 “The Mathematics of Emergent Effects”, 2012  2020
DFG project “Geodesic Paths in Shape Space”
in the research network of the FWF S117 “Geometry + Simulation”, 2012  2020
DFG Collaborative Research Center SFB 1060 “The Mathematics of Emergent Effects”
Deputy coordinator, since 2014
Conference “Panorama of Mathematics” (Bonn),
Organizer, 2015
GIF project “A Functional Map Approach to Shape Spaces”
by the GermanIsraeli Foundation for Scientific Research and Development, jointly with Miri BenChen, 2017  2020
Series of Oberwolfach Workshops on “Image and Surface Processing”
Organizer, 2005, 2007, 2011, 2016


Research Area A I'm studying a general, time discrete Riemannian calculus on different shape spaces, e.g. spaces of 2D surfaces, which physically behave like viscous shells, or spaces of images, where the metric encodes the cost of viscous transport and intensity changes along transport paths. The discretization is based on a proper approximation of the squared Riemannian distance by a functional on consecutive objects along a discrete path in shape space. The available tools of the derived discrete geodesic calculus are a discrete path energy, discrete geodesics, discrete logarithm, discrete exponential map, discrete parallel transport, discrete covariant derivative, and finally a discrete curvature tensor on shape space.
Application examples are image warping, extrapolation of image sequences, and pattern or texture transfer. In the context of Hilbert manifolds, we could establish a comprehensive convergence theory based on convergence, finite element and ODE estimates on Hilbert spaces could be developed [1,2]. We also study generalized spaces of images, where an image is no longer based on a pointwise image intensity value but a high dimensional texture and structure description. I'm investigating time discrete Riemannian splines in shape spaces with applications to smooth image key frame interpolation, compression and video processing [3].Furthermore, we integrated concepts from optimal transport and we studied a pure optimal transport model with source term [4]. Finally, I´m interested in the numerical approximation of optimal transport on discrete metric measure spaces.  Research Area B I´m investigating elastic shape optimization with a particular emphasis on twoscale models and on risk averse stochastic optimization [5]. Here, the minimization of compliance type cost functionals leads to microstructured shapes and via an additional interfacial cost a particular scale can be selected. In fact, in the limit of vanishing interfacial cost branching patterns represent optimal elastic shapes.
Furthermore, branching patterns arise naturally at material interfaces, at boundary incompatibilities, and in compressed thin films, and are determined by the competition of elastic and interfacial energies. Moreover, I'm studying the formation of optimal branching and folding patterns in elastic materials. For simple model problems we studied rigorous a posteriori error estimates and for the two scale models error estimates based on the dual weighted residual approach [6]. Furthermore, we could improve the upper energy bound for branching patterns at austenitemartensite interfaces [7]. We use phase field models [8] for the numerical approximation of optimal branching structures with the aim to better understand branchingtype patterns observed in natural elastic structures, as for example bones and graphene sheets. 


[ 1] Martin Rumpf, Benedikt Wirth
Variational time discretization of geodesic calculus IMA J. Numer. Anal. , 35: (3): 10111046 2015[ 2] B. Berkels, A. Effland, M. Rumpf
Time discrete geodesic paths in the space of images SIAM J. Imaging Sci. , 8: (3): 14571488 2015[ 3] Behrend Heeren, Martin Rumpf, Max Wardetzky, Benedikt Wirth
TimeDiscrete Geodesics in the Space of Shells Comput. Graph. Forum , 31: (5): 17551764 2012 DOI: 10.1111/j.14678659.2012.03180.x[ 4] Jan Maas, Martin Rumpf, Carola Schönlieb, Stefan Simon
A generalized model for optimal transport of images including dissipation and density modulation ESAIM Math. Model. Numer. Anal. , 49: (6): 17451769 2015[ 5] Benedict Geihe, Martin Lenz, Martin Rumpf, Rüdiger Schultz
Risk averse elastic shape optimization with parametrized fine scale geometry Math. Program. , 141: (12, Ser. A): 383403 2013[ 6] Benedict Geihe, Martin Rumpf
A posteriori error estimates for sequential laminates in shape optimization Discrete Contin. Dyn. Syst. Ser. S , 9: (5): 13771392 2016[ 7] Patrick Dondl, Behrend Heeren, Martin Rumpf
Optimization of the branching pattern in coherent phase transitions C. R. Math. Acad. Sci. Paris , 354: (6): 639644 2016[ 8] Patrick Penzler, Martin Rumpf, Benedikt Wirth
A phasefield model for compliance shape optimization in nonlinear elasticity ESAIM Control Optim. Calc. Var. , 18: (1): 229258 2012[ 9] Sergio Conti, Harald Held, Martin Pach, Martin Rumpf, Rüdiger Schultz
Risk averse shape optimization SIAM J. Control Optim. , 49: (3): 927947 2011[ 10] U. Clarenz, U. Diewald, G. Dziuk, M. Rumpf, R. Rusu
A finite element method for surface restoration with smooth boundary conditions Comput. Aided Geom. Design , 21: (5): 427445 2004[ 11] Günther Grün, Martin Rumpf
Nonnegativity preserving convergent schemes for the thin film equation Numer. Math. , 87: (1): 113152 2000[ 12] M. Flucher, M. Rumpf
Bernoulli's freeboundary problem, qualitative theory and numerical approximation J. Reine Angew. Math. , 486: : 165204 1997[ 13] Benjamin Berkels, Alexander Effland, Martin Rumpf
A Posteriori Error Control for the Binary MumfordShah Model Mathematics of Computation 2015 DOI: doi.org/10.1090/mcom/3138[14] Benjamin Berkels, Sebastian Bauer, Svenja Ettl, Oliver Arold, Joachim Hornegger, Martin Rumpf
Joint Surface Reconstruction and 4D Deformation Estimation from Sparse Data and Prior Knowledge for MarkerLess Respiratory Motion Tracking Medical Physics , 40: (9): 091703 2013 DOI: 10.1118/1.4816675





• Computing and Visualization in Science (since 1999)
• SIAM Journal on Imaging Science (since 2007)
• SIAM Journal on Numerical Analysis (since 2015)
• Journal of Mathematical Imaging and Vision (since 2015)


2003  Plenary lecture, GAMM annual meeting, Padua / Abano Terme, Italy  2004  Plenary lecture, SIAM Conference on Image Science, Salt Lake City, UT, USA  2005  Plenary lecture, EQUADIFF, Bratislava, Slovakia  2006  Plenary lecture, Curves and Surface, Avignon, France  2008  Lecture course, CIME summer school, Cetraro, Italy  2010  Lecture course, CNA summer school, Pittsburgh, PA, USA  2013  Plenary lecture, SSVM, Graz, Austria  2015  Lecture course, CRC summer school, Barcelona, Spain  2016  Geometry Summit, Berlin 


2002  Chair in Mathematics, University of Zürich, Switzerland  2003  Chair, MATHEON, FU Berlin  2012  Director position of the Weierstrass Institute Berlin combined with a chair at the HU Berlin 


Olga Wilderotter (2001): “Adaptive FiniteElementeMethode für singuläre parabolische Probleme”,
now Professor, HS Karlsruhe
Ulrich Weikard (2002): “Numerische Loesungen der CahnHilliardGleichung und der CahnLarcheGleichung”,
now Senior Economist, DekaBank Deutsche Girozentrale
Tobias Preußer (2003): “Anisotropic Geometric Diffusion in Image and ImageSequence Processing”,
now Professor, Jacobs University Bremen, and Member of Management Board, and Head of Modelling & Simulation, Fraunhofer MEVIS, Bremen
Robert Strzodka (2004): “Hardware Efficient PDE Solvers in Quantized Image Processing”,
now Professor, University of Heidelberg
Benedikt Wirth (2010): “Variational Methods in Shape Space”,
now Associate Professor, University of Münster
Benjamin Berkels (2012): “Joint Methods in imaging based on diffuse image representations”,
now Professor, RWTH Aachen


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