S. Bartels et al.
Numerical solution of nonsmooth problems and application to damage evolution and optimal insulation
Foundations of Computational Mathematics
Barcelona, Spain, July 17, 2017
Abstract. Nonsmooth minimization problems and singular partial differential equations arise in the description of inelastic material behavior, image processing, and modeling of non-Newtonian fluids. The numerical discretization and iterative solution is often based on regularizing or stabilizing terms. In this talk we address the influence of such modifications on error estimates and the robustness of iterative solution methods. In particular, we present an unconditional stability result for semi-implicit discretizations of a class of singular flows and devise a variant of the alternating direction of multipliers method with variable step sizes. The results and methods are illustrated by numerical experiments for a damage evolution model and a problem of optimal insulation leading to a break of symmetry.
S. Bartels, P. Reiter, J. Riege
Approximation of self-avoiding inextensible curves
8th Workshop on Numerical Methods for Evolution Equations
Heraklion, Crete, September 23, 2016
Abstract. The prediction of the behavior of elastic curves arises in knot theory and the modeling of filament networks in biological cells. When large deformations are considered or when one is interested in determining homotopy classes, injectivity of deformations has to be imposed. A practical realization consists in including an appropriate potential in the energy functional. Its numerical treatment is difficult due to the singular and nonlocal nature. We present a discretization of a tangent-point potential that leads to a simple and stable scheme and which allows for the use of large step sizes.
S. Bartels, P. Schön
Adaptive approximation of the Monge-Kantorovich problem
Brunel, UK, June 15, 2016
Abstract. Optimal transportation problems define high-dimensional linear programs. An efficient approach to their numerical solution is based on reformulations as nonlinear partial differential equations. If transportation cost is proportional to distance this leads to the Monge--Kantorovich problem which is a constrained minimization problem on Lipschitz continuous functions. We discuss the iterative solution via splitting methods and devise an adaptive mesh refinement strategy based on an a~posteriori error estimate for the primal-dual gap.