S. Bartels and P. Reiter
Approximating gradient flow evolutions of self-avoiding inextensible curves and elastic knots
Thematic Programme on Numerical analysis of complex PDE models in the sciences
Erwin-Schrödinger-Institut, Vienna, Austria, July 18, 2018
Abstract. We discuss a semi-implict scheme that allows for minimizing the bending energy of curves within certain isotopy classes. To this end we consider a weighted sum of a bending energy and a tangent-point functional. The latter is defined via the iterated integral of a negative power of the tangent-point radius which is for a given pair of points the radius of the circle that is tangent to the curve in one point and intersects the curve in the other point. Evolutions are defined via gradient flows for the total energy within classes of arclength-parametrized curves. The numerical scheme is obtained by a semi-implicit discretization which treats the nonlinear and nonlocal tangent-point functional explicitly. This avoids the occurrence of fully populated matrices and allows for a straightforward parallelization of its computation. Based on estimates for the second derivative of the tangent-point functional and a uniform bi-Lipschitz radius, we prove a stability result implying energy decay during the evolution as well as maintenance of arclength parametrization. We present some numerical experiments exploring the energy landscape, targeted to the question how to obtain global minimizers of the bending energy in knot classes, so-called elastic knots.
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.