

A Projected Gradient Flow for the Generalized Integral Menger Curvature in the Hilbert Case
Jan Knappmann, Henrik Schumacher, Daniel Steenebrügge, Heiko von der Mosel (2021)
We establish longtime existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case, and provide $C^{1,1}$bounds in time for the solution that only depend on the initial curve. The selfavoidance property of integral Menger curvature guarantees that the knot class of the initial curve is preserved under the flow, and the projection ensures that each curve along the flow is parametrized with the same speed as the initial configuration. Finally, we describe how to simulate this flow numerically with substantially higher efficiency than in the corresponding numerical $L^2$ gradient descent or other optimization methods.
arxiv



Repulsive Curves
Christopher Yu, Henrik Schumacher, Keenan Crane (2020)
Curves play a fundamental role across computer graphics, physical simulation, and mathematical visualization, yet most tools for curve design do nothing to prevent crossings or selfintersections. This paper develops efficient algorithms for (self)repulsion of plane and space curves that are wellsuited to problems in computational design. Our starting point is the socalled tangentpoint energy, which provides an infinite barrier to selfintersection. In contrast to local collision detection strategies used in, e.g., physical simulation, this energy considers interactions between all pairs of points, and is hence useful for global shape optimization: local minima tend to be aesthetically pleasing, physically valid, and nicely distributed in space. A reformulation of gradient descent, based on a SobolevSlobodeckij inner product enables us to make rapid progress toward local minimaindependent of curve resolution. We also develop a hierarchical multigrid scheme that significantly reduces the perstep cost of optimization. The energy is easily integrated with a variety of constraints and penalties (e.g., inextensibility, or obstacle avoidance), which we use for applications including curve packing, knot untangling, graph embedding, noncrossing spline interpolation, flow visualization, and robotic path planning.
arxiv



Sobolev Gradients for the Möbius Energy
Philipp Reiter, Henrik Schumacher (2020)
Aiming at optimizing the shape of closed embedded curves within prescribed isotopy classes, we use a gradientbased approach to approximate stationary points of the Möbius energy. The gradients are computed with respect to Sobolev inner products similar to the $W^{3/2,2}$inner product. This leads to optimization methods that are significantly more efficient and robust than standard techniques based on $L^2$gradients.
arxiv



Variational Methods for Discrete Geometric Functionals
Henrik Schumacher, Max Wardetzky (2020)
While consistent discrete notions of curvatures and differential operators have been widely studied, the question of whether the resulting minimizers converge to their smooth counterparts still remains open for various geometric functionals. Building on tools from variational analysis, and in particular using the notion of Kuratowski convergence, we offer a general framework for treating convergence of minimizers of (discrete) geometric functionals. We show how to apply the resulting machinery to minimal surfaces and Euler elasticae.
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Computing the conformal barycenter
Jason Cantarella, Henrik Schumacher (2020)
The conformal barycenter of a point cloud on the sphere at infinity of the Poincaré ball model of hyperbolic space is a hyperbolic analogue of the geometric median of a point cloud in Euclidean space. It was defined by Douady and Earle as part of a construction of a conformally natural way to extend homeomorphisms of the circle to homeomorphisms of the disk, and it plays a central role in Millson and Kapovich's model of the configuration space of cyclic linkages with fixed edge lengths. In this paper we consider the problem of computing the conformal barycenter. Abikoff and Ye have given an iterative algorithm for measures on $\mathbb{S}^d$ which is guaranteed to converge. We analyze Riemannian versions of Newton's method computed in the intrinsic geometry of the Poincare ball model. We give NewtonKantorovich (NK) conditions under which we show that Newton's method with fixed step size is guaranteed to converge quadratically to the conformal barycenter for measures on any $\mathbb{S}^d$ (including infinitedimensional spheres). For measures given by $n$ atoms on a finite dimensional sphere which obey the NK conditions, we give an explicit linear bound on the computation time required to approximate the conformal barycenter to fixed error. We prove that our NK conditions hold for all but exponentially few $n$ atom measures. For all measures with a unique conformal barycenter we show that a regularized Newton's method with line search will always converge (eventually superlinearly) to the conformal barycenter. Though we do not have hard time bounds for this algorithm, experiments show that it is extremely efficient in practice and in particular much faster than the AbikoffYe iteration.
arxiv code



Variational convergence of discrete elasticae
Sebastian Scholtes, Henrik Schumacher, Max Wardetzky (2020)
We discuss a discretization of the EulerBernoulli bending energy and of Euler elasticae under clamped boundary conditions by polygonal lines. We show Hausdorff convergence of the set of almost minimizers of the discrete bending energy to the set of smooth Euler elasticae under mesh refinement in (i) the $W^{1,\infty}$topology for piecewiselinear interpolation; and in (ii) the $W^{2,p}$topology, $p \in [2,\infty [$, using a suitable smoothing operator to create $W^{2,p}$curves from polygons.
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Elastic energy regularization for inverse obstacle scattering problems
J. Eckhardt, R. Hiptmair, T. Hohage, H. Schumacher, M. Wardetzky (2019)
This paper is motivated by inverse problems in which the boundary curve of a smooth bounded domain has to be reconstructed from indirect measurements. As a classical example we study acoustic inverse obstacle scattering problems for cylindrical soundsoft scatterers using farfield measurements of scattered timeharmonic waves. By introducing a shape manifold as a solution set we allow the reconstruction of general, not necessarily starshaped, curves. The bending energy is used as a stabilizing term in Tikhonov regularization to gain independence of the parametrization. Moreover, we discuss how selfintersections can be avoided by penalization with the Möbius energy and prove the regularizing property of our approach as well as convergence rates under variational source conditions. In a second part of the paper a discrete setting is introduced, and we describe a numerical method for finding the minimizer of the Tikhonov functional on the shapemanifold. Numerical examples demonstrate that our method can reconstruct nonstarshaped obstacles.
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Variational convergence of discrete minimal surfaces
Henrik Schumacher, Max Wardetzky (2019)
Building on and extending tools from variational analysis and relying on certain a~priori assumptions, we prove Kuratowski convergence of sets of simplicial area minimizers to minimizers of the smooth DouglasPlateau problem under simplicial refinement. This convergence is with respect to a topology that is finer than the topology of uniform convergence of both positions and surface normals.
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Maximal discrete sparsity in parabolic optimal control with measures
Evelyn Herberg, Michael Hinze, Henrik Schumacher (2019)
We consider variational discretization of a parabolic optimal control problem governed by spacetime measure controls. For the state discretization we use a PetrovGalerkin method employing piecewise constant states and piecewise linear and continuous test functions in time. For the space discretization we use piecewise linear and continuous functions. As a result the controls are composed of Dirac measures in spacetime, centered at points on the discrete spacetime grid. We prove that the optimal discrete states and controls converge strongly in $L^q$ and weakly$*$ in $\mathcal{M}$, respectively, to their smooth counterparts, where $q \in (1,\min\{2,1+2/d\}]$ is the spatial dimension. Furthermore, we compare our approach to [Casas, Kunisch  Parabolic control problems in spacetime measure spaces], where the corresponding control problem is discretized employing a discontinuous Galerkin method for the state discretization and where the discrete controls are piecewise constant in time and Dirac measures in space. Numerical experiments highlight the features of our discrete approach.
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Pseudogradient flows of geometric energies
Henrik Schumacher (2018)
For differentiable functions on Riemannian manifolds, the gradient vector field and its induced gradient flow are welldefined and wellunderstood concepts. Unfortunately, many nonquadratic, infinitedimensional optimization problems cannot be formulated on Riemannian manifolds in a convenient way. We introduce a category of infinitedimensional Banach manifolds that allows us to define a generalized gradient. Moreover, we show shorttime existence of the induced flows and apply their discretizations to the numerical minimization of various geometric energies of immersed curves and surfaces.
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On $H^2$gradient Flows for the Willmore Energy
Henrik Schumacher (2017)
We show that the concept of $H^2$gradient flow for the Willmore energy and other functionals that depend at most quadratically on the second fundamental form is welldefined in the space of immersions of Sobolev class $W^{2,p}$ from a compact, $n$dimensional manifold into Euclidean space, provided that $p \geq 2$ and $p>n$. We also discuss why this is not true for Sobolev class $H^2= W^{2,2}$. In the case of equality constraints, we provide sufficient conditions for the existence of the projected $H^2$gradient flow and demonstrate its usability for optimization with several numerical examples.
arxiv
