A speed preserving Hilbert gradient flow for generalized integral Menger curvature

Advances in Calculus of Variations, vol. 16, pp. 597–635

Abstract

Abstract We establish long-time existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case and provide C 1 , 1 C^{1,1} -bounds in time for the solution that only depend on the initial curve. The self-avoidance 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 L^{2} gradient descent or other optimization methods.

Authors 4

1. Jan Knappmann Aachen Institut für Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Mathematik , RWTH Aachen University , Templergraben 55, 52062 Aachen , Germany

2. H. Ralph Schumacher Aachen Institut für Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Mathematik , RWTH Aachen University , Templergraben 55, 52062 Aachen , Germany

3. Daniel Steenebrügge corresponding Aachen Institut für Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Mathematik , RWTH Aachen University , Templergraben 55, 52062 Aachen , Germany

4. Heiko von der Mosel Aachen Institut für Mathematik

   RWTH Aachen University

   Affiliation as printed

   Institut für Mathematik , RWTH Aachen University , Templergraben 55, 52062 Aachen , Germany