Abstract: | We consider the problem of finding (possibly non connected)
discrete surfaces spanning a finite set of discrete boundary curves in the
three-dimensional space and minimizing (globally) a discrete energy involving
mean curvature. Although we consider a fairly general class of
energies, our main focus is on the Willmore energy, i.e. the total squared
mean curvature. Most works in the literature have been devoted to the
approximation of a surface evolving by the Willmore flow and, in particular,
to the approximation of the so-called Willmore surfaces, i.e., the
critical points of the Willmore energy. Our purpose is to address the delicate
task of approximating global minimizers of the energy under boundary
constraints. The main contribution of this work is to translate the
nonlinear boundary value problem into an integer linear program, using
a natural formulation involving pairs of elementary triangles chosen in
a pre-specified dictionary and allowing self-intersection. The reason for
such strategy is the well-known existence of algorithms that can compute
global minimizers of a large class of linear optimization problems, however
at a significant computational and memory cost. The case of integer
linear programming is particularly delicate and usual strategies consist
in relaxing the integral constraint x ∈ {0, 1} into x ∈ [0, 1] which is easier
to handle. Our work focuses essentially on the connection between the
integer linear program and its relaxation. We prove that:
– One cannot guarantee the total unimodularity of the constraint matrix,
which is a sufficient condition for the global solution of the
relaxed linear program to be always integral, and therefore to be a
solution of the integer program as well;
– Furthermore, there are actually experimental evidences that, in some
cases, solving the relaxed problem yields a fractional solution.
These facts indicate that the problem cannot be tackled with classical
linear programming solvers, but only with pure integer linear solvers.
Nevertheless, due to the very specific structure of the constraint matrix here, we strongly believe that it should be possible in the future to design
ad-hoc integer solvers that yield high-definition approximations to
solutions of several boundary value problems involving mean curvature,
in particular the Willmore boundary value problem. |