离散Willmore边值问题的线性规划方法及其推广
On a linear programming approach to the discrete Willmore boundary value problem and generalizations
摘要：
我们考虑的问题，发现（可能是非连接）离散表面跨越一组有限的离散边界曲线在三维空间和最小化（全球）的离散能量涉及平均曲率。虽然我们认为一个相当普遍的一类能源，我们的主要重点是Willmore能源，即总平方平均曲率。我们的目的是解决的微妙的任务，近似全球最小的边界约束下的能量。这项工作的主要贡献是翻译成一个整数线性规划的非线性边值问题，使用一个自然的配方，涉及对在预先指定的字典中选择的基本三角形，并允许自交。我们的工作主要集中在整数线性规划及其松弛之间的联系上。我们证明：- 我们不能保证约束矩阵的总单模性，这是松弛线性规划的全局解总是积分的充分条件，因此也是整数规划的解; -此外，实际上有实验证据表明，在某些情况下，解决松弛问题会产生分数解。由于这里的约束矩阵的非常具体的结构，我们强烈认为，它应该是可能的，在未来的设计特设整数求解器，产生高清晰度的近似解决方案的几个边值问题，涉及平均曲率，特别是Willmore边值问题。
英文摘要：
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 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. 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. 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.
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