Handling Convexity-Like Constraints in Variational Problems
Handling Convexity-Like Constraints in Variational Problems
复制标题
处理变分问题中的类凸约束
DOI：
10.1137/130938359
复制
发表时间：
2014
期刊：
影响因子：
--
通讯作者：
É. Oudet
中科院分区：
文献类型：
--
作者：
Q. Mérigot;É. Oudet
我们给出了构造凸函数空间的有限维近似的一般框架，它也适用于$c-凸函数空间和凸体的支撑函数空间。我们给出了近似空间与容许集之间距离的精确估计。这一框架适用于区域网格上的分段线性函数和其他有限维空间(如张量积样条)对凸函数的逼近。我们展示了这些离散化如何很好地适用于凸约束下的变分问题的数值求解。我们的实现依赖于近邻算法，可以很容易地并行化，从而使其适用于大规模的二维和三维问题。对变分中的三个问题：三维去噪、委托代理问题和最优化问题的数值求解，说明了该方法的通用性和有效性。
We provide a general framework to construct finite-dimensional approximations of the space of convex functions, which also applies to the space of $c$-convex functions and to the space of support functions of convex bodies. We give precise estimates of the distance between the approximation space and the admissible set. This framework applies to the approximation of convex functions by piecewise-linear functions on a mesh of the domain and by other finite-dimensional spaces such as tensor-product splines. We show how these discretizations are well suited for the numerical solution of problems of calculus of variations under convexity constraints. Our implementation relies on proximal algorithms and can be easily parallelized, thus making it applicable to large-scale problems in dimension two and three. We illustrate the versatility and the efficiency of our approach on the numerical solution of three problems in calculus of variation: three-dimensional denoising, the principal agent problem, and optimizat...