最优输运monge - ampantere方程的单调离散化

IF 1.9 3区 数学 Q2 Mathematics
G. Bonnet, J. Mirebeau
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引用次数: 8

摘要

本文设计了monge - ampantere方程第二边值问题的单调有限差分离散方法,该方法的主要应用是最优输运问题。通过添加一个常数,证明了一类解集稳定的退化椭圆型方程单调数值格式解的存在性,并证明了所引入的monge - ampantere方程的格式属于该类。我们证明了该方案的收敛性,尽管只在二次最优传输的情况下。该方案基于将monge - amp算子重新表述为半线性算子的最大值。在二维中,我们建议使用源自低维点阵几何的sell公式来选择离散化的参数。我们表明,这种方法产生了一个在离散算子中出现的最大值的封闭形式公式,这使得该方案能够特别有效地求解。本文给出了将该格式应用于二次最优输运问题和非成像光学中的远场折射问题的一些数值结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Monotone discretization of the Monge-Ampère equation of optimal transport
We design a monotone finite difference discretization of the second boundary value problem for the Monge-Ampère equation, whose main application is optimal transport. We prove the existence of solutions to a class of monotone numerical schemes for degenerate elliptic equations whose sets of solutions are stable by addition of a constant, and we show that the scheme that we introduce for the Monge-Ampère equation belongs to this class. We prove the convergence of this scheme, although only in the setting of quadratic optimal transport. The scheme is based on a reformulation of the Monge-Ampère operator as a maximum of semilinear operators. In dimension two, we recommend to use Selling's formula, a tool originating from low-dimensional lattice geometry, in order to choose the parameters of the discretization. We show that this approach yields a closed-form formula for the maximum that appears in the discretized operator, which allows the scheme to be solved particularly efficiently. We present some numerical results that we obtained by applying the scheme to quadratic optimal transport problems as well as to the far field refractor problem in nonimaging optics.
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来源期刊
CiteScore
2.70
自引率
5.30%
发文量
27
审稿时长
6-12 weeks
期刊介绍: M2AN publishes original research papers of high scientific quality in two areas: Mathematical Modelling, and Numerical Analysis. Mathematical Modelling comprises the development and study of a mathematical formulation of a problem. Numerical Analysis comprises the formulation and study of a numerical approximation or solution approach to a mathematically formulated problem. Papers should be of interest to researchers and practitioners that value both rigorous theoretical analysis and solid evidence of computational relevance.
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