Approximation of transport maps from local and non-local Monge-Ampere equations

许多与光学系统设计、经济学、气象学和资源最优运输有关的重要问题都可以归结为求解一类称为蒙格-安培方程的问题。目前还没有在最有趣和最具挑战性的情况下求解这些方程的方法。在最简单的问题中，现有方法的成本高得令人望而却步，而且会产生巨大的错误，严重限制了它们在现代应用中的有效性。这个项目将引入新的数学和计算技术来求解一般的(非局部)Monge-Ampere型方程，这将导致软件的开发，这些软件可以有效和准确地解决当前光学、经济学和气象学中的几个问题。该项目将包括研究生的培训。该项目的目标是设计、分析和实现求解一大类局部和非局部Monge-Ampere方程的收敛的数值方法。这个项目将引入一个新的Monge-Ampere算子的积分改写形式，它将允许使用高阶求积格式，在离散水平上保持椭圆性，并可证明收敛到正确的弱解。我们还将以一种突出椭圆结构的方式重新表述相关的非定域项，并且可以使用离散版本的狄拉克增量函数来近似。我们将为这些数值方法引入新的误差界，这些误差界将被提供给新的解梯度的宽模板近似。由此产生的方法将被证明以超线性精度近似解决方案和相应的交通地图。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Many important problems related to the design of optical systems, economics, meteorology, and the optimal transportation of resources can be reduced to the problem of solving a type of equation known as a Monge-Ampere equation. There are currently no methods available for solving these equations in the most interesting and challenging settings. In the simplest problems, existing methods are prohibitively expensive and produce large errors that severely limit their usefulness in modern applications. This project will introduce new mathematical and computational techniques for solving general (non-local) Monge-Ampere type equations, which will lead to the development of software that can efficiently and accurately solve several current problems in optics, economics, and meteorology. The project will include training of graduate students.The goal of this project is to design, analyze, and implement convergent numerical methods for solving a large class of local and non-local Monge-Ampere equations. This project will introduce a new integral reformulation of the Monge-Ampere operator, which will allow for the use of higher-order quadrature schemes that preserve ellipticity at the discrete level and provably converge to the correct weak solution. We will also reformulate relevant non-local terms in a way that highlights the elliptic structure and can be approximated using a discrete version of the Dirac delta function. We will introduce new error bounds for these numerical methods, which will be fed into new wide-stencil approximations of the solution gradient. The resulting methods will provably approximate both the solution and corresponding transport map with superlinear accuracy.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.