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

许多与光学系统设计、经济学、气象学和资源最佳运输有关的重要问题都可以归结为求解一种称为蒙格-安培方程的方程。 目前还没有方法可以在最有趣和最具挑战性的环境中求解这些方程。 在最简单的问题，现有的方法是昂贵的，并产生很大的误差，严重限制了它们在现代应用中的有用性。 该项目将引入新的数学和计算技术来解决一般（非本地）Monge-Ampere型方程，这将导致软件的开发，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，分析和实施收敛的数值方法来解决一大类局部和非局部Monge-Ampere方程。该项目将引入一个新的积分重新表述的蒙格-安培算子，这将允许使用高阶求积方案，保持椭圆在离散水平和可证明收敛到正确的弱解。我们还将重新制定相关的非局部项的方式，突出了椭圆结构，可以近似使用离散版本的狄拉克δ函数。我们将为这些数值方法引入新的误差界，这些误差界将被馈送到解梯度的新的宽模板近似中。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.