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

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.