Axel Georg Rosenkrantz Turnquist, Assistant Professor, Beijing Institute of Mathematical Sciences and Applications

李文博，副研究员

Finite-Difference Methods for Computing Optimal Transport on the Sphere: Tangent Plane Methods and Volumetric Methods

7月9日（周三）11:00-12:00，南楼714

We first introduce the Monge problem of optimal transport on the unit sphere \(\mathbb{S}^{2}\subset\mathbb{R}^{3}\). Solving the Monge problem yields a pushforward map \(m\) from a source probability measure \(\mu\) to a target probability measure \(\nu\) on \(\Gamma\) that minimizes a cost functional, which is constructed from a choice of cost function. For many choices of the cost function, the pushforward map \(m\) can be expressed as \(m=h(\nabla u(x))\), where \(h:\mathcal{T}_{\mathbb{S}^{2}}\rightarrow\mathbb{S}^{2}\) and \(u\) is a \(c\)-convex function, a natural generalization of the notion of convexity. Such a function \(u\) can be shown to solve a Monge-Ampere-type PDE on the sphere.

We present two finite-difference methods to solve these Monge-Ampere-type PDEs on the unit sphere for different cost functions, the first being a provably convergent method where computations are performed on local tangent planes. The finite-difference methods here are constructed to have certain properties, most notably monotonicity, which allow for one to prove local uniform convergence to a continuous viscosity solution of the PDE.

The second method requires us to first properly extend the optimal transport problem to a tubular neighborhood \(T_{\epsilon}\) of \(\mathbb{S}^{2}\), by carefully defining an extended cost function and extended source and target probability measures. We then define the corresponding Monge-Ampere PDE on the tubular region \(T_{\epsilon}\), where all derivatives in the PDE are Euclidean. This allows for the construction of simple finite-difference methods for solving the resulting PDE, whose solution can be used to solve the original optimal transport problem on the sphere.