Convergence Rates for Discretized Monge-Ampere Equations and Quantitative Stability of Optimal Transport

• In recent works-both experimental and theoretical-it has been shown how to use computational geometry to efficiently construct approximations to the optimal transport map between two given probability measures on Euclidean space, by discretizing one of the measures. Here we provide a quantitative convergence analysis for the solutions of the corresponding discretized Monge-Ampere equations. This yields H-1-converge rates, in terms of the corresponding spatial resolution h, of the discrete approximations of the optimal transport map, when the source measure is discretized and the target measure has bounded convex support. Periodic variants of the results are also established. The proofs are based on new quantitative stability results for optimal transport maps, shown using complex geometry.

Subject headings

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

NATURVETENSKAP  -- Matematik -- Beräkningsmatematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Computational Mathematics (hsv//eng)

NATURVETENSKAP  -- Fysik -- Annan fysik (hsv//swe)

NATURAL SCIENCES  -- Physical Sciences -- Other Physics Topics (hsv//eng)

NATURVETENSKAP  -- Matematik -- Matematisk analys (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Mathematical Analysis (hsv//eng)

Keyword

Monge-Ampere equations

Optimal transport

Numerical analysis

Complex

differential geometry

Computer Science

Mathematics

Optimal transport

Publication and Content Type

ref (subject category)

art (subject category)