| 1. |
- Berman, Robert, 1976
(författare)
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Convergence Rates for Discretized Monge-Ampere Equations and Quantitative Stability of Optimal Transport
- 2021
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Ingår i: Foundations of Computational Mathematics. - : Springer Science and Business Media LLC. - 1615-3375 .- 1615-3383. ; 21, s. 1099-1140
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 2. |
- Berman, Robert, 1976
(författare)
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The probabilistic vs the quantization approach to Kahler-Einstein geometry
- 2023
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Ingår i: Mathematische Annalen. - 0025-5831 .- 1432-1807.
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Tidskriftsartikel (refereegranskat)abstract
- In the probabilistic construction of Kahler-Einstein metrics on a complex projective algebraic manifold X-involving random point processes on X-a key role is played by the partition function. In this work a new quantitative bound on the partition function is obtained. It yields, in particular, a new direct analytic proof that X admits a Kahler-Einstein metrics if it is uniformly Gibbs stable. The proof makes contact with the quantization approach to Kahler-Einstein geometry.
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| 3. |
- Berman, Robert, 1976
(författare)
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The Sinkhorn algorithm, parabolic optimal transport and geometric Monge-Ampere equations
- 2020
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Ingår i: Numerische Mathematik. - : Springer Science and Business Media LLC. - 0029-599X .- 0945-3245. ; 145, s. 771-836
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Tidskriftsartikel (refereegranskat)abstract
- We show that the discrete Sinkhorn algorithm-as applied in the setting of Optimal Transport on a compact manifold-converges to the solution of a fully non-linear parabolic PDE of Monge-Ampere type, in a large-scale limit. The latter evolution equation has previously appeared in different contexts (e.g. on the torus it can be be identified with the Ricci flow). This leads to algorithmic approximations of the potential of the Optimal Transport map, as well as the Optimal Transport distance, with explicit bounds on the arithmetic complexity of the construction and the approximation errors. As applications we obtain explicit schemes of nearly linear complexity, at each iteration, for optimal transport on the torus and the two-sphere, as well as the far-field antenna problem. Connections to Quasi-Monte Carlo methods are exploited.
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| 4. |
- Berman, Robert, 1976
(författare)
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The spherical ensemble and quasi-Monte-Carlo designs
- 2023
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Ingår i: Constructive Approximation. - 0176-4276 .- 1432-0940.
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Tidskriftsartikel (refereegranskat)abstract
- The spherical ensemble is a well-known ensemble of N repulsive points on the twodimensional sphere, which can realized in various ways (as a random matrix ensemble, a determinantal point process, a Coulomb gas, a Quantum Hall state...). Here we show that the spherical ensemble enjoys remarkable convergence properties from the point of view of numerical integration. More precisely, it is shown that the numerical integration rule corresponding to N nodes on the two-dimensional sphere sampled in the spherical ensemble is, with overwhelming probability, nearly a quasi-Monte-Carlo design in the sense of Brauchart-Saff-Sloan-Womersley for any smoothness parameter s = 2. The key ingredient is a new explicit sub-Gaussian concentration of measure inequality for the spherical ensemble.
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