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Here we provide a quantitative convergence analysis for the solutions of the corresponding discretized Monge\u2013Amp\u00e8re equations. This yields<jats:inline-formula><jats:alternatives><jats:tex-math>$$H^{1}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mi>H<\/mml:mi><mml:mn>1<\/mml:mn><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-converge rates, in terms of the corresponding spatial resolution<jats:italic>h<\/jats:italic>, 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.<\/jats:p>","DOI":"10.1007\/s10208-020-09480-x","type":"journal-article","created":{"date-parts":[[2020,12,14]],"date-time":"2020-12-14T21:02:37Z","timestamp":1607979757000},"page":"1099-1140","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Convergence Rates for Discretized Monge\u2013Amp\u00e8re Equations and Quantitative Stability of Optimal Transport"],"prefix":"10.1007","volume":"21","author":[{"given":"Robert J.","family":"Berman","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,12,14]]},"reference":[{"key":"9480_CR1","unstructured":"Alexandrov, A. D.: Convex polyhedra. Translated from the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze and A. B. Sossinsky. Springer Monographs in Mathematics. 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