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Obtaining solutions to optimal transportation (OT) problems is typically intractable when marginal spaces are continuous. Recent research has focused on approximating continuous solutions with discretization methods based on i.i.d. sampling, and this has shown convergence as the sample size increases. However, obtaining OT solutions with large sample sizes requires intensive computation effort, which can be prohibitive in practice. In this paper, we propose an algorithm for calculating discretizations with a given number of weighted points for marginal distributions by minimizing the (entropy-regularized) Wasserstein distance and providing bounds on the performance. The results suggest that our plans are comparable to those obtained with much larger numbers of i.i.d. samples and are more efficient than existing alternatives. Moreover, we propose a local, parallelizable version of such discretizations for applications, which we demonstrate by approximating adorable images. 

Volatile chemical products (VCPs) and other non-combustion-related sourceshave become important for urban air quality, and bottom-up calculationsreport emissions of a variety of functionalized compounds that remainunderstudied and uncertain in emissions estimates. Using a new instrumentalconfiguration, we present online measurements of oxygenated organiccompounds in a US megacity over a 10 d wintertime sampling period, whenbiogenic sources and photochemistry were less active. Measurements wereconducted at a rooftop observatory in upper Manhattan, New York City, USAusing a Vocus chemical ionization time-of-flight mass spectrometer, withammonium (NH4+) as the reagent ion operating at 1 Hz. The range ofobservations spanned volatile, intermediate-volatility, and semi-volatileorganic compounds, with targeted analyses of ∼150 ions, whoselikely assignments included a range of functionalized compound classes suchas glycols, glycol ethers, acetates, acids, alcohols, acrylates, esters,ethanolamines, and ketones that are found in various consumer, commercial,and industrial products. Their concentrations varied as a function of winddirection, with enhancements over the highly populated areas of the Bronx,Manhattan, and parts of New Jersey, and included abundant concentrations ofacetates, acrylates, ethylene glycol, and other commonly used oxygenatedcompounds. The results provide top-down constraints on wintertime emissionsof these oxygenated and functionalized compounds, with ratios to commonanthropogenic marker compounds and comparisons of their relative abundancesto two regionally resolved emissions inventories used in urban air qualitymodels.