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Register a new userFunction Approximation via The Subsampled Poincar e Inequality
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Added by
Yifan Chen
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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Function approximation and recovery via some sampled data have long been studied in a wide array of applied mathematics and statistics fields. Analytic tools, such as the Poincare inequality, have been handy for estimating the approximation errors in different scales. The purpose of this paper is to study a generalized Poincar e inequality, where the measurement function is of subsampled type, with a small but non-zero lengthscale that will be made precise. Our analysis identifies this inequality as a basic tool for function recovery problems. We discuss and demonstrate the optimality of the inequality concerning the subsampled lengthscale, connecting it to existing results in the literature. In application to function approximation problems, the approximation accuracy using different basis functions and under different regularity assumptions is established by using the subsampled Poincare inequality. We observe that the error bound blows up as the subsampled lengthscale approaches zero, due to the fact that the underlying function is not regular enough to have well-defined pointwise values. A weighted version of the Poincar e inequality is proposed to address this problem; its optimality is also discussed.
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There is an intimate connection between numerical upscaling of multiscale PDEs and scattered data approximation of heterogeneous functions: the coarse variables selected for deriving an upscaled equation (in the former) correspond to the sampled information used for approximation (in the latter). As such, both problems can be thought of as recovering a target function based on some coarse data that are either artificially chosen by an upscaling algorithm, or determined by some physical measurement process. The purpose of this paper is then to study that, under such a setup and for a specific elliptic problem, how the lengthscale of the coarse data, which we refer to as the subsampled lengthscale, influences the accuracy of recovery, given limited computational budgets. Our analysis and experiments identify that, reducing the subsampling lengthscale may improve the accuracy, implying a guiding criterion for coarse-graining or data acquisition in this computationally constrained scenario, especially leading to direct insights for the implementation of the Gamblets method in the numerical homogenization literature. Moreover, reducing the lengthscale to zero may lead to a blow-up of approximation error if the target function does not have enough regularity, suggesting the need for a stronger prior assumption on the target function to be approximated. We introduce a singular weight function to deal with it, both theoretically and numerically. This work sheds light on the interplay of the lengthscale of coarse data, the computational costs, the regularity of the target function, and the accuracy of approximations and numerical simulations.
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