Mathematics > Numerical Analysis
[Submitted on 26 Sep 2023 (v1), last revised 2 Oct 2023 (this version, v2)]
Title:A Sparse Fast Chebyshev Transform for High-Dimensional Approximation
View PDFAbstract:We present the Fast Chebyshev Transform (FCT), a fast, randomized algorithm to compute a Chebyshev approximation of functions in high-dimensions from the knowledge of the location of its nonzero Chebyshev coefficients. Rather than sampling a full-resolution Chebyshev grid in each dimension, we randomly sample several grids with varied resolutions and solve a least-squares problem in coefficient space in order to compute a polynomial approximating the function of interest across all grids simultaneously. We theoretically and empirically show that the FCT exhibits quasi-linear scaling and high numerical accuracy on challenging and complex high-dimensional problems. We demonstrate the effectiveness of our approach compared to alternative Chebyshev approximation schemes. In particular, we highlight our algorithm's effectiveness in high dimensions, demonstrating significant speedups over commonly-used alternative techniques.
Submission history
From: M. Harper Langston [view email][v1] Tue, 26 Sep 2023 00:15:09 UTC (6,016 KB)
[v2] Mon, 2 Oct 2023 16:38:27 UTC (6,016 KB)
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