"A Walk Around the Torus: SFTs in Many Dimensions"Gross, CraigThis talk presents two ways of using efficient sparse Fourier transforms in one-dimension to produce sparse Fourier approximations in many dimensions. Both techniques rely on applying various simple transformations to the signal, mapping the many dimensions down to one via reconstructing rank-1 lattices, and then synchronizing the results from each transformation. Both results depend only linearly on the dimension and sparsity in runtime complexity with near-optimal error guarantees; they differ by trading off faster runtime for better error guarantees. These theoretical complexity and error bounds are also reflected in numerical experiments for function reconstruction and approximation. Time permitting, we will also discuss an application to sparse spectral methods for solving high dimensional PDEs. This is joint work with Mark Iwen (Michigan State University), Lutz Kämmerer (TU Chemnitz), and Toni Volkmer (TU Chemnitz). |
| http://univie.ac.at/projektservice-mathematik/e/talks/Gross_2021-06_HighDimSFT.pdf |
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