Sigmoid functions and multiscale resolution of singularities

03/03/2023

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by   Daan Huybrechs, et al.

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In this short, conceptual paper we observe that essentially the same mathematics applies in three contexts with disparate literatures: (1) sigmoidal and RBF approximation of smooth functions, (2) rational approximation of analytic functions near singularities, and (3) hp mesh refinement for solution of PDEs. The relationship of (1) and (2) is as simple as the change of variables s = log(x), and our informal mnemonic for this relationship is “sigmoid = log(ratapprox).”