Computational Mathematics and Scientific Computing Seminar

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On the Evaluation of Sums of Periodic Gaussians

Speaker: Andrei Osipov, Shaw Research

Location: Warren Weaver Hall 1302

Date: March 30, 2018, 10 a.m.

Synopsis:

Discrete sums of the form

\(\sum_{k=1}^N q_k \cdot \exp\left( -\frac{t – s_k}{2 \cdot \sigma^2} \right)\)

where \(\sigma>0\) and \(q_1, \dots, q_N\) are real numbers and \(s_1, \dots, s_N\) and \(t\) are vectors in \(R^d\), are frequently encountered in numerical computations across a variety of fields.

We describe an algorithm for the evaluation of such sums under periodic boundary conditions, provide a rigorous error analysis, and discuss its implications on the computational cost and choice of parameters. While the algorithm itself was introduced before (and is closely related to a class of algorithms for the evaluation of non-uniform discrete Fourier Transforms), the error analysis and its consequences appear to be novel.

We illustrate our results via numerical experiments.