# Numerical Analysis and Scientific Computing Seminar

## 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.