Smolyak's algorithm: A powerful black box for the acceleration of scientific computations

TL;DR

This paper discusses Smolyak's algorithm, a versatile method for accelerating high-dimensional scientific computations, highlighting its general framework and broad applications across various scientific fields.

Contribution

It provides a unified, application-independent framework for Smolyak's algorithm, summarizing fundamental results and assumptions for diverse scientific applications.

Findings

Effective in high-dimensional integral computation

Applicable in finance, chemistry, and physics

Enhances efficiency of solving differential equations

Abstract

We provide a general discussion of Smolyak's algorithm for the acceleration of scientific computations. The algorithm first appeared in Smolyak's work on multidimensional integration and interpolation. Since then, it has been generalized in multiple directions and has been associated with the keywords: sparse grids, hyperbolic cross approximation, combination technique, and multilevel methods. Variants of Smolyak's algorithm have been employed in the computation of high-dimensional integrals in finance, chemistry, and physics, in the numerical solution of partial and stochastic differential equations, and in uncertainty quantification. Motivated by this broad and ever-increasing range of applications, we describe a general framework that summarizes fundamental results and assumptions in a concise application-independent manner.