Smolyak's algorithm: A powerful black box for the acceleration of scientific computations
For researchers in scientific computing, this offers a concise, unified perspective on a widely used acceleration technique, but it is a survey/tutorial rather than a novel contribution.
This paper provides a general framework for Smolyak's algorithm, unifying its variants (sparse grids, combination technique, multilevel methods) across applications in high-dimensional integration, PDEs, and uncertainty quantification. It summarizes fundamental results and assumptions in an application-independent manner.
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.