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

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

​R. Tempone, S. Wolfers, Smolyak’s algorithm: A powerful black box for the acceleration of scientific computations, submitted arXiv:1703.08872 Apr. 2017​​​​
R. Tempone, S. Wolfers
Smolyak algorithm, sparse grids, hyperbolic cross approximation, combination technique, multilevel methods
2017
​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 highdimensional 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.