Smolyak Method for Solving Dynamic Economic Models: Lagrange Interpolation, Anisotropic Grid and Adaptive Domain

First, we propose a more e¢ cient implementation of the Smolyak method for inter- polation, namely, we show how to avoid costly evaluations of repeated basis functions in the conventional Smolyak formula. Second, we extend the Smolyak method to include anisotropic constructions; this allows us to target higher quality of approximation in some dimensions than in others. Third, we show how to e¤ectively adapt the Smolyak hyper- cube to a solution domain of a given economic model. Finally, we advocate the use of low-cost .xed-point iteration, instead of conventional time iteration. In the context of one- and multi-agent growth models, we .nd that the proposed techniques lead to sub- stantial increases in accuracy and speed of a Smolyak-based projection method for solving dynamic economic models.