Sparse grids are numerical techniques to represent, integrate or interpolate high dimensional functions. They were originally developed by the Russian mathematician Sergey A. Smolyak, a student of Lazar Lyusternik, and are based on a sparse tensor product construction. Computer algorithms for efficient implementations of such grids were later developed by Michael Griebel, Christoph Zenger, and Dirk Pflüger.

Curse of dimensionality

The standard way of representing multidimensional functions are tensor or full grids. The number of basis functions or nodes (grid points) that have to be stored and processed depend exponentially on the number of dimensions.

The curse of dimensionality.

Smolyak's quadrature rule

Smolyak found a computationally more efficient method of integrating multidimensional functions based on a univariate can be written as a recursion formula with the tensor product.

is the level of the discretization

Further reading

External links

• A memory efficient data structure for regular sparse grids

• Finite difference scheme on sparse grids

• Visualization on sparse grids

• Datamining on sparse grids, J.Garcke, M.Griebel (pdf)