Sparse grids are numerical techniques to represent, integrate or interpolate high
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
al functions. They were originally developed by the
Russia
Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-eig ...
n
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
Sergey A. Smolyak, a student of
Lazar Lyusternik
Lazar Aronovich Lyusternik (also Lusternik, Lusternick, Ljusternik; ; 31 December 1899, in Zduńska Wola, Congress Poland, Russian Empire – 23 July 1981, in Moscow, Soviet Union) was a Soviet mathematician.
He is famous for his work in topol ...
, and are based on a sparse tensor product construction. Computer algorithms for efficient implementations of such grids were later developed by
Michael Griebel and
Christoph Zenger.
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
The curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces that do not occur in low-dimensional settings such as the three-dimensional physical space of everyday experience. T ...
is expressed in the order of the integration error that is made by a quadrature of level
, with
points. The function has regularity
, i.e. is
times differentiable. The number of dimensions is
.
Smolyak's quadrature rule
Smolyak found a computationally more efficient method of integrating multidimensional functions based on a univariate quadrature rule
. The
-dimensional Smolyak integral
of a function
can be written as a recursion formula with the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
.
The index to
is the level of the discretization. If a 1-dimension integration on level
is computed by the evaluation of
points, the error estimate for a function of regularity
will be
Further reading
*
*
*
External links
A memory efficient data structure for regular sparse gridsVisualization on sparse gridsDatamining on sparse grids, J.Garcke, M.Griebel (pdf)
{{Mathanalysis-stub
Numerical analysis