Dimension–Adaptive Tensor–Product Quadrature

@article{Gerstner2003DimensionAdaptiveTQ,
  title={Dimension–Adaptive Tensor–Product Quadrature},
  author={Thomas Gerstner and Michael Griebel},
  journal={Computing},
  year={2003},
  volume={71},
  pages={65-87}
}
We consider the numerical integration of multivariate functions defined over the unit hypercube. Here, we especially address the high–dimensional case, where in general the curse of dimension is encountered. Due to the concentration of measure phenomenon, such functions can often be well approximated by sums of lower–dimensional terms. The problem, however, is to find a good expansion given little knowledge of the integrand itself. The dimension–adaptive quadrature method which is developed and… 

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References

SHOWING 1-10 OF 51 REFERENCES

Numerical integration using sparse grids

The usage of extended Gauss (Patterson) quadrature formulas as the one‐dimensional basis of the construction is suggested and their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, Clenshaw–Curtis and Gauss rules is shown.

Classification with sparse grids using simplicial basis functions

It turns out that the method scales linearly with the number of given data points and is well suited for data mining applications where the amount of data is very large, but where the dimension of the feature space is moderately high.

Sparse grids

The basic features of sparse grids are described and the results of various numerical experiments for the solution of elliptic PDEs as well as for other selected problems such as numerical quadrature and data mining are reported.

Optimized Tensor-Product Approximation Spaces

Abstract. This paper is concerned with the construction of optimized grids and approximation spaces for elliptic differential and integral equations. The main result is the analysis of the

A Note on the Complexity of Solving Poisson's Equation for Spaces of Bounded Mixed Derivatives

A strong tractability result of the order O(e−1) is given and this paper provides a practically usable hierarchical basis finite element method of this complexity O( e−1), i.e., without logarithmic terms growing exponentially in d, at least for the authors' sparse grid setting with its underlying smoothness requirements.

Nonlinear approximation

This is a survey of nonlinear approximation, especially that part of the subject which is important in numerical computation, and emphasis will be placed on approximation by piecewise polynomials and wavelets as well as their numerical implementation.

Additive models in high dimensions

A novel framework which includes the number of variables as an ingredient in the definition of the smoothness of the underlying functions is introduced, motivated by the effect of concentration of measure in high dimensional spaces and convergence of the additive decompositions is established.

A New Algorithm for Multi-Dimensional Adaptive Numerical Quadrature

We present an algorithm for multi-dimensional quadrature that is adaptive both in the refinement of the subdomains and in the order. The method is based on an extrapolation technique using sparse

Weighted Tensor Product Algorithms for Linear Multivariate Problems

It is shown that these multivariate problems defined over weighted tensor product Hilbert spaces of functions f of d variables are sometimes tractable even with a worst-case assurance.

Adaptive sparse grids

It is observed in first tests that these general adaptive sparse grids allow the identification of the ANOVA structure and thus provide comprehensible models, very important for data mining applications.
...