## Sparse grids and related approximation schemes for higher dimensional problems

Citations: | 28 - 12 self |

### BibTeX

@MISC{Griebel_sparsegrids,

author = {Michael Griebel},

title = {Sparse grids and related approximation schemes for higher dimensional problems},

year = {}

}

### OpenURL

### Abstract

The efficient numerical treatment of high-dimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorov’s theorem, the ANOVA decomposition and the sparse grid approach and discuss their prerequisites and properties. Moreover, we present energy-norm based sparse grids and demonstrate that, for functions with bounded mixed derivatives on the unit hypercube, the associated approximation rate in terms of the involved degrees of freedom shows no dependence on the dimension at all, neither in the approximation order nor in the order constant.

### Citations

749 | Constructive Approximation
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- 1993
(Show Context)
Citation Context ...tives. A better understanding of these results is possible with the help of harmonic analysis (Donoho 2000). Here, we resort to the approach of the L1-combination of L∞-atoms, see also (Triebel 1992, =-=DeVore 1998-=-). Consider the class of functions F(K) with integral representation f(x) = ∫ A(x, t)dµ(t) with ∫ d|µ|(t) ≤ K , (1.1) where for fixed t we call A(x, t) = At(x) an L∞-atom, if |At(x)| ≤ 1 holds. Then, ... |

505 |
Dynamic Asset Pricing Theory
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- 2001
(Show Context)
Citation Context ... theory of supergravitation are formulated in 10 or 11 dimensions, respectively, see (Green, Schwarz and Witten 1998) and the references cited therein. Sparse grids for higher dimensional problems 5 (=-=Duffie 1996-=-, Kwok 1998, Wilmott 1998, Reisinger 2003, Schwab 2003, Escobar and Seco 2005). Furthermore, homogenization with multiple scales (Allaire 1992, Cioranescu, Damlamian and Griso 2002, Matache 2002, Hoan... |

499 | Multivariate Adaptive Regression Splines
- Friedman
(Show Context)
Citation Context ... have finite order q ≤ 2 in the associated ANOVA decomposition. Sparse grids for higher dimensional problems 21 • In data mining it is found from multivariate adaptive regression splines (MARS), see (=-=Friedman 1991-=-), that even for really high-dimensional data there appear at most 5-7 dimensional interactions, i.e., q ≤ 7, and higher-order interactions are practically not significant. • The Brownian bridge repre... |

443 | Projection Pursuit Regression - Friedman, Stuetzle - 1981 |

404 | Chebyshev and Fourier Spectral Methods - Boyd - 1999 |

402 |
Universal approximation bounds for superposition of n sigmoidal functions
- Barron
- 1993
(Show Context)
Citation Context ...thness of the function f such that r = O(d), then, we directly obtain ||f − fM || = O(M−c) with constant c > 0. Of course, such an assumption is quite unrealistic. However, about thirteen years ago, (=-=Barron 1993-=-) found an interesting result: Denote by FL1 the class of functions with Fourier transforms in L1. Then, consider the class of functions of IR d with ∇f ∈ FL1. We expect for the best M -term approxima... |

331 | Regularization theory and neural networks architectures - Girosi, Jones, et al. - 1995 |

252 | Homogenization and two-scale convergence
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- 1992
(Show Context)
Citation Context ...d therein. Sparse grids for higher dimensional problems 5 (Duffie 1996, Kwok 1998, Wilmott 1998, Reisinger 2003, Schwab 2003, Escobar and Seco 2005). Furthermore, homogenization with multiple scales (=-=Allaire 1992-=-, Cioranescu, Damlamian and Griso 2002, Matache 2002, Hoang and Schwab 2003) as well as stochastic elliptic equations (Schwab and Todor 2003a, Schwab and Todor 2003b) result in highdimensional PDEs. N... |

182 |
Symmetric iterative interpolation processes
- Deslauriers, Dubuc
- 1989
(Show Context)
Citation Context ...Bungartz 1998, Bungartz and Griebel 2004) or function families with localization properties like wavelets (Daubechies 1992), prewavelets (Chui and Wang 1992, Griebel and Oswald 1995b) or interpolets (=-=Deslauriers and Dubuc 1989-=-, Donoho and Yu 1999) and related wavelet-like constructs, see (Cohen 2003, Bungartz and Griebel 2004) for a survey. But also multiscale finite element systems and frames (Oswald 1994, Griebel 1994, G... |

123 |
The jacknife estimate of variance
- Efron, Stein
- 1981
(Show Context)
Citation Context ...tions, fj1,j2 are two-dimensional functions, and so on. This type of decomposition goes back to (Hoeffding 1948) and is well known in statistics under the name ANOVA (analysis of variance), see also (=-=Efron and Stein 1981-=-). Note that (1.8) is a finite expansion of f into 2d different terms. Such a decomposition can be gained by a tensor product construction of a splitting of the one-dimensional function space into its... |

120 | Sparse grids - Bungartz, Griebel - 2004 |

101 | High-dimensional data analysis: the curses and blessings of dimensionality. Neural Comput, Aide-Memoire of a Lecture at - Donoho - 2000 |

100 | Numerical analysis of wavelet methods - Cohen - 2003 |

63 |
Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung
- Bungartz
- 1992
(Show Context)
Citation Context ...ts and triangle inequalities. Moreover, for this special choice of basis we are able to also derive estimates of the constants involved and their dependence on the dimension d. We now closely follow (=-=Bungartz 1992-=-, Bungartz and Griebel 1999, Bungartz 1998, Bungartz and Griebel 2004). 1.4.1 Hierarchical multilevel subspace splitting 1.4.1.1 Subspace decomposition Let Ω̄ := [0, 1]d denote the d-dimensional unit ... |

49 |
A general framework of compactly supported splines and wavelets
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- 1992
(Show Context)
Citation Context ...d 2000, Karniadakis and Sherwin 1999, Szabo and Babuska 1991, Bungartz 1998, Bungartz and Griebel 2004) or function families with localization properties like wavelets (Daubechies 1992), prewavelets (=-=Chui and Wang 1992-=-, Griebel and Oswald 1995b) or interpolets (Deslauriers and Dubuc 1989, Donoho and Yu 1999) and related wavelet-like constructs, see (Cohen 2003, Bungartz and Griebel 2004) for a survey. But also mult... |

44 | Periodic unfolding and homogenization - Cioranescu, Damlamian, et al. |

42 | Hyperbolic wavelet approximation - DeVore, Konyagin, et al. - 1998 |

40 | 2003): “Dimension-Adaptive TensorProduct Quadrature - Gerstner, Griebel |

36 |
Statistical Dynamics - Matter out of Equilibrium
- Balescu
- 1977
(Show Context)
Citation Context ...rons and nuclei. Then, problems in statistical mechanics lead to the Liouville equation or the Langevin equation and related phase space models where the dimension depends on the number of particles (=-=Balescu 1997-=-). Furthermore, reinforcement learning and stochastic optimal control in continuous time give raise to the HamiltonJacobi-Bellman equation in high dimensions (Sutton and Barto 1998, Munos 2000, Munos ... |

31 |
Approximation by trigonometric polynomials in a certain class of periodic functions of several variables
- Babenko
- 1960
(Show Context)
Citation Context ...ion instead of the hierarchical Faber basis. An example of a regular sparse grid is given for the two- and three-dimensional case in Figure 1.3. The basic concept can be traced back to (Smolyak 1963, =-=Babenko 1960-=-), see also (Gordon 1969, Gordon 1971, Delvos 1982, Delvos and Schempp 1989, DeVore, Konyagin and Temlyakov 1998). The dimension of the space V (1) n fulfills |V (1)n | = O(h−1n · | log2 hn|d−1) (1.67... |

29 | Liberating the weights - Dick, Sloan, et al. - 2004 |

26 |
Über stetige Funktionen
- Faber
- 1909
(Show Context)
Citation Context ... in Section 1.4. To this end, we refine the remainder space of the one-dimensional splitting, i.e., we equip it with a basis. We use the standard piecewise linear hierarchical basis in one dimension (=-=Faber 1909-=-, Yserentant 1986) as the simplest example of a onedimensional multiscale series expansion which involves interpolation by piecewise linears. Then the tensor product construction generates a basis for... |

21 |
d-Variate Boolean interpolation
- Delvos
- 1982
(Show Context)
Citation Context ...mple of a regular sparse grid is given for the two- and three-dimensional case in Figure 1.3. The basic concept can be traced back to (Smolyak 1963, Babenko 1960), see also (Gordon 1969, Gordon 1971, =-=Delvos 1982-=-, Delvos and Schempp 1989, DeVore, Konyagin and Temlyakov 1998). The dimension of the space V (1) n fulfills |V (1)n | = O(h−1n · | log2 hn|d−1) (1.67) 40 Michael Griebel Fig. 1.3. Lp-norm based spars... |

21 |
Numerical Integration using Sparse Grids” Numerical Algorithms
- Gerstner, Griebel
- 1998
(Show Context)
Citation Context ...n the first few levels of the discretization since the variance decays with the factor 2−1/2 from level to level, see also (Caflisch, Morokoff and Owen 1997, Morokoff 1998, Gerstner and Griebel 2003, =-=Gerstner and Griebel 1998-=-) where the Brownian bridge was used in high-dimensional integration problems. A further analysis in view of reproducing kernel Hilbert spaces with weights is given in (Leobacher, Scheicher and Larche... |

21 |
Representation properties of networks: Kolmogorov's theorem is irrelevant
- F, Poggio
- 1989
(Show Context)
Citation Context ...it turns out that the representing functions are quite bad, i.e., they are at best only continuous and highly non-smooth. This limits their practical use for approximation and interpolation purposes (=-=Girosi and Poggio 1989-=-), like e.g., for the discretization of PDEs within the Galerkin approach. In particular, the representing functions cannot be chosen to be differentiable. This even holds if one wants to represent an... |

20 | A note on the complexity of solving Poisson’s equation for spaces of bounded mixed derivatives
- Bungartz, Griebel
- 1999
(Show Context)
Citation Context ... inequalities. Moreover, for this special choice of basis we are able to also derive estimates of the constants involved and their dependence on the dimension d. We now closely follow (Bungartz 1992, =-=Bungartz and Griebel 1999-=-, Bungartz 1998, Bungartz and Griebel 2004). 1.4.1 Hierarchical multilevel subspace splitting 1.4.1.1 Subspace decomposition Let Ω̄ := [0, 1]d denote the d-dimensional unit interval. We consider multi... |

19 | The multiconfiguration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propagating wavepackets, Phys. Rep - Beck, Jäckle, et al. - 2000 |

17 |
Distributive lattices and the approximation of multivariate functions
- Gordon
- 1969
(Show Context)
Citation Context ...chical Faber basis. An example of a regular sparse grid is given for the two- and three-dimensional case in Figure 1.3. The basic concept can be traced back to (Smolyak 1963, Babenko 1960), see also (=-=Gordon 1969-=-, Gordon 1971, Delvos 1982, Delvos and Schempp 1989, DeVore, Konyagin and Temlyakov 1998). The dimension of the space V (1) n fulfills |V (1)n | = O(h−1n · | log2 hn|d−1) (1.67) 40 Michael Griebel Fig... |

16 |
Adaptive control processes: A guided tour
- Bellmann
- 1961
(Show Context)
Citation Context ...nal methods is limited to problems with up to three or four dimensions due to storage requirements and computational complexity. The reason is the so-called curse of dimensionality, a term coined in (=-=Bellmann 1961-=-). Here, the cost to compute and represent an approximation with a prescribed accuracy ε depends exponentially on the dimensionality d of the problem considered. We encounter complexities of the order... |

16 | Simulation of dilute polymer solutions using a Fokker-Planck equation, Computers and Fluids - Chauviere, Lozinski |

15 |
Boolean Methods in Interpolation and Approximation
- Delvos, Schempp
- 1989
(Show Context)
Citation Context ...ular sparse grid is given for the two- and three-dimensional case in Figure 1.3. The basic concept can be traced back to (Smolyak 1963, Babenko 1960), see also (Gordon 1969, Gordon 1971, Delvos 1982, =-=Delvos and Schempp 1989-=-, DeVore, Konyagin and Temlyakov 1998). The dimension of the space V (1) n fulfills |V (1)n | = O(h−1n · | log2 hn|d−1) (1.67) 40 Michael Griebel Fig. 1.3. Lp-norm based sparse grids: Two-dimensional ... |

9 |
On the representation of continuous functions of three variables by superpositions of continuous functions of two variables
- Arnold
- 1959
(Show Context)
Citation Context ... : [0, 1]d → IR. Each function f ∈ C([0, 1]d) has a representation f(x1, ..., xd) = 2d+1∑ i=1 fi( d∑ j=1 φi,j(xj)) (1.3) † Kolmogorov’s student Arnold showed even before in (Arnold 1957, Arnold 1958, =-=Arnold 1959-=-) that any f ∈ C([0, 1]3) can be represented as a superposition of continuous functions in two variables, and thus refuted Hilbert’s conjecture. 10 Michael Griebel where all {fi} and {φi,j} are one-di... |

9 | 2004), ‘On Kolmogorov’s representation of functions of several variables by functions of one variable - Coppejans |

8 | Deslauriers-Dubuc: Ten years after
- Donoho, Yu
- 1999
(Show Context)
Citation Context ...Griebel 2004) or function families with localization properties like wavelets (Daubechies 1992), prewavelets (Chui and Wang 1992, Griebel and Oswald 1995b) or interpolets (Deslauriers and Dubuc 1989, =-=Donoho and Yu 1999-=-) and related wavelet-like constructs, see (Cohen 2003, Bungartz and Griebel 2004) for a survey. But also multiscale finite element systems and frames (Oswald 1994, Griebel 1994, Griebel and Oswald 19... |

7 | 2004), Maschinelles Lernen durch Funktionsrekonstruktion mit verallgemeinerten dünnen Gittern, Dissertation, Institut für Numerische Simulation - Garcke |

6 | 2005), Dünngitterverfahren für hochdimensionale elliptische partielle Differentialgleichungen, Diplomarbeit, Institut für Numerische Simulation - Feuersänger |

5 | Implications and applications of Kolmogorov’s superposition theorem - Figueiredo - 1980 |

5 |
An improvement on the smoothness of the functions in Kolmogorov’s theorem on superpositions
- Fridman
- 1967
(Show Context)
Citation Context ...bert’s conjecture. 10 Michael Griebel where all {fi} and {φi,j} are one-dimensional continuous functions defined on IR and all {φi,j} are independent of the choice of f . An improvement was given in (=-=Fridman 1967-=-) where it was shown that the inner functions {φi,j} can be chosen to be Lipschitz continuous with exponent one. There have been various refinements of this result. A version with just one outer funct... |

4 |
On the representation of functions of several variables by superpositions of functions of fewer variables
- Arnold
- 1958
(Show Context)
Citation Context ...(x1, ..., xd) : [0, 1]d → IR. Each function f ∈ C([0, 1]d) has a representation f(x1, ..., xd) = 2d+1∑ i=1 fi( d∑ j=1 φi,j(xj)) (1.3) † Kolmogorov’s student Arnold showed even before in (Arnold 1957, =-=Arnold 1958-=-, Arnold 1959) that any f ∈ C([0, 1]3) can be represented as a superposition of continuous functions in two variables, and thus refuted Hilbert’s conjecture. 10 Michael Griebel where all {fi} and {φi,... |

4 | A partial differential equation for credit derivatives pricing. Working Paper - Escobar, Seco - 2005 |

3 |
Finite Elements of Higher Order on Sparse
- Bungartz
- 1998
(Show Context)
Citation Context ... on the respective higher dimensional problem under consideration), classical Fourier bases, (hierarchical) global polynomial systems (Boyd 2000, Karniadakis and Sherwin 1999, Szabo and Babuska 1991, =-=Bungartz 1998-=-, Bungartz and Griebel 2004) or function families with localization properties like wavelets (Daubechies 1992), prewavelets (Chui and Wang 1992, Griebel and Oswald 1995b) or interpolets (Deslauriers a... |

3 |
Approximation by functions of fewer variables, in On Numerical Approximation
- Golomb
- 1959
(Show Context)
Citation Context ... Hackbusch 2003) and the references cited therein. A similar decomposition is used in the MCTDH approach (Beck, Jäckle, Worth and Meyer 2000). The basic theory of this decomposition can be found in (=-=Golomb 1959-=-): In the case d=2 mainly the classical Hilbert-Schmidt theory appears, i.e., the functions fij are the unique solution of a system of two coupled linear integral equations which resemble the continuo... |

2 |
On functions of three variables’, Dokl
- Arnold
- 1957
(Show Context)
Citation Context ...cube, i.e., f(x1, ..., xd) : [0, 1]d → IR. Each function f ∈ C([0, 1]d) has a representation f(x1, ..., xd) = 2d+1∑ i=1 fi( d∑ j=1 φi,j(xj)) (1.3) † Kolmogorov’s student Arnold showed even before in (=-=Arnold 1957-=-, Arnold 1958, Arnold 1959) that any f ∈ C([0, 1]3) can be represented as a superposition of continuous functions in two variables, and thus refuted Hilbert’s conjecture. 10 Michael Griebel where all ... |

2 |
Theory of reproducing kernels
- Aronzaijn
- 1950
(Show Context)
Citation Context ...ion as well as truncations to finite order. This will be dealt with in the following. 1.3.2 Reproducing kernel Hilbert spaces The theory of reproducing kernel Hilbert spaces (RKHS) was introduced in (=-=Aronzaijn 1950-=-). It allows to describe function spaces in a concise and elegant way by means of so-called reproducing kernel functions. To this end, we assume that f : [0, 1]d → IR belongs to a Hilbert space H with... |

2 | Iserles (2003), On the foundations of computational mathematics - Baxter, A |

2 | Molenkamp (2005), ‘Algorithms for numerical analysis in high dimensions - Beylkin, M |

2 | Lötstedt and P. Sjöberg (2001), Problems of high dimension in molecular biology - Elf, P |

2 | Thess (2001), ‘Data mining with sparse grids - Garcke, Griebel, et al. |

1 | Molenkamp (2002), ‘Numerical operator calculus in higher dimensions - Beylkin, M |

1 | Valuation of mortgage backed Michael Griebel securities using Brownian bridges to reduce effective dimension - Caflisch, Morokoff, et al. - 1997 |