Multi-level Monte Carlo algorithms for infinite-dimensional integration on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="double-struck">N</mml:mi></mml:mrow></mml:msup></mml:math>

Hickernell, Fred J.; Müller-Gronbach, Thomas; Niu, Ben; Ritter, Klaus

doi:10.1016/j.jco.2010.02.002

Cited by 43 publications

(4 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Integration in the infinite-dimensional setting has been analysed in a number of recent papers [10,15,34,38,44], mostly in a Hilbert space setting. In a formal sense, there is little difficulty in considering integration with an infinite number of variables; for a function F of infinitely many variables y = (y 1 , y 2 , .…”

Section: Infinite-dimensional Integrationmentioning

confidence: 99%

Quasi-Monte Carlo Methods for High-Dimensional Integration: The Standard (Weighted Hilbert Space) Setting and Beyond

2011

View full text Add to dashboard Cite

This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1] s . It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost 1/N, where N is the number of points, independently of dimension) to so-called "product and order dependent" (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets.

show abstract

Section: Infinite-dimensional Integrationmentioning

confidence: 99%

Quasi-Monte Carlo Methods for High-Dimensional Integration: The Standard (Weighted Hilbert Space) Setting and Beyond

2011

View full text Add to dashboard Cite

show abstract

“…Mitigating or circumventing the CoD is the primary goal when it comes to constructing efficient high-dimensional numerical quadrature rules. A lot of progress has been made in this direction in the past 50 years, including sparse grid (SG) methods [1,[5][6][7], Monte Carlo (MC) methods [8,9], quasi-Monte Carlo (QMC) methods [10][11][12][13][14], and deep neural network (DNN) methods [15][16][17][18][19]. To some certain extent, these methods are effective for computing integrals in low and medium dimensions (i.e., d 100), but it is still a challenge for them to compute integrals in very high dimensions (i.e., d ≈ 1000).…”

Section: Introductionmentioning

confidence: 99%

An Efficient and Fast Sparse Grid Algorithm for High-Dimensional Numerical Integration

Zhong,

Feng

2023

Mathematics

View full text Add to dashboard Cite

This paper is concerned with developing an efficient numerical algorithm for the fast implementation of the sparse grid method for computing the d-dimensional integral of a given function. The new algorithm, called the MDI-SG (multilevel dimension iteration sparse grid) method, implements the sparse grid method based on a dimension iteration/reduction procedure. It does not need to store the integration points, nor does it compute the function values independently at each integration point; instead, it reuses the computation for function evaluations as much as possible by performing the function evaluations at all integration points in a cluster and iteratively along coordinate directions. It is shown numerically that the computational complexity (in terms of CPU time) of the proposed MDI-SG method is of polynomial order O(d3Nb)(b≤2) or better, compared to the exponential order O(N(logN)d−1) for the standard sparse grid method, where N denotes the maximum number of integration points in each coordinate direction. As a result, the proposed MDI-SG method effectively circumvents the curse of dimensionality suffered by the standard sparse grid method for high-dimensional numerical integration.

show abstract

“…Mitigating or circumventing the CoD has been the primary goal when it comes to constructing efficient high-dimensional numerical quadrature rules. A lot of progress has been made in this direction in the past fifty years, this includes sparse grid (SG) methods [1,[5][6][7], Monte Carlo (MC) methods [8,9], Quasi-Monte Carlo (QMC) methods [10][11][12][13][14], deep neural network (DNN) methods [15][16][17][18][19]. To some certain extent, those methods are effective for computing integrals in low and medium dimensions (i.e., d 100), but it is still a challenge for them to compute integrals in very high dimensions (i.e., d ≈ 1000).…”

Section: Introductionmentioning

confidence: 99%

An Efficient and Fast Sparse Grid Algorithm for High-Dimensional Numerical Integration

Zhong,

Feng

2023

Preprint

View full text Add to dashboard Cite

This paper is concerned with developing an efficient numerical algorithm for fast implementation of the sparse grid method for computing the $d$-dimensional integral of a given function. The new algorithm, called the MDI-SG ({\em multilevel dimension iteration sparse grid}) method, implements the sparse grid method based on a dimension iteration/reduction procedure, it does not need to store the integration points, nor does it compute the function values independently at each integration point, instead, it reuses the computation for function evaluations as much as possible by performing the function evaluations at all integration points in a cluster and iteratively along coordinate directions. It is shown numerically that the computational complexity (in terms of CPU time) of the proposed MDI-SG method is of polynomial order $O(Nd^3 )$ or better, compared to the exponential order $O(N(\log N)^{d-1})$ for the standard sparse grid method, where $N$ denotes the maximum number of integration points in each coordinate direction. As a result, the proposed MDI-SG method effectively circumvents the curse of dimensionality suffered by the standard sparse grid method for high-dimensional numerical integration.

show abstract

Multi-level Monte Carlo algorithms for infinite-dimensional integration on RN

Cited by 43 publications

References 17 publications

Quasi-Monte Carlo Methods for High-Dimensional Integration: The Standard (Weighted Hilbert Space) Setting and Beyond

Quasi-Monte Carlo Methods for High-Dimensional Integration: The Standard (Weighted Hilbert Space) Setting and Beyond

An Efficient and Fast Sparse Grid Algorithm for High-Dimensional Numerical Integration

An Efficient and Fast Sparse Grid Algorithm for High-Dimensional Numerical Integration

Contact Info

Product

Resources

About