High-dimensional integration: The quasi-Monte Carlo way

Dick, Josef; Kuo, Frances Y.; Sloan, Ian H.

doi:10.1017/s0962492913000044

Cited by 581 publications

(637 citation statements)

References 243 publications

(258 reference statements)

Supporting

Mentioning

634

Contrasting

Unclassified

Order By: Relevance

“…Quasi-Monte Carlo (QMC) approaches have been found to be useful for estimating multidimensional integrals that otherwise would face the curse of dimensionality suffered by the quadrature approaches described above (see Dick et al 2013). (A QMC approach was not considered in Section 3.1 as Equation (6) is not a multidimensional integral.…”

Section: Quasi-monte Carlomentioning

confidence: 99%

Non-Parametric Integral Estimation Using Data Clustering in Stochastic dynamic Programming: An Introduction Using Lifetime Financial Modelling

Khemka

Butt

2017

Risks

View full text Add to dashboard Cite

This paper considers an alternative way of structuring stochastic variables in a dynamic programming framework where the model structure dictates that numerical methods of solution are necessary. Rather than estimating integrals within a Bellman equation using quadrature nodes, we use nodes directly from the underlying data. An example of the application of this approach is presented using individual lifetime financial modelling. The results show that data-driven methods lead to the least losses in result accuracy compared to quadrature and Quasi-Monte Carlo approaches, using historical data as a base. These results hold for both a single stochastic variable and multiple stochastic variables. The results are significant for improving the computational accuracy of lifetime financial models and other models that employ stochastic dynamic programming.

show abstract

Section: Quasi-monte Carlomentioning

confidence: 99%

Non-Parametric Integral Estimation Using Data Clustering in Stochastic dynamic Programming: An Introduction Using Lifetime Financial Modelling

Khemka

Butt

2017

Risks

View full text Add to dashboard Cite

show abstract

“…Le Maître and Kino [13] and recent reviews [5,12, and extensive references] provided details of these three stochastic approximation methods. All three approaches lead to tens of thousands of independent scattering realizations with fixed scatterers.…”

Section: C315mentioning

confidence: 99%

Scattering by stochastic boundaries: hybrid low- and high-order quantification algorithms

Ganesh

Hawkins

2016

ANZIAMJ

View full text Add to dashboard Cite

We present an efficient framework for simulating the average wave scattering properties of two dimensional randomly shaped particles with statistical properties similar to model aerosols particles that are important in atmospheric science applications. Our framework is based on an efficient high order discretisation of the spatial dimensions and parallel implementations for the large number of stochastic dimensions. We demonstrate our framework by simulating the mean (and higher order moments) of the far field of the model particles. We use tens of thousands of Monte Carlo, quasi-Monte Carlo and sparse grid generalised polynomial chaos realisations of the random particle model.

show abstract

“…The reproducing kernel has the properties that K(·, y) ∈ H K for all y ∈ [0, 1] s and f (y) = f, K(·, y) H K for all f ∈ H K and all y ∈ [0, 1] s , where ·, · H K is the inner product in H K . One class of reproducing kernel Hilbert spaces that has been studied intensively, are the Korobov spaces [6]. Those consist of functions on [0, 1] s which have square integrable derivatives up to order α in each variable.…”

Section: Classical Korobov Spacesmentioning

confidence: 99%

Numerical integration in log-Korobov and log-cosine spaces

et al. 2015

Self Cite

View full text Add to dashboard Cite

Quasi-Monte Carlo (QMC) rules are equal weight quadrature rules for approximating integrals over the s-dimensional unit cube [0, 1] s . One line of research studies the integration error of functions in the unit ball of so-called Korobov spaces, which are Hilbert spaces of periodic functions on [0, 1] s with square integrable partial mixed derivatives of order α. Using Parseval's identity, this smoothness can be defined for all real numbers α > 1/2. In this setting, the condition α > 1/2 is necessary as otherwise the Korobov space contains discontinuous functions for which function evaluation is not well defined.This paper is concerned with more precise endpoint estimates of the integration error using QMC rules for Korobov spaces with α arbitrarily close to 1/2. To obtain such estimates we introduce a log-scale for functions with smoothness close to 1/2, which we call log-Korobov spaces. We show that lattice rules can be used to obtain an integration error of order O(N −1/2 (log N ) −µ(1−λ)/2 ) for any 1/µ < λ ≤ 1, where µ > 1 is a certain power in the log-scale.A second result is concerned with tractability of numerical integration for weighted Korobov spaces with product weights (γ j ) j∈N . Previous results have shown that if ∞ j=1 γ τ j < ∞ for some 1/(2α) < τ ≤ 1 one can obtain error bounds which are independent of the dimension. In this paper we give a more refined estimate for the case where τ is close to 1/(2α), namely we show dimension independent error bounds under the condition that ∞ j=1 γ j max{1, log γ −1 j } µ(1−λ) < ∞ for some 1/µ < λ ≤ 1. The essential tool in our analysis is a log-scale Jensen's inequality.The results described above also apply to integration in log-cosine spaces using tent-transformed lattice rules.

show abstract

High-dimensional integration: The quasi-Monte Carlo way

Cited by 581 publications

References 243 publications

Non-Parametric Integral Estimation Using Data Clustering in Stochastic dynamic Programming: An Introduction Using Lifetime Financial Modelling

Non-Parametric Integral Estimation Using Data Clustering in Stochastic dynamic Programming: An Introduction Using Lifetime Financial Modelling

Scattering by stochastic boundaries: hybrid low- and high-order quantification algorithms

Numerical integration in log-Korobov and log-cosine spaces

Contact Info

Product

Resources

About