Lattice Rule | ScienceGate

Bounds on the Lattice Rule Criterion R

Mathematics of Computation ◽

10.2307/2153255 ◽

1993 ◽

Vol 61 (204) ◽

pp. 821

Author(s):

Stephen Joe

Keyword(s):

Lattice Rule

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The number of lattice rules having given invariants

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700012144 ◽

1992 ◽

Vol 46 (3) ◽

pp. 479-495 ◽

Cited By ~ 2

Author(s):

Stephen Joe ◽

David C. Hunt

Keyword(s):

Normal Form ◽

Quadrature Rule ◽

Unit Cube ◽

Numerical Results ◽

Smith Normal Form ◽

Lattice Rules ◽

Dimensional Unit ◽

Lattice Rule ◽

Positive Integers ◽

Theoretical Results

A lattice rule is a quadrature rule used for the approximation of integrals over the s-dimensional unit cube. Every lattice rule may be characterised by an integer r called the rank of the rule and a set of r positive integers called the invariants. By exploiting the group-theoretic structure of lattice rules we determine the number of distinct lattice rules having given invariants. Some numerical results supporting the theoretical results are included. These numerical results are obtained by calculating the Smith normal form of certain integer matrices.

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p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering

Algorithms ◽

10.3390/a13050110 ◽

2020 ◽

Vol 13 (5) ◽

pp. 110

Author(s):

Philippe Blondeel ◽

Pieterjan Robbe ◽

Cédric Van hoorickx ◽

Stijn François ◽

Geert Lombaert ◽

...

Keyword(s):

Finite Element ◽

Monte Carlo ◽

Variance Reduction ◽

Civil Engineering ◽

Galerkin Finite Element ◽

Quasi Monte Carlo ◽

Stability Of Slopes ◽

Galerkin Finite Element Methods ◽

Significant Cost Reduction ◽

Lattice Rule

Civil engineering applications are often characterized by a large uncertainty on the material parameters. Discretization of the underlying equations is typically done by means of the Galerkin Finite Element method. The uncertain material parameter can be expressed as a random field represented by, for example, a Karhunen–Loève expansion. Computation of the stochastic responses, i.e., the expected value and variance of a chosen quantity of interest, remains very costly, even when state-of-the-art Multilevel Monte Carlo (MLMC) is used. A significant cost reduction can be achieved by using a recently developed multilevel method: p-refined Multilevel Quasi-Monte Carlo (p-MLQMC). This method is based on the idea of variance reduction by employing a hierarchical discretization of the problem based on a p-refinement scheme. It is combined with a rank-1 Quasi-Monte Carlo (QMC) lattice rule, which yields faster convergence compared to the use of random Monte Carlo points. In this work, we developed algorithms for the p-MLQMC method for two dimensional problems. The p-MLQMC method is first benchmarked on an academic beam problem. Finally, we use our algorithm for the assessment of the stability of slopes, a problem that arises in geotechnical engineering, and typically suffers from large parameter uncertainty. For both considered problems, we observe a very significant reduction in the amount of computational work with respect to MLMC.

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