Dirk Nuyens

Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces

We reformulate the original component-by-component algorithm for rank-1 lattices in a matrix-vector notation so as to highlight its structural properties. For function spaces similar to a weighted Korobov space, we derive a technique which has construction cost O(snlog(n)), in contrast with the original algorithm which has construction cost O(sn 2 ). Herein s is the number of dimensions and n the number of points (taken prime). In contrast to other approaches to speed up construction, our fast algorithm computes exactly the same quantity as the original algorithm. The presented algorithm can also be used to construct randomly shifted lattice rules in weighted Sobolev spaces.

Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications

We devise and implement quasi-Monte Carlo methods for computing the expectations of nonlinear functionals of solutions of a class of elliptic partial differential equations with random coefficients. Our motivation comes from fluid flow in random porous media, where relevant functionals include the fluid pressure/velocity at any point in space or the breakthrough time of a pollution plume being transported by the velocity field. Our emphasis is on situations where a very large number of random variables is needed to model the coefficient field. As an alternative to classical Monte Carlo, we here employ quasi-Monte Carlo methods, which use deterministically chosen sample points in an appropriate (usually high-dimensional) parameter space. Each realization of the PDE solution requires a finite element (FE) approximation in space, and this is done using a realization of the coefficient field restricted to a suitable regular spatial grid (not necessarily the same as the FE grid). In the statistically homogeneous case the corresponding covariance matrix can be diagonalized and the required coefficient realizations can be computed efficiently using FFT. In this way we avoid the use of a truncated Karhunen-Loeve expansion, but introduce high nominal dimension in parameter space. Numerical experiments with 2-dimensional rough random fields, high variance and small length scale are reported, showing that the quasi-Monte Carlo method consistently outperforms the Monte Carlo method, with a smaller error and a noticeably better than O(N^-^1^/^2) convergence rate, where N is the number of samples. Moreover, the rate of convergence of the quasi-Monte Carlo method does not appear to degrade as the nominal dimension increases. Examples with dimension as high as 10^6 are reported.

Constructing Embedded Lattice Rules for Multivariate Integration

Lattice rules are a family of equal-weight cubature formulae for approximating high-dimensional integrals. By now it is well established that good generating vectors for lattice rules having

Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points

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Fast Component-by-Component Construction, a Reprise for Different Kernels

points can be constructed component-by-component for integrands belonging to certain weighted function spaces, and that they can achieve the optimal rate of convergence. Although the lattice rules constructed this way are extensible in dimension, they are not extensible in

Higher Order QMC Petrov--Galerkin Discretization for Affine Parametric Operator Equations with Random Field Inputs

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Lattice rules for nonperiodic smooth integrands

; thus when

Constructing lattice rules based on weighted degree of exactness and worst case error

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A Belgian view on lattice rules

is changed the generating vector needs to be constructed anew. In this paper we introduce a new algorithm for constructing good generating vectors for embedded lattice rules which can be used for a range of

Application of Quasi-Monte Carlo Methods to Elliptic PDEs with Random Diffusion Coefficients: A Survey of Analysis and Implementation

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