1 citations · 1 across the 16 of their papers we have counts for
19 papers
A higher order lattice-based method for high-dimensional numerical integration using periodization
Felix Bartel, Alexander D. Gilbert, Michael Griebel +2
We present a novel numerical integration method for non-periodic functions over the high-dimensional unit cube by a specially crafted (unequally-)weighted quadrature rule…
Finite element analysis of density estimation using preintegration for elliptic PDE with random input
Alexander D. Gilbert
This paper analyses the finite element component of the error when using preintegration to approximate the cdf and pdf for uncertainty quantification (UQ) problems involving ellipt…
Quasi-Monte Carlo methods for uncertainty quantification of tumor growth modeled by a parametric semi-linear parabolic reaction-diffusion equation
Alexander D. Gilbert, Frances Y. Kuo, Dirk Nuyens +3
We study the application of a quasi-Monte Carlo (QMC) method to a class of semi-linear parabolic reaction-diffusion partial differential equations used to model tumor growth. Mathe…
Minimal Subsampled Rank-1 Lattices for Multivariate Approximation with Optimal Convergence Rate
Felix Bartel, Alexander D. Gilbert, Frances Y. Kuo +1
In this paper we show error bounds for randomly subsampled rank-1 lattices. We pay particular attention to the ratio of the size of the subset to the size of the initial lattice, w…
Multilevel lattice-based kernel approximation for elliptic PDEs with random coefficients
Alexander D. Gilbert, Michael B. Giles, Frances Y. Kuo +2
This paper introduces a multilevel kernel-based approximation method to estimate efficiently solutions to elliptic partial differential equations (PDEs) with periodic random coeffi…
Density estimation for elliptic PDE with random input by preintegration and quasi-Monte Carlo methods
Alexander D. Gilbert, Frances Y. Kuo, Abirami Srikumar
In this paper, we apply quasi-Monte Carlo (QMC) methods with an initial preintegration step to estimate cumulative distribution functions and probability density functions in uncer…
A complex-projected Rayleigh quotient iteration for targeting interior eigenvalues
Nils Friess, Alexander D. Gilbert, Robert Scheichl
We introduce a new Projected Rayleigh Quotient Iteration aimed at improving the convergence behaviour of classic Rayleigh Quotient iteration (RQI) by incorporating approximate info…
Multilevel Monte Carlo methods for stochastic convection-diffusion eigenvalue problems
Tiangang Cui, Hans De Sterck, Alexander D. Gilbert +2
We develop new multilevel Monte Carlo (MLMC) methods to estimate the expectation of the smallest eigenvalue of a stochastic convection-diffusion operator with random coefficients.…
Theory and construction of Quasi-Monte Carlo rules for option pricing and density estimation
Alexander D. Gilbert, Frances Y. Kuo, Ian H. Sloan +1
In this paper we propose and analyse a method for estimating three quantities related to an Asian option: the fair price, the cumulative distribution function, and the probability…
Preintegration is not smoothing when monotonicity fails
Alexander D. Gilbert, Frances Y. Kuo, Ian H. Sloan
Preintegration is a technique for high-dimensional integration over -dimensional Euclidean space, which is designed to reduce an integral whose integrand contains kinks or jumps…
Analysis of preintegration followed by quasi-Monte Carlo integration for distribution functions and densities
Alexander D. Gilbert, Frances Y. Kuo, Ian H. Sloan
In this paper, we analyse a method for approximating the distribution function and density of a random variable that depends in a non-trivial way on a possibly high number of indep…
Equivalence between Sobolev spaces of first-order dominating mixed smoothness and unanchored ANOVA spaces on
Alexander D. Gilbert, Frances Y. Kuo, Ian H. Sloan
We prove that a variant of the classical Sobolev space of first-order dominating mixed smoothness is equivalent (under a certain condition) to the unanchored ANOVA space on $\mathb…
Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems II: Efficient algorithms and numerical results
Alexander D. Gilbert, Robert Scheichl
Stochastic PDE eigenvalue problems often arise in the field of uncertainty quantification, whereby one seeks to quantify the uncertainty in an eigenvalue, or its eigenfunction. In…
Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems I: Regularity and error analysis
Alexander D. Gilbert, Robert Scheichl
Stochastic PDE eigenvalue problems are useful models for quantifying the uncertainty in several applications from the physical sciences and engineering, e.g., structural vibration…
Bounding the spectral gap for an elliptic eigenvalue problem with uniformly bounded stochastic coefficients
Alexander D. Gilbert, Ivan G. Graham, Robert Scheichl +1
A key quantity that occurs in the error analysis of several numerical methods for eigenvalue problems is the distance between the eigenvalue of interest and the next nearest eigenv…
Hiding the weights -- CBC black box algorithms with a guaranteed error bound
Alexander D. Gilbert, Frances Y. Kuo, Ian H. Sloan
The component-by-component (CBC) algorithm is a method for constructing good generating vectors for lattice rules for the efficient computation of high-dimensional integrals in the…
Analysis of quasi-Monte Carlo methods for elliptic eigenvalue problems with stochastic coefficients
Alexander D. Gilbert, Ivan G. Graham, Frances Y. Kuo +2
We consider the forward problem of uncertainty quantification for the generalised Dirichlet eigenvalue problem for a coercive second order partial differential operator with random…
Efficient implementations of the Multivariate Decomposition Method for approximating infinite-variate integrals
Alexander D. Gilbert, Frances Y. Kuo, Dirk Nuyens +1
In this paper we focus on efficient implementations of the Multivariate Decomposition Method (MDM) for approximating integrals of -variate functions. Such -variate…
Small Superposition Dimension and Active Set Construction for Multivariate Integration Under Modest Error Demand
Alexander D. Gilbert, Greg W. Wasilkowski
Constructing active sets is a key part of the Multivariate Decomposition Method. An algorithm for constructing optimal or quasi-optimal active sets is proposed in the paper. By num…