86 citations · 242 across the 30 of their papers we have counts for
46 papers
Fourier Neural Operators with rank-1 lattice points and hyperbolic cross
Jakob Dilen, Alexander Keller, Frances Y. Kuo +1
The \emph{Fourier neural operator} (FNO) is a neural network architecture that learns mappings between function spaces. Its efficient implementation is based on the multi-dimension…
Lattice-based Deep Neural Networks: Regularity and Tailored Regularization
Alexander Keller, Frances Y. Kuo, Dirk Nuyens +1
This survey article is concerned with the application of lattice rules to Deep Neural Networks (DNNs), lattice rules being a family of quasi-Monte Carlo methods. They have demonstr…
Regularity and tailored regularization of Deep Neural Networks, with application to parametric PDEs in uncertainty quantification
Alexander Keller, Frances Y. Kuo, Dirk Nuyens +1
In this paper we consider Deep Neural Networks (DNNs) with a smooth activation function as surrogates for high-dimensional functions that are somewhat smooth but costly to evaluate…
Quasi-Monte Carlo methods for uncertainty quantification of wave propagation and scattering problems modelled by the Helmholtz equation
Ivan G. Graham, Frances Y. Kuo, Dirk Nuyens +2
We analyse and implement a quasi-Monte Carlo (QMC) finite element method (FEM) for the forward problem of uncertainty quantification (UQ) for the Helmholtz equation with random coe…
Random-prime--fixed-vector randomised lattice-based algorithm for high-dimensional integration
Frances Y. Kuo, Dirk Nuyens, Laurence Wilkes
We show that a very simple randomised algorithm for numerical integration can produce a near optimal rate of convergence for integrals of functions in the -dimensional weighted…
Comparison of Two Search Criteria for Lattice-based Kernel Approximation
Frances Y. Kuo, Weiwen Mo, Dirk Nuyens +2
The kernel interpolant in a reproducing kernel Hilbert space is optimal in the worst-case sense among all approximations of a function using the same set of function values. In thi…
Lattice-based kernel approximation and serendipitous weights for parametric PDEs in very high dimensions
Vesa Kaarnioja, Frances Y. Kuo, Ian H. Sloan
We describe a fast method for solving elliptic partial differential equations (PDEs) with uncertain coefficients using kernel interpolation at a lattice point set. By representing…
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…
Uncertainty quantification for random domains using periodic random variables
Harri Hakula, Helmut Harbrecht, Vesa Kaarnioja +2
We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently in…
Constructing Embedded Lattice-based Algorithms for Multivariate Function Approximation with a Composite Number of Points
Frances Y. Kuo, Weiwen Mo, Dirk Nuyens
We approximate -variate periodic functions in weighted Korobov spaces with general weight parameters using function values at lattice points. We do not limit to be a pri…
Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration
Philipp A. Guth, Vesa Kaarnioja, Frances Y. Kuo +2
We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints un…
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…
Lattice field computations via recursive numerical integration
Tobias Hartung, Karl Jansen, Frances Y. Kuo +3
We investigate the application of efficient recursive numerical integration strategies to models in lattice gauge theory from quantum field theory. Given the coupling structure of…
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…
Lattice meets lattice: Application of lattice cubature to models in lattice gauge theory
Tobias Hartung, Karl Jansen, Frances Y. Kuo +3
High dimensional integrals are abundant in many fields of research including quantum physics. The aim of this paper is to develop efficient recursive strategies to tackle a class o…
Fast approximation by periodic kernel-based lattice-point interpolation with application in uncertainty quantification
Vesa Kaarnioja, Yoshihito Kazashi, Frances Y. Kuo +2
This paper deals with the kernel-based approximation of a multivariate periodic function by interpolation at the points of an integration lattice -- a setting that, as pointed out…
Quasi-Monte Carlo finite element analysis for wave propagation in heterogeneous random media
M. Ganesh, Frances Y. Kuo, Ian H. Sloan
We propose and analyze a quasi-Monte Carlo (QMC) algorithm for efficient simulation of wave propagation modeled by the Helmholtz equation in a bounded region in which the refractiv…
A quasi-Monte Carlo Method for an Optimal Control Problem Under Uncertainty
Philipp A. Guth, Vesa Kaarnioja, Frances Y. Kuo +2
We study an optimal control problem under uncertainty, where the target function is the solution of an elliptic partial differential equation with random coefficients, steered by a…
Fast component-by-component construction of lattice algorithms for multivariate approximation with POD and SPOD weights
Ronald Cools, Frances Y. Kuo, Dirk Nuyens +1
In a recent paper by the same authors, we provided a theoretical foundation for the component-by-component (CBC) construction of lattice algorithms for multivariate approxima…
Lattice algorithms for multivariate approximation in periodic spaces with general weight parameters
Ronald Cools, Frances Y. Kuo, Dirk Nuyens +1
This paper provides the theoretical foundation for the construction of lattice algorithms for multivariate approximation in the worst case setting, for functions in a periodi…
Function integration, reconstruction and approximation using rank-1 lattices
Frances Y. Kuo, Giovanni Migliorati, Fabio Nobile +1
We consider rank-1 lattices for integration and reconstruction of functions with series expansion supported on a finite index set. We explore the connection between the periodic Fo…
Uncertainty quantification using periodic random variables
Vesa Kaarnioja, Frances Y. Kuo, Ian H. Sloan
Many studies in uncertainty quantification have been carried out under the assumption of an input random field in which a countable number of independent random variables are each…
Derandomised lattice rules for high dimensional integration
Yoshihito Kazashi, Frances Y. Kuo, Ian H. Sloan
We seek shifted lattice rules that are good for high dimensional integration over the unit cube in the setting of an unanchored weighted Sobolev space of functions with square-inte…