86 citations · 301 across the 37 of their papers we have counts for
48 papers
A machine-learned probability distribution in the phase space of turbulent channel flow for synthetic turbulence and flow reconstruction
Frederik Aerts, Dirk Nuyens, Johan Meyers
Although a complete characterisation of the probability distribution in the phase space of turbulent flows remains elusive, accurately sampling this distribution is essential for b…
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…
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…
Error bounds for function approximation using generated sets
Ronald Cools, Dirk Nuyens, Laurence Wilkes
This paper explores the use of "generated sets" for function approximation in reproducing kernel Hilbert spaces which consist of mul…
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…
A randomised lattice rule algorithm with pre-determined generating vector and random number of points for Korobov spaces with
Dirk Nuyens, Laurence Wilkes
In previous work (Kuo, Nuyens, Wilkes, 2023), we showed that a lattice rule with a pre-determined generating vector but random number of points can achieve the near optimal converg…
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…
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…
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…
Scaled lattice rules for integration on achieving higher-order convergence with error analysis in terms of orthogonal projections onto periodic spaces
Dirk Nuyens, Yuya Suzuki
We introduce a new method to approximate integrals which simply scales lattice rules from the unit cube $[0,1]^…
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…
Digit-by-digit and component-by-component constructions of lattice rules for periodic functions with unknown smoothness
Adrian Ebert, Peter Kritzer, Dirk Nuyens +1
Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-1 lattice rule to approximate an -dimensional integr…
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…
Enhanced Multi-Index Monte Carlo by means of Multiple Semi-Coarsened Multigrid for Anisotropic Diffusion Problems
Pieterjan Robbe, Dirk Nuyens, Stefan Vandewalle
In many models used in engineering and science, material properties are uncertain or spatially varying. For example, in geophysics, and porous media flow in particular, the uncerta…
Rank-1 lattices and higher-order exponential splitting for the time-dependent Schrödinger equation
Yuya Suzuki, Dirk Nuyens
In this paper, we propose a numerical method to approximate the solution of the time-dependent Schrödinger equation with periodic boundary condition in a high-dimensional setting.…
MDFEM: Multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM
Dong T. P. Nguyen, Dirk Nuyens
We introduce the multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficient where is a Gaussian random field defined by…
Constructing QMC finite element methods for elliptic PDEs with random coefficients by a reduced CBC construction
Adrian Ebert, Peter Kritzer, Dirk Nuyens
In the analysis of using quasi-Monte Carlo (QMC) methods to approximate expectations of a linear functional of the solution of an elliptic PDE with random diffusion coefficient the…
MDFEM: Multivariate decomposition finite element method for elliptic PDEs with uniform random diffusion coefficients using higher-order QMC and FEM
Dong T. P. Nguyen, Dirk Nuyens
We introduce the multivariate decomposition finite element method (MDFEM) for solving elliptic PDEs with uniform random diffusion coefficients. We show that the MDFEM can be used t…
Strang splitting in combination with rank- and rank- lattices for the time-dependent Schrödinger equation
Yuya Suzuki, Gowri Suryanarayana, Dirk Nuyens
We approximate the solution for the time dependent Schrödinger equation (TDSE) in two steps. We first use a pseudo-spectral collocation method that uses samples of functions on ran…