arXiv subjects

Laurence Wilkes

Publications and source records attributed to Laurence Wilkes.

4 recordsLinked to original sources

arXiv2025

Error bounds for function approximation using generated sets

This paper explores the use of "generated sets" $\{ \{ k \boldsymbolζ \} : k = 1, \ldots, n \}$ for function approximation in reproducing kernel Hilbert spaces which consist of multi-dimensional functions with an absolutely convergent Fourier series. The algorithm is a least squares algorithm that samples the function at the points of a generated set. We show that there exist $\boldsymbolζ \in [0,1]^d$ for which the worst-case $L_2$ error has the optimal order of convergence if the space has polynomially converging approximation numbers. In fact, this holds for a significant portion of the generators. Additionally we show that a restriction to rational generators is possible with a slight increase of the bound. Furthermore, we specialise the results to the weighted Korobov space, where we derive a bound applicable to low values of sample points, and state tractability results.

math.NA↗

arXiv2023

A randomised lattice rule algorithm with pre-determined generating vector and random number of points for Korobov spaces with $0 < α\le 1/2$

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 convergence of $O(n^{-α-1/2+ε})$, $ε> 0$, for the worst case expected error, commonly referred to as the randomised error, for numerical integration of high-dimensional functions in the Korobov space with smoothness $α> 1/2$. Compared to the optimal deterministic rate of $O(n^{-α+ε})$, $ε> 0$, such a randomised algorithm is capable of an extra half in the rate of convergence. In this paper, we show that a pre-determined generating vector also exists in the case of $0 < α\le 1/2$. Also here we obtain the near optimal convergence of $O(n^{-α-1/2+ε})$, $ε> 0$; or in more detail, we obtain $O(\sqrt{r} \, n^{-α-1/2+1/(2r)+ε'})$ which holds for any choices of $ε' > 0$ and $r \in \mathbb{N}$ with $r > 1/(2α)$.

math.NA↗

arXiv2023

Random-prime--fixed-vector randomised lattice-based algorithm for high-dimensional integration

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 $d$-dimensional weighted Korobov space. This algorithm uses a lattice rule with a fixed generating vector and the only random element is the choice of the number of function evaluations. For a given computational budget $n$ of a maximum allowed number of function evaluations, we uniformly pick a prime $p$ in the range $n/2 < p \le n$. We show error bounds for the randomised error, which is defined as the worst case expected error, of the form $O(n^{-α- 1/2 + δ})$, with $δ> 0$, for a Korobov space with smoothness $α> 1/2$ and general weights. The implied constant in the bound is dimension-independent given the usual conditions on the weights. We present an algorithm that can construct suitable generating vectors \emph{offline} ahead of time at cost $O(d n^4 / \ln n)$ when the weight parameters defining the Korobov spaces are so-called product weights. For this case, numerical experiments confirm our theory that the new randomised algorithm achieves the near optimal rate of the randomised error.

math.NA↗

arXiv2022

Shape of the asymptotic maximum sum-free sets in integer lattice grids

We determine the shape of all sum-free sets in $\{1,2,\ldots,n\}^2$ of size close to the maximum $\frac{3}{5}n^2$, solving a problem of Elsholtz and Rackham. We show that all such asymptotic maximum sum-free sets lie completely in the stripe $\frac{4}{5}n-o(n)\le x+y\le\frac{8}{5}n+ o(n)$. We also determine for any positive integer $p$ the maximum size of a subset $A\subseteq \{1,2,\ldots,n\}^2$ which forbids the triple $(x,y,z)$ satisfying $px+py=z$.

math.CO↗