arXiv subjects

Nicholas Hale

Publications and source records attributed to Nicholas Hale.

10 recordsLinked to original sources

arXiv2026

Summation-by-parts operators for general function spaces: optimal nodes

Gauss-Lobatto quadrature nodes and weights are optimal for closed summation-by-parts (SBP) formulations based on polynomial approximation spaces in the sense that for a prescribed function space they yield an SBP operator of minimal dimension. We show that the same principle extends to general (possibly non-polynomial) function spaces: an associated generalised Gauss-Lobatto quadrature provides the optimal nodes and weights for the SBP formulation. We present an algorithm for computing these quadrature rules, demonstrate their accuracy and efficiency across a range of function spaces, and illustrate their use in solving initial boundary value problems.

math.NA↗

arXiv2026

Construction of Laguerre pseudospectral differentiation matrices

In this paper, we present a stable and efficient approach for constructing Laguerre pseudospectral differentiation matrices. The proposed method reformulates the off-diagonal entries and computes all required quantities simultaneously using an existing fast algorithm that also generates the collocation nodes. For the diagonal entries, a closed-form expression is employed to improve numerical accuracy. This construction avoids the catastrophic cancellation present in classical formulations and yields an all-in-one procedure for generating differentiation matrices. Numerical experiments demonstrate improved robustness and sustained high accuracy for significantly larger numbers of collocation points compared to standard implementations.

math.NA↗

arXiv2025

FASTGEN: Fast and Cost-Effective Synthetic Tabular Data Generation with LLMs

Synthetic data generation has emerged as an invaluable solution in scenarios where real-world data collection and usage are limited by cost and scarcity. Large language models (LLMs) have demonstrated remarkable capabilities in producing high-fidelity, domain-relevant samples across various fields. However, existing approaches that directly use LLMs to generate each record individually impose prohibitive time and cost burdens, particularly when large volumes of synthetic data are required. In this work, we propose a fast, cost-effective method for realistic tabular data synthesis that leverages LLMs to infer and encode each field's distribution into a reusable sampling script. By automatically classifying fields into numerical, categorical, or free-text types, the LLM generates distribution-based scripts that can efficiently produce diverse, realistic datasets at scale without continuous model inference. Experimental results show that our approach outperforms traditional direct methods in both diversity and data realism, substantially reducing the burden of high-volume synthetic data generation. We plan to apply this methodology to accelerate testing in production pipelines, thereby shortening development cycles and improving overall system efficiency. We believe our insights and lessons learned will aid researchers and practitioners seeking scalable, cost-effective solutions for synthetic data generation.

cs.LG↗

arXiv2024

A spectral collocation method for functional and delay differential equations

A framework for Chebyshev spectral collocation methods for the numerical solution of functional and delay differential equations (FDEs and DDEs) is described. The framework combines interpolation via the barycentric resampling matrix with a multidomain approach used to resolve isolated discontinuities propagated by non-smooth initial data. Geometric convergence is demonstrated for several examples of linear and nonlinear FDEs and DDEs with various delay types, including discrete, proportional, continuous, and state-dependent delay. The framework is a natural extension of standard spectral collocation methods based on polynomial interpolants and can be readily incorporated into existing spectral discretisations, such as in Chebfun/Chebop, allowing the automated and efficient solution of a wide class of nonlinear functional and delay differential equations.

math.NA↗

arXiv2024

Approximate solutions to a nonlinear functional differential equation

A functional differential equation related to the logistic equation is studied by a combination of numerical and perturbation methods. Parameter regions are identified where the solution to the nonlinear problem is approximated well by known series solutions of the linear version of the equation. The solution space for a particular class of functions is then mapped out using a continuation approach.

math.CA↗

arXiv2020

The ultraspherical spectral element method

We introduce a novel spectral element method based on the ultraspherical spectral method and the hierarchical Poincaré-Steklov scheme for solving second-order linear partial differential equations on polygonal domains with unstructured quadrilateral or triangular meshes. Properties of the ultraspherical spectral method lead to almost banded linear systems, allowing the element method to be competitive in the high-polynomial regime ($p > 5$). The hierarchical Poincaré-Steklov scheme enables precomputed solution operators to be reused, allowing for fast elliptic solves in implicit and semi-implicit time-steppers. The resulting spectral element method achieves an overall computational complexity of $\mathcal{O}(p^4/h^3)$ for mesh size $h$ and polynomial order $p$, enabling $hp$-adaptivity to be efficiently performed. We develop an open-source software system, ultraSEM, for flexible, user-friendly spectral element computations in MATLAB.

math.NA↗

arXiv2018

Hedging and Pricing European-type, Early-Exercise and Discrete Barrier Options using Algorithm for the Convolution of Legendre Series

This paper applies an algorithm for the convolution of compactly supported Legendre series (the CONLeg method) (cf. Hale and Townsend 2014a), to pricing/hedging European-type, early-exercise and discrete-monitored barrier options under a Levy process. The paper employs Chebfun (cf. Trefethen et al. 2014) in computational finance and provides a quadrature-free approach by applying the Chebyshev series in financial modelling. A significant advantage of using the CONLeg method is to formulate option pricing and option Greek curves rather than individual prices/values. Moreover, the CONLeg method can yield high accuracy in option pricing and hedging when the risk-free smooth probability density function (PDF) is smooth/non-smooth. Finally, we show that our method can accurately price/hedge options deep in/out of the money and with very long/short maturities. Compared with existing techniques, the CONLeg method performs either favourably or comparably in numerical experiments.

q-fin.CP↗

arXiv2017

An ultraspherical spectral method for linear Fredholm and Volterra integro-differential equations of convolution type

The Legendre-based ultraspherical spectral method for ordinary differential equations is combined with a formula for the convolution of two Legendre series to produce a new technique for solving linear Fredholm and Volterra integro-differential equations with convolution-type kernels. When the kernel and coefficient functions are sufficiently smooth then the method is spectrally-accurate and the resulting almost-banded linear systems can be solved with linear complexity.

math.NA↗

arXiv2016

A fast and spectrally convergent algorithm for rational-order fractional integral and differential equations

A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for ordinary differential equations [Olver & Townsend 2013], and involves constructing two different bases, one for the domain of the operator and one for the range of the operator. The bases are constructed from direct sums of suitably weighted ultraspherical or Jacobi polynomial expansions, for which explicit representations of fractional integrals and derivatives are known, and are carefully chosen so that the resulting operators are banded or almost-banded. Geometric convergence is demonstrated for numerous model problems when the variable coefficients and right-hand side are sufficiently smooth.

math.NA↗

arXiv2015

A fast FFT-based discrete Legendre transform

An $\mathcal{O}(N(\log N)^2/\log\!\log N)$ algorithm for computing the discrete Legendre transform and its inverse is described. The algorithm combines a recently developed fast transform for converting between Legendre and Chebyshev coefficients with a Taylor series expansion for Chebyshev polynomials about equally-spaced points in the frequency domain. Both components are based on the FFT, and as an intermediate step we obtain an $\mathcal{O}(N\log N)$ algorithm for evaluating a degree $N-1$ Chebyshev expansion at an $N$-point Legendre grid. Numerical results are given to demonstrate performance and accuracy.

math.NA↗