Nicholas Hale

Nicholas Hale, Sheehan Olver

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.

Nicholas Hale

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.

Nicholas Hale, Charis Harley, Prince Nchupang et al.

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.

Nicholas Hale, Enrique Thomann, JAC Weideman

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.

Emma Nel, Nicholas Hale

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.

Anh Nguyen, Sam Schafft, Nicholas Hale et al.

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.

Tat Lung Chan, Nicholas Hale

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.