Daniel Fortunato

Daniel Fortunato, Alex Townsend

Poisson's equation is the canonical elliptic partial differential equation. While there exist fast Poisson solvers for finite difference and finite element methods, fast Poisson solvers for spectral methods have remained elusive. Here, we derive spectral methods for solving Poisson's equation on a square, cylinder, solid sphere, and cube that have an optimal complexity (up to polylogarithmic terms) in terms of the degrees of freedom required to represent the solution. Whereas FFT-based fast Poisson solvers exploit structured eigenvectors of finite difference matrices, our solver exploits a separated spectra property that holds for our spectral discretizations. Without parallelization, we can solve Poisson's equation on a square with 100 million degrees of freedom in under two minutes on a standard laptop.

Daniel Fortunato, Chris H. Rycroft, Robert Saye

An efficient $hp$-multigrid scheme is presented for local discontinuous Galerkin (LDG) discretizations of elliptic problems, formulated around the idea of separately coarsening the underlying discrete gradient and divergence operators. We show that traditional multigrid coarsening of the primal formulation leads to poor and suboptimal multigrid performance, whereas coarsening of the flux formulation leads to optimal convergence and is equivalent to a purely geometric multigrid method. The resulting operator-coarsening schemes do not require the entire mesh hierarchy to be explicitly built, thereby obviating the need to compute quadrature rules, lifting operators, and other mesh-related quantities on coarse meshes. We show that good multigrid convergence rates are achieved in a variety of numerical tests on 2D and 3D uniform and adaptive Cartesian grids, as well as for curved domains using implicitly defined meshes and for multi-phase elliptic interface problems with complex geometry. Extension to non-LDG discretizations is briefly discussed.

Ruben Ohana, Michael McCabe, Lucas Meyer et al. · cambridge

Machine learning based surrogate models offer researchers powerful tools for accelerating simulation-based workflows. However, as standard datasets in this space often cover small classes of physical behavior, it can be difficult to evaluate the efficacy of new approaches. To address this gap, we introduce the Well: a large-scale collection of datasets containing numerical simulations of a wide variety of spatiotemporal physical systems. The Well draws from domain experts and numerical software developers to provide 15TB of data across 16 datasets covering diverse domains such as biological systems, fluid dynamics, acoustic scattering, as well as magneto-hydrodynamic simulations of extra-galactic fluids or supernova explosions. These datasets can be used individually or as part of a broader benchmark suite. To facilitate usage of the Well, we provide a unified PyTorch interface for training and evaluating models. We demonstrate the function of this library by introducing example baselines that highlight the new challenges posed by the complex dynamics of the Well. The code and data is available at https://github.com/PolymathicAI/the_well.

Giordano d'Aloisio, Sophie Fortz, Carol Hanna et al.

Background: Quantum computing is a rapidly growing new programming paradigm that brings significant changes to the design and implementation of algorithms. Understanding quantum algorithms requires knowledge of physics and mathematics, which can be challenging for software developers. Aims: In this work, we provide a first analysis of how LLMs can support developers' understanding of quantum code. Method: We empirically analyse and compare the quality of explanations provided by three widely adopted LLMs (Gpt3.5, Llama2, and Tinyllama) using two different human-written prompt styles for seven state-of-the-art quantum algorithms. We also analyse how consistent LLM explanations are over multiple rounds and how LLMs can improve existing descriptions of quantum algorithms. Results: Llama2 provides the highest quality explanations from scratch, while Gpt3.5 emerged as the LLM best suited to improve existing explanations. In addition, we show that adding a small amount of context to the prompt significantly improves the quality of explanations. Finally, we observe how explanations are qualitatively and syntactically consistent over multiple rounds. Conclusions: This work highlights promising results, and opens challenges for future research in the field of LLMs for quantum code explanation. Future work includes refining the methods through prompt optimisation and parsing of quantum code explanations, as well as carrying out a systematic assessment of the quality of explanations.

Michael Chiwere, Daniel Fortunato, Grady B. Wright

Semi-implicit semi-Lagrangian (SISL) methods are commonly used for the shallow water equations (SWE) because they allow for larger time steps than those permitted by the Courant-Friedrichs-Lewy (CFL) stability condition in Eulerian schemes. In these methods, the semi-Lagrangian treatment of advection is typically performed using lower-order interpolation, such as tensor-product Lagrange interpolation with cubic or quintic polynomials. However, operational SISL schemes routinely employ spectrally accurate spatial discretizations, such as spherical harmonics or the double Fourier sphere (DFS) method, for computing horizontal derivatives of the prognostic variables. This creates a mismatch in numerical accuracy, making the use of low-order interpolation less clearly justified. In this work, we present the first numerical investigation of spectrally accurate interpolation in SISL schemes for the SWE. Our approach builds upon the recently developed DFS-based SWE model, incorporating a spectral interpolation scheme that is accelerated using the nonuniform fast Fourier transform (NUFFT) to maintain the same overall computational complexity as the original model. Using several standard SWE test cases, we evaluate the accuracy, conservation, and numerical diffusion of the new model, particularly over long integration times. Compared to an equivalent SISL model with low-order interpolation, the new model achieves higher accuracy, improved mass and energy conservation, and reduced numerical diffusion, demonstrating the potential benefits of incorporating spectrally accurate interpolation into SISL schemes.

Daniel Fortunato, José Campos, Rui Abreu

Coverage criteria play a central role in assessing test adequacy in classical software, yet their effectiveness for quantum programs remains poorly understood and largely unexplored. In this paper, we propose six quantum-tailored criteria - condition, decision, and path coverage, and their probabilistic variants - adapted from their classical counterparts. We present QaCoCo, a tool that computes these criteria for circuit-based quantum programs. We empirically evaluate these criteria on a large and diverse set of 540 circuits and analyze the coverage achieved. Our results show that while circuits frequently achieve high condition and decision coverage (97.56% and 97.63%, on average), path coverage remains limited (71.84%), particularly in the presence of multi-controlled gates, which induce extreme path explosion and coverage imbalance. Moreover, to account for the probabilistic nature of quantum circuits, we introduce probabilistic coverage, which augments structural coverage with a confidence measure (88.87%, 88.65%, and 37.18% for condition, decision, and path coverage, respectively, on average). Finally, through mutation testing, we find weak or no correlation between fault detection and structural coverage, consistent with observations in classical computing.