arXiv subjects

Daniel Fortunato

Publications and source records attributed to Daniel Fortunato.

12 recordsLinked to original sources

arXiv2026

Spectral interpolation in semi-implicit semi-Lagrangian methods for shallow water equations on the sphere

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.

math.NA↗

arXiv2026

Probabilistic Condition, Decision and Path Coverage of Circuit-based Quantum Programs

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.

quant-ph↗

arXiv2025

Hardware Acceleration for HPS Algorithms in Two and Three Dimensions

We provide a flexible, open-source framework for hardware acceleration, namely massively-parallel execution on general-purpose graphics processing units (GPUs), applied to the hierarchical Poincar\'e--Steklov (HPS) family of algorithms for building fast direct solvers for linear elliptic partial differential equations. To take full advantage of the power of hardware acceleration, we propose two variants of HPS algorithms to improve performance on two- and three-dimensional problems. In the two-dimensional setting, we introduce a novel recomputation strategy that minimizes costly data transfers to and from the GPU; in three dimensions, we modify and extend the adaptive discretization technique of Geldermans and Gillman [2019] to greatly reduce peak memory usage. We provide an open-source implementation of these methods written in JAX, a high-level accelerated linear algebra package, which allows for the first integration of a high-order fast direct solver with automatic differentiation tools. We conclude with extensive numerical examples showing our methods are fast and accurate on two- and three-dimensional problems.

math.NA↗

arXiv2025

A fully adaptive, high-order, fast Poisson solver for complex two-dimensional geometries

We present a new framework for the fast solution of inhomogeneous elliptic boundary value problems in domains with smooth boundaries. High-order solvers based on adaptive box codes or the fast Fourier transform can efficiently treat the volumetric inhomogeneity, but require care to be taken near the boundary to ensure that the volume data is globally smooth. We avoid function extension or cut-cell quadratures near the boundary by dividing the domain into two regions: a bulk region away from the boundary that is efficiently treated with a truncated free-space box code, and a variable-width boundary-conforming strip region that is treated with a spectral collocation method and accompanying fast direct solver. Particular solutions in each region are then combined with Laplace layer potentials to yield the global solution. The resulting solver has an optimal computational complexity of $O(N)$ for an adaptive discretization with $N$ degrees of freedom. With an efficient two-dimensional (2D) implementation we demonstrate adaptive resolution of volumetric data, boundary data, and geometric features across a wide range of length scales, to typically 10-digit accuracy. The cost of all boundary corrections remains small relative to that of the bulk box code. The extension to 3D is expected to be straightforward in many cases because the strip ``thickens'' an existing boundary quadrature.

math.NA↗

arXiv2024

The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning

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.

cs.LG↗

arXiv2024

Exploring LLM-Driven Explanations for Quantum Algorithms

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.

cs.CL↗

arXiv2022

Corner cases of the tau method: symmetrically imposing boundary conditions on hypercubes

Polynomial spectral methods produce fast, accurate, and flexible solvers for broad ranges of PDEs with one bounded dimension, where the incorporation of general boundary conditions is well understood. However, automating extensions to domains with multiple bounded dimensions is challenging because of difficulties in imposing boundary conditions at shared edges and corners. Past work has included various workarounds, such as the anisotropic inclusion of partial boundary data at shared edges or approaches that only work for specific boundary conditions. Here we present a general system for imposing boundary conditions for elliptic equations on hypercubes. We take an approach based on the generalized tau method, which allows for a wide range of boundary conditions for many different spectral schemes. The generalized tau method has the distinct advantage that the specified polynomial residual determines the exact algebraic solution; afterwards, any stable numerical scheme will find the same result. We can, therefore, provide one-to-one comparisons to traditional collocation and Galerkin methods within the tau framework. As an essential requirement, we add specific tau corrections to the boundary conditions, in addition to the bulk PDE, which produce a unique set of compatible boundary data at shared subsurfaces. Our approach works with general boundary conditions that commute on intersecting subsurfaces, including Dirichlet, Neumann, Robin, and any combination of these on all boundaries. The boundary tau corrections can be made hyperoctahedrally symmetric and easily incorporated into existing solvers. We present the method explicitly for the Poisson equation in two and three dimensions and describe its extension to arbitrary elliptic equations (e.g. biharmonic) in any dimension.

math.NA↗

arXiv2022

Generation and motion of interfaces in a mass-conserving reaction-diffusion system

Reaction-diffusion models with nonlocal constraints naturally arise as limiting cases of coupled bulk-surface models of intracellular signalling. In this paper, a minimal, mass-conserving model of cell-polarization on a curved membrane is analyzed in the limit of slow surface diffusion. Using the tools of formal asymptotics and calculus of variations, we study the characteristic wave-pinning behavior of this system on three dynamical timescales. On the short timescale, generation of an interface separating high- and low-concentration domains is established under suitable conditions. Intermediate timescale dynamics is shown to lead to a uniform growth or shrinking of these domains to sizes which are fixed by global parameters. Finally, the long time dynamics reduces to area-preserving geodesic curvature flow that may lead to multi-interface steady state solutions. These results provide a foundation for studying cell polarization and related phenomena in biologically relevant geometries.

nlin.PS↗

arXiv2022

A high-order fast direct solver for surface PDEs

We introduce a fast direct solver for variable-coefficient elliptic partial differential equations on surfaces based on the hierarchical Poincaré-Steklov method. The method takes as input an unstructured, high-order quadrilateral mesh of a surface and discretizes surface differential operators on each element using a high-order spectral collocation scheme. Elemental solution operators and Dirichlet-to-Neumann maps tangent to the surface are precomputed and merged in a pairwise fashion to yield a hierarchy of solution operators that may be applied in $\mathcal{O}(N \log N)$ operations for a mesh with $N$ elements. The resulting fast direct solver may be used to accelerate high-order implicit time-stepping schemes, as the precomputed operators can be reused for fast elliptic solves on surfaces. On a standard laptop, precomputation for a 12th-order surface mesh with over 1 million degrees of freedom takes 17 seconds, while subsequent solves take only 0.25 seconds. We apply the method to a range of problems on both smooth surfaces and surfaces with sharp corners and edges, including the static Laplace-Beltrami problem, the Hodge decomposition of a tangential vector field, and some time-dependent nonlinear reaction-diffusion systems.

math.NA↗

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

Efficient operator-coarsening multigrid schemes for local discontinuous Galerkin methods

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.

math.NA↗

arXiv2017

Fast Poisson solvers for spectral methods

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.

math.NA↗