arXiv subjects

Robert I. Saye

Publications and source records attributed to Robert I. Saye.

6 recordsLinked to original sources

arXiv2026

Cascading Smoothers for Multigrid

Multigrid methods are among the most effective frameworks for solving large-scale sparse systems. However, achieving their hallmark linear scaling and rapid convergence crucially depends on an effective smoother algorithm, whose design is often highly problem-dependent. This paper develops a new approach, referred to as \textit{cascading smoothers} due to their operation as an ordered sequence of single-step block-diagonal smoothers. Each level in the cascade is optimised to maximally damp the output of prior steps via a Frobenius norm minimisation of the corresponding error propagators. In particular, we develop an additive (resp., multiplicative) formulation analogous to Jacobi (resp., Gauss-Seidel). Applied within a standard multigrid V-cycle, we show they are remarkably effective across a wide array of problems, including finite difference, finite element, and discontinuous Galerkin discretisations applied to Poisson, elliptic interface, and Stokes systems as well as multiphase variants. In every case, cascading smoothers closely match or significantly outperform their optimally-damped classical counterparts, yet require no parameter tuning apart from a few discrete solver choices. Additionally, the approach is highly parallelisable and robust to geometric and operator complexities such as unstructured meshes and high-contrast coefficients.

math.NA↗

arXiv2025

A comparative study of efficient multigrid solvers for high-order local discontinuous Galerkin methods: Poisson, elliptic interface, and multiphase Stokes problems

We design and investigate a variety of multigrid solvers for high-order local discontinuous Galerkin methods applied to elliptic interface and multiphase Stokes problems. Using the template of a standard multigrid V-cycle, we consider a variety of element-wise block smoothers, including Jacobi, multi-coloured Gauss-Seidel, processor-block Gauss-Seidel, and with special interest, smoothers based on sparse approximate inverse (SAI) methods. In particular, we develop SAI methods that: (i) balance the smoothing of velocity and pressure variables in Stokes problems; and (ii) robustly handles high-contrast viscosity coefficients in multiphase problems. Across a broad range of two- and three-dimensional test cases, including Poisson, elliptic interface, steady-state Stokes, and unsteady Stokes problems, we examine a multitude of multigrid smoother and solver combinations. In every case, there is at least one approach that matches the performance of classical geometric multigrid algorithms, e.g., 4 to 8 iterations reduce the residual by 10 orders of magnitude. We also discuss their relative merits with regard to simplicity, robustness, computational cost, and parallelisation.

math.NA↗

arXiv2025

Efficient multigrid solvers for mixed-degree local discontinuous Galerkin multiphase Stokes problems

We design and investigate efficient multigrid solvers for multiphase Stokes problems discretised via mixed-degree local discontinuous Galerkin methods. Using the template of a standard multigrid V-cycle, we develop a smoother analogous to element-wise block Gauss-Seidel, except the diagonal block inverses are replaced with an approximation that balances the smoothing of the velocity and pressure variables, factoring in the unequal scaling of the various Stokes system operators, and optimised via two-grid local Fourier analysis. We evaluate the performance of the multigrid solver across an extensive range of two- and three-dimensional test problems, including steady-state and unsteady, standard-form and stress-form, single-phase and high-contrast multiphase Stokes problems, with multiple kinds of boundary conditions and various choices of polynomial degree. In the lowest-degree case, i.e., that of piecewise constant pressure fields, we observe reliable multigrid convergence rates, though not especially fast. However, in every other case, we see rapid convergence rates matching those of classical Poisson-style geometric multigrid methods; e.g., 5 iterations reduce the Stokes system residual by 5 to 10 orders of magnitude.

math.NA↗

arXiv2022

A Connected Component Labeling Algorithm for Implicitly-Defined Domains

A connected component labeling algorithm is developed for implicitly-defined domains specified by multivariate polynomials. The algorithm operates by recursively subdividing the constraint domain into hyperrectangular subcells until the topology thereon is sufficiently simple; in particular, we devise a topology test using properties of Bernstein polynomials. In many cases the algorithm produces a certificate guaranteeing its correctness, i.e., two points yield the same label if and only if they are path-connected. To robustly handle various kinds of edge cases, the algorithm may assign identical labels to distinct components, but only when they are exactly or nearly touching, relative to a user-controlled length scale. A variety of numerical experiments assess the effectiveness of the overall approach, including statistical analyses on randomly generated multi-component geometry in 2D and 3D, as well as specific examples involving cusps, self-intersections, junctions, and other kinds of singularities.

math.NA↗

arXiv2022

On two conjectures concerning the ternary digits of powers of two

Erdős conjectured that 1, 4, and 256 are the only powers of two whose ternary representations consist solely of 0s and 1s. Sloane conjectured that, except for $\{2^0,2^1,2^2,2^3,2^4,2^{15}\}$, every other power of two has at least one 0 in its ternary representation. In this paper, numerical results are given in strong support of these conjectures. In particular, we verify both conjectures for all $2^n$ with $n \leq 2 \cdot 3^{45} \approx 5.9 \times 10^{21}$. Our approach makes use of a simple recursive construction of numbers $2^n$ having prescribed patterns in their trailing ternary digits.

math.NT↗

arXiv2021

High-Order Quadrature on Multi-Component Domains Implicitly Defined by Multivariate Polynomials

A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly defined geometry as the graph of an implicitly defined, multi-valued height function, and applies a dimension reduction approach needing only one-dimensional quadrature. In particular, we explore the use of Gauss-Legendre and tanh-sinh methods and demonstrate that the quadrature algorithm inherits their high-order convergence rates. Under the action of $h$-refinement with $q$ fixed, the quadrature schemes yield an order of accuracy of $2q$, where $q$ is the one-dimensional node count; numerical experiments demonstrate up to 22nd order. Under the action of $q$-refinement with the geometry fixed, the convergence is approximately exponential, i.e., doubling $q$ approximately doubles the number of accurate digits of the computed integral. Complex geometry is automatically handled by the algorithm, including, e.g., multi-component domains, tunnels, and junctions arising from multiple polynomial level sets, as well as self-intersections, cusps, and other kinds of singularities. A variety of numerical experiments demonstrates the quadrature algorithm on two- and three-dimensional problems, including: randomly generated geometry involving multiple high-curvature pieces; challenging examples involving high degree singularities such as cusps; adaptation to simplex constraint cells in addition to hyperrectangular constraint cells; and boolean operations to compute integrals on overlapping domains.

math.NA↗