arXiv subjects

Michael O'Neil

Publications and source records attributed to Michael O'Neil.

At least 19 recordsLinked to original sources

arXiv2026

Linear-cost Polyharmonic Spline Interpolation of Arbitrary Degree

We introduce a simple and performant approach for rapidly and accurately performing polyharmonic spline (PHS) interpolation using a combination of the fast multipole method (FMM) from computational electrostatics and the Vecchia approximation from the Gaussian process and sparse approximate inverse literatures. Using basic properties about Hadamard products and low-rank matrices, we demonstrate that an FMM with two kernels, the logarithmic and distance kernels, results in fast PHS interpolation for all orders. Furthermore, we demonstrate the exceptional performance of sparse inverse approximation methods with the Mat\'ern covariance model for preconditioning. Combined with careful management of disallowed subspaces, we describe a procedure for obtaining prediction weights using preconditioned conjugate gradient that converges in less than $15$ iterations, even for problem sizes with over one million points. As a result, thin-plate spline interpolation---a particularly popular method that does not require parameter tuning---that matches the fully dense $\mathcal{O}(n^3)$ computation in accuracy can be done at the cost of approximately $50-60$ FMMs. A high-performance software library for odd-order PHS interpolation in two dimensions is made available as a companion to this work.

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arXiv2026

A Butterfly-Accelerated Manifold Harmonic Transform

The eigenfunctions of the Laplacian are a natural basis of functions for many tasks in computational mathematics. On the circle and sphere, the eigenfunctions are given by complex periodic exponentials and spherical harmonics, respectively, and much work has been done to develop fast algorithms for analyzing and synthesizing data in these bases. In this work, we generalize these special-case transforms to Laplace-Beltrami eigenfunctions of arbitrary surfaces, referred to as manifold harmonics. The resulting fast algorithm for computing linear combinations of the manifold harmonics is based on a butterfly factorization, which hierarchically compresses the transform matrix by constructing nested low-rank approximations of carefully selected submatrices. Several numerical examples are provided which demonstrate the speedups and reduction in memory requirements achieved by our algorithm for a variety of geometries, discretizations, and applications. In addition, a detailed analysis of the algorithm is given in the case that the underlying manifold is the flat periodic square.

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arXiv2025

Fast Direct Solvers

This survey describes a class of methods known as "fast direct solvers". These algorithms address the problem of solving a system of linear equations $\boldsymbol{Ax}=\boldsymbol{b}$ arising from the discretization of either an elliptic PDE or of an associated integral equation. The matrix $\boldsymbol{A}$ will be sparse when the PDE is discretized directly, and dense when an integral equation formulation is used. In either case, industry practice for large scale problems has for decades been to use iterative solvers such as multigrid, GMRES, or conjugate gradients. A direct solver, in contrast, builds an approximation to the inverse of $\boldsymbol{A}$, or alternatively, an easily invertible factorization (e.g. LU or Cholesky). A major development in numerical analysis in the last couple of decades has been the emergence of algorithms for constructing such factorizations or performing such inversions in linear or close to linear time. Such methods must necessarily exploit that the matrix $\boldsymbol{A}^{-1}$ is "data-sparse", typically in the sense that it can be tessellated into blocks that have low numerical rank. This survey provides a unifying context to both sparse and dense fast direct solvers, introduces key concepts with a minimum of notational overhead, and provides guidance to help a user determine the best method to use for a given application.

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arXiv2025

Fast adaptive high-order integral equation methods for electromagnetic scattering from smooth perfect electric conductors

Many integral equation-based methods are available for problems of time-harmonic electromagnetic scattering from perfect electric conductors. Among the many challenges that arise in such calculations are the avoidance of spurious resonances, robustness of the method to scatterers of non-trivial topology or multiscale features, stability under mesh refinement, ease of implementation with high-order basis functions, and behavior in the static limit. Since three-dimensional scattering is a challenging, large-scale problem, many of these issues have been historically difficult to investigate. It is only with the advent of fast algorithms for matrix-vector multiplies coupled with modern iterative methods that a careful study of these issues can be carried out effectively. Our focus here is on comparing the behavior of several integral equation formulations with regard to the issues noted above, namely: the well-known, standard electric, magnetic, and combined field integral equations with standard RWG basis functions, and the more modern non-resonant charge-current and decoupled potential integral equation. Numerical results are provided to demonstrate the behavior of each of these schemes. Furthermore, we provide some analytical properties and comparisons with the electric charge-current integral equation and the augmented regularized combined source integral equation.

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arXiv2024

A Nonuniform Fast Hankel Transform

We describe a fast algorithm for computing discrete Hankel transforms of moderate orders from $n$ nonuniform points to $m$ nonuniform frequencies in $O((m+n)\log\min(n,m))$ operations. Our approach combines local and asymptotic Bessel function expansions with nonuniform fast Fourier transforms. The order of each expansion is adjusted automatically according to error analysis to obtain any desired precision $\varepsilon$. Several numerical examples are provided which demonstrate the speed and accuracy of the algorithm in multiple regimes and applications.

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arXiv2024

On the construction of scattering matrices for irregular or elongated enclosures using Green's representation formula

Multiple scattering methods are widely used to reduce the computational complexity of acoustic or electromagnetic scattering problems when waves propagate through media containing many identical inclusions. Historically, this numerical technique has been limited to situations in which the inclusions (particles) can be covered by nonoverlapping disks in two dimensions or spheres in three dimensions. This allows for the use of separation of variables in cylindrical or spherical coordinates to represent the solution to the governing partial differential equation. Here, we provide a more flexible approach, applicable to a much larger class of geometries. We use a Green's representation formula and the associated layer potentials to construct incoming and outgoing solutions on rectangular enclosures. The performance and flexibility of the resulting scattering operator formulation in two-dimensions is demonstrated via several numerical examples for multi-particle scattering in free space as well as in layered media. The mathematical formalism extends directly to the three dimensional case as well, and can easily be coupled with several commercial numerical PDE software packages.

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arXiv2024

A parametrix method for elliptic surface PDEs

Elliptic problems along smooth surfaces embedded in three dimensions occur in thin-membrane mechanics, electromagnetics (harmonic vector fields), and computational geometry. In this work, we present a parametrix-based integral equation method applicable to several forms of variable coefficient surface elliptic problems. Via the use of an approximate Green's function, the surface PDEs are transformed into well-conditioned integral equations. We demonstrate high-order numerical examples of this method applied to problems on general surfaces using a variant of the fast multipole method based on smooth interpolation properties of the kernel. Lastly, we discuss extensions of the method to surfaces with boundaries.

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arXiv2023

An interface formulation of the Laplace-Beltrami problem on piecewise smooth surfaces

The Laplace-Beltrami problem on closed surfaces embedded in three dimensions arises in many areas of physics, including molecular dynamics (surface diffusion), electromagnetics (harmonic vector fields), and fluid dynamics (vesicle deformation). In particular, the Hodge decomposition of vector fields tangent to a surface can be computed by solving a sequence of Laplace-Beltrami problems. Such decompositions are very important in magnetostatic calculations and in various plasma and fluid flow problems. In this work we develop $L^2$-invertibility theory for the Laplace-Beltrami operator on piecewise smooth surfaces, extending earlier weak formulations and integral equation approaches on smooth surfaces. Furthermore, we reformulate the weak form of the problem as an interface problem with continuity conditions across edges of adjacent piecewise smooth panels of the surface. We then provide high-order numerical examples along surfaces of revolution to support our analysis, and discuss numerical extensions to general surfaces embedded in three dimensions.

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arXiv2023

FMM-accelerated solvers for the Laplace-Beltrami problem on complex surfaces in three dimensions

The Laplace-Beltrami problem on closed surfaces embedded in three dimensions arises in many areas of physics, including molecular dynamics (surface diffusion), electromagnetics (harmonic vector fields), and fluid dynamics (vesicle deformation). Using classical potential theory, the Laplace-Beltrami operator can be pre-/post-conditioned with an integral operator whose kernel is translation invariant, resulting in well-conditioned Fredholm integral equations of the second-kind. These equations have the standard~$1/r$ kernel from potential theory, and therefore the equations can be solved rapidly and accurately using a combination of fast multipole methods (FMMs) and high-order quadrature corrections. In this work we detail such a scheme, presenting two alternative integral formulations of the Laplace-Beltrami problem, each of whose solution can be obtained via FMM acceleration. We then present several applications of the solvers, focusing on the computation of what are known as harmonic vector fields, relevant for many applications in electromagnetics. A battery of numerical results are presented for each application, detailing the performance of the solver in various geometries.

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arXiv2023

FMM-LU: A fast direct solver for multiscale boundary integral equations in three dimensions

We present a fast direct solver for boundary integral equations on complex surfaces in three dimensions using an extension of the recently introduced recursive strong skeletonization scheme. For problems that are not highly oscillatory, our algorithm computes an ${LU}$-like hierarchical factorization of the dense system matrix, permitting application of the inverse in $\mathcal O(n)$ time, where $n$ is the number of unknowns on the surface. The factorization itself also scales linearly with the system size, albeit with a somewhat larger constant. The scheme is built on a level-restricted adaptive octree data structure, and therefore it is compatible with highly nonuniform discretizations. Furthermore, the scheme is coupled with high-order accurate locally-corrected Nyström quadrature methods to integrate the singular and weakly-singular Green's functions used in the integral representations. Our method has immediate applications to a variety of problems in computational physics. We concentrate here on studying its performance in acoustic scattering (governed by the Helmholtz equation) at low to moderate frequencies, and provide rigorous justification for compression of submatrices via proxy surfaces.

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arXiv2022

Efficient reduced-rank methods for Gaussian processes with eigenfunction expansions

In this work we introduce a reduced-rank algorithm for Gaussian process regression. Our numerical scheme converts a Gaussian process on a user-specified interval to its Karhunen-Loève expansion, the $L^2$-optimal reduced-rank representation. Numerical evaluation of the Karhunen-Loève expansion is performed once during precomputation and involves computing a numerical eigendecomposition of an integral operator whose kernel is the covariance function of the Gaussian process. The Karhunen-Loève expansion is independent of observed data and depends only on the covariance kernel and the size of the interval on which the Gaussian process is defined. The scheme of this paper does not require translation invariance of the covariance kernel. We also introduce a class of fast algorithms for Bayesian fitting of hyperparameters, and demonstrate the performance of our algorithms with numerical experiments in one and two dimensions. Extensions to higher dimensions are mathematically straightforward but suffer from the standard curses of high dimensions.

stat.CO↗

arXiv2021

Accurate and efficient numerical calculation of stable densities via optimized quadrature and asymptotics

Stable distributions are an important class of infinitely-divisible probability distributions, of which two special cases are the Cauchy distribution and the normal distribution. Aside from a few special cases, the density function for stable distributions has no known analytic form, and is expressible only through the variate's characteristic function or other integral forms. In this paper we present numerical schemes for evaluating the density function for stable distributions, its gradient, and distribution function in various parameter regimes of interest, some of which had no pre-existing efficient method for their computation. The novel evaluation schemes consist of optimized generalized Gaussian quadrature rules for integral representations of the density function, complemented by various asymptotic expansions near various values of the shape and argument parameters. We report several numerical examples illustrating the efficiency of our methods. The resulting code has been made available online.

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arXiv2021

Fast multipole methods for evaluation of layer potentials with locally-corrected quadratures

While fast multipole methods (FMMs) are in widespread use for the rapid evaluation of potential fields governed by the Laplace, Helmholtz, Maxwell or Stokes equations, their coupling to high-order quadratures for evaluating layer potentials is still an area of active research. In three dimensions, a number of issues need to be addressed, including the specification of the surface as the union of high-order patches, the incorporation of accurate quadrature rules for integrating singular or weakly singular Green's functions on such patches, and their coupling to the oct-tree data structures on which the FMM separates near and far field interactions. Although the latter is straightforward for point distributions, the near field for a patch is determined by its physical dimensions, not the distribution of discretization points on the surface. Here, we present a general framework for efficiently coupling locally corrected quadratures with FMMs, relying primarily on what are called generalized Gaussian quadratures rules, supplemented by adaptive integration. The approach, however, is quite general and easily applicable to other schemes, such as Quadrature by Expansion (QBX). We also introduce a number of accelerations to reduce the cost of quadrature generation itself, and present several numerical examples of acoustic scattering that demonstrate the accuracy, robustness, and computational efficiency of the scheme. On a single core of an Intel i5 2.3GHz processor, a Fortran implementation of the scheme can generate near field quadrature corrections for between 1000 and 10,000 points per second, depending on the order of accuracy and the desired precision. A Fortran implementation of the algorithm described in this work is available at https://gitlab.com/fastalgorithms/fmm3dbie.

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arXiv2019

Efficient high-order singular quadrature schemes in magnetic fusion

Several problems in magnetically confined fusion, such as the computation of exterior vacuum fields or the decomposition of the total magnetic field into separate contributions from the plasma and the external sources, are best formulated in terms of integral equation expressions. Based on Biot-Savart-like formulae, these integrals contain singular integrands. The regularization method commonly used to address the computation of various singular surface integrals along general toroidal surfaces is low-order accurate, and therefore requires a dense computational mesh in order to obtain sufficient accuracy. In this work, we present a fast, high-order quadrature scheme for the efficient computation of these integrals. Several numerical examples are provided demonstrating the computational efficiency and the high-order accurate convergence. A corresponding code for use in the community has been publicly released.

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arXiv2019

A fast boundary integral method for high-order multiscale mesh generation

In this work we present an algorithm to construct an infinitely differentiable smooth surface from an input consisting of a (rectilinear) triangulation of a surface of arbitrary shape. The original surface can have non-trivial genus and multiscale features, and our algorithm has computational complexity which is linear in the number of input triangles. We use a smoothing kernel to define a function $Φ$ whose level set defines the surface of interest. Charts are subsequently generated as maps from the original user-specified triangles to $\mathbb R^3$. The degree of smoothness is controlled locally by the kernel to be commensurate with the fineness of the input triangulation. The expression for~$Φ$ can be transformed into a boundary integral, whose evaluation can be accelerated using a fast multipole method. We demonstrate the effectiveness and cost of the algorithm with polyhedral and quadratic skeleton surfaces obtained from CAD and meshing software.

math.NA↗

arXiv2019

Taylor States in Stellarators: A Fast High-order Boundary Integral Solver

We present a boundary integral equation solver for computing Taylor relaxed states in non-axisymmetric solid and shell-like toroidal geometries. The computation of Taylor states in these geometries is a key element for the calculation of stepped pressure stellarator equilibria. The integral representation of the magnetic field in this work is based on the generalized Debye source formulation, and results in a well-conditioned second-kind boundary integral equation. The integral equation solver is based on a spectral discretization of the geometry and unknowns, and the computation of the associated weakly-singular integrals is performed with high-order quadrature based on a partition of unity. The resulting scheme for applying the integral operator is then coupled with an iterative solver and suitable preconditioners. Several numerical examples are provided to demonstrate the accuracy and efficiency of our method, and a direct comparison with the leading code in the field is reported.

math.NA↗

arXiv2018

An FFT-accelerated direct solver for electromagnetic scattering from penetrable axisymmetric objects

Fast, high-order accurate algorithms for electromagnetic scattering from axisymmetric objects are of great importance when modeling physical phenomena in optics, materials science (e.g. meta-materials), and many other fields of applied science. In this paper, we develop an FFT-accelerated separation of variables solver that can be used to efficiently invert integral equation formulations of Maxwell's equations for scattering from axisymmetric penetrable (dielectric) bodies. Using a standard variant of M\"uller's integral representation of the fields, our numerical solver rapidly and directly inverts the resulting second-kind integral equation. In particular, the algorithm of this work (1) rapidly evaluates the modal Green's functions, and their derivatives, via kernel splitting and the use of novel recursion formulas, (2) discretizes the underlying integral equation using generalized Gaussian quadratures on adaptive meshes, and (3) is applicable to geometries containing edges. Several numerical examples are provided to demonstrate the efficiency and accuracy of the aforementioned algorithm in various geometries.

math.NA↗

arXiv2017

A high-order wideband direct solver for electromagnetic scattering from bodies of revolution

The generalized Debye source representation of time-harmonic electromagnetic fields yields well-conditioned second-kind integral equations for a variety of boundary value problems, including the problems of scattering from perfect electric conductors and dielectric bodies. Furthermore, these representations, and resulting integral equations, are fully stable in the static limit as $\omega \to 0$ in multiply connected geometries. In this paper, we present the first high-order accurate solver based on this representation for bodies of revolution. The resulting solver uses a Nystr\"om discretization of a one-dimensional generating curve and high-order integral equation methods for applying and inverting surface differentials. The accuracy and speed of the solvers are demonstrated in several numerical examples.

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