Michael O'Neil

Fast Direct Methods for Gaussian Processes

A number of problems in probability and statistics can be addressed using the multivariate normal (Gaussian) distribution. In the one-dimensional case, computing the probability for a given mean and variance simply requires the evaluation of the corresponding Gaussian density. In the n-dimensional setting, however, it requires the inversion of an n x n covariance matrix, C, as well as the evaluation of its determinant, det(C). In many cases, such as regression using Gaussian processes, the covariance matrix is of the form C = σ2I + K, where K is computed using a specified covariance kernel which depends on the data and additional parameters (hyperparameters). The matrix C is typically dense, causing standard direct methods for inversion and determinant evaluation to require O(n3) work. This cost is prohibitive for large-scale modeling. Here, we show that for the most commonly used covariance functions, the matrix C can be hierarchically factored into a product of block low-rank updates of the identity matrix, yielding an O(n log2 n) algorithm for inversion. More importantly, we show that this factorization enables the evaluation of the determinant det(C), permitting the direct calculation of probabilities in high dimensions under fairly broad assumptions on the kernel defining K. Our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single CPU core. The combination of nearly optimal scaling in terms of problem size with high-performance computing resources will permit the modeling of previously intractable problems. We illustrate the performance of the scheme on standard covariance kernels.

A fast, high-order solver for the Grad-Shafranov equation

Abstract We present a new fast solver to calculate fixed-boundary plasma equilibria in toroidally axisymmetric geometries. By combining conformal mapping with Fourier and integral equation methods on the unit disk, we show that high-order accuracy can be achieved for the solution of the equilibrium equation and its first and second derivatives. Smooth arbitrary plasma cross-sections as well as arbitrary pressure and poloidal current profiles are used as initial data for the solver. Equilibria with large Shafranov shifts can be computed without difficulty. Spectral convergence is demonstrated by comparing the numerical solution with a known exact analytic solution. A fusion-relevant example of an equilibrium with a pressure pedestal is also presented.

Fast algorithms for Quadrature by Expansion I: Globally valid expansions

Abstract The use of integral equation methods for the efficient numerical solution of PDE boundary value problems requires two main tools: quadrature rules for the evaluation of layer potential integral operators with singular kernels, and fast algorithms for solving the resulting dense linear systems. Classically, these tools were developed separately. In this work, we present a unified numerical scheme based on coupling Quadrature by Expansion , a recent quadrature method, to a customized Fast Multipole Method (FMM) for the Helmholtz equation in two dimensions. The method allows the evaluation of layer potentials in linear-time complexity, anywhere in space, with a uniform, user-chosen level of accuracy as a black-box computational method. Providing this capability requires geometric and algorithmic considerations beyond the needs of standard FMMs as well as careful consideration of the accuracy of multipole translations. We illustrate the speed and accuracy of our method with various numerical examples.

Slow Passage through Resonance in Mathieu's Equation:

We investigate slow passage through the 2:1 resonance tongue in Mathieus equation. Using numerical integration, we find that amplification or de-amplification can occur. The amount of amplification (or de-amplification) depends on the speed of travel through the tongue and the initial conditions. We use the method of multiple scales to obtain a slow flow approximation. The Wentzel-Kramers-Brillouin (WKB) method is then applied to the slow flow equations to obtain an analytic approximation.

A Consistency Condition for the Vector Potential in Multiply-Connected Domains

A classical problem in electromagnetics concerns the representation of the electric and magnetic fields in the low-frequency or static regime, where topology plays a fundamental role. For multiply-connected conductors, at zero frequency, the standard boundary conditions on the tangential components of the magnetic field do not uniquely determine the vector potential. We describe a (gauge-invariant) consistency condition that overcomes this nonuniqueness and resolves a long-standing difficulty in inverting the magnetic field integral equation.

Smoothed Corners and Scattered Waves

We introduce an arbitrary order, computationally efficient method to smooth corners on curves in the plane, as well as edges and vertices on surfaces in

Exact axisymmetric Taylor states for shaped plasmas

\mathbb R^3

Robust integral formulations for electromagnetic scattering from three-dimensional cavities

. The method is local, only modifying the original surface in a neighborhood of the geometric singularity, and preserves desirable features like convexity and symmetry. The smoothness of the final surface is an explicit parameter in the method, and the bandlimit of the smoothed surface is proportional to its smoothness. Several numerical examples are provided in the context of acoustic scattering. In particular, we compare scattered fields from smoothed geometries in two dimensions with those from polygonal domains. We observe that significant reductions in computational cost can be obtained if merely approximate solutions are desired in the near- or far-field. Provided that it is sub-wavelength, the error of the scattered field is proportional to the size of the geometry that is modified.

An integral equation-based numerical solver for Taylor states in toroidal geometries

We present a general construction for exact analytic Taylor states in axisymmetric toroidal geometries. In this construction, the Taylor equilibria are fully determined by specifying the aspect ratio, elongation, and triangularity of the desired plasma geometry. For equilibria with a magnetic X-point, the location of the X-point must also be specified. The flexibility and simplicity of these solutions make them useful for verifying the accuracy of numerical solvers and for theoretical studies of Taylor states in laboratory experiments.

Quadrature by expansion

Abstract Scattering from large, open cavity structures is of importance in a variety of electromagnetic applications. In this paper, we propose a new well conditioned integral equation for scattering from general open cavities embedded in an infinite, perfectly conducting half-space. The integral representation permits the stable evaluation of both the electric and magnetic field, even in the low-frequency regime, using the continuity equation in a post-processing step. We establish existence and uniqueness results, and demonstrate the performance of the scheme in the cavity-of-revolution case. High-order accuracy is obtained using a Nystrom discretization with generalized Gaussian quadratures.