Sparse spectral methods for integral equations and equilibrium measures

In this thesis, we introduce new numerical approaches to two important types of integral equation problems using sparse spectral methods. First, linear as well as nonlinear Volterra integral and integro-differential equations and second, power-law integral equations on d-dimensional balls involved in the solution of equilibrium measure problems. These methods are based on ultraspherical spectral methods and share key properties and advantages as a result of their joint starting point: By working in appropriately weighted orthogonal Jacobi polynomial bases, we obtain recursively generated banded operators allowing us to obtain high precision solutions at low computational cost. This thesis consists of three chapters in which the background of the above-mentioned problems and methods are respectively introduced in the context of their mathematical theory and applications, the necessary results to construct the operators and obtain solutions are proved and the method's applicability and efficiency are showcased by comparing them with current state-of-the-art approaches and analytic results where available. The first chapter gives a general scope introduction to sparse spectral methods using Jacobi polynomials in one and higher dimensions. The second chapter concerns the numerical solution of Volterra integral equations. The introduced method achieves exponential convergence and works for general kernels, a major advantage over comparable methods which are limited to convolution kernels. The third chapter introduces an approximately banded method to solve power law kernel equilibrium measures in arbitrary dimensional balls. This choice of domain is suggested by the radial symmetry of the problem and analytic results on the supports of the resulting measures. For our method, we obtain the crucial property of computational cost independent of the dimension of the domain, a major contrast to particle simulations which are the current standard approach to these problems and scale extremely poorly with both the dimension and the number of particles.Open Acces

Sharp mixed norm spherical restriction

Let $d\geq 2$ be an integer and let $2d/(d-1) < q \leq \infty$. In this paper we investigate the sharp form of the mixed norm Fourier extension inequality \begin{equation*} \big\|\widehat{f\sigma}\big\|_{L^q_{{\rm rad}}L^2_{{\rm ang}}(\mathbb{R}^d)} \leq {\bf C}_{d,q}\, \|f\|_{L^2(\mathbb{S}^{d-1},{\rm d}\sigma)}, \end{equation*} established by L. Vega in 1988. Letting $\mathcal{A}_d \subset (2d/(d-1), \infty]$ be the set of exponents for which the constant functions on $\mathbb{S}^{d-1}$ are the unique extremizers of this inequality, we show that: (i) $\mathcal{A}_d$ contains the even integers and $\infty$; (ii) $\mathcal{A}_d$ is an open set in the extended topology; (iii) $\mathcal{A}_d$ contains a neighborhood of infinity $(q_0(d), \infty]$ with $q_0(d) \leq \left(\tfrac{1}{2} + o(1)\right) d\log d$. In low dimensions we show that $q_0(2) \leq 6.76\,;\,q_0(3) \leq 5.45 \,;\, q_0(4) \leq 5.53 \,;\, q_0(5) \leq 6.07$. In particular, this breaks for the first time the even exponent barrier in sharp Fourier restriction theory. The crux of the matter in our approach is to establish a hierarchy between certain weighted norms of Bessel functions, a nontrivial question of independent interest within the theory of special functions.Comment: 21 page

Numerical Computing with Functions on the Sphere and Disk

A new low rank approximation method for computing with functions in polar and spherical geometries is developed. By synthesizing a classic procedure known as the double Fourier sphere (DFS) method with a structure-preserving variant of Gaussian elimination, approximants to functions on the sphere and disk can be constructed that (1) preserve the bi-periodicity of the sphere, (2) are smooth over the poles of the sphere (and origin of the disk), (3) allow for the use of FFT-based algorithms, and (4) are near-optimal in their underlying discretizations. This method is used to develop a suite of fast, scalable algorithms that exploit the low rank form of approximants to reduce many operations to essentially 1D procedures. This includes algorithms for differentiation, integration, and vector calculus. Combining these ideas with Fourier and ultraspherical spectral methods results in an optimal complexity solver for Poisson\u27s equation, which can be used to solve problems with 108 degrees of freedom in just under a minute on a laptop computer. All of these algorithms have been implemented and are publicly available in the open-source computing system called Chebfun [21]

Computing with functions in the ball

A collection of algorithms in object-oriented MATLAB is described for numerically computing with smooth functions defined on the unit ball in the Chebfun software. Functions are numerically and adaptively resolved to essentially machine precision by using a three-dimensional analogue of the double Fourier sphere method to form "ballfun" objects. Operations such as function evaluation, differentiation, integration, fast rotation by an Euler angle, and a Helmholtz solver are designed. Our algorithms are particularly efficient for vector calculus operations, and we describe how to compute the poloidal-toroidal and Helmholtz--Hodge decomposition of a vector field defined on the ball.Comment: 23 pages, 9 figure

Multidomain spectral method for the Gauss hypergeometric function

We present a multidomain spectral approach for Fuchsian ordinary differential equations in the particular case of the hypergeometric equation. Our hybrid approach uses Frobenius’ method and Moebius transformations in the vicinity of each of the singular points of the hypergeometric equation, which leads to a natural decomposition of the real axis into domains. In each domain, solutions to the hypergeometric equation are constructed via the well-conditioned ultraspherical spectral method. The solutions are matched at the domain boundaries to lead to a solution which is analytic on the whole compactified real line R∪∞ , except for the singular points and cuts of the Riemann surface on which the solution is defined. The solution is further extended to the whole Riemann sphere by using the same approach for ellipses enclosing the singularities. The hypergeometric equation is solved on the ellipses with the boundary data from the real axis. This solution is continued as a harmonic function to the interior of the disk by solving the Laplace equation in polar coordinates with an optimal complexity Fourier–ultraspherical spectral method. In cases where logarithms appear in the solution, a hybrid approach involving an analytical treatment of the logarithmic terms is applied. We show for several examples that machine precision can be reached for a wide class of parameters, but also discuss almost degenerate cases where this is not possible

Integrals of Bernstein polynomials: An application for the solution of high even-order differential equations

AbstractA new explicit formula for the integrals of Bernstein polynomials of any degree for any order in terms of Bernstein polynomials themselves is derived. A fast and accurate algorithm is developed for the solution of high even-order boundary value problems (BVPs) with two point boundary conditions but by considering their integrated forms. The Bernstein–Petrov–Galerkin method (BPG) is applied to construct the numerical solution for such problems. The method is then tested on examples and compared with other methods. It is shown that the BPG yields better results