Logarithmic and Riesz Equilibrium for Multiple Sources on the Sphere --- the Exceptional Case

We consider the minimal discrete and continuous energy problems on the unit sphere $\mathbb{S}^d$ in the Euclidean space $\mathbb{R}^{d+1}$ in the presence of an external field due to finitely many localized charge distributions on $\mathbb{S}^d$, where the energy arises from the Riesz potential $1/r^s$ ($r$ is the Euclidean distance) for the critical Riesz parameter $s = d - 2$ if $d \geq 3$ and the logarithmic potential $\log(1/r)$ if $d = 2$. Individually, a localized charge distribution is either a point charge or assumed to be rotationally symmetric. The extremal measure solving the continuous external field problem for weak fields is shown to be the uniform measure on the sphere but restricted to the exterior of spherical caps surrounding the localized charge distributions. The radii are determined by the relative strengths of the generating charges. Furthermore, we show that the minimal energy points solving the related discrete external field problem are confined to this support. For $d-2\leq s<d$, we show that for point sources on the sphere, the equilibrium measure has support in the complement of the union of specified spherical caps about the sources. Numerical examples are provided to illustrate our results.Comment: 23 pages, 4 figure

Solving parabolic equations on the unit sphere via Laplace transforms and radial basis functions

We propose a method to construct numerical solutions of parabolic equations on the unit sphere. The time discretization uses Laplace transforms and quadrature. The spatial approximation of the solution employs radial basis functions restricted to the sphere. The method allows us to construct high accuracy numerical solutions in parallel. We establish $L_2$ error estimates for smooth and nonsmooth initial data, and describe some numerical experiments.Comment: 26 pages, 1 figur

Application of quasi-Monte Carlo methods to PDEs with random coefficients -- an overview and tutorial

This article provides a high-level overview of some recent works on the application of quasi-Monte Carlo (QMC) methods to PDEs with random coefficients. It is based on an in-depth survey of a similar title by the same authors, with an accompanying software package which is also briefly discussed here. Embedded in this article is a step-by-step tutorial of the required analysis for the setting known as the uniform case with first order QMC rules. The aim of this article is to provide an easy entry point for QMC experts wanting to start research in this direction and for PDE analysts and practitioners wanting to tap into contemporary QMC theory and methods.Comment: arXiv admin note: text overlap with arXiv:1606.0661

Matching Schur complement approximations for certain saddle-point systems

The solution of many practical problems described by mathematical models requires approximation methods that give rise to linear(ized) systems of equations, solving which will determine the desired approximation. This short contribution describes a particularly effective solution approach for a certain class of so-called saddle-point linear systems which arises in different contexts

Higher order quasi-monte carlo integration for holomorphic, parametric operator equations

We analyze the convergence of higher order quasi-Monte Carlo (QMC) quadratures of solution functionals to countably parametric, nonlinear operator equations with distributed uncertain parameters taking values in a separable Banach space X admitting an unconditional Schauder basis. Such equations arise in numerical uncertainty quantification with random field inputs. Unconditional bases of X render the random inputs and the solutions of the forward problem countably parametric, deterministic. We show that these parametric solutions belong to a class of weighted Bochner spaces of functions of countably many variables, with a particular structure of the QMC quadrature weights: up to a (problem-dependent, and possibly large) finite dimension, product weights can be used, and beyond this dimension, weighted spaces with so-called smoothness-driven product-and-order dependent (SPOD) weights recently introduced in [F. Y. Kuo, Ch. Schwab, I. H. Sloan, SIAM J. Numer. Anal., 50 (2012), pp. 3351-3374] can be used to describe the solution regularity. The regularity results in the present paper extend those in [J. Dick, F. Y. Kuo, Q. T. Le Gia, D. Nuyens, Ch. Schwab, SIAM J. Numer. Anal., 52 (2014), pp. 2676-2702] established for affine-parametric, linear operator families; they imply, in particular, efficient constructions of (sequences of) QMC quadrature methods there, which are applicable to these problem classes. We present a hybridized version of the fast component-by-component construction of a certain type of higher order digital net. We prove that this construction exploits the product nature of the QMC weights with linear scaling with respect to the integration dimension up to a possibly large, problem-dependent finite dimension, and the SPOD structure of the weights with quadratic scaling with respect to the weights beyond this dimension

Stability and preconditioning for a hybrid approximation on the sphere

This paper proposes a new preconditioning scheme for a linear system with a saddle-point structure arising from a hybrid approximation scheme on the sphere, an approximation scheme that combines (local) spherical radial basis functions and (global) spherical polynomials. In principle the resulting linear system can be preconditioned by the block-diagonal preconditioner of Murphy, Golub and Wathen. Making use of a recently derived inf-sup condition and the Brezzi stability and convergence theorem for this approximation scheme, we show that in this context the Schur complement in the above preconditioner is spectrally equivalent to a certain non-constant diagonal matrix. Numerical experiments with a non-uniform distribution of data points support the theoretically proved quality of the new preconditioner. © 2011 Springer-Verlag

Multilevel higher-order quasi-Monte Carlo Bayesian estimation

We propose and analyze deterministic multilevel (ML) approximations for Bayesian inversion of operator equations with uncertain distributed parameters, subject to additive Gaussian measurement data. The algorithms use a ML approach based on deterministic, higher-order quasi-Monte Carlo (HoQMC) quadrature for approximating the high-dimensional expectations, which arise in the Bayesian estimators, and a Petrov-Galerkin (PG) method for approximating the solution to the underlying partial differential equation (PDE). This extends the previous single-level (SL) approach from [J. Dick, R. N. Gantner, Q. T. Le Gia and Ch. Schwab, Higher order quasi-Monte Carlo integration for Bayesian estimation, Report 2016-13, Seminar for Applied Mathematics, ETH Zürich (in review)]. Compared to the SL approach, the present convergence analysis of the ML method requires stronger assumptions on holomorphy and regularity of the countably-parametric uncertainty-to-observation maps of the forward problem. As in the SL case and in the affine-parametric case analyzed in [J. Dick, F. Y. Kuo, Q. T. Le Gia and Ch. Schwab, Multi-level higher order QMC Galerkin discretization for affine parametric operator equations, SIAM J. Numer. Anal. 54 (2016) 2541-2568], we obtain sufficient conditions which allow us to achieve arbitrarily high, algebraic convergence rates in terms of work, which are independent of the dimension of the parameter space. The convergence rates are limited only by the spatial regularity of the forward problem, the discretization order achieved by the PG discretization, and by the sparsity of the uncertainty parametrization. We provide detailed numerical experiments for linear elliptic problems in two space dimensions, with s = 1024 parameters characterizing the uncertain input, confirming the theory and showing that the ML HoQMC algorithms can outperform, in terms of error versus computational work, both multilevel Monte Carlo (MLMC) methods and SL HoQMC methods, provided the parametric solution maps of the forward problems afford sufficient smoothness and sparsity of the high-dimensional parameter spaces

Fully discrete needlet approximation on the sphere

Spherical needlets are highly localized radial polynomials on the sphere Sd⊂Rd+1, d≥2, with centers at the nodes of a suitable cubature rule. The original semidiscrete spherical needlet approximation of Narcowich, Petrushev and Ward is not computable, in that the needlet coefficients depend on inner product integrals. In this work we approximate these integrals by a second quadrature rule with an appropriate degree of precision, to construct a fully discrete needlet approximation. We prove that the resulting approximation is equivalent to filtered hyperinterpolation, that is to a filtered Fourier–Laplace series partial sum with inner products replaced by appropriate cubature sums. It follows that the Lp-error of discrete needlet approximation of order J for 1≤p≤∞ and s>d/p has for a function f in the Sobolev space Wps(Sd) the optimal rate of convergence in the sense of optimal recovery, namely O(2−Js). Moreover, this is achieved with a filter function that is of smoothness class C⌊[Formula presented]⌋, in contrast to the usually assumed C∞. A numerical experiment for a class of functions in known Sobolev smoothness classes gives L2 errors for the fully discrete needlet approximation that are almost identical to those for the original semidiscrete needlet approximation. Another experiment uses needlets over the whole sphere for the lower levels together with high-level needlets with centers restricted to a local region. The resulting errors are reduced in the local region away from the boundary, indicating that local refinement in special regions is a promising strategy