Showing 1–18 of 18 results
Johann S. Brauchart, Peter J. Grabner
We extend the notion of hyperuniformity to the projective spaces $\mathbb{RP}^{d-1}$, $\mathbb{CP}^{d-1}$, $\mathbb{HP}^{d-1}$, and $\mathbb{OP}^2$. We show that hyperuniformity implies uniform distribution and present examples of deterministic point sets as well as point processes which exhibit hyperuniform behaviour.
Johann S. Brauchart, Josef Dick
Stolarsky [Proc. Amer. Math. Soc. 41 (1973), 575--582] showed a beautiful relation that balances the sums of distances of points on the unit sphere and their spherical cap $\mathbb{L}_2$-discrepancy to give the distance integral of the uniform measure on the sphere a potential-theoretical quantity (Bj{ö}rck [Ark. Mat. 3 (1956), 255--269]). Read differently it expresses the worst-case numerical integration error for functions from the unit ball in a certain Hilbert space setting in terms of the $\mathbb{L}_2$-discrepancy and vice versa (first author and Womersley [Preprint]). In this note we give a simple proof of the invariance principle using reproducing kernel Hilbert spaces.
Johann S. Brauchart, Edward B. Saff, Ian H. Sloan, Rob S. Womersley
We study equal weight numerical integration, or Quasi Monte Carlo (QMC) rules, for functions in a Sobolev space $H^s(S^d)$ with smoothness parameter $s>d/2$ defined over the unit sphere $S^d$ in $R^{d+1}$. Focusing on $N$-point sets that achieve optimal order QMC error bounds (as is the case for efficient spherical designs), we are led to introduce the concept of QMC designs: these are sequences of $N$-point node sets $X_N$ on $S^d$ such that the worst-case error of the corresponding QMC rules satisfy a bound of order $O(N^{-s/d})$ as $N\to\infty$ with an implied constant that depends on the $H^s(S^d)$-norm. We provide methods for generation and numerical testing of QMC designs. As a consequence of a recent result of Bondarenko et al. on the existence of spherical designs with appropriate number of points, we show that minimizers of the $N$-point energy for the reproducing kernel for $H^s(S^d)$, $s>d/2$, form a sequence of QMC designs for $H^s(S^d)$. Furthermore, without appealing to the Bondarenko et al. result, we prove that point sets that maximize the sum of suitable powers of the Euclidean distance between pairs of points form a sequence of QMC designs for $H^s(S^d)$ with $s\in(d/2,d/2+1)$. Numerical experiments suggest that many familiar sequences of point sets on the sphere (equal area, spiral, minimal [Coulomb or log.] energy, and Fekete points) are QMC designs for appropriate values of $s$. For comparison purposes we show that sets of random points that are independently and uniformly distributed on the sphere do not constitute QMC designs for any $s>d/2$. If $(X_N)$ is a sequence of QMC designs for $H^s(S^d)$, we prove that it is also a sequence of QMC designs for $\mathbb{H}^{s'}(S^d)$ for all $s'\in(d/2,s)$. This leads to the question of determining the supremum of such $s$, for which we provide estimates based on computations for the aforementioned sequences.
Johann S. Brauchart, Josef Dick
We study numerical integration on the unit sphere $\mathbb{S}^2 \subset \mathbb{R}^3$ using equal weight quadrature rules, where the weights are such that constant functions are integrated exactly. The quadrature points are constructed by lifting a $(0,m,2)$-net given in the unit square $[0,1]^2$ to the sphere $\mathbb{S}^2$ by means of an area preserving map. A similar approach has previously been suggested by Cui and Freeden [SIAM J. Sci. Comput. 18 (1997), no. 2]. We prove three results. The first one is that the construction is (almost) optimal with respect to discrepancies based on spherical rectangles. Further we prove that the point set is asymptotically uniformly distributed on $\mathbb{S}^2$. And finally, we prove an upper bound on the spherical cap $L_2$-discrepancy of order $N^{-1/2} (\log N)^{1/2}$ (where $N$ denotes the number of points). This slightly improves upon the bound on the spherical cap $L_2$-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Comm. Pure Appl. Math. 39 (1986), 149--186]. Numerical results suggest that the $(0,m,2)$-nets lifted to the sphere $\mathbb{S}^2$ have spherical cap $L_2$-discrepancy converging with the optimal order of $N^{-3/4}$.
Johann S. Brauchart, Peter J. Grabner
This survey discusses recent developments in the context of spherical designs and minimal energy point configurations on spheres. The recent solution of the long standing problem of the existence of spherical $t$-designs on $\mathbb{S}^d$ with $\mathcal{O}(t^d)$ number of points by A. Bondarenko, D. Radchenko, and M. Viazovska attracted new interest to this subject. Secondly, D. P. Hardin and E. B. Saff proved that point sets minimising the discrete Riesz energy on $\mathbb{S}^d$ in the hypersingular case are asymptotically uniformly distributed. Both results are of great relevance to the problem of describing the quality of point distributions on $\mathbb{S}^d$, as well as finding point sets, which exhibit good distribution behaviour with respect to various quality measures.
Johann S. Brauchart, Josef Dick, Edward B. Saff, Ian H. Sloan, Yu Guang Wang, Robert S. Womersley
We prove that the covering radius of an $N$-point subset $X_N$ of the unit sphere $S^d \subset R^{d+1}$ is bounded above by a power of the worst-case error for equal weight cubature $\frac{1}{N}\sum_{\mathbf{x} \in X_N}f(\mathbf{x}) \approx \int_{S^d} f \, \mathrm{d} σ_d$ for functions in the Sobolev space $\mathbb{W}_p^s(S^d)$, where $σ_d$ denotes normalized area measure on $S^d.$ These bounds are close to optimal when $s$ is close to $d/p$. Our study of the worst-case error along with results of Brandolini et al. motivate the definition of Quasi-Monte Carlo (QMC) design sequences for $\mathbb{W}_p^s(S^d)$, which have previously been introduced only in the Hilbert space setting $p=2$. We say that a sequence $(X_N)$ of $N$-point configurations is a QMC-design sequence for $\mathbb{W}_p^s(S^d)$ with $s > d/p$ provided the worst-case equal weight cubature error for $X_N$ has order $N^{-s/d}$ as $N \to \infty$, a property that holds, in particular, for a sequence of spherical $t$-designs in which each design has order $t^d$ points. For the case $p = 1$, we deduce that any QMC-design sequence $(X_N)$ for $\mathbb{W}_1^s(S^d)$ with $s > d$ has the optimal covering property; i.e., the covering radius of $X_N$ has order $N^{-1/d}$ as $N \to \infty$. A significant portion of our effort is devoted to the formulation of the worst-case error in terms of a Bessel kernel, and showing that this kernel satisfies a Bernstein type inequality involving the mesh ratio of $X_N$. As a consequence we prove that any QMC-design sequence for $\mathbb{W}_p^s(S^d)$ is also a QMC-design sequence for $\mathbb{W}_{p^\prime}^s(S^d)$ for all $1 \leq p < p^\prime \leq \infty$ and, furthermore, if $(X_N)$ is a quasi-uniform QMC-design sequence for $\mathbb{W}_p^s(S^d)$, then it is also a QMC-design sequence for $\mathbb{W}_p^{s^\prime}(S^d)$ for all $s > s^\prime > d/p$.
Johann S. Brauchart, Peter J. Grabner
We study integrals of the form \begin{equation*} \int_{-1}^1(C_n^{(λ)}(x))^2(1-x)^α(1+x)^β\, dx, \end{equation*} where $C_n^{(λ)}$ denotes the Gegenbauer-polynomial of index $λ>0$ and $α,β>-1$. We give exact formulas for the integrals and their generating functions, and obtain asymptotic formulas as $n\to\infty$.
Johann S. Brauchart, Peter J. Grabner, Ian H. Sloan, Robert S. Womersley
Spherical needlets were introduced by Narcowich, Petrushev, and Ward to provide a multiresolution sequence of polynomial approximations to functions on the sphere. The needlet construction makes use of integration rules that are exact for polynomials up to a given degree. The aim of the present paper is to relax the exactness of the integration rules by replacing them with QMC designs as introduced by Brauchart, Saff, Sloan, and Womersley (2014). Such integration rules (generalised here by allowing non-equal cubature weights) provide the same asymptotic order of convergence as exact rules for Sobolev spaces $\mathbb{H}^s$, but are easier to obtain numerically. With such rules we construct ``generalised needlets''. The paper provides an error analysis that allows the replacement of the original needlets by generalised needlets, and more generally, analyses a hybrid scheme in which the needlets for the lower levels are of the traditional kind, whereas the new generalised needlets are used for some number of higher levels. Numerical experiments complete the paper.
Dmitriy Bilyk, Johann S. Brauchart
We start by providing a very simple and elementary new proof of the classical bound due to J. Beck which states that the spherical cap $\mathbb{L}_2$-discrepancy of any $N$ points on the unit sphere $\mathbb S^d$ in $\mathbb{R}^{d+1}$, $d\geq2$, is at least of the order $N^{-\frac12-\frac{1}{2d}}$. The argument used in this proof leads us to many further new results: estimates of the discrepancy in terms of various geometric quantities, an easy proof of {point-independent} upper estimates for the sum of positive powers of Euclidean distances between points on the sphere, lower bounds for the discrepancy of rectifiable curves and sets of arbitrary Hausdorff dimension. Moreover, refinements of the proof also allow us to obtain explicit values of the constants in the lower discrepancy bound on $\mathbb{S}^d$. The value of the obtained asymptotic constant falls within $3\%$ of the conjectured optimal constant on $\mathbb S^2$ (and within up to $7\%$ on $\mathbb S^4$, $\mathbb S^8$, $\mathbb S^{24}$).
Johann S. Brauchart, Alexander B. Reznikov, Edward B. Saff, Ian H. Sloan, Yu Guang Wang, Robert S. Womersley
Geometric properties of $N$ random points distributed independently and uniformly on the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$ with respect to surface area measure are obtained and several related conjectures are posed. In particular, we derive asymptotics (as $N \to \infty$) for the expected moments of the radii of spherical caps associated with the facets of the convex hull of $N$ random points on $\mathbb{S}^{d}$. We provide conjectures for the asymptotic distribution of the scaled radii of these spherical caps and the expected value of the largest of these radii (the covering radius). Numerical evidence is included to support these conjectures. Furthermore, utilizing the extreme law for pairwise angles of Cai et al., we derive precise asymptotics for the expected separation of random points on $\mathbb{S}^{d}$.
Rachele Nerattini, Johann S. Brauchart, Michael K. -H. Kiessling
Smale's 7-th problem concerns N-point configurations on the 2-dim sphere which minimize the logarithmic pair-energy V_0(r) = -ln r averaged over the pairs in a configuration; here, r is the chordal distance between the points forming a pair. More generally, V_0(r) may be replaced by the standardized Riesz pair-energy V_s(r)= (r^{-s} -1)/s, which becomes - ln r in the limit s to 0, and the sphere may be replaced by other compact manifolds. This paper inquires into the concavity of the map from the integers N>1 into the minimal average standardized Riesz pair-energies v_s(N) of the N-point configurations on the 2-sphere for various real s. It is known that v_s(N) is strictly increasing for each real s, and for s<2 also bounded above, hence "overall concave." It is (easily) proved that v_{-2}(N) is even locally strictly concave, and that so is v_s(2n) for s<-2. By analyzing computer-experimental data of putatively minimal average Riesz pair-energies v_s^x(N) for s in {-1,0,1,2,3} and N in {2,...,200}, it is found that {v}_{-1}^x(N) is locally strictly concave, while v_s^x(N) is not always locally strictly concave for s in {0,1,2,3}: concavity defects occur whenever N in C^{x}_+(s) (an s-specific empirical set of integers). It is found that the empirical map C^{x}_+(s), with s in {-2,-1,0,1,2,3}, is set-theoretically increasing; moreover, the percentage of odd numbers in C^{x}_+(s), s in {0,1,2,3}, is found to increase with s. The integers in C^{x}_+(0) are few and far between, forming a curious sequence of numbers, reminiscent of the "magic numbers" in nuclear physics. It is conjectured that the "magic numbers" in Smale's 7-th problem are associated with optimally symmetric optimal-energy configurations.
Johann S. Brauchart, Peter D. Dragnev, Edward B. Saff
For a positively charged insulated d-dimensional sphere we investigate how the distribution of this charge is affected by proximity to a nearby positive or negative point charge when the system is governed by a Riesz s-potential 1/r^s, s>0, where r denotes Euclidean distance between point charges. Of particular interest are those distances from the point charge to the sphere for which the equilibrium charge distribution is no longer supported on the whole of the sphere (i.e. spherical caps of negative charge appear). Arising from this problem attributed to A. A. Gonchar are sequences of polynomials of a complex variable that have some fascinating properties regarding their zeros.
Johann S. Brauchart, Peter D. Dragnev, Edward B. Saff
We consider the minimal energy problem on the unit sphere $\mathbb{S}^d$ in the Euclidean space $\mathbb{R}^{d+1}$ in the presence of an external field $Q$, where the energy arises from the Riesz potential $1/r^s$ (where $r$ is the Euclidean distance and $s$ is the Riesz parameter) or the logarithmic potential $\log(1/r)$. Characterization theorems of Frostman-type for the associated extremal measure, previously obtained by the last two authors, are extended to the range $d-2 \leq s < d - 1.$ The proof uses a maximum principle for measures supported on $\mathbb{S}^d$. When $Q$ is the Riesz $s$-potential of a signed measure and $d-2 \leq s <d$, our results lead to explicit point-separation estimates for $(Q,s)$-Fekete points, which are $n$-point configurations minimizing the Riesz $s$-energy on $\mathbb{S}^d$ with external field $Q$. In the hyper-singular case $s > d$, the short-range pair-interaction enforces well-separation even in the presence of more general external fields. As a further application, we determine the extremal and signed equilibria when the external field is due to a negative point charge outside a positively charged isolated sphere. Moreover, we provide a rigorous analysis of the three point external field problem and numerical results for the four point problem.
Johann S. Brauchart, Josef Dick, Lou Fang
In this paper we introduce a reproducing kernel Hilbert space defined on $\mathbb{R}^{d+1}$ as the tensor product of a reproducing kernel defined on the unit sphere $\mathbb{S}^{d}$ in $\mathbb{R}^{d+1}$ and a reproducing kernel defined on $[0,\infty)$. We extend Stolarsky's invariance principle to this case and prove upper and lower bounds for numerical integration in the corresponding reproducing kernel Hilbert space. The idea of separating the direction from the distance from the origin can also be applied to the construction of quadrature methods. An extension of the area-preserving Lambert transform is used to generate points on $\mathbb{S}^{d-1}$ via lifting Sobol' points in $[0,1)^{d}$ to the sphere. The $d$-th component of each Sobol' point, suitably transformed, provides the distance information so that the resulting point set is normally distributed in $\mathbb{R}^{d}$. Numerical tests provide evidence of the usefulness of constructing Quasi-Monte Carlo type methods for integration in such spaces. We also test this method on examples from financial applications (option pricing problems) and compare the results with traditional methods for numerical integration in $\mathbb{R}^{d}$.
Johann S. Brauchart, Peter J. Grabner, Wöden B. Kusner, Jonas Ziefle
The concept of hyperuniformity has been introduced by Torquato and Stillinger in 2003 as a notion to detect structural behaviour intermediate between amorphous disorder and crystalline order. The present paper studies a generalisation of this concept to the unit sphere. It is shown that several well studied determinantal point processes are hyperuniform, one recently introduced process, the projective ensemble, is shown not to be hyperuniform.
Johann S. Brauchart
The electrostatic interpretation of zeros of Jacobi polynomials, due to Stieltjes and Schur, enables us to obtain the complete asymptotic expansion as $n \to \infty$ of the minimal logarithmic potential energy of $n$ point charges restricted to move in the interval $[-1,1]$ in the presence of an external field generated by endpoint charges. By the same methods, we determine the complete asymptotic expansion of the logarithmic energy $\sum_{j\neq k} \log(1/| x_j - x_k |)$ of Fekete points, which, by definition, maximize the product of all mutual distances $\prod_{j\neq k} | x_j - x_k |$ of $N$ points in $[-1,1]$ as $N \to \infty$. The results for other compact intervals differ only in the quadratic and linear term of the asymptotics. Explicit formulas and their asymptotics follow from the discriminant, leading coefficient, and special values at $\pm 1$ of Jacobi polynomials. For all these quantities we derive complete Poincaré-type asymptotics.
Johann S. Brauchart, Peter D. Dragnev, Edward B. Saff, Robert S. Womersley
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.
Johann S. Brauchart, Josef Dick, Friedrich Pillichshammer
We prove that the information complexity (i.e., the inverse) of the classical spherical cap $L_2$ discrepancy on the $d$-dimensional sphere $\mathbb{S}^d$ decreases with dimension $d$, indicating a ``blessing of dimensionality'' for the associated numerical integration problem. We then introduce a modified spherical cap $L_2$ discrepancy that emphasizes large caps (close to hemispheres). For this variant, the problem does not become easier with increasing $d$. We also establish a Stolarsky invariance principle which connects the modified spherical cap $L_2$ discrepancy to numerical integration in the Sobolev space $H^{(d+1)/2}(\mathbb{S}^d)$, represented by the reproducing kernel $K(\boldsymbol{x}, \boldsymbol{y}) = 1 - \tfrac{1}{\sqrt{2}} \|\boldsymbol{x} - \boldsymbol{y}\|$. Stolarsky's invariance principle then implies that the worst-case integration error in this space grows polynomially with $d$.