Constructive approximation

Constructive approximation was the focus of my PhD research, and continues to interest me. The subject area includes constructive function theory, and wider aspects of approximation theory, and uses the theories of special functions, orthogonal polynomials, and reproducing kernel Hilbert spaces, amongst others.

My PhD thesis and a number of my subsequent papers focus on sequences of finite sets of points on the sphere, on the torus, on products of spheres, and on compact connected Riemannian manifolds in general.

PhD thesis

My PhD thesis describes a partition of the unit sphere into regions of equal area and bounded diameter, the associated spherical codes obtained from the partition, and the properties of these codes, in terms of separation, quadrature discrepancy and energy. It is accompanied by the EQSP Matlab Toolbox, open source software that implements the partition and produces the associated spherical codes.

Thesis

Distributing points on the sphere: Partitions, separation, quadrature and energy , UNSW, 2007. (Citations). 

Clean PDF file. Accompanying Thesis/Dissertation Sheet.

Open source software

• EQSP: Recursive Zonal Equal Area Sphere Partitioning Toolbox. (Citations).

Sphere partitions and associated spherical codes

My PhD thesis gave rise to some presentations and two papers describing the equal area partition of the unit sphere and the properties of the associated spherical codes. The first of these two papers [1],  is my most cited paper with 224 citations in Scopus, 179 in Web of Science, and 43 in MathSciNet (as at 2024-08-20). The second [2] contains proofs of statements in the first paper.

During visits to the University of Padova I identified scope for collaboration on investigations into the discrepancy and energy properties of approximate Fekete points, and discrete Leja points on the sphere. This led to a joint paper with Alvise Sommariva and Marco Vianello on the use of non-negative least squares algorithms to obtain positive weight interpolatory quadrature rules on the sphere [4]. On an invitation from Shayne Waldron of the University of Auckland, I presented this work at a meeting on Tight Frames and Approximation in New Zealand in February 2018.

Presentations

• A partition of the unit sphere into regions of equal area and small diameter, AustMS, 2004.

• Partitions of the unit sphere into regions of equal area and small diameter, Macquarie U, 2005.

  • Movie of the partition EQ(3,99) [MPG file], 2006.

• Spherical codes with good separation, discrepancy and energy, HDA, ICIAM, AustMS, 2007. HDA talk handout, 2007.

• Applications of equal area partitions of the unit sphere, ORNL, 2014; ECMWF, 2016; BoM 2017; QAMT Annual Conference 2021 (video).

• Approximate Fekete points and discrete Leja points based on equal area partitions of the unit sphere, CTAC, ANU, 2014; Bergamo, 2015.

• Diameter bounded equal measure partitions of Ahlfors regular metric measure spaces, AustMS, 2015. Point Distributions Webinar, FSU, 2020.

• Optimal polynomial meshes and Caratheodory-Tchakaloff submeshes on the sphere, Tight Frames and Approximation, 2018.

• Grids and meshes on the sphere, ACCESS-NRI, Canberra, 2024.

• Applicability of an equal area partition of the unit sphere, Joint Meeting of NZMS, AustMS, AMS, Auckland, 2024.

• Applications of equal area partitions of the sphere to the mathematics of planet Earth, Constructive Functions 2025, Nashville (remote).

Posters

• Applicability of an equal area partition of the unit sphere, DWCAA24, Canazei, 2024; Joint Meeting of NZMS, AustMS, AMS, Auckland, 2024.

Publications and preprints

1. Paul Leopardi, "A partition of the unit sphere into regions of equal area and small diameter", Electronic Transactions on Numerical Analysis, Volume 25, 2006, pp. 309-327. MR 2280380, (Citations).  Preprint: UNSW Applied Mathematics Report AMR05/18, May 2005, revised June 2006.

   • Describes the algorithm used in the EQSP software package, which partitions a finite dimensional unit sphere into regions of equal area and small diameter.

2. Paul Leopardi, "Diameter bounds for equal area partitions of the unit sphere", Electronic Transactions on Numerical Analysis, Volume 35, 2009, pp. 1-16. Preprint: December 2007, revised January 2009.

   • Proves diameter bounds for the sphere partition described in [1], and a modified version of the construction of Feige and Schechtman.

3. Giacomo Gigante and Paul Leopardi, "Diameter bounded equal measure partitions of Ahlfors regular metric measure spaces", Discrete and Computational Geometry, 57 (2), 2017, pp. 419–430. Preprint: arXiv:1510.05236 [math.NA].

   • Combines the Feige and Schechtman construction with David's and Christ's dyadic cubes to yield a partition algorithm for connected Ahlfors regular metric measure spaces of finite measure.

4. Paul Leopardi, Alvise Sommariva, Marco Vianello, "Optimal polynomial meshes and Caratheodory-Tchakaloff submeshes on the sphere", Dolomites Research Notes on Approximation, Volume 10 Special Issue, 2017, pp. 18-24. Preprint: arXiv:1612.04952 [math.NA].

   • Using the notion of Dubiner distance, gives an elementary proof that good covering point configurations on the 2-sphere are optimal polynomial meshes, and extracts Caratheodory-Tchakaloff submeshes for compressed least squares fitting.

5. Paul Leopardi, "The applicability of equal area partitions of the unit sphere", Journal of Approximation Software, 1(2), 2024. Preprint: arXiv:2408.13434 [math.NA].

   • Examines the idea of applicability of mathematical constructions and algorithms, by using the citations to [1] as a case study.

Quadrature and energy for finite subsets of spheres and compact manifolds

I have investigated the relationship between discrepancy, separation and energy of finite sets of points on spheres and on compact manifolds, resulting in three papers. My paper [3] concerning the discrepancy and energy of sets of points on compact Riemannian manifolds has led to a new collaboration with researchers at the University of Bergamo.

Presentations

• The Coulomb energy of spherical designs on S^2, Vanderbilt U, 2003.

• The Riesz energy of point sets on the unit sphere under weak-star convergence, Auckland, 2005.

• Discrepancy, separation and energy on spheres and on compact connected Riemannian manifolds, DWCAA, ANU, 2012.

• Discrepancy, separation and Riesz energy of finite point sets on compact connected Riemannian manifolds, Vanderbilt U, ESI Vienna, Bergamo, 2014.

Publications and preprints

1. Kerstin Hesse and Paul Leopardi, "The Coulomb energy of spherical designs on S^2", Advances in Computational Mathematics, Volume 28, Number 4 / May, 2008 , pp. 331-354. DOI 10.1007/s10444-007-9026-7 , MR 2390282, (Citations). Preprint: UNSW Applied Mathematics Report AMR04/34, December 2004, revised January 2006.

   • Gives bounds for the Coulomb energy of a sequence of well separated spherical designs on the unit sphere, including a conjectured bound comparable to the minimum Coulomb energy.

2. Paul Leopardi, "Discrepancy, separation and Riesz energy of finite point sets on the unit sphere", Advances in Computational Mathematics, Volume 39, Issue 1, July 2013, pp. 27-43. DOI 10.1007/s10444-011-9266-4 . Preprint: revised November 2012 (Proof of Lemma 4.6 was incorrect, "Stolarsky" was spelled incorrectly, minor reformatting).

   • Shows that a sequence of spherical codes with a well behaved upper bound on discrepancy and a well behaved lower bound on separation, satisfies an upper bound on Riesz s-energy.

3. Paul Leopardi, "Discrepancy, separation and Riesz energy of finite point sets on compact connected Riemannian manifolds", Dolomites Research Notes on Approximation, Volume 6, July 2014, pp. 120-129. Preprint: arXiv:1403.6550 [math.NA].

   • Proves that, for a smooth compact connected d-dimensional Riemannian manifold M, if 0 <= s <= d then an asymptotically equidistributed sequence of finite subsets of M that is also well-separated yields a sequence of Riesz s-energies that converges to the energy double integral.

Jacobi polynomials, quadrature and finite subsets of spheres

Two of my papers [1,2] deal with properties of Jacobi polynomials related to positive quadrature on the unit sphere.

During my PhD research, I became interested in the matrices used for polynomial interpolation and interpolatory quadrature on the sphere, as a new ensemble of random matrices. I gave a presentation on this in 2009, and have talked to a number of researchers about this subject.

Presentations

• Positive quadrature on the sphere and conjectures on monotonicities of Jacobi polynomials, DWCAA, AustMS, 2006.

• Porting a sphere optimization program from LAPACK to ScaLAPACK, CTAC, 2008.

• Polynomial interpolation on the sphere, reproducing kernels and random matrices, MASCOS Workshop on Stochastics and Special Functions, U Queensand, 2009.

Publications and preprints

1. Paul Leopardi, "Positive weight quadrature on the sphere and monotonicities of Jacobi polynomials", DWCAA06 proceedings, Numerical Algorithms, Volume 45, Numbers 1-4 / August, 2007, pp. 75-87. DOI 10.1007/s11075-007-9073-7 , MR 2355973, (Citations). Preprint: UNSW Applied Mathematics Report AMR06/41, December 2006, revised Febrary 2007.

   • Examines the relationship, for a positive weight quadrature rule on the unit sphere, between the the total quadrature weight on any spherical cap and the area of that cap. Uses conjectures from [2] to give improved estimates.

2. Walter Gautschi and Paul Leopardi, "Conjectured inequalities for Jacobi polynomials and their largest zeros", DWCAA06 proceedings, Numerical Algorithms, Volume 45, Numbers 1-4 / August, 2007, pp. 217-230. DOI 10.1007/s11075-007-9067-5 , MR 2355984, (Citations). Preprint: UNSW Applied Mathematics Report AMR07/2, February 2007.

   • Describes new conjectures on monotonicities of the values and the zeros of functions related to Jacobi polynomials with fixed \alpha and \beta and increasing degree.

3. Paul Leopardi and Rob Womersley, "Porting a sphere optimization program from LAPACK to ScaLAPACK", ANZIAM Journal, 50 (CTAC 2008), November 2008, pp. C204-C219. Draft: Paul Leopardi, "Conversion of a Sphere Optimization Program from LAPACK to ScaLAPACK", (unpublished draft, 2002). Preprint: Revised October 2008. Figure 1: TG, TF. Figure 2: TI, TT.

   • Describes methods used to parallelize code used in optimization on the sphere, and analyzes performance of the code in relation to the topology of the computer cluster used for testing.

Integer sequences

• A129337: Maximal possible degree of a Chebyshev-type quadrature formula with n nodes, in the case of the constant weight function on [ -1,1], May 2007.

Sparse grid quadrature on compact manifolds

My work with Markus Hegland on sparse grid quadrature has led to two joint papers [1,2]. The latter paper proves that a type of greedy algorithm gives an optimal solution to a particular type of constrained knapsack problem, under circumstances which occur in the numerical examples given.

Presentations

• Quadrature using sparse grids on products of spheres, HDA, MASCOS AGM, ANZIAM NSW/ACT Branch Mini Meeting, 2009.

• The rate of convergence of sparse grid quadrature on products of spheres, MCQMC, AustMS, 2010.

• The rate of convergence of sparse grid quadrature on the torus, CTAC 2010.

• Sparse grid quadrature as a knapsack problem, HDA 2011.

• Sparse grid quadrature on products of spheres, ANU, University of Canterbury, 2012; University of Newcastle, 2015.

Publications and preprints

1. Markus Hegland and Paul Leopardi, "The rate of convergence of sparse grid quadrature on the torus", ANZIAM Journal, 52 (CTAC 2010), June 2011, pp. C500--C517. Preprint: January 2011, revised June 2011.

   • Describes a dimension adaptive algorithm for sparse grid quadrature on reproducing kernel Hilbert spaces on the unit torus, and compares this algorithm to the WTP algorithm of Wasilkowski and Wozniakowski.

2. Markus Hegland and Paul Leopardi, "Sparse grid quadrature on products of spheres", Numerical Algorithms,  Volume 70, Issue 3, 2015, pp. 485-517. DOI:10.1007/s11075-015-9958-9. Preprint: arXiv:1202.5710 [math.NA].

   • Describes sparse grid quadrature on products of spheres, giving the initial and asymptotic rates of convergence.