Paul Leopardi

Visiting Fellow

$Real representation matrix for neutral Clifford algebra R_{2,2}$
Email address	paul.leopardi {AT} gmail.com
Postal address	Mathematical Sciences Institute Building 27 Australian National University Canberra ACT Australia 0200
Phone	+61 2 6125 1229

Brief history

Academic: BSc (Hons in Computer Science), University of New South Wales, 1983-10-05. MCom (Information Systems), University of New South Wales, 1990-05-04. Master of Science and Technology by coursework in Mathematics, University of New South Wales, 2002-04-09. Doctor of Philosophy in Applied Mathematics, University of New South Wales, 2007-05-17, supervised by Professor Ian Sloan and Associate Professor Rob Womersley. Grad. Cert. in Higher Education, Australian National University, 2012-07-12.
Working: Telecom Australia 1983-1986 - Computer Systems Officer. Memorex-Telex 1986-1990 - Systems Engineer. Travel Industries Automated Systems 1990-1995 - Systems Analyst. Accenture 1995-2001 - Consultant. UNSW 2001-2002 - Research assistant programmer, Mathematics. University of Sydney 2005-2007 - Scientific Computing Officer, Physics. ANU 2007-2012 - Postdoctoral Research Fellow, Mathematics. ANU 2012-2014 - Research Fellow, Bioinformation Science.

Research interests

Constructive approximation: sparse grids, approximation and quadrature on the sphere and compact manifolds. Clifford algebras: Clifford analysis, new constructions for Hadamard matrices. Combinatorics and statistics: random number generation and testing, combinatorics and statistics of words in sequences. Numerical analysis: object-oriented numerical analysis, parallel linear algebra using ScaLAPACK.

Teaching

2014 Jan-Feb: AMSI Summer School - Bioinformatics [tutor - proofreading and marking] (convened by Conrad Burden).
MATH2307/MATH6100 Bioinformatics and Biological Modelling 2013 Semester 2 (convened by Conrad Burden).
2013, 2012: Reading course and seminar on Clifford algebras.
MATH3512/MATH6112 Matrix Computations
- 2011 Semester 2.
MATH3512/MATH6112 Matrix Computations and Optimization
- 2010 Semester 2 (with Richard Brent).
- 2009 Semester 2 (with Richard Brent and Vikram Sunkara).
Approximation Theory
- 2008 Semester 1: MATH3352 - Approximation Theory (convened by Markus Hegland).
- 2008 Jan-Feb: ICE-EM/AMSI Summer School - Approximation Theory (convened by Markus Hegland).

PhD thesis

Distributing points on the sphere: Partitions, separation, quadrature and energy , UNSW, 2007. (Citations).
Accompanying Thesis/Dissertation Sheet.

Publications and preprints

Paul Leopardi, "Conversion of a Sphere Optimization Program from LAPACK to ScaLAPACK", (unpublished draft, 2002).
Describes methods used to parallelize code used in optimization on the sphere.
Paul Leopardi, "A generalized FFT for Clifford algebras", Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 11, Number 5, 2005, pp. 663-688.
MR 2130632. (Citations).
Preprint: UNSW Applied Mathematics Report AMR04/17, March 2004.
Describes algorithms used in the GluCat C++ software library, for the real representations of real Clifford algebras, having the same order of complexity as the generalized FFTs on finite groups.
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.
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 [5] to give improved estimates.
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.
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.
Paul Leopardi and Rob Womersley, "Porting a sphere optimization program from LAPACK to ScaLAPACK", ANZIAM Journal, 50 (CTAC 2008), November 2008, pp. C204-C219.
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.
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 [3], and a modified version of the construction of Feige and Schechtman.
Paul Leopardi, "Testing the tests: using random number generators to improve empirical tests", Monte Carlo and Quasi-Monte Carlo Methods 2008, Pierre L' Ecuyer, Art B. Owen (Eds.) Springer, 2009 pp. 501--512. ISBN: 978-3-642-04106-8, MR 2743916, (Citations).
Preprint: Revised July 2009.
Examines implementations of the overlapping serial tests of Marsaglia and Zaman, and improves them, using accurate calculation of the mean and variance of the number of missing words in a random string.
Paul Leopardi, "Approximating the square root and logarithm functions in Clifford algebras: what to do in the case of negative eigenvalues?", (extended abstract) AGACSE 2010, June 2010.
Describes how the Clifford algebras over the real numbers can be treated as real matrices, except in the case of negative real eigenvalues, when the square root and logarithm functions may take values in a larger Clifford algebra.
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.
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, (published online December 2011).
DOI 10.1007/s10444-011-9266-4 .
Preprint: June 2010, revised December 2011, 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.
Paul Leopardi, "Can compatible discretization, finite element methods, and discrete Clifford analysis be fruitfully combined?", Clifford Analysis, Clifford Algebras and their applications (CACAA), Volume 7, Issue 1, 2012, pp. 57-64. ISSN 2050- 0300 (print), 2050-0319 (online).
Preprint: January 2012.
Conference paper: 9th International Conference on Clifford Algebras and their Applications (ICCA 9), July 2011.
Preprint: May 2011, revised July 2011.
Describes work in progress, towards the formulation, implementation and testing of compatible discretization of differential equations, using a combination of Finite Element Exterior Calculus and discrete Geometric Calculus / Clifford analysis.
Paul Leopardi, "Constructions for Hadamard matrices, Clifford algebras, and their relation to amicability - anti-amicability graphs", Australasian Journal of Combinatorics, Volume 58(2) (2014), pp. 214–248.
Preprint: Revised January 2014.
Supplementary material can be found in the Hadamard directory, and the Hadamard-fractious Github repository.
Describes how the pattern of commuting and anticommuting pairs of basis elements of a real Clifford algebra, and their representation theory, can be used in the construction of Hadamard matrices.
Markus Hegland and Paul Leopardi, "Sparse grid quadrature on products of spheres", submitted to Numerical Algorithms, April 2014, resubmitted with corrections, October 2014.
Preprint: arXiv:1202.5710 [math.NA].
Describes sparse grid quadrature on products of spheres, giving the initial and asymptotic rates of convergence.
Conrad Burden, Paul Leopardi and Sylvain Fôret, "Word match counts between Markovian sequences", "Biomedical Engineering Systems and Technologies 6th International Joint Conference, BIOSTEC 2013", Springer series on Communications in Computer and Information Science, 2014.
Preprint of conference paper: ("The distribution of short word match counts between Markovian sequences"), presented at International Conference on Bioinformatics Models, Methods and Algorithms (Bioinformatics 2013). November 2012.
Preprint of revised and extended paper: June 2013.
Examines the D2 statistic, which counts the number of word matches between two given sequences, under the assumptions of periodic boundary conditions and Markovian dependence. Includes the calculation of the mean of D2 for all Markov orders and the variance for Markov order 1.
Conrad Burden, Paul Leopardi and Sylvain Fôret, "The distribution of word matches between Markovian sequences with periodic boundary conditions", Journal of Computational Biology, 21(1), pp. 41--63, January 2014. DOI 10.1089/cmb.2012.0277 .
Preprint: Revised May 2013.
Further examines the D2 statistic, which counts the number of word matches between two given sequences, under the assumptions of periodic boundary conditions and Markovian dependence. Includes the calculation of the mean of D2 for all Markov orders and the variance for all Markov orders up to and including the word length. Also includes a comparison of synthetic data with DNA data from human chromosome 1.
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.
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.
Paul Leopardi and Ari Stern, "The abstract Hodge-Dirac operator and its stable discretization", in preparation.
Preprint: arXiv:1401.1576 [math.NA].
Adapts the techniques of finite element exterior calculus to study and discretize the abstract Hodge-Dirac operator, a square root of the abstract Hodge-Laplace operator considered by Arnold, Falk, and Winther.
Paul Leopardi, "Twin bent functions and Clifford algebras", submitted to the proceedings of the Workshop on Algebraic Design Theory and Hadamard Matrices (ADTHM 2014), September 2014.
Preprint: September 2014.
Examines a pair of bent functions on and their relationship to a necessary condition for the existence of an automorphism of an edge-coloured graph, whose colours are defined by the properties of a canonical basis for the real representation of a real Clifford algebra.

Presentations

Constructive approximation

The Coulomb energy of spherical designs on S^2, 2003.
A partition of the unit sphere into regions of equal area and small diameter, 2004.
Partitions of the unit sphere into regions of equal area and small diameter, 2005.
Movie of the partition EQ(3,99) [MS MPEG4 V2 AVI file], [MPEG4 file], [MPEG4 Quicktime MOV file], 2006.
The Riesz energy of point sets on the unit sphere under weak-star convergence, 2005.
Positive quadrature on the sphere and conjectures on monotonicities of Jacobi polynomials, DWCAA06, AustMS, 2006.
Spherical codes with good separation, discrepancy and energy, HDA, ICIAM, AustMS, 2007. HDA talk handout, 2007.
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, 2009.
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.
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.
Applications of equal area partitions of the unit sphere, ORNL, 2014.
Approximate Fekete points and discrete Leja points based on equal area partitions of the unit sphere, CTAC, ANU, 2014.

Clifford algebras

Practical computation with Clifford algebras, 2002.
A generic library of universal Clifford algebra templates, (poster) 2002.
Quick introduction to Clifford algebras, 2003, (Citations).
A generalized FFT for Clifford algebras, 2003.
Martin Albrecht, Synergy Effects: a Sage introduction, updated and edited by Paul Leopardi, CLUG, 2008.
Approximating functions in Clifford algebras, ANZMC 2008.
Approximation of the square root and logarithm functions in Clifford algebras: what to do in case of negative eigenvalues? (poster) DWCAA09, 2009.
Approximating functions in Clifford algebras: What to do with negative eigenvalues? AustMS 2009, (AGACSE 2010 short version), (AGACSE 2010 long version).
Can compatible discretization, finite element methods, and discrete Clifford analysis be fruitfully combined? ICCA 9, 2011.
Is a dual mesh really necessary? ICIAM 2011.
An abstract Hodge-Dirac operator and its stable discretization, IWOTA 2012, CTAC 2012, ICCA 10, 2014.
Visualization as its own reward: The mathematics of conformal chaos, EViMS 2, ANU, 2014.

Combinatorics and statistics

Testing the tests: using pseudorandom number generators to improve empirical tests, MCQMC, 2008.
Accurate computation of the variance of the number of missing words in a random string, 4ICC, 2008.
A conjecture on the alphabet size needed to produce all correlation classes of pairs of words, 34 ACCMCC, 2010.
Amicability graphs and Clifford algebras, Hadamard Maximal Determinant Workshop, 2010.
Constructions for Hadamard matrices, Clifford algebras, and their relation to amicability - anti-amicability graphs, 35 ACCMCC, 2011.
New constructions for Hadamard matrices, CARMA Seminar, University of Newcastle, 2012.
Determining the distribution of word matches between Markovian sequences, CTAC 2012.
Alignment-free comparison of biological sequences, ANZIAM 2013.
Skew, bent and fractious: a confession, AustMS, 2013.
Representations of Clifford algebras: skew, bent and fractious, 37 ACCMCC, 2013.
Twin bent functions and Clifford algebras, ADTHM 2014 (revised), Newcastle, 2014.
Twin strongly regular graphs: some questions, relayed by Robert Craigen at Algebraic design theory with Hadamard matrices: applications, current trends and future directions (14w2199), 2014.

Software

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.
A152139: Correlation classes of pairs of different words, November 2008.
A152959: Number of correlation classes for pairs of different words in an alphabet of size 4, December 2008.

Research project proposals and grants

2010-2013: ARC Discovery DP1095941: Solving problems in physics and engineering involving differential and integral equations by means of mutivariate approximation theory and Clifford algebras (not funded).
2011-2014: ARC Discovery DP110101392: Approximating the solution of problems in physics and engineering by means of discrete geometric calculus (not funded).
2012-2015: ARC DECRA DE120100028: High performance computation in Grassmann and Clifford algebras, with applications to number theory and partial differential equations (not funded).
2013-2015: ARC DECRA DE130100402: New approaches to the discretization of partial differential equations (not funded).
2013-2015: ARC Discovery DP130101924: New constructions for Hadamard matrices (not funded).
2013 NCI ANU Time Grant xc2: New constructions for Hadamard matrices

Conference, session and user group organization

Clifford minisymposia at ICIAM 2003, 7-11 July 2003 (Sydney, Australia).
8th Australian Space Science Conference (ASSC), 29 September - 1 October 2008 (ANU).
Public lecture: Life on Mars: Phoenix and Beyond, 1 October 2008 (ANU).
HDA09 - Third Workshop on High-dimensional Approximation, 16-20 February 2009 (UNSW, Sydney, Australia).
34 ACCMCC - Australasian Conference on Combinatorial Mathematics and Combinatorial Computing, 6-10 December 2010 (ANU).
Thematic session on Continuous and discrete Clifford analysis at IWOTA 2012, 16-20 July 2012 (UNSW, Sydney, Australia).
NECTAR Early Career Academics Retreat, 4-5 June 2013 (Mt Stromlo, Australia).
Canberra Python Users Group (2013) (Google+ Community) (Wiki) (with Jay Larson,Ed Schofield and Padarn Wilson).
Session on Reproducibility in Computational Scientific Research, Mathematics of Plate Earth (MPE) Australia 2013, 8-12 July 2013 (Melbourne, Australia).

Citations

My Google Scholar Citations

Mentions in acknowledgements and elsewhere

J. Richardson, "The Blockhandler and the Bitfield Package", J. Symbolic Computation (1992) 14, 93-101, [Acknowledgements, p. 100].
M. Ganesh, I. G. Graham, "A high-order algorithm for obstacle scattering in three dimensions", J. Comput. Phys. 198 (2004), no. 1, 211-242, [Acknowledgements, p. 239].
D. P. Hardin, E. B. Saff, "Discretizing manifolds via minimum energy points", Notices Amer. Math. Soc. 51 (2004), no. 10, 1186-1194, [mention, p. 1189].
I. H. Sloan, R. S. Womersley, "Extremal systems of points and numerical integration on the sphere", Adv. Comput. Math. 21 (2004), no. 1-2, 107-125, [Acknowledgements, p. 124].
P. J. Forrester, N. S. Witte, "Discrete Painlevé equations for a class of PVI tau-functions given as U(N) averages", Nonlinearity 18 (2005), no. 5, 2061-2088, [Acknowledgements, p. 2087].
G. Bard, "Matrix Inversion (or LUP-Factorization) via the Method of Four Russians, in Theta(n3/log n) Time", (preprint), 2008, [Acknowledgements, p. 6].
N. J. Higham, Functions of matrices: theory and computation, SIAM, 2008, [reference to GluCat, p. 47 (not seen by Google Scholar)].
Sacha van Albada, "Mean-Field Analysis of Basal Ganglia and Thalamocortical Dynamics", PhD thesis, University of Sydney, 2009, [Acknowledgements, p. v].
R. Koekoek, P. A. Lesky, R. F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their Q-Analogues, Springer Monographs in Mathematics, 2010, [reference 244, p. 221 (not seen by Google Scholar).].
Jorg Arndt, "Generating Random Permutations", PhD thesis, ANU, 2010, [Acknowledgements, p. vii].
Congpei An, "Distribution of Points on the Sphere and Spherical Designs", PhD thesis, The Hong Kong Polytechnic University, 2011, [Acknowledgements, p. iii].
Shi Bai, "Polynomial Selection for the Number Field Sieve", PhD thesis, ANU, 2011, [Acknowledgements, p. vii].
M. Holst, A. Stern, "Geometric Variational Crimes: Hilbert Complexes, Finite Element Exterior Calculus, and Problems on Hypersurface", Foundations of Computational Mathematics, (2012) 12:263-293, DOI: 10.1007/s10208-012-9119-7, [Acknowledgements].
M. Holst, A. Stern, "Semilinear Mixed Problems on Hilbert Complexes and Their Numerical Approximation", Foundations of Computational Mathematics, (2012) 12:363-387, DOI: 10.1007/s10208-011-9110-8, [Acknowledgements].
Vikram Sunkara, "Analysis and Numerics of the Chemical Master Equation", PhD thesis, ANU, 2013, [Acknowledgements, p. vii].

Mathematical Sciences Institute