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 |
Academic: BSc (Hons in Computer Science), University of New South
Wales, 1983. MCom (Information Systems), University of New South Wales,
1990. Master of Science and Technology by coursework in Mathematics,
University of New South Wales, 2002.
Doctor of Philosophy in Applied Mathematics, University of New South Wales, 2007,
supervised by
Professor Ian Sloan and
Associate Professor Rob Womersley.
Grad. Cert. in Higher Education, Australian National University, 2012.
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.
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.
Distributing points on the sphere: Partitions, separation, quadrature and energy
, UNSW, 2007.
(Citations).
Accompanying
Thesis/Dissertation Sheet.
Describes methods used to parallelize code used in optimization on the sphere.
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.
Describes the algorithm used in the EQSP software package, which partitions a finite dimensional unit sphere into regions of equal area and small diameter.
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.
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.
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.
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.
Proves diameter bounds for the sphere partition described in [3], and a modified version of the construction of Feige and Schechtman.
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.
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.
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.
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.
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.
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.
Describes sparse grid quadrature on products of spheres, giving the initial and asymptotic rates of convergence.
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.
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.
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.
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.
Updated: 4 July 2014/ Responsible Officer: Director, MSI / Page Contact: Paul Leopardi