Abstracts

Approximation and Applications

A Bayesian take on RBFs
by
Colin Fox
University of Otago

Radial basis functions (RBFs) provide a convenient representation of (implicity defined) boundaries and explicit functions, and are therefore useful as mid-level representations of unknowns in statistical inference. In the presence of measurement noise, or other uncertainty, these high-dimensional representations necessarily induce a bias in quantities of interest, or `statistics', and often provide fickle estimates. These problems are easy to demonstrate, and just as easy to fix through the design of the Bayesian's `prior' distribution.

Date received: November 3, 2008

On some aggregation/disaggregation based approximations
by
Markus Hegland
Australian National University

Originally, aggregation and disaggregation were considered as acceleration techniques similar to multigrid methods for the solution of linear systems of equations. In some recent work we have demonstrated that these methods can also be used for the numerical solution of the chemical master equations.

In the simplest case, one would like to approximate smooth summable sequences by piecewise polynomials. I will show how the chemical master equations provide a notion of smoothness and will derive approximation error bounds using convolution and sampling theorems. If time permits, I will discuss some issues arising in the multi- and high-dimensional case.

Date received: October 21, 2008

Approximating functions in Clifford algebras
by
Paul Leopardi
Australian National University

As is well known, the Clifford algebras over the real field R can be represented as matrix algebras over R or the complex field C. This means that various functions defined over matrices, such as square root and logarithm, can also be defined over the Clifford algebras. The talk will discuss the similarities and differences between matrix and Clifford functions, and look at specific approximation algorithms, notably the Denman-Beavers square root and the Cheng-Higham-Kenney-Laub logarithm.

Date received: October 30, 2008

Geodesic interpolating splines
by
Stephen Marsland
Massey University, Palmerston North

Spline interpolation is a common problem in many areas of data analysis, including statistic sand image and signal processing. There has been some interest in constructing splines that are guaranteed diffeomorphic, e.g., that distort space in a smooth, invertible way. I will demonstrate different methods of computing such splines and show that the ability to reach a metric on the diffeomorphism group from this provides some benefits for data analysis.

Date received: October 29, 2008

Preserving Energy resp. dissipation in numerical partial differential equations, using the "Average Vector Field" method.
by
David McLaren
La Trobe University
Coauthors: E. Celledoni, V. Grimm, R.I. McLachlan, B. Owren, G.R.W. Quispel

We give a method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly. The same method, applied to PDEs with constant dissipative structure, also preserves the correct monotonic decrease of energy. The method is illustrated in many examples. In the Hamiltonian case they are: the sine-Gordon, Korteweg-de Vries, nonlinear Schrödinger, and Maxwell equations. In the dissipative case the examples are: the Allen-Cahn, Cahn-Hilliard, Ginzburg-Landau, and Heat equations.

Date received: October 16, 2008

Fast algorithms for integral transforms based on domain decomposition and composition equations
by
Garry Newsam
Defence Science & Technology Organisation, Australia

The standard decomposition of the unit interval I ≡ [-1/2, 1/2] into the union of two equal subintervals induces a corresponding decomposition of a function f on I as:

f(x)    =   f-(x+1/4)  +  f+(x-1/4)     .

(1)

Applying the Fourier transform to this equation gives a simple composition equation for the transform of f in terms of the transforms of f_- and f₊, i.e.

F(w)    =   e2 pi w/ 4 F-(w)  +  e-2 pi w/ 4 F+(w)     .

(2)

This equation in turn provides the basis of a simple fast hierarchical algorithm for computing the Fourier transform: given initial evaluations over a coarse grid of the transform of the function over some defining set of N subintervals, an approximation to the transform of the full function is obtained by interpolating successive pairs of functions F_-, F₊ on to a common grid of twice the resolution and then combining them as in (2). The paper will detail the algorithm and provide an error analysis that shows the transform of a function defined by N data points can be computed to within accuracy e in O(N logN loge^-1) operations.

The algorithm is readily generalised to any other operator (e.g. the Radon transform) that commutes in a simple way with the shift operator, and to nonuniform domain decompositions. Its main advantage over existing fast algorithms is that in the course of computation it evaluates the transform not just on the full interval but also on a complete pyramid of subintervals: this allows operations such as feature detection to be carried out at all length scales rather than just a single scale.

Date received: October 14, 2008

High order of convergence using lattice sequences for numerical integration
by
Dirk Nuyens
University of New South Wales
Coauthors: Fred J. Hickernell, Peter Kritzer and Frances Y. Kuo

We study the worst case integration error of combinations of quadrature rules in a reproducing kernel Hilbert space. We show that the error, with respect to the total number N of function evaluations used, cannot decrease faster than O(N^-1) in the case where several quasi-Monte Carlo rules are combined to a compound quasi-Monte Carlo rule. However, if the errors of the quadrature rules constituting the compound rule have an order of convergence O(N^-a) for a > 1 then, by introducing weights, this order of convergence can be shown to be recovered for the compound rule. We apply our results to the case of lattice sequences.

Date received: November 24, 2008

On the construction of equiangular tight frames
by
Shayne Waldron
University of Auckland

Tight frames are optimal for signal reconstruction when there is one erasure, and equiangular tight frames are optimal when there are two erasures. We give a survey of the known methods for constructing real and complex equiangular tight frames. This includes the identification of real equiangular frames with graphs, and Zauner's conjecture on the existence of an equiangular tight frame of n=d² vectors for C^d.

Date received: November 10, 2008