Presenter: Markus Hegland, ANU
Email contact: Markus.Hegland@anu.edu.au
Title: Adaptive Sparse Grids
Abstract: Sparse grids, as studied by Zenger and Griebel in the last 10 years have been a very successful in the solution of partial differential equations, integral equations and classification problems with up to around 10 dimensions. We will discuss a general framework which allows to extend the method to the case of nominal and mixed variables and to very high dimensions and applications to nonparametric regression as used in data mining.
AMS Classification: 93E14
Presenter: Vanessa Robins, Applied Maths, RSPhysSE, ANU
Email contact: Vanessa.Robins@anu.edu.au
Title: Foundations for computing topology from finite approximations
Abstract: There is growing interest in extracting topological information from data in a diverse range of fields ranging from structure-function modelling of protein molecules to deducing dynamics from embedded time series data to image processing. The desired topological properties include the number of connected components and number of "holes" - i.e. the Betti numbers of the homology groups. These give an elementary description of structure, and are independent of geometric measures like volume or surface area. Efficient numerical algorithms for computing Betti numbers are based on constructions in computational geometry such as minimal spanning trees and alpha shapes. This talk will address the fundamental problem of extrapolating the topology of an object from a finite scattered-point approximation to it. Our approach is to coarse-grain the data at a sequence of resolutions and deduce the limiting structure. The theory is based on an inverse system of approximating spaces, as in Cech homology and shape theory. It leads to the concept of persistent Betti numbers which count holes that are genuine topological features, while excluding those introduced by the coarse-graining.
AMS Classification: 55-04
Presenter: D. E. Taylor
Email contact: D.Taylor@maths.usyd.edu.au
Title: Computing in groups of Lie type: generalized row reduction.
Abstract: In computational algebra systems, such as GAP and Magma, groups are represented in several ways: as permutation groups, as matrix groups or as finitely presented groups. Following the pioneering work of Chevalley and Steinberg there is another way to represent the groups of Lie type, namely by the Steinberg presentation in which the generators are parametrized by the elements of a field. In joint work with Arjeh Cohen and Scott Murray this approach has been implemented in Magma. As part of this work we have produced algorithms to convert between a matrix representation of an element and the corresponding word in the Steinberg generators. For groups of type $A_n$ this is none other than row-reduction and the PLU algorithm. In this talk I will give an overview of this algorithm for all Cartan types.
AMS Classification: 20G15
Presenter: John Lloyd Computer Sciences Laboratory Research School of Information Sciences and Engineering Australian National University
Email contact: jwl@discus.anu.edu.au
Title: Computational Logic applied to Machine Learning
Abstract: Inductive learning is concerned with inducing general theories from specific facts. The theories should be comprehensible and have predictive power. There are many applications of inductive learning systems, ranging from systems that predict carcinogenicity of chemical molecules, that learn definitions of functions in programming languages, and that assign credit ratings.
In this talk, I will discuss the application of higher-order logic to inductive learning, concentrating on the key ideas of representation of individuals and predicate construction. I will also outline some applications of a decision-tree learning system based on these ideas.
AMS Classification: 68Q32
Presenter: M.R. James Department of Engineering ANU
Email contact: Matthew.James@anu.edu.au
Title: Max-Plus Approximation Methods in Partially Observed Robust Control
Abstract: In this talk we discuss the use of max-plus approximation methods in the so-called nonlinear $H_\infty$ control. A key issue is the numerical approximation of a quantity called the "information state" which solves a first order nonlinear Hamilton Jacobi equation. Using the max-plus linearity of the solution operator for the information state, we show how the information state can be represented as a max-plus expansion relative to what we call a monotone approximate basis. The forward time propagation of the information state is expressed as max-plus matrix multiplication. This leads to a natural approximation of the optimal value function and the optimal feedback function. The issue of real-time calculation of the information state is addressed, and a Lie-Trotter type splitting scheme is proposed.
AMS Classification: 93B36
Presenter: Ian H. Sloan University of New South Wales
Email contact: sloan@maths.unsw.edu.au
Title: Numerical Integration in Hundreds of Dimensions
Abstract: Nowadays numerical integrals with hundreds or even thousands of variables often arise, and are (especially in mathematical finance) evaluated numerically, using some form of quasi-Monte Carlo method. One way of understanding the apparent success of such calculations is to assume that the integrands become easier and easier in successive coordinate directions. Wo\'zniakowski and I quantified this assumption by associating numerical weights $\gamma_1, \gamma_2, \ldots$ with successive coordinate directions, with $\gamma_1 \ge \gamma_2 \ge \cdots > 0,$ and showed that the worst case error in some classes of functions is independent of $d$ if and only if $\sum \gamma_j < \infty.$ The original proofs were not constructive, but recently Kuo and Joe at Waikato and I have found a way to construct explicit high-dimensional rules that achieve the theoretical error bounds if $\sum \gamma_j < \infty.$ This talk will introduce some of these ideas.
AMS Classification: 65D30
Presenter: R. S. Womersley(*) and I. H. Sloan
Email contact: R.Womersley@unsw.edu.au
Title: Optimization methods for choosing good points on the sphere for polynomial approximation and interpolatory cubature.
Abstract: We look at several optimization problems that arise when choosing points $\mathbf{x}_j, j = 1,\ldots,m$ on the unit sphere $S^{r-1} \subset \mathbb{R}^r$ which are good for both polynomial interpolation and interpolatory cubature.
In particular we look at extremal fundamental systems which maximize a determinant, and at extremal spherical designs which maximize a determinant while providing equal weight cubature rules which are exact for polynomials of degree at most $n$. For $n = 128$ this involves the maximization of the determinant of a $16,641$ by $16,641$ symmetric positive definite matrix which is a nonlinear function of some $32,279$ variables, proving a major computational challenge.
AMS Classification: 90C90, 65D05, 65D32
Presenter: Irfan Altas School of Information Studies Charles Sturt University
Email contact: ialtas@csu.edu.au
Title: High Order Difference Scheme for Biharmonic Equation
Abstract:
In this talk, we consider several finite difference approximations for the three- dimensional biharmonic equation. A symbolic algebra package is utilized to derive a family of finite difference approximations for the biharmonic equation on a 27 point compact stencil. The unknown solution and its first derivatives are carried as unknowns at selected grid points. This formulation allows us to incorporate the Dirichlet boundary conditions automatically and there is no need to define special formulas near the boundaries, as is the case with the standard discretizations of biharmonic equations.
As finite difference approximations for three- dimensional biharmonic equations are not available in the literature, we exhibit the standard second order finite difference approximations that require 25 grid points. We also exhibit two compact formulations of the 3D biharmonic equations; these compact formulas are defined on a 27 point compact cubic grid. The fourth order approximations are used to solve a set of test problems and produce high accuracy numerical solutions.
The system of linear equations is solved using a variety of iterative methods. We employ multigrid, Krylov and classical iterative methods to solve the system of equations.
AMS Classification: 65M06
Presenter: Patrick McLean
Email contact: p_mclean@maths.utas.edu.au
Title: Sinc Approximation to Steady State Waves
Abstract: This seminar involves analytical and approximate methods of solution of a singular integrodifferential equation arising from modelling steady fluid flow over a submerged disturbance. Analytical methods of solution involve reduction to a differential equation; while approximate methods employed are Galerkin methods using sinc functions. The asymptotic behaviour of the exact solution and the error in the approximatation are investigated.
AMS Classification: 41
Registered for B.H. Neumann Prize
Presenter: Jamie Sethian Mathematics Department University of California, Berkeley
Email contact: sethian@math.berkeley.edu
Title: Computing Multiple Arrivals to the Eikonal Equation Using Liouville Transforms
Abstract: In this talk, we show how to compute multiple arrivals to the Eikonal equation using Liouville transforms. Computing multiple solutions allows us to find secondary arrivals, which often contain more energy than the primary arrivals. The technique is performed by casting the result in terms of a higher dimensional representation whose structure describes the arrivals, and which can be computed using very fast finite difference techniques. We demonstrate the results on a variety of problems. This work is joint with Sergey Fomel.
AMS Classification: 49L25
Presenter: Swanhild Bernstein Bauhaus-University Weimar Mathematical Optimization Coudraystr. 13B D-99421 Weimar Germany
Email contact: sbernstein@sfb.uni-weimar.de
Title: The use of genetic algorithms in finite element model identification
Abstract: We precent a procedure to identify parameters of large structures fromgiven vibration data. The structures are modeled by finite elements and the parameters are identified by an optimization procedure based on genetic algorithms. We use frequency response functions in an output error formulation to state the objective function for our optimization problem. To demonstrate the basic features we consider a simply supported beam and a plate.
AMS Classification: 62P35
School of Mathematical and Computing Sciences Victoria University of Wellington PO Box 600 Wellington New Zealand
Email contact: Aleksandar.Stojmirovic@mcs.vuw.ac.nz
Title: Challenges of indexing into large datasets with the aim of similarity search.
Abstract: We survey the existing indexing schemes into large datasets with the purpose of similarity-based information retrieval and analyse the problem of dimensionality curse in the context of an interplay between the phenomenon of concentration of measure and complexity. In particular, we stress the difference between `inner' similarity search (when the query points all belong to the indexed dataset) and the `outer' search (when the space of potential query points is so large as to defy attempts at creating a precomputed index structure). We illustrate the main points using the authors' attempt to index into datasets of protein fragments currently under way.
AMS Classification: 68P10
Registered for B.H. Neumann Prize
Presenter: Stephen Roberts School of Mathematical Sciences ANU
Email contact: stephen.roberts@anu.edu.au
Title: An application of Sparse Grids to the estimation of probability density functions
Abstract: In this talk we will provide an overview of the approximation properties of sparse grids and then show how sparse grids can be used to provide an efficient method for the estimation of high dimensional probability density functions (pdf's). We take a simple approach and obtain an estimate of a pdf using an L^2 + Mixed Sobolov semi-norm projection method. This method in combination with the work of Markus Hegland on adaptive sparse grids has the potential to provide a efficient method for estimating pdf's in very high dimensions.
AMS Classification: 93E14
Presenter: *Phil Williams Markus Hegland Steve Roberts
Email contact: phil@csl.anu.edu.au
Title: An extension of Light and Wayne's basis function interpolaton theory to hat functions
Abstract: The goal of my Masters thesis was to develop a theoretical foundation for the analysis of the scalable smoothing algorithm which I have developed with the assistance of my supervisors. This theoretical work applies in any dimension but the smoother is only practical up to about 8 imensions. In future work we intend to improve the smoother using the theory of adaptive, sparse grids. Currently the smoother minimizes a functional on a regular rectangular grid. The functional consists of the sum of a smoothing seminorm term and a constraining least squares term. The minimal smoother is known as a basis function smoother and consists of a linear combination of basis functions translated by the data points plus a polynomial. The basis function is defined directly in terms of the components of the seminorm.
In fact, the function spaces used for smoothing are the same as those used for the minimal interpolation problem, where the functional only has a seminorm term. My thesis only studies the minimal seminorm interpolant problem.
In the seventies Duchon developed interpolation theory using seminorms constructed from a positive weight function. These weight functions generated thin plate spline basis functions. For these interpolants he was also able to obtain rates of pointwise convergence as the data becomes denser.
In the 90s Will Light and Henry Wayne extended the class of weight functions used to form the seminorm.
Unfortunately, our smoothing algorithms involve using tensor product hat basis functions, and these cannot be generated using the weight functions of Light and Wayne. However, we have been able to devise an extension of their class of weight functions to overcome this limitation and so allow the theory of Light and Wayne to be applied. Pointwise convergence results for the hat basis function interpolant have thereby been obtained.
AMS Classification: 41A05