High-order, High-accuracy Solution of a Nonlinear PDE Arising in a Two-dimensional Heat Transfer Model

Host Institution:

University of South Australia

Title of Seminar:

High-order, High-accuracy Solution of a Nonlinear PDE Arising in a Two-dimensional Heat Transfer Model

Speaker's Name:

Professor Robert M. Corless

Speaker's Institution:

Western Applied Mathematics, The University of Western Ontario

Time and Date:

Monday 9 December 2013, 2.30pm (ACDT) - 3.00pm AEDT

Seminar Abstract:

A classical nonlinear PDE used for modelling heat transfer between concentric cylinders by fluid convection and also for modelling porous flow can be solved by hand using a low-order perturbation method. Extending this solution to higher order using computer algebra is surprisingly hard owing to exponential growth in the size of the series terms, naively computed. In the mid-1990's, so-called "Large Expression Management" tools were invented to allow construction and use of so-called "computation sequences" or "straight-line programs" to extend the solution to 11th order. The cost of the method was O(N^8) in memory, high but not exponential.
Twenty years of doubling of computer power allows this method to get 15 terms. A new method, which reduces the memory cost to O(N^4), allows us to compute to N=30. At this order, singularities can reliably be detected using the quotient-difference algorithm. This allows confident investigation of the solutions, for different values of the Prandtl number.

This work is joint with Yiming Zhang (PhD Oct 2013).

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Statistical Analyses dealing with Water & Environment

Host Institution:

La Trobe University

Title of Seminar:

Statistical Analyses dealing with  Water & Environment

Speaker's Name:

Dr Mark James Fielding

Speaker's Institution:

DHI Water & Environment

Time and Date:

Monday 16 December, 11.00am (AEDT)

Seminar Abstract:

With varied methods used for multivariate extreme value analyses, a number of different techniques are compared, toward
the development of a more standard approach. Issues will be addressed with the determination of univariate and
bivariate extreme levels, with applications in extreme rainfall.

Time permitting, other areas of statistical analyses in Water & Environment may touched upon. Including Gaussian process emulation of nonlinear computer outputs.

 

 

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On the classification of Quantum groups

Host Institution:

La Trobe University

Title of Seminar:

On the classification of Quantum groups

Speaker's Name:

Professor Alexander Stolin

Speaker's Institution:

University of Gothenburg, Sweden

Time and Date:

Friday 15 November, 2.00pm (AEDT)

Seminar Abstract:

We will explain an approach to classification of certain quantum groups. Let g be a complex simple Lie algebra. A quantum group is a Hopf algebra over C[[h]], which has g as its classical limit. To obtain it we, roughly speaking, set h=0. More precisely, being a classical limit of a Hopf algebra, g becomes a Lie bialgebra. It is well-known that simple Lie algebras are classified by means of the so-called Dynkin diagrams. In 1982, Belavin and Drinfeld classified the corresponding Lie bialgebras by means of the Belavin-Drinfeld triples, which can be described as two isomorphic subdiagrams of the Dynkin diagram of g (they are called triples because the isomorphism between the subdiagrams matters). In order to classify the corresponding quantum groups we introduced further combinatorial data, which we called Belavin-Drinfeld cohomologies. There are two types of the BD-cohomologies, namely nontwisted and twisted versions.In my talk, I will explain how to describe BD-cohomologies for special linear and orthogonal algebras. No prior knowledge except for the standard facts about simple Lie algebras is needed.

This is a joint work Iulia Pop (Chalmers University of Technology, Sweden).

 

 

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An approach to solving decomposable optimization problems with coupling constraints

Host Institution:

University of South Australia

Title of Seminar:

An approach to solving decomposable optimization problems with coupling constraints

Speaker's Name:

Oleg Burdakov

Speaker's Institution:

Linko¨ping University, Sweden

Time and Date:

Tuesday 10 December 2013, 2.30pm (ACDT) - 3.00pm AEDT

Seminar Abstract:

We consider a problem of minimising f1(x)+f2(y) over x ∈ X ⊆ Rn and y ∈ Y ⊆ Rm subject to a number of extra coupling constraints of the form g1(x)g2(y) ≥ 0. Due to these constraints, the problem may have a large number of local minima. For any feasible combination of signs of g1(x) and g2(y), the coupled problem is decomposable, and the resulting two problems are assumed to be easily solved. An approach to solving the coupled problem is presented. We apply it to solving coupled monotonic regression problems arising in experimental psychology.

Co-authors: John C. Dunn and Mike Kalish

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Integrable-like behavior in the Fermi-Pasta-Ulam model

Host Institution:

La Trobe University

Title of Seminar:

Integrable-like behavior in the Fermi-Pasta-Ulam model

Speaker's Name:

Dr Heleni Christodoulidi

Speaker's Institution:

Patras University, Greece

Time and Date:

Monday 18 November, 4.00pm (AEDT)

Seminar Abstract:

In 1950’s Fermi, motivated by fundamental questions of statistical mechanics, started a numerical experiment in collaboration with Pasta and Ulam to test the ergodic properties of nonlinear dynamical systems. The chosen so-called FPU system was a one dimensional chain of N nonlinear coupled oscillators, described by a quadratic potential of nearby particle interactions plus a cubic perturbation. Fermi’s ergodic hypothesis states that a system under an arbitrarily small perturbing force becomes generically ergodic. Starting with the longest wavelength normal mode, the FPU system showed a non-ergodic behavior. Many pioneer works followed for the explanation of this paradox. The most prominent of them have been the work of Zabusky and Kruskal (1965), with evidence of connection between the FPU system in the thermodynamic limit and the pde Korteweg-de Vries, and the work of Flaschka et al. (1982), where the authors discovered a similar behavior of the FPU model in the Toda chain. Recent developments show a more complete picture of the problem and its explanation.

 

 

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