Columbia University,

**Jochen Brüning **

Humboldt University of Berlin, *bruening@mathematik.hu-berlin.de*

University of Auckland,

In this talk I will introduce the beautifully symmetric properties of toroidal harmonics and discuss how one may obtain toroidal Fourier expansions, for free-space Green's functions, whose Fourier coefficients are given in terms of these harmonics. Toroidal Fourier expansions might be obtained in all the rotationally invariant coordinate systems which allow separation of variables for a homogeneous 3-variable partial differential equation which admits a free-space Green's function. In this regard, I will focus and present explicit formulae as applied to the free-space Green's function for the 3-variable Laplace equation and for the 3-variable biharmonic equation. Finally, I will conclude with a discussion of some results and conjecture into the possibility of generalising this notion to various other 3-variable partial differential equations which admit a free-space Green's function.

University of Auckland,

The aim of this work is the development of a method that will allow the automated calibration of computer models of geothermal fields. The mathematical method used is Markov chain Monte Carlo (MCMC) sampling from the posterior distribution. The technique is applied to calibrating a simple single-layer model of the feed-zone of a well, using discharge test measurements of flowing enthalpy and well-head pressure, and secondly to calibrating a large 3D natural state model using pre-exploitation measurements of temperature versus depth in several wells.

Berkeley Lab,

Australian National University,

We will discuss simple conditions equivalent to the discreteness of the spectrum of the laplacian on a class of riemannian manifolds and connections with brownian motion on the manifold.

Texas A&M University,

**Jari Kaipio **

University of Kaipio, *jpkaipio@physics.uku.fi*

University of Waikato,

University of Kaipio,

UCLA,

**Pavel Kurasov **

Stockholm University, *pak@math.su.se*

Massey University,

Bringing images into alignment (so that they look 'the same') promises to be a useful method for automated diagnosis from medical images. More importantly, it contains several interesting mathematical problems. In my formulation of the problem it requires solving the Euler equations on the diffeomorphism group, and then interpolating the resultant deformation across the entire image. I will present an overview of the problem, detail some of the work we've done so far, and suggest areas that still need investigation.

University of Auckland,

**Vladimir Oleinik **

University of Auckland, *vladimir@phatware.co.nz*

A set of periodic problems, each composed of shifts of a suitable basic non-periodic one, is considered. The existence of an isolated eigenvalue of the non-periodic problem is assumed. In this case the corresponding periodic problem has a spectral band (an isolated interval of continuous spectrum) which lies in a neighbouhood of the eigenvalue. Moreover the width of the band vanishes as the period goes to infinity. Examples include the Schrödinger operator, metric graphs and plane waveguides. The first one is of primary importance for solid state physics.

University of Auckland,

University of Auckland,

TBA

Graz University of Technology,

I will discuss principles of capacitive and magnetic position sensing, probabilistic design of sensors and instrumentation, and application to multiphase flow measurement.

Electrical Capacitance Tomography (ECT) is a non-invasive technique that aims at estimating the permittivity distribution within closed objects from boundary measurements. Such objects are, for instance, pipelines in the oil industry or chambers and vessels in the production of food, chemicals and pharmaceuticals. ECT is able to provide information about internal states with low-cost requirements and is therefore well suited for industrial applications in order to determine process parameters like void fraction. In this talk different regularized deterministic methods (e.g. Gauss-Newton-based permittivity reconstruction, level set method) and sampling-based statistical approaches (particle filter, MCMC) to solve the inverse ECT problem from measured electrical capacitance data will be addressed. A key difference between regularization and Bayesian methods in inverse problems is that sampling methods present averages over all solutions to the inverse problem consistent with the measured data while regularization gives single point estimates based on a data-misfit criterion. This may lead to a significant difference in robustness of process parameters of interest calculated from the inverse problem solutions. The single most likely solution, found by a regularized minimization of misfit to the measured data, may be unrepresentative of the bulk of feasible solutions in a high-dimensional non-linear problem such as it occurs in ECT. The advantages of sampling methods, like comprehensive information about the parameter space and treatment of arbitrary noise distributions come at the cost of increased computational effort. The talk presents a synopsis of the performance of the different classes of algorithms in the context of industrial process tomography.

**Steve Taylor **

University of Auckland, *taylor@math.auckland.ac.nz*

Consider a PDE that involves a time variable t. The solution of the PDE depends on the initial condition (a given function of the space variables, say at t=0) and boundary conditions. Boundary control of such a PDE involves finding appropriate boundary conditions so that the final value of the solution (say at t=T>0) is a specified function of the space variables. In this talk we consider links between boundary control and certain smoothing properties of partial differential operators.

University of Auckland,

We discuss operators on R

Graz University of Technology,

This talk will cover the application of condition monitoring of large diesel engines. Important issues are the mathematical modelling, thermodynamics of the processes, and how to perform fault diagnosis. Measurements are made using electrical capacitance tomography (ECT). I will discuss motivation for using this modality, implementation details, measurement noise, numerical modeling, and robust inversion using statistical methods implemented via Markov chain Monte Carlo (MCMC) sampling and particle filters.

Electrical Capacitance Tomography (ECT) is a non-invasive technique that aims at estimating the permittivity distribution within closed objects from boundary measurements. Such objects are, for instance, pipelines in the oil industry or chambers and vessels in the production of food, chemicals and pharmaceuticals. ECT is able to provide information about internal states with low-cost requirements and is therefore well suited for industrial applications in order to determine process parameters like void fraction. In this talk different regularized deterministic methods (e.g. Gauss-Newton-based permittivity reconstruction, level set method) and sampling-based statistical approaches (particle filter, MCMC) to solve the inverse ECT problem from measured electrical capacitance data will be addressed. A key difference between regularization and Bayesian methods in inverse problems is that sampling methods present averages over all solutions to the inverse problem consistent with the measured data while regularization gives single point estimates based on a data-misfit criterion. This may lead to a significant difference in robustness of process parameters of interest calculated from the inverse problem solutions. The single most likely solution, found by a regularized minimization of misfit to the measured data, may be unrepresentative of the bulk of feasible solutions in a high-dimensional non-linear problem such as it occurs in ECT. The advantages of sampling methods, like comprehensive information about the parameter space and treatment of arbitrary noise distributions come at the cost of increased computational effort. The talk presents a synopsis of the performance of the different classes of algorithms in the context of industrial process tomography.

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On 03 Jan 2007, 19:02.