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Inverse Problems

Participation in INI programmes is by invitation only. Anyone wishing to apply to participate in the associated workshop(s) should use the relevant workshop application form.

25th July 2011 to 21st December 2011

Organisers: Malcolm Brown (Cardiff), Thanasis Fokas (Cambridge), Yaroslav Kurylev (University College London),
Bill Lionheart (Manchester) and William Symes (Rice)

Scientific Advisors: Simon Arridge (University College London), Christer Bennewitz (Lund), Margaret Cheney (Rensselaer Polytechnic Institute), Chris Farmer (Oxford), Jari Kaipio (Auckland), Andreas Kirsch (Karlsruhe), Lassi Päivärinta (Helsinki) and Gunther Uhlmann (Washington)

Programme Theme

Many important real world problems give rise to an Inverse problem (IP). These include medical imaging, non-destructive testing, oil and gas exploration, land-mine detection and process control. For example, in the exploration for oil and gas, one needs to assess the structure of the interior of the earth from observations made at the surface. Typically, an explosion is created and the resulting shockwaves together with their reflections are used to build a model of the structure of the earth. In magnetoencephalography one needs to determine the electric current in the neurones from the measurement of the magnetic field outside the head. In the field of medical imaging IP forms an important tool in diagnostic investigations. For example, PET and SPECT are two modern imaging techniques whose success is dependent on solving IPs.

At their simplest level IP are concerned with obtaining information about the interior of a body from data which is available at its surface. Mathematically, this is a parameter identification problem: given a set of data representing the behaviour of solutions, identify the unknown parameters of the model.

The mathematical machinery needed for solving various IPs is mainly founded in mathematical analysis and uses tools from functional analysis, function theory, conformal maps, spectral theory, theory of PDEs, integral equations, and micro-local and global analysis. In recent years tools from differential geometry, stochastic analysis, etc., are becoming important. Moreover, in order to realise the solution to many applied problems in a useful way, the tools of numerical analysis and scientific computing are needed.

We intend this programme to help cross-fertilise ideas between scientists by providing them with the opportunity to work on important problems with experts from other groups. We intend to focus on the following topics during the period of the programme: Inverse spectral problems, Analytic and geometric methods for IP, Stochastic methods, Numerical methods, and Application of IP to industry, medicine, finance, biology, and exploration seismology; and plan to organise several workshops which focus on these highlighted areas, as well as a tutorial meeting.

Final Scientific Report: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute The Leverhulme Trust London Mathematical Society Microsoft Research NM Rothschild and Sons