WORKSHOP: Multi-scale and high-contrast PDE: from modelling, to mathematical analysis, to inversion

The goal of this proposal is to ask for support from the Council towards the cost of a workshop on PDE which will be held in Oxford between June 28th and July 1st 2011.The mathematical analysis of PDE modelling materials presenting multiple scales have been an active area of research for more than 40 years. The study of the corresponding imaging, or reconstruction, problem is a more recent one. If the material parameters of the PDE present high contrast ratio, the solutions of the PDE become particularly challenging to analyse and to compute. Similar problems occur in time dependent equations in the frequency domain for high frequency. On the other hand, very high frequency regimes, or very contrasted materials, were considered first in imaging, as well-differentiated areas are, at first sight, simpler to locate by ad-hoc methods. Over the last decade the analysis of the inversion problem at moderate frequencies, the rigorous derivation of asymptotics at very high frequencies, and the regularity properties of solutions of elliptic PDE in very heterogeneous media have received a lot of attention.Part of the attention is due to the fact that these problems are particularly challenging. For another part, it is because of the numerous applications of these results in material sciences and in bio-medical imaging. Recently, emerging bio-medical imaging methods based on the observation of non-linear interactions of coupled physical phenomena (such as for example vibro-acoustography) have also become the subject of active research. Progresses on the mathematical understanding of the direct and inverse problems associated to these hybrid imaging methods are crucial to obtain enhanced imaging possibilities, beyond what is obtained by the integration of different imaging modalities taken separately. The focus of this workshop will be to stimulate collaborations between the participants, in the hope of achieving significant progress in (a) complete understanding of the direct problem with high contrast or high frequencies, (b) unified approaches to the inverse problem for both small and large contrast or frequencies, and (c) mathematical modelling of emerging experimental measurement methods. With this goal in mind, we wish to bring together senior experts and young researchers interested in the mathematical problems associated with imaging of multi-scale, or high contrast materials. All the mathematicians participating in the workshop are actively working on different aspects on these problems. Their expertise comprises heterogeneous random media, regularity theory for linear and non-linear PDE with very contrasted coefficients, mathematical invisibility (or cloaking), imaging and numerical reconstruction, numerical methods for high frequency elliptic problems, and emerging biomedical imaging methods. We have also invited an experimental physicist, whose recent work is devoted to new imaging methods for liquid crystals. The mathematical challenges associated with the mathematical formulation and understanding of these experiments and other hybrid measurement methods could be one of the applications of theoretical developments we hope this workshop will produce.

The problem of accurate imaging by non-invasive methods is of paramount importance in many applications, from medicine to material science. The research in medical imaging modalities is one of the celebrated strength of the UK, where CT, MRI, Impedance and Optical tomography were developed, leading to several Nobel prizes. The Inverse Problem journal is established in the UK, together with the Inverse Problem International Association. The mathematics of imaging is a growth area internationally. Several new journals (such as SIAM Imaging Science) have been created recently. The problem of accurate imaging by non-invasive methods is of paramount importance in many applications, from medicine to material sciences. The talks during the workshop will explain what is understood of this problem for high frequency problems, or in very contrasted media. The workshop should be most beneficial to mathematicians, but possible outcomes are very relevant to engineers and practitioners of inverse problems. The motivation to organise this workshop is originally due to the high impact of two preceding events organized in Seoul and Paris. The first one, entitled Workshop in Seoul: Inverse Problems, Multi-Scale Analysis and Homogenization was held in Seoul National University, Korea in 2005. The focus was the convergence of two previously distinct fields, homogenization theory and inverse problems. Among the outcomes of this workshop was a strengthened collaboration between physicists, working on time reversal experiments in material sciences. Amongst the mathematical achievements, the Poly-Szeg conjecture was solved, as a consequence of collaborations initiated by this workshop. A new imaging method, combining sonographic and electric measurements was also invented (and patented) as the result of a fruitful collaboration between participants. The second one, entitled Imaging Microstructure : Mathematical and Computational Challenges was held in Institut Henri Poincar, Paris in 2008. The focus of this meeting was on the mathematical challenges that appear in certain area of imaging. In this workshop, many new collaborations were developed, in particular between pure and applied mathematicians, as the workshop allowed the participants to realise a clear convergence of interests. The peer-reviewed proceedings of the workshop where published in the Contemporary Mathematics series of the AMS. During that workshop, it became apparent that the next frontier in imaging was high frequency, and/or high contrast. This workshop is focused on this challenging problem, which has deep implications both in the area of pure mathematics and in its applications to inverse problems as they appear in industry. In the two workshops mentioned above, several of the contributed papers where collaborations between either main speakers, or main speakers and attendees, and were the result of collaborations started during the workshop. We anticipate a similar outcome from this workshop. We plan to edit and publish proceedings form this workshop, either in the Modelling and Simulation in Medical Imaging Book Series of Imperial College Press, or in Contemporary Mathematics series of the AMS.