Multi-Scale and High-Contrast PDE: From Modelling, to Mathematical Analysis, to Inversion
About this Title
Habib Ammari, Ecole Normale Supérieure, Paris, France, Yves Capdeboscq, Mathematical Institute, Oxford, United Kingdom and Hyeonbae Kang, Inha University, Incheon, Korea, Editors
Publication: Contemporary Mathematics
Publication Year 2012: Volume 577
ISBNs: 978-0-8218-6929-1 (print); 978-0-8218-9100-1 (online)
This volume contains the proceedings of the conference “Multi-Scale and High-Contrast PDE: From Modelling, to Mathematical Analysis, to Inversion”, held June 28–July 1, 2011, at the University of Oxford.
The mathematical analysis of PDE modelling materials, or tissues, presenting multiple scales has 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, then the solution to the PDE becomes particularly challenging to analyze, or compute. Similar difficulties occur in time dependent equations in high frequency regimes. Over the last decade the analysis of the inversion problem at moderate frequencies, the rigorous derivation of asymptotics at high frequencies, and the regularity properties of solutions of elliptic PDE in highly heterogeneous media have received a lot of attention.
The focus of this volume is on recent progress towards a complete understanding of the direct problem with high contrast or high frequencies, and unified approaches to the inverse and imaging problems for both small and large contrast or frequencies. The volume also includes contributions on the inverse problem, both on its analysis and on numerical reconstructions. It offers the reader a good overview of current research and direction for further pursuit on multiscale problems, both in PDE and in signal processing, and in the analysis of the equations or the computation of their solutions. Special attention is devoted to new models and problems coming from physics leading to innovative imaging methods.
Graduate students and research mathematicians interested in applications of PDE to material sciences.
Table of Contents
- Habib Ammari, Josselin Garnier, Vincent Jugnon, Hyeonbae Kang, Hyundae Lee and Mikyoung Lim – Enhancement of near-cloaking. Part III: Numerical simulations, statistical stability, and related questions
- Oleg D. Lavrentovich – Looking at the world through liquid crystal glasses
- Enrique Zuazua – A remark on the observability of conservative linear systems
- Yves Capdeboscq, George Leadbetter and Andrew Parker – On the scattered field generated by a ball inhomogeneity of constant index in dimension three
- Eric Bonnetier and Faouzi Triki – Pointwise bounds on the gradient and the spectrum of the Neumann-Poincaré operator: The case of 2 discs
- Bertram Düring and Carola-Bibiane Schönlieb – A high-contrast fourth-order PDE from imaging: numerical solution by ADI splitting
- Maarten de Hoop, Ennio Fedrizzi, Josselin Garnier and Knut Sølna – Imaging with noise blending
- Guillaume Bal and Olivier Pinaud – Correlations of heterogeneous wave fields propagating in homogeneous media