**Contemporary Mathematics**

Volume: 577;
2012;
144 pp;
Softcover

MSC: Primary 35; 65; 76; 93;

**Print ISBN: 978-0-8218-6929-1
Product Code: CONM/577**

List Price: $66.00

AMS Member Price: $52.80

MAA Member Price: $59.40

**Electronic ISBN: 978-0-8218-9100-1
Product Code: CONM/577.E**

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# Multi-Scale and High-Contrast PDE: From Modelling, to Mathematical Analysis, to Inversion

Share this page *Edited by *
*Habib Ammari; Yves Capdeboscq; Hyeonbae Kang*

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.

#### Readership

Graduate students and research mathematicians interested in applications of PDE to material sciences.

# Table of Contents

## Multi-Scale and High-Contrast PDE: From Modelling, to Mathematical Analysis, to Inversion

- Preface vii8 free
- Enhancement of near-cloaking. Part III: Numerical simulations, statistical stability, and related questions 110 free
- Looking at the world through liquid crystal glasses 2534
- 1. Introduction 2534
- 2. Polarizing microscopy 2736
- 3. LC-PolScope: Variable and universal compensators 3039
- 4. Fluorescence Confocal Polarizing Microscopy 3342
- 5. Coherent Anti-Stokes Raman Scattering Microscopy 3847
- 6. Fermat Principle and Extraordinary Mode in Liquid Crystals with Gradient Properties 4049
- 7. Conclusion 4352
- Acknowledgement 4453
- References 4453

- A remark on the observability of conservative linear systems 4756
- On the scattered field generated by a ball inhomogeneity of constant index in dimension three 6170
- Pointwise bounds on the gradient and the spectrum of the Neumann-Poincaré operator: The case of 2 discs 8190
- A high-contrast fourth-order PDE from imaging: numerical solution by ADI splitting 93102
- Imaging with noise blending 105114
- 1. Introduction 105114
- 2. Multi-offset sources 109118
- 3. Stationary random sources 110119
- 4. Incoherence by blending 113122
- 5. Applications 116125
- 6. Numerical illustrations 120129
- 7. Conclusion 121130
- Appendix A. Statistical stability for a continuum of point sources 122131
- Appendix B. Resolution analysis 122131
- References 123132

- Correlations of heterogeneous wave fields propagating in homogeneous media 125134