Preface

Recent developments in inverse problems, multi-scale analysis, and effective

medium theory reveal that these fields share several fundamental concepts.

Over the last 10 years or so, a considerable amount of interesting work has

been dedicated to the imaging of small inclusions. To image small inclusions with

a good resolution one needs to design stable and accurate algorithms for the nu-

merical computations of solutions to partial differential equations in the presence of

small-size features. It is known that small-size features cause difficulties in the nu-

merical solution by the finite element or finite difference methods. This is because

such features require refined meshes in their neighborhoods, with their attendant

problems. Some of the promising techniques developed in the field of multi-scale

analysis are very useful in this context.

On the other hand, the derivation of optimal bounds for the volume fraction of

the small conductivity inclusions are direct analogues of the corresponding estimates

for the effective conductivity matrix known in the theory of composite materials.

The main purpose of this volume is to highlight the benefits of sharing new, deep

ideas among the fields of inverse problems, multi-scale analysis, and effective medi-

um theory. It provides exposition of fresh techniques for solving inverse problems,

emphasizing their connection with multi-scale analysis and the mathematical theory

of composite materials.

The mathematical problems that appear in these three areas are of practical

importance and pose significant challenges to pure and applied mathematicians.

These problems have attracted a lot of attention of researchers lately. The meth-

ods involved come from a wide range of areas of pure and applied mathematics,

ranging from potential theory to partial differential equations, to scattering theory,

to complex analysis, to numerical methods. The topics of this volume are addressed

from analytic, numerical, and physics perspectives.

Due to the character of its topic, this volume is of interest not only to math-

ematicians working in these areas, but also to physicists and engineers who could

communicate with mathematicians on these issues. We envision that these pro-

ceedings will stimulate much needed progress in the directions described above.

This is the proceedings of the research conference "Workshop in Seoul: Inverse

Problems, Multi-Scale Analysis, and Homogenization" held at Seoul National Uni-

versity, June 22-24, 2005. We believe that our objective to bring together experts in

these related fields and to share their ideas was successfully achieved. This was only

possible thanks to the enthusiastic participation of wonderful speakers and authors

of this volume. We thank them all. We also thank the staffs at the mathematics

department and RIM of Seoul National University, without whom the workshop

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