This form of publication is not a joke or a charade-type game: it imitates informal conversations between colleagues when the "back of the envelope" is used to illustrate ideas and approaches. In mathematics, this style has proven to be very efficient and seems to be consistent with mainstream tone of the Web. We decided to keep this informal style in our Web discussion of the serious problem of detectability which we formulate below. The standard mathematical text is placed in the Epilogue.

Andrej and Elena Cherkaev 

Please send your comments/suggestions to Andrej and Elena cherk@math.utah.edu

Characters:

Red Spot - hides in the Domain. 

Green Current - seeks Red Spot.

Inverse Problem - states the problem.

Optimi-Zation - solves the problem.

 Game Theory - gives the mixed strategy for Green Current.

Optimal Composites - provide an asylum for Red Spot who finally dissolves itself. 

Choir: Technical Diagnostics and Tomography - represent applications.

Prologue

Consider the problem of detection of a material's damage by simple electrical measurements. The question addressed is whether or not a conducting body contains damaged regions inside? Suppose that damaged material forms inclusions of a lower conductivity, and nothing is known about inclusions location.
A boundary current is applied to detect the presence of inclusions and the spent energy is measured. The sensitivity (resolution) of measurements is limited. An inclusion is detected if the difference between the spent energies for the body with and without inclusions is greater then the noise level of measurements. We want
- to find a current which detects the worst position of inclusions and
- to establish bounds for detectability.
The problem of detection naturally arises in technical diagnostics and tomography where the presence of cracks, corrosion, or other defects in a sample is to be determined.
__________

Rules of the game:

The problem is formulated as a game between a spot of damaged material 
which hides and the boundary current which seeks to find it.
 
I. Red player places inclusions (damaged material) of fixed total volume somewhere inside the domain.

II. Green player detects them by injecting arbitrary boundary currents of fixed energy. 

III. The goal of Red is to minimize the energy difference between spot-free sample and the real one, the goal of Green is to maximize it. 

IV. Green has no information about the Red's strategy. 
 
The cost of the game is equal to the resolution. The best "Green" strategy ensures the detection of defects of given size. 

An optimal location of a small inclusion and an optimal field density:

We analyse first the optimal location of an infinitesimal inclusion.
The measured response is proportional to the field density in the neighborhood of the inclusion. Therefore:

I. The `worst' inclusion hides itself where the field density is minimal.

II. An optimal current maximizes the minimum throughout the domain of its absolute value, hence the density of an optimal field is constant everywhere.

Green strategy

 
The best strategy for the Green is to make the field inside the tested body uniform.

The class of Green Currents always contains Neumann data which provide a constant field inside any homogeneous domain (see the picture).

The constant fields density is independent of the direction of an optimal current. This direction can be arbitrary chosen.

Red strategy:

A shape of the most hidden inclusions.

Hidden inclusions are elongated along the lines of the field: additional portions of the inclusion material hide in the shadow of seed inclusions.

Formation of an optimal laminated composite from the hidden inclusions.

In order to hide itself the inclusion disperses and forms laminates oriented along the field. 

One can check that these laminates fit the necessary conditions of optimality: the current is constant, the inclusions lead to minimal disturbance of the energy.

Solution: Inclusions form a laminate composite!

 
The most unfortunate distribution of inclusions: homogeneous laminates oriented along the field. The effective properties of laminates lead to an analytical lower bound for detectability which is independent of the shape of the domain.

Reformulation of the problem:

Green can do better either by applying a current in another direction or by applying several currents. The problem is reformulated as following:
either a constant current with random orientation is applied and the mean of detectability is maximized, or two orthogonal constant currents are applied.

Strategy for the reformulated problem

In the reformulated (relaxed) problem, Red uses Laminates of Second Rank which can be adjusted to minimize the sum of energies due to two orthogonal currents by changing the free parameter of inclusion density in the inner layers

Saddle point

The reformulated problem possesses a saddle point solution.

Optimal solution of the game is achieved by choosing the degree of anisotropy of the composite and the magnitude of one of the applied currents. 

Solution of the relaxed problem

Green strategy : the magnitudes of two orthogonal currents are equal

Red strategy: The worst location of inclusions is a uniform isotropic composite of the minimal conductivity.

This solution gives an analytical lower bound for detectability in the reformulated (relaxed) problem.

Epilogue

(seen the movie - read the paper!)

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ELENA  ANDREJ