Integral representations in electromagnetic inversion

The image reconstruction is based on the numerical solution of an inverse scattering EM problem. This problem requires the development of intelligent methods for EM forward modeling and inversion in inhomogeneous media. These problems arc extremely difficult, especially in three-dimensional cases. 

Difficulties arise even in forward modeling because of the huge size of the numerical problem to be solved for adequate representation of the complex 3-D distribution of EM parameters in the media. As a re.sult, computer simulation time and memory requirements could be excessive even for practically realistic models. Additional difficulties are related to EM imaging which is based on EM inverse problems. These problems are nonlinear and ill-posed, because, in general cases, the solutions can be unstable and/or nonunique. In order to overcome these difficulties one should 

The methods for solving multi-dimensional EM inverse problems are usually based on optimization of the model parameters by applying different inversion schemes. The key problem in the optimization technique, as we demonstrated in Chapter 5, is the calculation of the Prechet derivative (sensitivity matrix), which usually requires much computational time. 

A powerful tool for EM modeling and inversion is the integral equation (IE) method and the corresponding linear and nonlinear approximations, introduced in the previous chapter. One important advantage which the IE method has over the finite difference (FD) and finite element (FE) methods is its greater suitability for inversion. Integral equation formulation readily contains a sensitivity matrix, which can be recomputed at each inversion iteration at little expense. With finite differences, however, this matrix has to be established anew on each iteration at a cost at least equal to the cost of the full forward simulation. 

In this chapter we will outline the basic principles of electromagnetic inversion and imaging based on integral representations of the electromagnetic field developed in Chapter 9. 

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