The measured data are simulated by solving the direct problem, in which a modified zonal method together with the Edwards exponential wide-band model is employed.
Using an initial guess from the Born approximation, each update is obtained via continuation on the spatial frequency of a two-parameter family of plane waves by solving one direct problem and one adjoint problem of the Maxwell equation.
In Examples 2 and 3, since the direct problem with the source f ( x ) does not have an analytical solution, the data g ( x ) is obtained by solving the direct problem.
The computed temperatures are obtained by solving the direct heat conduction problem with the finite difference method.
The numerical examples were constructed in the following way: First we selected the source function f ( x ), and we obtained the exact data function g ( x ) by solving the direct problem.
Using a set of predefined eigenvalues obtained by solving repeatedly the direct problem, a radial basis neural network is designed with the purpose to find the appropriate components of the anisotropy matrix, related to the Laplace operator, and thus solving the associated inverse problem.
The cost function is computed at each vertex of a simplex defined in the space of optimization parameters by solving direct problem thanks to a finite element method.
The dataset for reconstruction was computed by solving direct problems, Eq. (6), when the direction of the incident direct radiation, s0, was rotated by 5° around the z-axis over 360° starting from its initial direction s0 = (1,0,0) T. The source plane r0 · s0 = const was parallel to the z-axis (e z · s0 = 0) and set at some distance from the scattering domain.
The optimal control problem is solved by a direct method, transcribing it into a nonlinear programming problem.
