\documentclass[12pt, a4paper]{article}

\title{Numerical Identification of Unknown Coefficient in Helmholtz
Equation via Method of Variational Imbedding}

\author{ Tchavdar  Marinov  \\
{\small\it Dept. of Mathematics, Technical University of
Varna,}\\[-0.1cm]
{\small\it Varna, 9010, BULGARIA }
}
\date{}

\begin{document}


\maketitle

We consider the problem of identification of the refraction index
$n(x,y)$ in inhomogeneous medium when the wave amplitude inside
the region $D\>$ is governed by Helmholtz equation $\Delta u + n(x,y)
u = 0$ and when over-posed boundary data is available:
$$
\left.
u\right|_{\partial D}
= \varphi \  \hbox{and}\
\left.
\frac{\partial u}{\partial \nu}\right|_{\partial D}
= \psi
$$
%
Following the previous authors' works on coefficient identification
in parabolic equations, we ``imbed'' the inverse problem into a
fourth-order elliptic boundary value problem for Euler-Lagrange
equation being the necessary condition for minimization with respect
to function $u$ of the quadratic functional of the original equation,
namely
$$
\Delta\Delta u + \Delta [n(x,y) u] + n(x,y)\Delta u + n^2(x,y) u
=0 \>.
$$
%
It is well-posed with the two boundary conditions under consideration.
The Euler-Lagrange equation for $n(x,y)$ provides an explicit equation
for the unknown refraction index.

The equivalence of the two problems is demonstrated and iterative
procedure is devised for solving the ``imbedding'' b.v.p. A
featuring example is elaborated numerically where the over-posed date
is obtained from the solution of the ``direct'' problem for Helmholtz
equation with given refraction index.  Consecutively, the same
coefficient is identified by means of Method of Variational Imbedding.


\end{document}