Numerical Solution Of Inverse Problem For Elliptic Pdes

This work is concerned with computing the solution of the following inverse problem: Finding u and 𝜌on D such that: $$\nabla \cdot (\rho \nabla u) = 0,\quad \hbox{on}\ D;$$ $$u = g,\quad \hbox{on}\ \partial D;\qquad \rho u_n = f,\quad \hbox{on}\ \partial D;$$ $$\rho (x_0, y_0) = \rho_0,\quad \hbox{for a given point}\ (x_0, y_0) \in D$$ where f and g are two given continuous functions defined on the boundary of D , and D is a given bounded region of R 2 . The solution is found using a development of the direct variational method. The two unknown functions are represented by linear combinations of certain classes of functions and using multiobjective optimization to minimize the two objective functionals F and H , where $$F = \vint \vint_D \rho (x,y) \nabla u\cdot \nabla u\,\hbox{d}x\,\hbox{d}y\quad \hbox{and}\quad H = \vint_{\partial D} (\rho u_n - f)^2 \hbox{d}s$$ A computer program is written and implemented and tested for data formed by numerical simulation.

Keywords:

• Inverse Problem
• Elliptic Pdes
• Multiobjective Optimization
• Direct Variational Method