# Finite difference scheme for solving nonlinear least-squares problem

I am dealing with following problem:

$$\min_{u,\gamma}\Bigg\{ \frac{1}{1000} \iint_{S_2} {\gamma (x,y)^2 dxdy} + \iint_{S_2} {[u(x,y) - u_0 (x,y)]^2 dxdy} + \iint_{S_2} {[\Delta u(x,y) - \gamma (x,y) u(x,y)]^2 dxdy}\Bigg\}$$

where the unknown functions $u(x, y)$ and $\gamma(x, y)$ are defined on the unit square $S_2$ and the function $u_0(x, y)$ is defined on $S_2$ by $u_0(x, y) = \sin(6\pi x) \sin(2\pi y)$.

This problem should be discretized using 5-point finite differences.

I have no experience discretizing such problems. Could somebody help me out, or point me to right literature to look for some examples?

Note: this problem comes from the paper Recursive trust-region methods for multiscale nonlinear optimization (page 10)

• Could you try to make the equation easier to read using LaTeX? – fibonatic Jul 25 '15 at 2:26
• I think there might be a bug in the middle term. As written, that's identically zero. Presumably one of those $u$ terms must be a $u_0$, but I'll let you fix it. – Bill Barth Jul 25 '15 at 15:06
• you are right, I was missing subscript in one term – SmallElephant Jul 25 '15 at 15:07
• What kind of operation does $\Delta$ perform on $u(x,y)$? – fibonatic Jul 27 '15 at 3:36
• @fibonatic do u have any suggestion? I am mostly curious how to deal with last term – SmallElephant Jul 27 '15 at 18:01