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I am trying to use finite difference method to solve the minimizing problem $$ J[u]=\min_{u\in BV(Q)}\{\|u-f\|_{L^1(Q)}+|u|_{BV(Q)}\} $$ where $Q=(0,1)\times (0,1)$ is a uint square and $|\cdot|_{BV}$ is the standard BV semi-norm.

I am really new for this area. (before I am only working on the theoretical level of mathematical analysis, never study numerical method before...). So, could you guys recommend me some notes or book to help me to get start on finite difference method, particular for the problem I mentioned? Also, if it possible, could you give me some reference which contains the actual code, maybe matlab code, for the problem similar to what I mentioned above?

Thank you!

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  • $\begingroup$ Is there any possibility to replace $||u-f||_{L_1(Q)}$ by $||u-f||^2_{L_2(Q)}$? Also $|u|_{BV(Q)}$ with $||u'||^2_{L_2(Q)}$. Thus the problem becomes minimization of a quadratic functional, which is a lot simpler $\endgroup$ – uranix Jun 24 '15 at 20:46
  • $\begingroup$ @uranix The use of BV norm is exactly what we want. Theoretically it will behave much better then quadratic functional :) $\endgroup$ – spatially Jun 24 '15 at 20:53
  • $\begingroup$ there's no big deal to discretize the problem. Just take a look at Galerkin and Ritz methods. The deal is to solve the resuling nonsmooth minimization problem $\endgroup$ – uranix Jun 24 '15 at 20:58
  • $\begingroup$ @uranix Thx! I will check it out~ $\endgroup$ – spatially Jun 24 '15 at 20:59
  • $\begingroup$ That's not exactly a finite difference method, but a finite element one. $\endgroup$ – uranix Jun 24 '15 at 21:00

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