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I have a linear minimization problem as, $$ \inf_u E(u)=‎ \inf_u \frac{\mu}{2} \int_{\Omega}(u-f)^{2}d\Omega+\frac{c}{2}||u||^{2}-\langle u, g \rangle, $$ where $f$ and $g$ are fixed and c is a positive scalar. Since this minimization problem is linear, I tried as follows, $$ ‎\nabla E(u)=\mu(u-f)+cu-g=0, $$ ‎As a result,‎‎ $$ ‎‎‎u=\frac{1}{\mu+c}(\mu f+g).‎ $$ What I have tried is ofcourse traditional and I need an iterative scheme to find u because it is an image denoising problem and I need to find the denoised image. I appreciate it if someone could please help me how to solve this problem with an iterative scheme and whether my own method can be transformed to an iterative one?

eli
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    The expression to minimize does not look like a typical image denoising, which usually has a $||\nabla u||^2$ term. For the expression you present, there is no need for iterations. You have found the fixed point directly. – user619894 Nov 29 '20 at 17:04
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    @ user619894 Yes, but not all image denoising problems have a $||\nabla u||^2$ term e.g. tikhonov regularization. What I have tried does not produce a denoised image and I am seeking for how to gain a denoised image which can be obtained by an iterative scheme. The fixed ponit I have gained is not a denoised image. – eli Nov 29 '20 at 17:58
  • @eli You already found the unique answer to the above problem in closed form, so if it does not look sufficiently denoised, then you may need to try defining a different problem (one whose solutions will look more denoised). Regularizing by the norm of the gradient of the image is a common method, but certainly not the only one. – Zim Nov 29 '20 at 23:11
  • @Zim But I can not change the minimization problem. Am I allowed to use iterative methods like accelerated gradient or Euler lagrange equation for such problem? – eli Nov 29 '20 at 23:32
  • @eli why can't you change the problem? I mean, you could use a gradient descent algorithm to solve it, but you already know what the answer is going to be. If you must keep the exact same problem structure, perhaps editing your parameters would yield better performance – Zim Nov 30 '20 at 00:30
  • @Zim Many thanks. I will change the parameters. Maybe I gain a better denoised result. – eli Nov 30 '20 at 00:47

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