Hi @AlexanderChervov! I'm working on a problem similar to this, except that my h=v=0 but, the derivative of my matrix S is multiplied point-wise with a weight matrix W_x and W_y. I can follow the same procedure to solve the optimization problem, but when I apply FFT, I got F*(dx)F(W_x o dx(S)) where o is point wise multiplication. How can I isolate S to have a closed form formula for S? in other words, what is the DFT of point wise multiplication? thank you in advance!

You can get using the "Gateau" definition of derivative. Namely, if you have a function J, you can get it's Gateau derivative in point S, using the formula: lim (epsilon ->0) 1/epsilon * [J(S + epsilon * M) - J(S)]