# How to solve a linear system on a very large scale

I have a very large scale of linear system equations
enter the description of the image here where $$N_ {i}$$ denotes the neighborhood of a pixel $$i$$. $$C_ {i}$$ and $$psi_ {i}$$ and $$omega_ {i, j}$$ are known but $$widetilde {C_ {i}}$$ and $$widetilde {C_ {j}}$$ are unknown. And $$C_ {i}$$,
$$widetilde {C_ {i}}$$ and $$widetilde {C_ {j}}$$ can be considered as a number. I know it's a linear system on a very large scale, but I do not know how to solve it. It is the treatment of images and the algorithm of image processing