In this paper we propose the gradient descent type methods to solve convex optimization problems in Hilbert space. We apply it to solve ill-posed Cauchy problem for Poisson equation and make a comparative analysis with Landweber iteration and steepest descent method. The theoretical novelty of the paper consists in the developing of new stopping rule for accelerated gradient methods with inexact gradient (additive noise). Note that up to the moment of stopping the method ``doesn't feel the noise''. But after this moment the noise start to accumulate and the quality of the solution becomes worse for further iterations.
