On the construction of a gradient method of quadratic optimization, optimal from the point of view of minimizing the distance to the exact solution

Problems of quadratic optimization in Hilbert space often arise when solving ill-posed problems for differential equations. In this case, the target value of the functional is known. In addition, the structure of the functional allows calculating the gradient by solving well-posed problems, which allows applying first-order methods. This article is devoted to the construction of the $m$-moment minimum error method -- an effective method that minimizes the distance to the exact solution. The convergence and optimality of the constructed method are proved, as well as the impossibility of uniform convergence of methods operating in Krylov subspaces. Numerical experiments are carried out demonstrating the efficiency of applying the $m$-moment minimum error method to solving various ill-posed problems: the initial-boundary value problem for the Helmholtz equation, the retrospective Cauchy problem for the heat equation, and the inverse problem of thermoacoustics.