On Constructing a Gradient Quadratic Optimization Method that is Optimal in Terms of Distance to the Exact Solution

Quadratic optimization problems 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 paper is devoted to the construction of the $$m$$ -moment minimal error method (MEM), which is an effective method that minimizes the distance to the exact solution. The convergence and optimality of the constructed method are proved, and the impossibility of uniform convergence of methods operating in Krylov subspaces is proved as well. Numerical experiments are carried out demonstrating the efficiency of applying the $$m$$ -moment minimal 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.