Abstract:
Quadratic optimization problems in Hilbert space often arise when solving ill-posed problems for differential equations. At the same time, the target value of the functional is known. In addition, the functional structure makes it possible to calculate the gradient by solving correct problems, which allows applying first-order methods. This article is devoted to the construction of the $m$-moment method of minimal errors, an effective method that minimizes the distance to an accurate 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 being conducted to demonstrate the effectiveness of applying the $m$-moment minimum error method to solving various incorrect problems: the initial boundary value problem for the Helmholtz equation, the retrospective Cauchy problem for the heat equation, and the inverse thermoacoustics problem.
Key words:incorrect and inverse problems, quadratic optimization, optimization in hilbert space, minimizing the distance to the exact solution, initial boundary value problem for the Helmholtz equation, Cauchy retrospective problem for the heat equation, inverse thermoacoustics problem.
