Asymptotic optimality of projection methods (Q1589197)

The aim of the paper is the study of asymptotic optimality for the numerical projection methods for the approximation of the exact solution for operator equations. The method for the investigation of this problem is based on ideas of B. G. Gabdulchaiev. Let \[ Kx= y\quad (x\in X, y\in Y)\tag{1} \] be an operator equation on the real or complex normed linear spaces \(X\) and \(Y\), \(K\) a linear bijection from \(X\) to \(Y\). Let further for a fixed integer \(n\) \[ P_n Kx_n= P_n y\quad (x_n\in X)\tag{2} \] be a projection of this equation, where \(P_n\) is a linear surjection from \(Y\) onto \(Y_n\). Equation (2) is said to be a projection method for solving (1). Let the equation (2) have the unique solution \(x^*_n\). The problem is to find such a method for which the distance of the solution \(x^*_n\) from the solution \(x^*\) of (1) is minimal on given set \(F\subset X\) of solutions. Let us denote \[ V_n(F)= \inf_{X_n Y_n P_n} \sup_{x^*{\i}F} \|x^*- x^*_n\|. \] The projection method (2) is said to be asymptotic optimal for the set of solutions \(F\) if \(\sup_{x^*\in F} \|x^*- x^*_n\|\sim V_n(F)\) for \(n\to\infty\). The theorems of the paper clarify the relations between the distance from the asymptotic optimal solution and the exact one and the diameter of the set \(F\).