On the approximation to solutions of operator equations by the least squares method

Abstract

We consider the equation Au = f , where A is a linear operator with compact inverse A −1 in a separable Hilbert space H. For the approximate solution u n of this equation by the least squares method in a coordinate system {e k } k∈N that is an orthonormal basis of eigenvectors of a self-adjoint operator B similar to A (D(B) = D(A)), we give a priori estimates for the asymptotic behavior of the expressions r n = u n − u and R n = Au n − f as n → ∞. A relationship between the order of smallness of these expressions and the degree of smoothness of u with respect to the operator B is established.Let H be a separable Hilbert space over C, let ( · , · ) and · be the corresponding inner product and norm in H, and let A be an invertible closed linear operator with D(A −1 ) = H. (The symbol D( · ) stands for the domain of an operator.)We consider the equation(1) and seek an approximate solution u n by the least squares method, i.e., in the formwhere {e k } k∈N is a given linearly independent system of vectors in D(A) (a so-called coordinate system) and α k ∈ C are some numbers such that R 2 n = Au n − f 2 assumes the least value. The parameters α k are uniquely determined by the system of equations n k=1 α k (Ae k , Ae i ) = (f, Ae i ), i= 1, . . . , n.If the system {e k } k∈N is complete in the Hilbert space H 1 = D(A) with inner product (x, y) 1 = (Ax, Ay), then (e.g., see [1])However, studying the dependence of the asymptotic behavior of r n and R n as n → ∞ on the choice of {e k } k∈N , the properties of the parameters of Eq. (1), and the solution u proves to be a rather difficult problem, which has not yet been solved in the general case. See [2, 3] for some particular results in the case of ordinary differential equations and [4] for Eq. (1) with a positive definite self-adjoint operator A with discrete spectrum. The present paper deals with the solution of this problem for the case in which {e k } k∈N is an orthonormal basis of eigenvectors of a positive definite self-adjoint operator B similar to A, i.e., such that D(B) = D(A). The results presented here are definitive in that the conditions guaranteeing a certain order of smallness of the expressions r n and R n and stated in terms of the degree of smoothness of the solution u with respect to the operator B are necessary and sufficient. We also note that the dependence of the degree of smoothness of u *