Regularization of Linear Approximate Schemes by the Gradient Descent

• Published in: SIAM journal on numerical analysis Vol. 39; no. 1; pp. 250 - 263
• Main Authors: Izmailov, Alexey F., Karmanov, Vladimir G., Tretyakov, Alexey A.
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 2002
• Subjects: Approximation; Banach spaces; Error rates; Estimation theory; Functional analysis; Hilbert space; Hilbert spaces; Ill posed problems; Iterative solutions; Linear transformations; Norms; Research grants
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/S0036142999364522

For a linear operator equation, we consider a general approximate scheme based on the following construction: the operator of the equation is approximated by continuous operators between Hibert spaces. Unlike traditional studies, a scheme is not supposed to be stable. Nevertheless, as it was shown by the authors earlier, assigning the number of iterations of the gradient descent (which in this context is also referred to as implicit iteration method, Landweber method, or Richardson method) for the approximate equation in a correspondence with the approximation error allows us to define the approximation to the solution of the original equation. In the present paper, we deal with so-called sourcewise representable solutions. Assuming the existence of such solutions, we get an opportunity to relax significantly the approximation condition which had been used earlier and to omit the continuity condition for the operator of the original equation.