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In this paper, quadrature methods for solving Volterra integral equations of the first kind with smooth kernels under the presence of noise in the right-hand sides are considered, with the quadrature methods generated by linear multistep methods. The regularizing properties of an a priori choice of the step size are analyzed, and the smoothness of the involved functions is carefully taken into consideration. The balancing principle as an adaptive choice of the step size is also studied. It is considered in a version which sometimes requires less amount of computational work than the standard version of this principle. Numerical results are included.

We consider an ill-posed problem Ta = f*in Hilbert spaces and suppose that the linear bounded operator Tis approximately available, with a known estimate for the operator perturbation at the solution. As a numerical scheme the CGNR-method is considered, that is, the classical method of conjugate gradients by Hestenes and Stiefel applied to the associated normal equations. Two a posteriori stopping rules are introduced, and convergence results are provided for the corresponding approximations, respectively. As a specific application, a parameter estimation problem is considered.

Summary For the solution of linear ill-posed problems some gradient methods like conjugate gradients and steepest descent have been examined previously in the literature. It is shown that even though these methods converge in the case of exact data their instability makes it impossible to base a-priori parameter choice regularization methods upon them.

Abstract: In this paper, for the numerical solution of linear accretive Volterra integral equations of the first kind in Hilbert spaces we consider the Galerkin scheme for Lavrentiev’sm-times iterated method, i.e., for each parameter choice for Lavrentiev’sm-times iterated method the arisingm stabilized equations are discretized by the Galerkin scheme. An associated discrepancy principle as parameter choice strategy for this finite-dimensional version of Lavrentiev’sm-times iterated method is proposed, and corresponding convergence results are provided.

We consider a linear ill-posed operator equation Ax = y in Hilbert spaces. An algorithm Rε:Y→X for solving this equation with given inexact right-hand side yε, such that [image omitted] , is called order optimal if it provides best possible error estimates under the assumption that the minimal norm solution x* of this operator equation fulfils some smoothness condition. It is shown that if such an algorithm is slightly modified to [image omitted]  then it is a regularization method, i.e., we have [image omitted]  without additional conditions on x*.We apply this result to show that the method of conjugate gradients for solving linear ill-posed equations together with a stopping rule yields a regularization method.

Summary.: For the numerical solution of (non-necessarily well-posed) linear equations in Banach spaces we consider a class of iterative methods which contains well-known methods like the Richardson iteration, if the associated resolvent operator fulfils a condition with respect to a sector. It is the purpose of this paper to show that for given noisy right-hand side the discrepancy principle (being a stopping rule for the iteration methods belonging to the mentioned class) defines a regularization method, and convergence rates are proved under additional smoothness conditions on the initial error. This extends similar results obtained for positive semidefinite problems in Hilbert spaces. Then we consider a class of parametric methods which under the same resolvent condition contains the method of the abstract Cauchy problem, and (under a weaker resolvent condition) the iterated method of Lavrentiev. A modified discrepancy principle is formulated for them, and finally numerical illustrations are presented.