Convergence of the grid method for the Fredholm equation of the first kind with Tikhonov regularization

Abstract

The paper describes a grid method for solving an ill-posed problem for the Fredholm equation of the first kind using the A. N. Tikhonov regularizer. The convergence theorem for this method was formulated and proved. A procedure for thickening grids with a simultaneous increase in digit capacity of calculations is proposed.

Full Text

1. Introduction A large number of applied tasks are ill-posed. A number of methods have been developed to solve them. Firstly, these are parametric methods in which the solution is represented as a decomposition over some basis, and the regularized equation is reduced to the problem of optimizing the coefficients of the decomposition (see, for example [1-3]). The success of this approach strongly depends on the successful choice of the basis. Such methods are difficult to study; finding estimates of accuracy and conditionality in calculations with finite digit numbers is particularly difficult. Most of the proofs are carried out for exact calculations with infinite digit capacity, i.e., without round-off errors. Secondly, iterative methods with simple or implicit iterations [4, 5] are often used to obtain an approximate analytical solution. The number of iterations is also a regularizing parameter [6]. This looks tempting, since there is no need to introduce additional stabilizing terms and thereby increase the discrepancy. On the other hand, in the general case, iterations have to be implemented numerically. The finite-difference approximation of the corresponding quadratures introduces some systematic error in the operator and the right part. To reduce it, it is necessary to perform calculations on thickening grids. The third approach is represented by various grid methods (finite-difference or finite-element), in which the solution is calculated in a set of discrete grid nodes, that is, essentially replaced by a piecewise constant function. In this approach, the initial problem is reduced to a system of algebraic equations that can be solved by any direct or iterative method [7, 8]. Yu. L. Gaponenko showed that finite-difference approximation makes the problem correct, i.e., self-regulation takes place [9, 10]. The study of finite element approximations (for specific applied problems) was carried out, for example, in [11, 12]. However, the proofs and convergence estimates are valid for calculations with infinite digit capacity, since they do not take into account rounding errors. The central point of all regularizing algorithms is the justification of convergence and the evaluation of the actual accuracy, that is, the difference between the exact solution and the approximate one found. A review of the literature on this issue is given in [13]. Known a posteriori estimates are majorant and often greatly overestimate the error (up to 10 times or more). Quite often, they require specific information and solutions that are not easy to obtain in complex application tasks [14]. Another important issue is the choice of the regularization parameter. This problem is not trivial, since in most applied calculations the error level is fixed and does not tend to zero [15]. The best known solution to this question is the well-known generalized residual principle [16]. In the present paper, we describe a grid method for solving an ill-posed problem for the Fredholm equation of the first kind using the Tikhonov regularizer of the zeroth order. For this method, we formulate and prove convergence theorem which takes into account finite digit capacity of calculations. For its practical implementation, we propose procedure of simultaneous grid thickening and increase of digit capacity. 2. Method We consider the Fredholm equation of the first kind