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On the regularization of projection methods for solving ill-posed problems. (English) Zbl 0675.65053

Let X and Y be real Hilbert spaces and A: \(X\to Y\) a bounded linear operator with possibly non-closed range R(A). For given \(y\in R(A)\) we consider the equation (1) \(Ax=y\) (a standard example is an integral equation of the first kind) and use projection methods to get a discretized version of (1). For that, let \(X_ h\) resp. \(Y_ h\) be linear finite-dimensional subspaces of X resp. Y and \(Q_ h: Y\to Y\) the orthogonal projection onto \(Y_ h\). The resulting Petrov-Galerkin- equation is \((2)\quad A_ hx_ h=y_ h\) with \(A_ h: X_ h\to Y_ h\), \(x_ h\mapsto Q_ hAx_ h\) and \(y_ h=Q_ hy.\) Even if the right-hand side y of (1) is known exactly a solution \(x^*_ h\) of (2) is not necessarily a good approximation to a solution \(x^*\) of (1). In this paper some methods (e.g., the method of Tikhonov or the method of Landweber), which are commonly used to solve equation (2), are investigated. An appropriate parameter choice (for iteration procedures this corresponds to a stopping rule) is given such that one obtains a good approximation to \(x^*\) even if y is only known approximately. For this parameter choice, which is comparable to the discrepancy principle of Ivanov-Morozov, convergence is proved. Using some results on fractional powers of nonnegative selfadjoint operators convergence rates for smooth (in some sense) solutions \(x^*\) of (1) are obtained.
Reviewer: R.Plato

MSC:

65J10 Numerical solutions to equations with linear operators
65R20 Numerical methods for integral equations
47A50 Equations and inequalities involving linear operators, with vector unknowns
45B05 Fredholm integral equations
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References:

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