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Parameter determination for Tikhonov regularization problems in general form. (English) Zbl 1391.65100
Summary: Tikhonov regularization is one of the most popular methods for computing an approximate solution of linear discrete ill-posed problems with error-contaminated data. A regularization parameter \(\lambda > 0\) balances the influence of a fidelity term, which measures how well the data are approximated, and of a regularization term, which dampens the propagation of the data error into the computed approximate solution. The value of the regularization parameter is important for the quality of the computed solution: a too large value of \(\lambda > 0\) gives an over-smoothed solution that lacks details that the desired solution may have, while a too small value yields a computed solution that is unnecessarily, and possibly severely, contaminated by propagated error. When a fairly accurate estimate of the norm of the error in the data is known, a suitable value of \(\lambda\) often can be determined with the aid of the discrepancy principle. This paper is concerned with the situation when the discrepancy principle cannot be applied. It then can be quite difficult to determine a suitable value of \(\lambda\). We consider the situation when the Tikhonov regularization problem is in general form, i.e., when the regularization term is determined by a regularization matrix different from the identity, and describe an extension of the COSE method for determining the regularization parameter \(\lambda\) in this situation. This method has previously been discussed for Tikhonov regularization in standard form, i.e., for the situation when the regularization matrix is the identity. It is well known that Tikhonov regularization in general form, with a suitably chosen regularization matrix, can give a computed solution of higher quality than Tikhonov regularization in standard form.

MSC:
65F22 Ill-posedness and regularization problems in numerical linear algebra
65F10 Iterative numerical methods for linear systems
65R30 Numerical methods for ill-posed problems for integral equations
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