On the choice of Lagrange multipliers in the iterated Tikhonov method for linear ill-posed equations in Banach spaces. (English) Zbl 1475.65109
Summary: This article is devoted to the study of nonstationary Iterated Tikhonov (nIT) type methods [M. Hanke and C. W. Groetsch, J. Optim. Theory Appl. 98, No. 1, 37–53 (1998; Zbl 0910.47005); H. W. Engl et al., Regularization of inverse problems. Dordrecht: Kluwer Academic Publishers (1996; Zbl 0859.65054)] for obtaining stable approximations to linear ill-posed problems modelled by operators mapping between Banach spaces. Here we propose and analyse an a posteriori strategy for choosing the sequence of regularization parameters for the nIT method, aiming to obtain a pre-defined decay rate of the residual. Convergence analysis of the proposed nIT type method is provided (convergence, stability and semi-convergence results). Moreover, in order to test the method’s efficiency, numerical experiments for three distinct applications are conducted: (i) a 1D convolution problem (smooth Tikhonov functional and Banach parameter-space); (ii) a 2D deblurring problem (nonsmooth Tikhonov functional and Hilbert parameter-space); (iii) a 2D elliptic inverse potential problem.
MSC:
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
| 65M30 | Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs |
| 65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |
| 65J10 | Numerical solutions to equations with linear operators |
| 65K10 | Numerical optimization and variational techniques |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35R30 | Inverse problems for PDEs |
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