Abstract
The total least squares (TLS) method is a successful approach for linear problems if both the right-hand side and the operator are contaminated by some noise. For ill-posed problems, a regularisation strategy has to be considered to stabilise the computed solution. Recently a double regularised TLS method was proposed within an infinite dimensional setup and it reconstructs both function and operator, reflected on the bilinear forms Our main focuses are on the design and the implementation of an algorithm with particular emphasis on alternating minimisation strategy, for solving not only the double regularised TLS problem, but a vast class of optimisation problems: on the minimisation of a bilinear functional of two variables.
Highlights
I R Bleyer and R RamlauIn [2], the authors described a new two-parameter regularisation scheme for solving an illposed operator equation
The total least squares (TLS) method is a successful approach for linear problems if both the right-hand side and the operator are contaminated by some noise
A double regularised TLS method was proposed within an infinite dimensional setup and it reconstructs both function and operator, reflected on the bilinear forms Our main focuses are on the design and the implementation of an algorithm with particular emphasis on alternating minimisation strategy, for solving the double regularised TLS problem, but a vast class of optimisation problems: on the minimisation of a bilinear functional of two variables
Summary
In [2], the authors described a new two-parameter regularisation scheme for solving an illposed operator equation. A cost functional with two penalisation terms based on the TLS (total least squares) technique is used. This approach presented in [2] focuses on linear operators that can be characterised by a function, as it is, e.g. the case for linear integral operators, where the kernel function determines the behaviour of the operator. The potential advantage is that the unknown solution is reconstructed, and a suitable characterising function and the governing operator describing the underlying data. The goal of this paper is the development of an efficient and convergent numerical scheme for the minimisation of the Tikhonov-type functional for the dbl-RTLS approach. For the convenience of the reader in appendix we display important concepts and fundamental results used throughout this article
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