Accurate conjugate gradient methods for families of shifted systems
Applied Numerical Mathematics - Numerical algorithms, parallelism and applications
The Lagrange method for the regularization of discrete ill-posed problems
Computational Optimization and Applications - Special issue: Numerical analysis of optimization in partial differential equations
Greedy Tikhonov regularization for large linear ill-posed problems
International Journal of Computer Mathematics - Fast Iterative and Preconditioning Methods for Linear and Non-Linear Systems
The Lagrange method for the regularization of discrete ill-posed problems
Computational Optimization and Applications
Computers & Mathematics with Applications
An iterative Lagrange method for the regularization of discrete ill-posed inverse problems
Computers & Mathematics with Applications
Efficient determination of the hyperparameter in regularized total least squares problems
Applied Numerical Mathematics
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Tikhonov--Phillips regularization is one of the best-known regularization methods for inverse problems. A posteriori criteria for determining the regularization parameter $\alpha$ require solving $$(*) (A^*A+\alpha I) x =A^* y^{\delta}$$ for different values of $\alpha$.We investigate two methods for accelerating the standard cg-algorithm for solving the family of systems (*). The first one utilizes a stopping criterion for the cg-iterations which depends on $\alpha$ and $\delta$. The second method exploits the shifted structure of the linear systems (*), which allows us to solve (*) simultaneously for different values of $\alpha$. We present numerical experiments for three test problems which illustrate the practical efficiency of the new methods. The experiments as well as theoretical considerations show that run times are accelerated by a factor of at least 3.