Preconditioning and regularization for ill-posed problems solved by conjugate gradients
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This work is dedicated to solving symmetric positive (semi)definite ill posed problems when a good symmetric positive (semi)definite Tikhonov regularization is known. The system to be solved can be written as the sum of the matrix associated with the inverse problem and of the regularization matrix weighted by some scalar parameter to be tuned by the user. We show that augmented conjugate gradient preconditioned by the regularization (i.e. by applying its inverse) provides an excellent framework to solve such systems as it favours regular solutions and progressively introduces details, it gives access at marginal cost to important indicators (error, norm of the solution, norm of the operator) to drive the solver. Moreover, a simple post-processing of Ritz eigen-elements makes it possible to analyse the system by Picard plots and L-curves, it even allows giving a relevant approximation of the solution for any regularization weight. The strategy will be illustrated by applications to the boundary completion of elliptic PDE which is a severely ill posed linear problem, and to digital images correlation where the preconditioned conjugate gradient is the inner solver (accelerated by recycling) for a Gauss-Newton outer solver. Details on the proposed strategy can be found at https://hal.science/hal-04602735v2.
