Abstract
Conjugated gradients on the normal equation (CGNE) is a popular method to regularise linear inverse problems. The idea of the method can be summarised as minimising the residuum over a suitable Krylov subspace. It is shown that using the same idea for the shift-and-invert rational Krylov subspace yields an order-optimal regularisation scheme.
Highlights
We consider the solution of the linear system
The linear system is assumed to be ill-posed, that is, the range R(T ) is not closed in Y. yδ is a perturbation of the exact data y, such that yδ − y ≤ δ. yδ is called the noisy data and δ the noise level
N (T )⊥ designates the orthogonal complement of the null space N (T ) of T . x+ can be characterised as the unique x+ ∈ N (T )⊥ that solves the normal equation
Summary
Where the operator T acts continuously between the Hilbert spaces X and Y. CGNE with the discrepancy principle as a stopping rule is an order-optimal regularisation scheme for all μ > 0 (cf Theorem 7.12 in [6],[24]). Due to its definition, CGNE is the fastest to satisfy the discrepancy principle with respect to all regularisation schemes that compute approximations in the Krylov subspace Km. The analysis of CGNE with respect to its regularisation properties is involved, since the operators Rm are nonlinear and not necessarily continuous (cf Theorem 7.6 in [6], [5]). Our proofs will follow closely or sometimes literally the corresponding proofs for CGNE in chapter 7 of [6]
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