Analysis of the Minimal Residual Method Applied to Ill Posed Optimality Systems

Abstract

We analyze the performance of the minimal residual (MINRES) method applied to linear Karush--Kuhn--Tucker systems arising in connection with inverse problems. Such optimality systems typically have a saddle point structure and have unique solutions for all $\alpha>0$, where $\alpha$ is the parameter employed in the Tikhonov regularization. Unfortunately, the associated spectral condition number is very large for small values of $\alpha$, which strongly indicates that their numerical treatment is difficult. Our main result shows that a broad range of linear ill posed optimality systems can be solved efficiently with the MINRES method. This result is obtained by carefully analyzing the spectrum of the associated saddle point operator: Except for a few isolated eigenvalues, the spectrum consists of three bounded intervals. Krylov subspace methods handle such problems very well. For severely ill posed cases, techniques based on Chebyshev polynomials are applied to prove that the number of iterations needed by...