Comparison of A-posteriori parameter choice rules for linear discrete ill-posed problems

Abstract

Tikhonov regularization is one of the most popular methods for computing approximate solutions of linear discrete ill-posed problems with error-contaminated data. A regularization parameter, μ>0, balances the influence of a fidelity term, which measures how well the data is approximated, and of a regularization term, which dampens the propagation of the data error into the computed approximate solution. The quality of the computed solution is affected by the value of the regularization parameter μ. The discrepancy principle is a popular a-posteriori rule for determining a suitable value of μ. It performs quite well when a fairly accurate estimate of the norm of the error in the data is known. A modification of the discrepancy principle, proposed independently by Gfrerer and Raus, also can be used to determine μ. Analysis of this modification in an infinite-dimensional Hilbert space setting suggests that it will determine a value of μ that yields an approximate solution of higher quality than the approximate solution obtained when using the (standard) discrepancy principle to compute μ. This paper compares these a-posteriori rules for determining μ when applied to the solution of many linear discrete ill-posed problems with different amounts of error in the data. Our comparison shows that in a discrete setting, the discrepancy principle generally gives a value of μ that yields a computed solution of higher quality than the value of μ furnished by the modified discrepancy principle.