Optimal rates for Lavrentiev regularization with adjoint source conditions

There are various ways to regularize ill-posed operator equations in Hilbert space. If the underlying operator is accretive then Lavrentiev regularization (singular perturbation) is an immediate choice. The corresponding convergence rates for the regularization error depend on the given smoothness assumptions, and for general accretive operators these may be both with respect to the operator or its adjoint. Previous analysis revealed different convergence rates, and their optimality was unclear, specifically for adjoint source conditions. Based on the fundamental study by T. Kato, Fractional powers of dissipative operators. J. Math. Soc. Japan, 13(3):247–274, 1961, we establish power type convergence rates for this case. By measuring the optimality of such rates in terms on limit orders we exhibit optimality properties of the convergence rates, for general accretive operators under direct and adjoint source conditions, but also for the subclass of nonnegative selfadjoint operators.

[1]  M. Krasnosel’skiǐ,et al.  Integral operators in spaces of summable functions , 1975 .

[2]  Tosio Kato,et al.  Fractional powers of dissipative operators , 1961 .

[3]  Jens Flemming,et al.  Sharp converse results for the regularization error using distance functions , 2011 .

[4]  C. W. Groetsch,et al.  The theory of Tikhonov regularization for Fredholm equations of the first kind , 1984 .

[5]  A. Mcintosh.  On the Comparability of A 1/2 and A ∗ 1/2 , 1972 .

[6]  Markus Haase,et al.  The Functional Calculus for Sectorial Operators , 2006 .

[7]  J. Conway.  A course in operator theory , 1999 .

[8]  U Tautenhahn,et al.  On the method of Lavrentiev regularization for nonlinear ill-posed problems , 2002 .

[9]  A. Mcintosh.  On the comparability of ^{1/2} and ^{∗1/2} , 1972 .

[10]  Y. Alber,et al.  Nonlinear Ill-posed Problems of Monotone Type , 2006 .

[11]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[12]  B. Hofmann,et al.  Lavrentiev's regularization method in Hilbert spaces revisited , 2015, 1506.01803.

[13]  Masahiro Yamamoto,et al.  Time-fractional diffusion equation in the fractional Sobolev spaces , 2015 .

[14]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[15]  Bernd Hofmann,et al.  Analysis of Profile Functions for General Linear Regularization Methods , 2007, SIAM J. Numer. Anal..

[16]  Hiroki Tanabe,et al.  Equations of evolution , 1979 .

[17]  Rudolf Gorenflo,et al.  Operator theoretic treatment of linear Abel integral equations of first kind , 1999 .

[18]  R. Plato.  On the discrepancy principle for iterative and parametric methods to solve linear ill-posed equations , 1996 .