One new strategy for a priori choice of regularizing parameters in Tikhonov’s regularization. (English) Zbl 0957.65052

Cheng, J.; Yamamoto, M.

Inverse Probl. 16, No. 4, L31-L38 (2000).

Ill-posed inverse problems in Banach spaces of the type \(K(f)= 0\) for given \(g\) and a possibly nonlinear, densely defined operator \(K\) are considered. The inverse \(K^{-1}\) is assumed to be not continuous or the range of \(K\) is not closed. The authors provide a strategy for choosing a priori the parameter in Tikhonov’s regularization method for solving the problem approximately. The resulting convergence is analyzed under the assumption and conditional stability.
The new strategy is demonstrated for some examples (determination of coefficients or sources in a wave equation, determination of the shape of a boundary, and an integral equation with analytic kernel of the first kind).

Reviewer: Etienne Emmrich (Berlin)

MSC:

65J15	Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
65J20	Numerical solutions of ill-posed problems in abstract spaces; regularization
65R32	Numerical methods for inverse problems for integral equations
45B05	Fredholm integral equations
35L05	Wave equation
47J25	Iterative procedures involving nonlinear operators
47J06	Nonlinear ill-posed problems
65M32	Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35R30	Inverse problems for PDEs
65J22	Numerical solution to inverse problems in abstract spaces

Keywords:

ill-posed inverse problems; Banach spaces; Tikhonov’s regularization method; convergence; conditional stability; wave equation; integral equation