New Trends in Applying LRM to Nonlinear Ill-Posed Equations

Abstract

Tautenhahn (2002) studied the Lavrentiev regularization method (LRM) to approximate a stable solution for the ill-posed nonlinear equation κ(u)=v, where κ:D(κ)⊆X⟶X is a nonlinear monotone operator and X is a Hilbert space. The operator in the example used in Tautenhahn’s paper was not a monotone operator. So, the following question arises. Can we use LRM for ill-posed nonlinear equations when the involved operator is not monotone? This paper provides a sufficient condition to employ the Lavrentiev regularization technique to such equations whenever the operator involved is non-monotone. Under certain assumptions, the error analysis and adaptive parameter choice strategy for the method are discussed. Moreover, the developed theory is applied to two well-known ill-posed problems—inverse gravimetry and growth law problems.

Keywords

• ill-posed nonlinear equation
• lavrentiev regularization
• adaptive parameter choice
• non-monotone operator
• iterative method