New Trends in Applying LRM to Nonlinear Ill-Posed Equations

Santhosh George, Ramya Sadananda, Jidesh Padikkal, Ajil Kunnarath and Ioannis K. Argyros ()
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Santhosh George: Department of Mathematical & Computational Sciences, National Institute of Technology Karnataka, Surathkal 575025, India
Ramya Sadananda: Department of Mathematical & Computational Sciences, National Institute of Technology Karnataka, Surathkal 575025, India
Jidesh Padikkal: Department of Mathematical & Computational Sciences, National Institute of Technology Karnataka, Surathkal 575025, India
Ajil Kunnarath: Department of Mathematical & Computational Sciences, National Institute of Technology Karnataka, Surathkal 575025, India
Ioannis K. Argyros: Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USA

Mathematics, 2024, vol. 12, issue 15, 1-19

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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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