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New Trends in Applying LRM to Nonlinear Ill-Posed Equations

Author

Listed:
  • 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)

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.

Suggested Citation

  • Santhosh George & Ramya Sadananda & Jidesh Padikkal & Ajil Kunnarath & Ioannis K. Argyros, 2024. "New Trends in Applying LRM to Nonlinear Ill-Posed Equations," Mathematics, MDPI, vol. 12(15), pages 1-19, July.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:15:p:2377-:d:1446302
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    References listed on IDEAS

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    1. Jidesh, P. & Shubha, Vorkady S. & George, Santhosh, 2015. "A quadratic convergence yielding iterative method for the implementation of Lavrentiev regularization method for ill-posed equations," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 148-156.
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