Multi-projection methods for Fredholm integral equations of the first kind

Abstract

We use piecewise polynomial basis functions to obtain the stable approximation solution of the Tikhonov regularized equation of the Fredholm integral equation of the first kind by utilizing multi-projection (multi-Galerkin and multi-collocation) methods. We evaluate the error bounds for the approximate solution with the exact solution in infinity norm. We provide an a priori parameter choice strategy under infinity norm. In addition to determining the regularization parameter, we discuss Arcangeli's discrepancy principle and calculate the convergence rates in infinity norm. We give test examples to validate the theoretical results.

Keywords:

• Ill-posed problems
• Fredholm integral equation of the first kind
• multi-projection methods
• Tikhonov regularization method
• piecewise polynomials

2020 AMS Subject Classifications:

• 65J10
• 65J20
• 65J22

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was supported by the INSPIRE fellowship, Department of Science and Technology, Government of India, New Delhi [grant number IF170638]. The second author was supported in part by OURIIP Seed Fund 2020, Odisha State Higher Education Council.