Logarithmic regularization of non-autonomous non-linear ill-posed problems in Hilbert spaces

Abstract

The regularization of non-autonomous non-linear ill-posed problems is established using a logarithmic approximation originally proposed by Boussetila and Rebbani, and later modified by Tuan and Trong. We first prove continuous dependence on modeling where the solution of the original ill-posed problem is estimated by the solution of an approximate well-posed problem. Finally, we illustrate the convergence via numerical experiments in $L^2$ spaces.

Keywords

• Non-linear ill-posed problem
• backward heat equation
• non-autonomous problem
• semigroup of linear operators
• regularization