Least squares formulation for ill-posed inverse problems and applications

Abstract

In this paper we propose a least squares formulation for ill-posed inverse problems. For example, ill-posed inverse problems in partial differential equations are those such that a solution of inverse problem exists for a smooth data but it does not depend continuously on data, or there is no solution for inverse problem, e.g. the Cauchy problem for elliptic equations and backward solution of parabolic equations. We develop the least squares formulation in which the sum of equations error over domain and data fitting criterion and Tikhonov regularization terms is minimized over entire solutions. In this way we can establish the existence and uniqueness of an inverse solution and establish the continuity of the inverse solution for noisy data in $L^2$. The method can be applied to a general class of non-linear inverse problems and an operator theoretic stability analysis is developed. We describe how one can apply the method for various PDE inverse problems. Numerical tests using backward heat equation and Cauchy problem for elliptic equations are presented to demonstrate the applicability and performance of the proposed method.

Keywords:

• Ill-posed inverse problem
• least squares formulation
• continuous dependency on data

2010 Mathematics Subject Classifications:

• 35R25
• 35R30
• 47A52
• 65N20
• 65N21

Disclosure statement

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