A truncated spectral regularization method for a source identification problem

Abstract

Abstract inverse source problem of identifying the source function f in the abstract Cauchy problem $u_t+Au=f(t),\, 0<t<\tau$ with $u(0)=\phi _0$ when the data, the final value, $u(\tau )=\phi _\tau$ is noisy is considered, where A is a densly defined self-adjont coercive unbounded operator on a Hilbert space H. This problem is known to be an ill-posed problem. A truncated spectral representation of a mild solution of the above problem is shown to be a regularized approximation, and error analysis is carried out when $\phi _\tau$ is noisy as well as exact, and stability estimate is given under appropriate parameter choice strategies.