Truncated spectral regularization for an ill-posed nonhomogeneous parabolic problem

The non homogeneous backward Cauchy problem ut+Au=f(t), u(τ)=ϕ for 0≤t<τ is considered, where A is a densely defined positive self-adjoint unbounded operator on a Hilbert space H with f∈L1([0,τ],H) and ϕ∈H is known to be an ill-posed problem. A truncated spectral representation of the mild solution of the above problem is shown to be a regularized approximation, and error analysis is considered when both ϕ and f are noisy. Error estimates are derived under appropriate choice of the regularization parameter. The results obtained unify and generalize many of the results available in the literature.