The paper presents a survey on the inverse problem for equations of parabolic type, mainly concentrating on the abstract backward Cauchy problem \(u'(t)+A u(t) = 0\) (\(0<t<T\)), \(u(T)=x\) in Hilbert and Banach spaces. Since inverse problems are generally ill-posed, solving them requires the use of regularization methods. The authors survey the quasi-reversibility (Lattes-Lions) method and its variations, where the idea is to replace the original problem with an approximate one, which is well-posed, and to use the solution of the latter problem in order to construct regularized (approximate) solutions to the former one. Particular attention is paid to the problem of identification of an unknown inhomogeneous term in an inhomogeneous version of the parabolic equation.
For the entire collection see [Zbl 1192.00079].