Identifying the diffusion coefficient by optimization from the final observation

This paper deals with the inverse problem of determining a pair (a, u) in the reaction -diffusion equation u(t) - (au(x))(x) = f (x, t), with initial and boundary conditions u(x, 0) = phi(x), u(x)vertical bar(x=0) = u(x)vertical bar(x=1) = 0, from the final measurement data u(x, T) = z(x), which has important application in a large fields of applied science. Based on the optimal control framework, the existence, uniqueness and stability of the minimizer for the cost functional are established. A necessary condition which is a couple system of a parabolic equation and a parabolic variational inequality is deduced. The gradient iteration algorithm is applied to the inverse problem and some numerical results are presented for various typical test examples. (C) 2012 Elsevier Inc. All rights reserved.