A gradient-based approach for identification of the zeroth-order coefficient \(p(x)\) in the parabolic equation \(u_t=(k(x)u_x)_x-p(x)u\) from Dirichlet-type measured output (Q2422500)

The author considers the inverse problem of identifying the unknown space dependent coefficient \(p(x)\) in the 1D diffusion equation \(u_t =(k(x)u_x)_x -p(x)u\) from the measured output \(f(t) = u(0,t)\) at the left boundary of a non-homogeneous rod. The author first proves a number of estimates for weak solutions of the forward problem. Then compactness of the input-output operator corresponding to the coefficient inverse problem is shown. All of this is used to solve the nonlinear inverse problem. Finally, an integral relationship between the change in the coefficient \(p(x)\) to the change of the output is derived. This relationship is used to obtain a formula for the gradient of an appropriate Tikhonov functional.