Simultaneous identification of two coefficients in a diffusion equation (Q2729317)

The article is devoted to the numerical study of the so-called two-coefficient inverse problem for a linear parabolic differential equation. The inverse problem reads as follows: Find two functions \(D(z)\) and \(B(z)\) such that the solution \(v(z,t)\) to the problem NEWLINE\[NEWLINE \begin{aligned} & v_t = D^2(z) + B(z)v\qquad\text{for } t > 0,\quad 0\leq z \leq H,\\ & v|_{t = 0} = 0,\quad v_z|_{z = 0} = \varphi_1, \quad v|_{z = H} = 0 \end{aligned} NEWLINE\]NEWLINE satisfies the additional condition \(v_{z = 0} = v_0(t)\).NEWLINENEWLINENEWLINEThis problem received great attention in the literature, as a prototype for several problems dealing with the coastal engineering.NEWLINENEWLINENEWLINETo study the inverse problem, the authors use the formal Laplace transform. Reformulating the original problem, the authors pose the inverse problem for the Helmholtz equation with parameter. As a result, the inverse problem is reduced to a system of two indirect equations and the system obtained involves only integral equations. The inversion algorithm proposed is used for determining the two coefficients \(D(z)\) and \(B(z)\) simultaneously. The iterative inversion procedure is based on the minimization of a suitable cost functional. The authors investigate the convergence of the iterative process only numerically. Regularization parameters are chosen experimentally without using any rigorous mathematical background. The authors note that a priori knowledge of the global variations of the coefficients \(D(z)\) and \(B(z)\) improves essentially the convergence of the inversion procedure described in the article under review. The authors expose numerical results to illustrate the performance of the algorithm.