Band 14, Heft 5

This article presents a mathematical and numerical analysis of the adjoint problem approach for inverse coefficient problems related to linear parabolic equations. Based on maximum principle a structure of the coefficient-to-data mapping is derived. The obtained integral identities permit one to prove the monotonicity and invertibility of the input-output mappings, as well as formulate the gradient of the cost functional via the solutions of the direct and adjoint problems. In the second part of the paper a numerical algorithm for determining the diffusion coefficient k = k ( x ) in the linear parabolic equation u t = ( k ( x ) u x ) x from the measured output data is presented. The main distinguished feature of the proposed algorithm is the use of a fine mesh for the numerical solution of the well-posed forward and backward parabolic problems, and a coarse mesh for the interpolation of unknown coefficient k = k ( x ). The nodal values of the unknown coefficient on the coarse mesh are recovered sequentially, solving on each step the well-posed forward and the sequence of backward initial value problems. This guarantees a compromise between the accuracy and stability of the solution of the considered inverse problem. An efficiency and applicability of the method is demonstrated on various numerical examples with noisy free and noisy data.

In this paper we consider an inverse problem for determining time-dependent heat conduction coefficient which vanishes at initial moment as a power t β . The case of strong degeneration ( β ≥ 1) is studied. To prove the existence of solution we employ the Schauder fixed point theorem. The uniqueness of the solution is established too.

We consider an abstract inverse Cauchy problem with a sectorial operator A in a Banach space. For solving the problem, a class of regularization methods is used and their rate of convergence is studied in terms of the regularization parameter α → 0+. It is shown that the methods provide an approximation error of order O ((– ln α ) −p ) ( p > 0) if and only if the initial discrepancy has a sourcewise representation with respect to the operator A −p .

In this paper we consider one-dimensional flow governed by Burgers' equation. We analyze two iterative methods for data assimilation problem for this equation. One of them so called adjoint optimization method, is based on minimization in L 2 -norm. We show that this minimization problem is ill-posed but the adjoint optimization iterative method is regularizing, and represents the well-known Landweber method in inverse problems. The second method is based on L 2 -minimization of the gradient. We prove that this problem always has a solution. We present numerical comparisons of these two methods.