We consider the inverse problem of recovering a 2D periodic structure from near-field measurements above the structure. First, following Bruckner et al (2001 Preprint No 682 Weierstrass Institute, Berlin; 2003 Proc. 3rd ISAAC Congress (Berlin, 2001) at press), the inverse problem is reformulated as an optimization problem which consists of two parts: a linear severely ill-posed problem and a nonlinear well-posed one. Then, unlike the work of Bruckner et al, the two problems are solved separately to diminish the computational effort by exploiting their special properties. Numerical results for exact and noisy data demonstrate the practicability of the inversion algorithm.

We consider a wave equation with point source termsWe discuss the inverse problem of determining point sources {N, α 1 , . . . , α N , x 1 , . . . , x N } or {x 1 , . . . , x N } from observation data u(η, t), 0 < t < T with given η ∈ (0, 1) and T > 0.We prove uniqueness and stability in determining point sources in terms of the norm in H 1 (0, T ) of observations. The uniqueness result requires that η is an irrational number and T 1, and our stability result needs further a priori (but reasonable) information of unknown {x 1 , . . . , x N }. Moreover, we establish two schemes for reconstructing {x 1 , . . . , x N } which are stable against errors in L 2 (0, T ).

The paper is devoted to the inverse problem of identifying the coefficient in the main term of an elliptic differential equation describing the filtration of ground water. Experience suggests that the gradient of the piezometric head, i.e. Darcy's velocity, may have discontinuities and the transmissivity coefficient is a piecewise constant function.To solve this problem we use a modification of a direct method of Vainikko. Starting with a weak formulation of the problem, a suitable discretization is obtained by the method of least error. If necessary, this method can be combined with Tikhonov's regularization.The main difficulty consists of generating distributed state observations from measurements of the ground-water level. For this step we propose an optimized data preparation procedure using additional information such as knowledge of the sought parameter values at some points and lower and upper bounds for the parameter.Numerical tests confirm the expected local behaviour of the method, i.e. locally sufficiently many measurements provide locally satisfactory results. Two numerical examples, one with simulated data and the other with real life data, are given.

No abstract

It is well known that projection schemes for certain linear ill-posed problems Ax = y can be regularized by a proper choice of the discretization level only, where no additional regularization is needed. The previous study of this self-regularization phenomenon was restricted to the case of so-called moderately ill-posed problems, i.e., when the singular values σk (A), k = 1, 2,..., of the operator A tend to zero with polynomial rate. The main accomplishment of the present paper is a new strategy for a discretization level choice that provides optimal order accuracy also for severely ill-posed problems, i.e., when σk (A) tend to zero exponentially. The proposed strategy does not require a priori information regarding the solution smoothness and the exact rate of σk (A).