Volume 10, Issue 6

- Existence and uniqueness of solution to the inverse problem Bu t + Lu = γ ( t ) ϕ + f ( t ), u (0) = u 0 , u ( T ) = uT are investigated. Operators B and L are selfadjoint in a Hilbert space E ; the spectrum of the operator L is semibounded; γ ( t ) is a scalar function. If a finite number of orthogonality conditions holds then the inverse problem is uniquely solvable. The method uses the representation as a series in eigenelements and associated elements of the pencil L − λB .

- This paper is concerned with the ill-posed problem of identifying a hydraulic transmissivity in an isotropic and confined aquifer in two space dimensions. To define a regularization by adaptive discretization of the parameter, we use refinement and coarsening indicators, which give the first order effect on the optimal data misfit of adding or removing degrees of freedom to a current set of parameters. The direct problem is discretized by a mixed hybrid finite element method. A combination of the direct problem discretization with the adaptive discretrization for the transmissivity allows us to prove convergence of the algorithm as the mesh size goes to zero, stability of the finite-dimensional approximation, and convergence for noisy data with an appropriate stopping rule.

- We consider the problem of identification of a heat source in a bounded domain Ω. Assuming that the point sources became inactive after the time T∗ , we prove that they can be identified by measurements of the heat flux on Γ 0 × (0 , T ), where Γ 0 is a part of the boundary of Ω, with non void interior, and T > T∗ . By a standard trandformation, we derive from these results a method to identify the polluting sources on the surface of a river, a lake . . .

- We prove stability estimates for an inverse elliptic problem consisting in the determination of an unknown boundary coefficient from an overdetermined data. We consider both the case of all boundary measurements and the case of single boundary measurement. After studying the problem in its generality we treat the special case in which the unknown boundary coefficient is supported in an hyperplane. We find in this latest case that the stability estimates are, in some sense, better than those in the general case.

- We study the inverse problem for a parabolic equation in which the unknowns are the solution, coefficient of the function, and right-hand side (exterior load). We prove existence and uniqueness theorems.

- We consider here the inverse problem, which consists in recovery of a nonlinear source in the equation of hyperbolic type with constant coefficients in main part of differential operator. The additional information for solution of the inverse problem is given in the form of final overdetermination. The iterative algorithm for recovery of the source is based on the theorem of existence and uniqueness of solution for the inverse problem that is proved in small. The convergence of the iterative method for solving the inverse problem is established.

- We consider a nonsymmetric first-order differential operator , where P is a 2 × 2 matrix whose components are in L 2 (0 , 1). We study an eigenvalue problem for A with boundary conditions at x = 0 , 1. We establish an asymptotic form of the eigenvalues and prove that the set of the root vectors forms a Riesz basis in {L 2 (0 , 1) } 2 . Moreover we show some characterization of L 2 -coefficients in an inverse eigenvalue problem. The key is a transformation formula.