Convergence rate for a convection parameter identified using Tikhonov regularization
by
Gabriel Dimitriu
University of Medicine and Pharmacy "Gr. T. Popa" Iasi, Department of Mathematics and Informatics, 16 Universitatii street, 6600 Iasi, Romania

In this study we present a convergence rate result for a parameter identification problem. To be precise, we show that the convergence rate of the convection parameter b in elliptic equation

-(aux)x+bux+cu=f   in    (0, 1),              (1)

with Dirichlet boundary conditions u(0)=u(1)=0 is O(delta^0.5), where delta is a norm bound for the noise in the data f. This parameter b represents the solution of the identification problem associated with (1) and regularised by Tikhonov method. We choose f and the coefficients (a, b, c) of the equation (1), from the following functional spaces: f in L²(0, 1), (a, b, c) in Q₁, a subset of the set Q=W^{1, 2}(0, 1)×W^{1, 2}(0, 1)×W^{1, 2}(0, 1) with Q endowed with the Hilbert-space product topology and

Q1={(a, b, c) in Q :    0 < a < a(x),     |a|W1, 2(0, 1) < K,     |b|W1, 2(0, 1) < K,     c(x) > c > 0    a.e. in (0, 1)}.

Here a, c and K are given constants. The result is based on the concept of minimum-norm solution with respect to a certain value b*, for equation (1) and a sequence of estimations using the first and the second Fréchet derivative of the mapping parameter --> solution, b --> u(b).

Date received: February 1, 2000