During my five-week visit to CMA I primarily worked in two problem areas: a) algorithms for the computation of functionals defined on the solution of a discrete ill-posed problem b) hyperbolic approximations for a Cauchy problem for the heat equation. Below I give a brief account of that work.

In this paper we present results for the inverse problem where the data is nodal positions. In the specific results to be stated here the solutions are spacially varying parameters, i.e., coefficients in differential operators of second order. We will also discuss future research goals for this type of inverse problem.

The identification problem of estimating certain functions in a system of linear ordinary differential equations from measured data of its state is considered. The approach consists in an imbedding of the problem into a family of parameter-dependent problems which can be solved at least numerically. The corresponding solutions are proved to converge to the unknown functions as the parameters tend to infinity. Stability results with respect to disturbances in the measmements and the initial data are developed as welL The method is applied to detennine mass exchange rates in a compartmental system of pharmaco--kinetic models.

It is well known that bounds on the asymptotic decay rate of the eigenvalues of the symmetric integral operator with kernel k(s,t) may be obtained from the smoothness of k(s,t) [1,4,7,9,11]. Some recent numerical results suggested that these bounds also applied to the eigenvalues of matrices with entries $K_{mn} = k(s_m,t_n)$ that occur in various discretizations of the continuous operator. This note shows that, in certain common situations, this is indeed the case.

In a recent paper [3] a method was proposed for solving the inverse SturmLiouville problem by finding a piecewise constant potential whose leading eigenvalues agree with the specified eigenvalues. The numerical evidence presented there indicated that the method worked well, but that the recovered solution was sensitive to perturbations in the specified eigenvalues. In this note the sensitivity of the recovered potential with respect to errors in the eigenvalues is investigated and a regularization technique for reducing the influence of such errors is proposed.