DEPARTMENT OF MECHANICS AND APPLIED
MATHEMATICS
INVERSE PROBLEMS
HEAD OF THE PROJECT Jaan JANNO, Ph.D.
DESCRIPTION
Two types of inverse problems are studied:
- problems for identification of kernels of parabolic and hyperbolic integrodifferential equations with given trace or domain measurements of solutions of these equations;
- nonlinear equations of convolution type
Our interest is concentrated both on theoretical (existence, uniqueness, stability) and numerical aspects of these problems.
APPLICATIONS
Constitutive relations of viscoelastic materials and materials with thermal memory contain integral terms over the past history of the material involving time- and sometimes also space-dependent kernels. In many engineering problems these kernels are a priori unknown or scarcely known. A nondestructive method to determine these kernels is to solve inverse problems for an integrodifferential equation describing the evolution of the medium under given boundary perturbation and make use of an additional information obtained by measurements of the solution of this equation.
Nonlinear equations of convolution type occur in several practical contexts, most important are the applications in spectroscopy and stochastics. For instance, in appearance potential spectroscopy (APS) one determines the density of unoccupied states u of a surface region of a body bombing the body by electrons and measuring the AP-spectrum signal f. Then u is the solution of the following autoconvolution equation
MAIN RESULTS
During last years a general method has been developed which enables to prove global (in time) solvability for certain classes of integrodifferential inverse problems and nonlinear convolution equations and get results about convergence of direct discretization methods for these problems. The essence of the method is Banach or Schauder fixed-point theorem in norms with exponential weights.
PARTNER
GRANTS
- ESF Grant No. 3131, 1998-2000
(in cooperation with A.Ravasoo, see also Non-Linear Acoustodiagnostics) - ESF Grant No. 3616, 1999-2002
- ESF Grant No. 4345, 2000-2003
- DAAD and DFG grants for research stays in Germany, 1994-2000
PUBLICATIONS
In 1995-1998 fifteen papers in refereed journals have appeared.