UNCERTAINTIES IN MATHEMATICAL MODELS
                AND INVERSE PROBLEMS

            Aleksey V. Nenarokomov', Ashley F. Emery",
                      Tushar D. Fadale'''

                 'Moscow Aviation Insitute
                      Moscow, Russia
                 "University of Washington
                       Seattle, WA, USA
               '''Applied Research Associates
                 Raleigh, North Carolina, USA

    A large class of promising methods for the analysis and
interpretation of data from transient thermophysical experiments is
based on the solution of the inverse heat conduction problems. In this
approach the coefficients (thermophysical characteristics) in the heat
conduction equation are determined from the known boundary and initial
conditions and from the data of transient temperature measurements at
a finite number of spatially distributed points of the analyzed body.

    The chief advantage of these methods is that they enable us to
conduct experimental investigations and tests under conditions as
close to nature as possible, or directly during operation of
engineering devices. Features of ill-posed inverse heat transfer problem
require special mathematical methods for solving them as well as a proper
technical organization of the studies. Only a rational combination of
physical combination of physical, mathematical and technical
fundamentals make it possible to effectively use these methods in
practice.

     Mathematical modeling data show that the error of determination
of the thermophysical characteristics from the solution of the inverse
problem may depend significantly on the spatial placement of the
temperature sensors in the investigated body. The proper setup of the
thermophysical experiment requires the solution of an optimal
experimental design problem, i.e. to place a fixed number of sensors
in the sample in such a way as to minimize the error of identification
of the required characteristics. The stated problem can be solved on the
basis of the fundamental principles of experimental design theory for
distributed-parameter system.

     The information matrix  characterizes the total sensitivity of
the analyzed system in the entire set of measurement points to the
variation all components of  unknown functions. The given
optimal measurement design problems entails finding  positions
of sensors for with the total sensitivity of the system in the adopted
sense will be a maximum.

     Various criteria are used for the optimization of the experimental
conditions. The so-called D-optimum design is widely used to ensure the
minimum error of estimation of the unknown functions. In this case the
measurement design can be determined from the condition of the maximum
of the determinant of the normalized information matrix.