**Numerical
and Computational Challenges in Science and Engineering Program**

## John T. Betts, The Boeing Company

*LECTURES*

Wenesday, July 31, 2002 at 4:00 pm, room 230

*Lecture on Computing Aerodynamic Models using Large Scale Parameter
Estimation*

The behavior of many physical processes can be described mathematically
by ordinary differential or differential-algebraic equations. Commonly
a finite number of parameters appear in the description of the system
dynamics. A parameter estimation problem arises when it is necessary
to compute values for these parameters based on observations of the
system dynamics. Methods for solving these so-called ``inverse problems''
have been used for many years. In fact, most techniques in use today
are based on ideas proposed by Gauss nearly 200 years ago, that he used
to solve orbit determination problems.

One approach to solving estimation problems is to discretize the dynamics
and then treat the values at mesh points as optimization variables.
A consequence of this discretization is that the original problem is
transcribed into a finite dimensional nonlinear programming problem.
Since the discrete variables directly optimize the approximate problem
this approach is referred to as the direct transcription method. Furthermore,
this nonlinear programming problem has two important properties that
can be exploited. First, it is possible to efficiently compute the (Hessian)
matrix of second derivatives, thereby overcoming one of the major limitations
of the Gauss algorithm. Second, the Hessian and Jacobian matrices are
sparse, and as a consequence very efficient linear algebra techniques
can be utilized. In this paper we describe a quadratically convergent
algorithm for solving parameter estimation problems. We illustrate how
the method can by used to construct aircraft aerodynamic models from
flight test data.

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