July 17, 2024

Numerical and Computational Challenges in Science and Engineering Program

John T. Betts, The Boeing Company


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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