Principles of Optimal Design Modeling and Computation

In designing, as in other endeavors, one learns by doing. In this sense, the present chapter, although at the end of the book, is the beginning of the action. The principles and techniques of the previous chapters will be summarized and organized into a problem-solving strategy that can provide guidance in practical design applications. Students in a design optimization course should fix these ideas by applying them to a term project. For the practicing designer, actual problems at the workplace can serve as first trials for this new knowledge, particularly if sufficient experience exists for verifying the first results.

The chapter begins with a review in Section 9.1 of some modeling implications derived from the discussion in previous chapters about how numerical algorithms work. Although the subject is quite extensive, our goal here is to highlight again the intimacy between modeling and computation that was explored first in Chapters 1 and 2. The reader should be convinced by now of the validity of this approach and experience a sense of closure on the subject.

Sections 9.2 and 9.3 deal with two extremely important practical issues: the computation of derivatives and model scaling. Local computation requires knowledge of derivatives. The accuracy by which derivatives are computed can have a profound influence on the performance of the algorithm. A closed-form computation would be best, and this has become dramatically easier with the advent of symbolic computation programs. When this is not possible, numerical approximation of the derivatives can be achieved using finite differences. The newer methods of automatic differentiation are a promising way to compute derivatives of functions defined implicitly by computer programs.

Scaling the model's functions and variables is probably the single most effective “trick” that one can do to tame an initially misbehaving problem. Although there are strong mathematical reasons why scaling works (and sometimes even why it should not work), effective scaling in practice requires a judicious trial and error approach.