Automatic Differentiation

Abstract

Computing gradients is a fundamental step in shape optimization. Gradients are needed in sensitivity analysis to measure the variation of performanced.dziinduced by a small perturbationdziof the i control parameter. They are also used in reliability analysis for computing a first-order random model from known stochastic distributions of the data. Finally, we have seen in the previous chapter that gradients are needed in most algorithms used in mathematical programming, and more generally in most multipoint optimization techniques. If the functions to be differentiated are outputs of computer programmes, these gradients can be computed automatically by differentiating each line of these computer programmes This is called automatic differentiation and is the topic of the present chapter. For complex cost functions used in shape optimization, automatic differentiation will have to be coupled with Lagrangian techniques, and this will be the purpose of the next chapter