Efficient Applications of Automatic Differnetiation for Shape Sensitivities

摘要

Abstractly, computer programs take a set of inputs and generates a set of outputs. Automatic Differentiation (AD) is a computational procedure which calculates the derivatives of output variables with respect to input variables. This information is especially useful for calcualting Jacobians and gradients for nonlinear equation solvers and optimization. However, when the software approximates the solution to a partial differential equation, and the input variable of interest affects the shape of the domain, the resulting code is inefficient. This is due to the fact that quantities such as the derivative of the mesh are cascaded throughout the code. We discuss a method based on the abstract form of the problem that circumvents mesh derivatives. This method derives an equaivalent problem which is solved on a fixed domain suchthat differnetiation produces the desired result without mesh derivatives. When this method can be applied, savings of up to 25