Using Automatic Differentiation Tools to Solve the Continuous Sensitivity Equation

Shape optimization problems frequently take the form of finding parameters that describe the shape of an object or a region in order to minimize a given design objective (such as weight or drag). In many practical situations, the behavior of states in the system can be modeled as the solution to a partial differential equation (PDE). Calculating the dependence of the state solution on these shape design parameters (the so-called state sensitivity) is of interest to design engineers who want to better understand their problem. Furthermore, these sensitivity variables can be used to evaluate the gradient of the objective function (and other constraint functions) at a given design, which can be readily coupled with an optimization algorithm in an attempt to find the optimal parameter values. The state sensitivity variables satisfy the continuous sensitivity equation (CSE), which can be formally derived by implicit differentiation of the state PDE and the corresponding boundary conditions [4]. This CSE is always linear and shares structure with the state PDE. In particular, these coupled PDEs share the same linearization and boundary condition type (Dirichlet, Neuman, etc.). Nearly all problems of interest require numerical techniques in order to approximate the solutions to these equations. Because these equations share the same structure, many computations which would be used to solve the state PDE alone can be reused in the solution of the CSE. Thus, obtaining the sensitivity variables can be performed for a fraction of the cost of computing the state variables. In many cases, the software to solve the state PDE is already available, representing years of development and testing. Thus, this software needs to be modified in order to solve the coupled system. This is not always straight-forward to implement. Design engineers are often not experts on the simulation software, which may contain “legacy code” (where the software developer is no longer available to consult on code modifications) or “spaghetti code” (where the structure of the code is fragile/poor), making the necessary modification very difficult. Automatic differentiation (AD) tools have been developed to simplify this process [1, 16]. By viewing the original code as a sequence of unary and binary operations, taking the design parameters and returning the (approximate) solution to the PDE, AD tools read in the original code and produce software to solve for the state and sensitivity variables simultaneously. We point out that we have alluded to two fundamentally different approaches for computing the sensitivity variables. In the first, we derive the CSE and then apply solution techniques, i.e. we differentiate-thenapproximate. In the second, we consider the traditional application of AD. Essentially, we approximate-thendifferentiate the state PDE. The operations of differentiation and approximation commute in many situations.