S. L. Lee and P. D. Hovland, "Derivative Techniques for Forward Sensitivity Analysis in Time-Dependent Problems," Preprint ANL/MCS-P1153-0404, April 2004. [pdf]
The complex-step derivative approximation technique is a highly accurate and convenient method for computing directional derivatives within simulation codes. The method is similar to finite-difference approximations and automatic differentiation techniques in terms of ease of implementation and accuracy, respectively. We examine the performance and accuracy of finite-difference, automatic differentiation, and complex-step techniques in analyzing the solutions to physical systems modeled as initial-value problems in ordinary differential equations (ODEs). In particular, we investigate the use of these derivative techniques in computing the sensitivity of the ODE solutions with respect to various model parameters. In doing so, we identify the strengths and weaknesses of the derivative techniques and find that the derivify implementation of automatic differentiation can be a simple, accurate, and cost-effective means of computing directional derivatives for forward sensitivity analysis.