Discrete Adjoint Gradients for Constraints Using Implicit Time-Integration

The adjoint variable method and the direct differentiation method are two key approaches used in sensitivity analysis to compute first-order gradients in optimization. Direct differentiation involves a straightforward differentiation of the cost function, constraints, and state equations with respect to the optimization variables. Therefore, this method can become computationally elaborate, particularly for large-scale optimization problems. In contrast, the adjoint variable method computes sensitivities by solving adjoint equations which size do not depend on the number of optimization variables. This approach eliminates the need to directly compute the sensitivity of the state equations and, therefore, significantly reduces the computational cost in sensitivity analysis of large-scale optimization problems. This work presents a discrete version of the adjoint gradient approach for equality and inequality constraints, which can be used for efficient gradient computation in sensitivity analysis or optimization of large-scale problems. The basic approach has been proposed by Lichtenecker and Nachbagauer and is extended in this paper to incorporate implicit one-step time integrators. With this extension, the sensitivity analysis for systems that are solved with implicit time integrators can be computed efficiently.