Adjoint-Based Local Sensitivity Analysis
This chapter introduces the adjoint operator equations for local sensitivity analysis, an intrusive method for sensitivity analysis. Section 6.1 introduces the adjoint operator and demonstrates that for a somewhat general class of QoIs, a QoI can be written as an inner product of the adjoint equations. Then using manipulations involving the definition of the adjoint we arrive at a general formula for the sensitivity to a given QoI to any parameter. This result requires the solution of a forward and adjoint problem and the estimation of some inner products (integrals). The benefit is that with the adjoint solution one can compute an arbitrary number of sensitivities to a single QoI. The possible downside is that each QoI requires a different adjoint solution; the number of adjoint solutions scales with the number of QoIs. In Sect. 6.2 the concept of an adjoint is extended to time-dependent and nonlinear operators.