# Adjoint-Based Local Sensitivity Analysis

## Abstract

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.

## Supplementary material

## References

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