Adjointbased

Adjoint-based methods refer to a class of techniques that use adjoint equations to compute sensitivities, gradients, and error estimates for complex mathematical models. The approach originated in control theory and variational calculus and is widely used in optimization, inverse problems, and uncertainty quantification. The central idea is to augment the forward model with a Lagrange multiplier, the adjoint variable, so that the total derivative of a scalar objective with respect to parameters can be obtained without differentiating the entire model with respect to every parameter.

In practice, one derives an adjoint equation from the forward model and the objective, solves the forward problem to obtain the state, then solves the adjoint problem to obtain the adjoint state. The gradient with respect to parameters is assembled from a combination of state, adjoint state, and possibly parameter-dependent terms. This typically yields a gradient that is accurate and inexpensive to compute even when the parameter space is high-dimensional, making adjoint methods attractive for PDE-constrained optimization and other large-scale problems.

There are continuous (analytical) and discrete (computed after discretization) adjoint formulations. Time-dependent problems require backward-in-time adjoint equations. Adjoint-based error estimation, or goal-oriented error control, uses the adjoint solution to assess and reduce the error in a quantity of interest.

Applications span computational fluid dynamics, structural optimization, electromagnetics, geophysics, and data assimilation. Challenges include constructing correct adjoint models, ensuring stability and consistency, and managing memory for large-scale or time-dependent problems. End users typically rely on specialized software or automatic differentiation to implement adjoint computations.