Adjoint state method

The adjoint state method is a numerical method for efficiently computing the gradient of a function or operator in a numerical optimization problem. It has applications in geophysics and seismic imaging.

The adjoint state space is chosen to simplify the physical interpretation of equation constraints.^[1] It may take the form of a Hilbert space.

Adjoint state techniques allow the use of integration by parts, resulting in a form which explicitly contains the physically interesting quantity. An adjoint state equation is introduced, including a new unknown variable.

The adjoint method formulates the gradient of a function towards it's parameters in a constraint optimization form. By using the dual form of this constraint optimization problem. It can be used to calculate the gradient very fast. A nice property is that the number of computations is independent of the number of parameters for which you want the gradient. The adjoint method is derived from the dual problem and is used i.e. in the landweber iteration method.

The name adjoint state method refers to the dual form of the problem. Where the adjoint matrix $A^*$ is used.(When non-complex $A^*==A^T$)

When the primal problem consists of calculating the product $s^T x$ and x must satisfy $Ax=b$, then the dual problem can be written as calculating the product $r^T b$ (which is equal to $s^T x$ ) and r must satisfy $A^* r = s$ in which $A^*$ is the adjoint matrix of the matrix $A$. And $r$ is called the adjoint state vector.

External Links

A well written explanation by Errico: What is an adjoint Model?

More technical explanation: A review of the adjoint-state method for computing the gradient of a functional with geophysical applications

MIT course

Deriving the adjoint state method from different fields:

MIT notes

References

1. Alain Sei & William Symes. Gradient Calculation of the Traveltime Cost Function Without Ray-tracing. Expanded Abstracts, 65th Annual Society of Exploration Geophysicists (SEG) Meeting and Exposition, pages 1351–1354 (Available online)