Adjoint state method
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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, seismic imaging and more recently in neural networks.
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 its 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 e.g. in the Landweber iteration method .
When the initial problem consists of calculating the product and must satisfy , the dual problem can be realized as calculating the product (), where must satisfy . And is called the adjoint state vector.
- Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, David Duvenaud Neural Ordinary Differential Equations Available online
- 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)
- A well written explanation by Errico: What is an adjoint Model?
- Another well written explanation with worked examples, written by Bradley 
- 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 
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