Lesson plan
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, seismic imaging, photonics and more recently in neural networks. The adjoint state space is chosen to simplify the physical interpretation of equation constraints. 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. The name adjoint state method refers to the dual form of the problem, where the adjoint matrix A^*=\overline A ^T is used. When the initial problem consists of calculating the product s^T x and x must satisfy Ax=b , the dual problem can be realized as calculating the product , where r must satisfy A^* r = s . And r is called the adjoint state vector.
Article summary
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, photonics and more recently in neural networks. The adjoint state space is chosen to simplify the physical interpretation of equation constraints. 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. The name adjoint state method refers to the dual form of the problem, where the adjoint matrix A^*=\overline A ^T is used. When the initial problem consists of calculating the product s^T x and x must satisfy Ax=b , the dual problem can be realized as calculating the product , where r must satisfy A^* r = s . And r is called the adjoint state vector.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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