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Adjoint state method — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Adjoint_state_method
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-15T14:25:04.083Z
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Adjoint state method — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Adjoint_state_method (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_2b1cd2bbbf04 (Metraipolis archive dc_2b1cd2bbbf04, sha256 19d79aedf6afcc0fd2cffbdb19ba87f039da272a6af9deb2171c7baa1cecd333)

BibTeX

@misc{dc2b1cd2bbbf04,
  title = {Adjoint state method — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Adjoint_state_method},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_2b1cd2bbbf04; sha256 19d79aedf6afcc0fd2cffbdb19ba87f039da272a6af9deb2171c7baa1cecd333},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Adjoint_state_method. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 ∗ = A ¯ T {\displaystyle A^{*}={\overline {A}}^{T}} is used. When the initial problem consists of calculating the product s T x {\displaystyle s^{T}x} and x {\displaystyle x} must satisfy A x = b {\displaystyle Ax=b} , the dual problem can be realized as calculating the product r T b {\displaystyle r^{T}b} ( = s T x {\displaystyle =s^{T}x} ), where r {\displaystyle r} must satisfy A ∗ r = s {\displaystyle A^{*}r=s} . And r {\displaystyle r} is called the adjoint state vector.

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; categoria «Numerical analysis»).
Fonte: Wikipedia in inglese, voce «Adjoint state method» — https://en.wikipedia.org/wiki/Adjoint_state_method
Wikidata Q4683172 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q4683172
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Adjoint_state_method?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1364581354, consultata il 2026-10-15T13:54:52.275Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 615ce3f844965f9448ff50d508c4a61829cc3b2a26f09c8a5040f0790372f9b2