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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.[1] It has applications in geophysics, seismic imaging, photonics and more recently in neural networks.[2]
The adjoint state space is chosen to simplify the physical interpretation of equation constraints.[3]
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[4] and is used e.g. in the Landweber iteration method.[5]
The name adjoint state method refers to the dual form of the problem, where the adjoint matrix is used.
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.
^Pollini, Nicolò; Lavan, Oren; Amir, Oded (2018-06-01). "Adjoint sensitivity analysis and optimization of hysteretic dynamic systems with nonlinear viscous dampers". Structural and Multidisciplinary Optimization. 57 (6): 2273–2289. doi:10.1007/s00158-017-1858-2. ISSN 1615-1488. S2CID 125712091.
^Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, David Duvenaud Neural Ordinary Differential Equations Available online
^Plessix, R-E. "A review of the adjoint-state method for computing the gradient of a functional with geophysical applications." Geophysical Journal International, 2006, 167(2): 495-503. free access on GJI website
^McNamara, Antoine; Treuille, Adrien; Popović, Zoran; Stam, Jos (August 2004). "Fluid control using the adjoint method" (PDF). ACM Transactions on Graphics. 23 (3): 449–456. doi:10.1145/1015706.1015744. Archived (PDF) from the original on 29 January 2022. Retrieved 28 October 2022.
^Lundvall, Johan (2007). "Data Assimilation in Fluid Dynamics using Adjoint Optimization" (PDF). Sweden: Linköping University of Technology. Archived (PDF) from the original on 9 October 2022. Retrieved 28 October 2022.
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