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.^[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 ${\displaystyle A^{*}={\overline {A}}^{T}}$ is used.

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

The original adjoint calculation method goes back to Jean Cea,^[6] with the use of the Lagrangian of the optimization problem to compute the derivative of a functional with respect to a shape parameter.

For a state variable ${\displaystyle u\in {\mathcal {U}}}$, an optimization variable ${\displaystyle v\in {\mathcal {V}}}$, an objective functional ${\displaystyle J:{\mathcal {U}}\times {\mathcal {V}}\to \mathbb {R} }$ is defined. The state variable ${\displaystyle u}$ is often implicitly dependent on ${\displaystyle v}$ through the (direct) state equation ${\displaystyle D_{v}(u)=0}$ (usually the weak form of a partial differential equation), thus the considered objective is ${\displaystyle j(v)=J(u_{v},v)}$. Usually, one would be interested in calculating ${\displaystyle \nabla j(v)}$ using the chain rule:

${\displaystyle \nabla j(v)=\nabla _{v}J(u_{v},v)+\nabla _{u}J(u_{v})\nabla _{v}u_{v}.}$

Unfortunately, the term ${\displaystyle \nabla _{v}u_{v}}$ is often very hard to differentiate analytically since the dependance is defined through an implicit equation. The Lagrangian functional can be used as a workaround for this issue. Since the state equation can be considered as a constraint in the minimization of ${\displaystyle j}$, the problem

${\displaystyle {\text{minimize}}\ j(v)=J(u_{v},v)}$

${\displaystyle {\text{subject to}}\ D_{v}(u_{v})=0}$

has an associate Lagrangian functional ${\displaystyle {\mathcal {L}}:{\mathcal {U}}\times {\mathcal {V}}\times {\mathcal {U}}\to \mathbb {R} }$ defined by

${\displaystyle {\mathcal {L}}(u,v,\lambda )=J(u,v)+\langle D_{v}(u),\lambda \rangle ,}$

where ${\displaystyle \lambda \in {\mathcal {U}}}$ is a Lagrange multiplier or adjoint state variable and ${\displaystyle \langle \cdot ,\cdot \rangle }$ is an inner product on ${\displaystyle {\mathcal {U}}}$. The method of Lagrange multipliers states that a solution to the problem has to be a stationary point of the lagrangian, namely

${\displaystyle {\begin{cases}d_{u}{\mathcal {L}}(u,v,\lambda ;\delta _{u})=d_{u}J(u,v;\delta _{u})+\langle \delta _{u},D_{v}^{*}(\lambda )\rangle =0&\forall \delta _{u}\in {\mathcal {U}},\\d_{v}{\mathcal {L}}(u,v,\lambda ;\delta _{v})=d_{v}J(u,v;\delta _{v})+\langle d_{v}D_{v}(u;\delta _{v}),\lambda \rangle =0&\forall \delta _{v}\in {\mathcal {V}},\\d_{\lambda }{\mathcal {L}}(u,v,\lambda ;\delta _{\lambda })=\langle D_{v}(u),\delta _{\lambda }\rangle =0\quad &\forall \delta _{\lambda }\in {\mathcal {U}},\end{cases}}}$

where ${\displaystyle d_{x}F(x;\delta _{x})}$ is the Gateaux derivative of ${\displaystyle F}$ with respect to ${\displaystyle x}$ in the direction ${\displaystyle \delta _{x}}$. The last equation is equivalent to ${\displaystyle D_{v}(u)=0}$, the state equation, to which the solution is ${\displaystyle u_{v}}$. The first equation is the so-called adjoint state equation,

${\displaystyle \langle \delta _{u},D_{v}^{*}(\lambda )\rangle =-d_{u}J(u_{v},v;\delta _{u})\quad \forall \delta _{u}\in {\mathcal {U}},}$

because the operator involved is the adjoint operator of ${\displaystyle D_{v}}$, ${\displaystyle D_{v}^{*}}$. Resolving this equation yields the adjoint state ${\displaystyle \lambda _{v}}$. The gradient of the quantity of interest ${\displaystyle j}$ with respect to ${\displaystyle v}$ is ${\displaystyle \langle \nabla j(v),\delta _{v}\rangle =d_{v}j(v;\delta _{v})=d_{v}{\mathcal {L}}(u_{v},v,\lambda _{v};\delta _{v})}$ (the second equation with ${\displaystyle u=u_{v}}$ and ${\displaystyle \lambda =\lambda _{v}}$), thus it can be easily identified by subsequently resolving the direct and adjoint state equations. The process is even simpler when the operator ${\displaystyle D_{v}}$ is self-adjoint or symmetric since the direct and adjoint state equations differ only by their right-hand side.

In a real finite dimensional linear programming context, the objective function could be ${\displaystyle J(u,v)=\langle Au,v\rangle }$, for ${\displaystyle v\in \mathbb {R} ^{n}}$, ${\displaystyle u\in \mathbb {R} ^{m}}$ and ${\displaystyle A\in \mathbb {R} ^{n\times m}}$, and let the state equation be ${\displaystyle B_{v}u=b}$, with ${\displaystyle B_{v}\in \mathbb {R} ^{m\times m}}$ and ${\displaystyle b\in \mathbb {R} ^{m}}$.

The lagrangian function of the problem is ${\displaystyle {\mathcal {L}}(u,v,\lambda )=\langle Au,v\rangle +\langle B_{v}u-b,\lambda \rangle }$, where ${\displaystyle \lambda \in \mathbb {R} ^{m}}$.

The derivative of ${\displaystyle {\mathcal {L}}}$ with respect to ${\displaystyle \lambda }$ yields the state equation as shown before, and the state variable is ${\displaystyle u_{v}=B_{v}^{-1}b}$. The derivative of ${\displaystyle {\mathcal {L}}}$ with respect to ${\displaystyle u}$ is equivalent to the adjoint equation, which is, for every ${\displaystyle \delta _{u}\in \mathbb {R} ^{m}}$,

${\displaystyle d_{u}[\langle B_{v}\cdot -b,\lambda \rangle ](u;\delta _{u})=-\langle A^{\top }v,\delta u\rangle \iff \langle B_{v}\delta _{u},\lambda \rangle =-\langle A^{\top }v,\delta u\rangle \iff \langle B_{v}^{\top }\lambda +A^{\top }v,\delta _{u}\rangle =0\iff B_{v}^{\top }\lambda =-A^{\top }v.}$

Thus, we can write symbolically ${\displaystyle \lambda _{v}=B_{v}^{-\top }A^{\top }v}$. The gradient would be

${\displaystyle \langle \nabla j(v),\delta _{v}\rangle =\langle Au_{v},\delta _{v}\rangle +\langle \nabla _{v}B_{v}:\lambda _{v}\otimes u_{v},\delta _{v}\rangle ,}$

where ${\displaystyle \nabla _{v}B_{v}={\frac {\partial B_{ij}}{\partial v_{k}}}}$ is a third order tensor, ${\displaystyle \lambda _{v}\otimes u_{v}=\lambda _{v}^{\top }u_{v}}$ is the dyadic product between the direct and adjoint states and ${\displaystyle :}$ denotes a double tensor contraction. It is assumed that ${\displaystyle B_{v}}$ has a known analytic expression that can be differentiated easily.

If the operator ${\displaystyle B_{v}}$ was self-adjoint, ${\displaystyle B_{v}=B_{v}^{\top }}$, the direct state equation and the adjoint state equation would have the same left-hand side. In the goal of never inverting a matrix, which is a very slow process numerically, a LU decomposition can be used instead to solve the state equation, in ${\displaystyle O(m^{3})}$ operations for the decomposition and ${\displaystyle O(m^{2})}$ operations for the resolution. That same decomposition can then be used to solve the adjoint state equation in only ${\displaystyle O(m^{2})}$ operations since the matrices are the same.

• Adjoint equation
• Backpropagation
• Method of Lagrange multipliers
• Shape optimization

• A well written explanation by Errico: What is an adjoint Model?
• Another well written explanation with worked examples, written by Bradley [1]
• More technical explanation: A review of the adjoint-state method for computing the gradient of a functional with geophysical applications
• MIT course [2]
• MIT notes [3]

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