“Adjoint equation”的意思、由来-开放百科全书

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Adjoint equation

释义

Example: Advection-Diffusion PDE
See also
References

An adjoint equation is a linear differential equation, usually derived from its primal equation using integration by parts. Gradient values with respect to a particular quantity of interest can be efficiently calculated by solving the adjoint equation. Methods based on solution of adjoint equations are used in wing shape optimization, fluid flow control and uncertainty quantification. For example this is an Itō stochastic differential equation. Now by using Euler scheme, we integrate the parts of this equation and get another equation, , here is a random variable, later one is an adjoint equation.

Example: Advection-Diffusion PDE

Consider the following linear, scalar advection-diffusion equation for the primal solution , in the domain with Dirichlet boundary conditions:

Let the output of interest be the following linear functional:

Derive the weak form by multiplying the primal equation with a weighting function and performing integration by parts:

where,

Then, consider an infinitesimal perturbation to which produces an infinitesimal change in as follows:

Note that the solution perturbation must vanish at the boundary, since the Dirichlet boundary condition does not admit variations on .

Using the weak form above and the definition of the adjoint given below:

we obtain:

Next, use integration by parts to transfer derivatives of into derivatives of :

The adjoint PDE and its boundary conditions can be deduced from the last equation above. Since is generally non-zero within the domain , it is required that be zero in , in order for the volume term to vanish. Similarly, since the primal flux is generally non-zero at the boundary, we require to be zero there in order for the first boundary term to vanish. The second boundary term vanishes trivially since the primal boundary condition requires at the boundary.

Therefore, the adjoint problem is given by:

Note that the advection term reverses the sign of the convective velocity in the adjoint equation, whereas the diffusion term remains self-adjoint.

References

{{cite journal

| last = Jameson
| first = Antony
| title = Aerodynamic Design via Control Theory
| journal = Journal of Scientific Computing
| volume = 3
| issue = 3
| year = 1988
}}{{DEFAULTSORT:Adjoint Equation}}{{Applied-math-stub}}{{Mech-engineering-stub}}

1 : Differential calculus

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Example: Advection-Diffusion PDE

See also

References