Properties of the adjoint equations

The adjoint equations have a reputation for being counterintuitive. When told the adjoint equations run backwards in time, this can strike the novice as bizarre. Therefore, it is worth taking some time to explore these properties, until it is obvious that the adjoint system should run backwards in time, and (more generally) reverse the propagation of information. In fact, these supposedly confusing properties are induced by nothing more exotic than simple transposition.

The adjoint reverses the propagation of information

A simple advection example

Suppose we are are solving a one-dimensional advection-type equation on a mesh with three nodes, at \(x_0=0\), \(x_1=0.5\), and \(x_2=1\). The velocity goes from left to right, and so we impose an inflow boundary condition at the left-most node \(x_0\). A simple sketch of the linear system that might describe this configuration could look as follows:

\[\begin{split}\begin{pmatrix} 1 & 0 & 0 \\ a & b & 0 \\ c & d & e \end{pmatrix} \begin{pmatrix} u_0 \\ u_1 \\ u_2 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix},\end{split}\]

where \(a, b, c, d\) and \(e\) are some coefficients of the matrix arising from a discretisation of the equation. The equation for \(u_1\) does not depend on \(u_2\), as information is flowing from left to right. The structure of the matrix dictates the propagation of information of the system: first \(u_0\) is set to the boundary condition value, then \(u_1\) may be computed, and then finally \(u_2\). The lower-triangular nature of the matrix reflects the rightward propagation of information.

Notice that \(u_0\) is prescribed: that is, the value of \(u_0\) does not depend on the values at any other nodes; all off-diagonal entries on the row for \(u_0\) are zero. Notice further that the value \(u_2\) is diagnostic: no other nodes depend on its value; all off-diagonal entries on its column are zero.

Now suppose that we take the adjoint of this system with respect to some functional \(J(u)\). The operator is linear (no entry in the matrix depends on \(u\)), and so the adjoint of this system is just its transpose:

\[\begin{split}\begin{pmatrix} 1 & a & c \\ 0 & b & d \\ 0 & 0 & e \end{pmatrix} \begin{pmatrix} \lambda_0 \\ \lambda_1 \\ \lambda_2 \end{pmatrix} = \begin{pmatrix} {\partial J}/{\partial u_0} \\ {\partial J}/{\partial u_1} \\ {\partial J}/{\partial u_2} \end{pmatrix},\end{split}\]

The adjoint of the advection equation

If the forward equation is \(u \cdot \nabla T\), where \(u\) is the advecting velocity and \(T\) is the advected tracer, then its corresponding adjoint term is \(-u \cdot \nabla \lambda\). The adjoint advection equation is itself an advection equation, with the reverse of the forward velocity.

where \(\lambda\) is the adjoint variable corresponding to \(u\). Observe that transposing the forward system yields an upper-triangular adjoint system: the adjoint propagates information from right to left, in the opposite sense to the propagation of the forward system. To solve this system, one would first solve for \(\lambda_2\), then compute \(\lambda_1\), and finally \(\lambda_0\).

Further notice that \(\lambda_2\) is now prescribed: it can be computed directly from the data, with no dependencies on the values of other adjoint variables; all of the off-diagonal entries in its row are zero. \(\lambda_0\) is now diagnostic: no other variables depend on its value; all off-diagonal entries in its column are zero.

Prescribed and diagnostic variables

This is a general pattern. Variables that are prescribed in the forward model are diagnostic in the adjoint; variables that are diagnostic in the forward model are prescribed in the adjoint.

A time-dependent example

Now consider a time-dependent system. For convenience, we assume the system is linear, but the result holds true in exactly the same way for nonlinear systems. We start with an initial condition \(f_0\) for \(u_0\) (where the subscript denotes the timestep, rather than the node). We then use this information to compute the value at the next timestep, \(u_1\). This information is then used to compute \(u_2\), and so on. This temporal structure can be represented as a block-structured matrix:

\[\begin{split}\begin{pmatrix} I & 0 & 0 \\ A & B & 0 \\ C & D & E \end{pmatrix} \begin{pmatrix} u_0 \\ u_1 \\ u_2 \end{pmatrix} = \begin{pmatrix} f_0 \\ f_1 \\ f_2 \end{pmatrix},\end{split}\]

where \(I\) is the identity operator, \(A, B, C, D\) and \(E\) are some operators arising from the discretisation of the time-dependent system, \(f_0\) is the initial condition for \(u_0\), and \(f_n\) is the source term for the equation for \(u_n\).

Again, the temporal propagation of information forward in time is reflected in the lower-triangular structure of the matrix. This reflects the fact that it is possible to timestep the system, and solve for parts of the solution \(u\) at a time. If the discrete operator were not lower-triangular, all timesteps of the solution \(u\) would be coupled, and would have to be solved for together.

Notice again that the value at the initial time \(u_0\) is prescribed, and the value at the final time \(u_2\) is diagnostic.

Now let us take the adjoint of this system. Since the operator has been assumed to be linear, the adjoint of this system is given by the block-structured matrix

\[\begin{split}\begin{pmatrix} I & A^* & C^* \\ 0 & B^* & D^* \\ 0 & 0 & E^* \end{pmatrix} \begin{pmatrix} \lambda_0 \\ \lambda_1 \\ \lambda_2 \end{pmatrix} = \begin{pmatrix} {\partial J}/{\partial u_0} \\ {\partial J}/{\partial u_1} \\ {\partial J}/{\partial u_2} \end{pmatrix},\end{split}\]

where \(\lambda\) is the adjoint variable corresponding to \(u\). Observe that the adjoint system is now upper-triangular: the adjoint propagates information from later times to earlier times, in the opposite sense to the propagation of the forward system. To solve this system, one would first solve for \(\lambda_2\), then compute \(\lambda_1\), and finally \(\lambda_0\).

Notice once more that the prescribed-diagnostic relationship applies. In the forward model, the initial condition is prescribed, and the solution at the final time is diagnostic. In the adjoint model, the solution at the final time is prescribed (a so-called terminal condition, rather than an initial condition), and the solution at the beginning of time is diagnostic. This is why when the adjoint of a continuous system is derived, the formulation always includes the specification of a terminal condition on the adjoint system.

The adjoint equation is linear

As noted in the previous section, the operator of the tangent linear system is the linearisation of the operator about the solution \(u\); therefore, the adjoint system is always linear in \(\lambda\).

Unconverged nonlinear iterations

Note that the nonlinear iteration has to converge for the linearisation about the solution at that timestep to be valid. If the model does not drive the nonlinear problem to convergence (perhaps it only does a fixed number of Picard iterations, say), then it is not consistent to see the nonlinear solve as one equation, and to trade it for a linear solve in the adjoint. In other words, if the nonlinear solve does not converge, then each iteration of the unconverged nonlinear solve induces a linear solve in the adjoint system, and so the adjoint will take approximately the same runtime as the forward model.

Converging your nonlinear problem is not only more accurate, it makes the adjoint relatively much more efficient!

This has two major effects. The first is a beneficial effect on the computation time of the adjoint run: while the forward model may be nonlinear, the adjoint is always linear, and so it can be much cheaper to solve than the forward model. For example, if the forward model employs a Newton solver for the nonlinear problem that uses on average \(5\) linear solves to converge to machine precision, then a rough estimate for the adjoint computation is that it will take \(1/5\) the runtime of the forward model.

The second major effect is on the storage requirements of the adjoint run. Unfortunately, this effect is not beneficial. The adjoint operator is a linearisation of the nonlinear operator about the solution \(u\): therefore, if the forward model is nonlinear, the forward solution must be available to assemble the adjoint system. If the forward model is steady, this is not a significant difficulty: however, if the forward model is time-dependent, the entire solution trajectory through time must be available.

The obvious approach to making the entire solution trajectory available is to store the value of every variable solved for. This approach is the simplest, and it is the most efficient option if enough storage is available on the machine to store the entire solution at once. However, for long simulations with many degrees of freedom, it is usually impractical to store the entire solution trajectory, and therefore some alternative approach must be implemented.

Summary

Now that the adjoint and tangent linear equations have been introduced, and some of their properties discussed, let us see in more detail the applications of these concepts. This is discussed in the next section.