Adjoint-based sensitivity analysis

We performed Adjoint-based Sensitivity Analysis (ABSA).⁹⁰ Suppose we have a general system consisting of N number of PDEs

where the solution x can be considered an implicit function of the parameters θ. Suppose we have a quantity of interest J(x), and we are interested to know its sensitivity to the parameters, i.e., ∂J∂θ. The direct way to do so is to perturb the parameters and see how they affect the quantity of interest. This method is not generally a bad approach, especially if θ is low-dimensional. However, it can quickly become impossible computationally if θ is a high-dimensional vector. Moreover, if the Equation 50 is time-dependent, it can significantly add to the computational burden, and our problem is a time and space-dependent PDE system with 82 parameters. A way to circumvent this issue is to use ABSA. The idea of this method is to obviate the dependence of ∂J∂θ on the solution x as much as possible. From Equation 50, we can immediately deduce that

or more generally,

holds for all vectors ψ∈RN. Now, if we slightly perturb θ, it will cause a slight perturbation in x and J, i.e.:

By the way of the chain rule, this can be rewritten using Equation 52 as:

So if we set (∗) equal to zero, we have achieved our goal. In fact, the equation

is known as the adjoint equation. Since ∂F∂x is non-singular by the implicit function theorem, it is guaranteed to give us a vector ψ which makes (∗) from the Equation 54 zero. Therefore, using the solution ψ from Equation 55 we can rewrite Equation 54 completely independent of x as follows:

Note that we still need to calculate x for a nominal set of parameters θ from Equation 50 to be able to get ψ from the adjoint equation. But, this needs to be done only once, and after that, the sensitivity calculation is independent of the solution. Computationally, we utilized a FEniCS package called dolfin-adjoint, which makes these calculations much easier.⁹¹