Evaluating the sensitivity of a scalar-valued objective function Q(ξ) with respect to the control variables, ξ, at a specific point, ξ0, can be rephrased as the problem of calculating the derivative ∂Q/∂ξ at ξ = ξ0. In the context of a multiphysics model, Q is usually not an explicit expression in the control variables ξ alone. Rather, Q(u(ξ), ξ) is also a function of the solution variables u, which are in turn implicitly functions of ξ. Therefore, by using the chain rule, the sensitivity can be computed as

The explicit partial derivative of the objective function with respect to the control variables or to the solution u are easy to compute using symbolic differentiation. In contrast, the remaining term, the partial derivative of the solution u with respect to the control variables, is more difficult to obtain. Since the multiphysics problem is a PDE, which after discretization is represented by a system of equations L(u(ξ), ξ) = 0, computing the total derivative of this expression with respect to ξ gives

Assuming that L(u(ξ), ξ) = 0 has a unique solution, the matrix ∂L/∂u is nonsingular, which implies

By inserting Equation 2-5 into Equation 2-4, the sensitivity can be rewritten as

Assuming that the PDE solution has N degrees of freedom and that there are n control variables ξi, ∂Q/∂u is an 1-by-N matrix, ∂L/∂u is an N-by-N matrix, and ∂L/∂ξ is an N-by-n matrix.

Evaluating the inverse of the N-by-N Jacobian matrix is too expensive. In order to avoid that step, an auxiliary linear problem can be introduced. This can be done in two different ways, each requiring at least one additional linear solution step (see Forward Sensitivity Methods and Adjoint Sensitivity Method below).

Assume that the Jacobian ∂L/∂u in Equation 2-4 above is incomplete and denote it by (∂L/∂u)incomplete.

Let the complete Jacobian be ∂L/∂u. Hence, the system to solve is

Then, the solution to the system in Equation 2-7 is approximated iteratively by

where n is the iteration number.

If the previous algorithm does not converge (that is, the estimated error is larger than the given tolerance), the sensitivity computations are repeated using the incomplete Jacobian and the warning Jacobian is incomplete. No convergence when attempting to use the complete Jacobian is written.

If the assemble of the complete Jacobian fails, the incomplete Jacobian is used and the warning Unable to assemble the complete Jacobian. Using incomplete Jacobian for sensitivity analysis is written.

The forward sensitivity method explicitly computes and returns the term ∂u/∂ξ in Equation 2-5 in order to compute the sensitivity in Equation 2-4 by solving n linear systems of equations

When this system is solved, a line is added to the progress window displaying the number (i) of the current sensitivity parameter being solved for.

To use the adjoint sensitivity method, introduce the N-by-1 adjoint solution u∗, which is defined as

Multiplying this relation from the right with the PDE Jacobian ∂L/∂u and transposing leads to a single linear system of equations