Sensitivity Analysis Using a Fixed Point Interval Iteration

Remark 1. In the traditional definition of the interval Gauss-Seidel operator, the interval vector [z] is equal to [x] (and hence does not appear explicitly in its definition). Using instead [z] = R n disactivates the intersection with the previous domain and can be useful for some applications (cf. Section 3).

Then, the Hansen-Sengupta operator [2] can be defined as follows

where [X] ∈ IR n×n and [x], [y], [z] ∈ IR n . The following theorem shows how the Hansen-Sengupta operator can be used to improve the enclosure and prove the existence of solutions (cf. [4]).

Remark 2. The interval evaluation of the derivatives can be replaced by Lipschitz interval matrices to release the differentiability hypothesis, and to slope matrices to improve the enclosure (though uniqueness of the solution is lost when slopes are used, cf. [4] for details).

A preconditioning is usually coupled to this kind of operator: The preconditioned system C • f (x) = 0, where C ∈ R n×n is nonsingular, is equivalently solved instead of f (x) = 0. The preconditioning matrix C is chosen so that C • f is close to the identity in the domain considered, hence improving the efficiency of the operator.

Functions with parameters are considered in this section. Let f : R p ×R n -→ R n be a function of n variables and p parameters. Parameters will be denoted by the vector a and variables by the vector x. The parametric Hansen-Sengupta operator is expressed applying its non-parametric version to different inputs. A more general parametric Hansen-Sengupta (which was dedicated to quantified parameters thanks to the usage of the Kaucher arithmetic) was proposed and used in [1].

Proof. Fix an arbitrary â ∈ [a] and define g : R n -→ R n by g(x) = f (â, x).

We are going to apply Theorem 1 to g. First,

Therefore, Theorem 1 can be applied to g and the domain [x], and shows that if

]. This holds for every â ∈ [a] and hence concludes the proof.

Using the mean-value extension to compute [y] and the usual inverse midpoint preconditioning gives rise the following parametric Hansen-Sengupta operator, denoted by

Experiments presented in the sequel will be carried out using the natural interval extensions of f , df da and df dx . Remark 3. In (5), the expression

The preconditioned parametric Hansen-Sengupta operator is compared to the preconditioned parametric Krawczyk operator

proposed in [5], where the same interval enclosure [y] : (6) and in (5) (this point is not detailed in [5]). As in (5)

) implies the existence of an unique solution to each system f (a,

is approximately represented on the left hand side graphic of Figure 1 solving the 2 × 2 system of equations for a finite set of parameters values inside [a].

Both operators ( 5) and ( 6) are used to improve the initial enclosure [x] by computing the sequences The final enclosure is also shown on the left hand side graphic of Figure 1. These results seem to show that the parametric Krawczyk operator is sharper than the parametric Hansen-Sengupta operator: they both compute the same final enclosure while the former proves the existence one step before. This is surprising since in their non parametric form the Hansen-Sengupta operator is proved to be sharper in general than the Krawczyk operator (cf. [4]). However, a closer study shows that the Hansen-Sengupta operator is actually sharper: The right hand side graphic of Figure 1 shows the ratio

As this graphic shows, the enclosure computed by the Hansen-Sengupta operator is alway sharper. The difference is sensible at the first iterations (reaching approximately 20% at step 5), and converges to 0 as k goes to infinity (the dashed line corresponds to 12 exp(-0.46k) for information about the convergence rate to 0).

A direct application of Theorem 2 requires an initial domain. However, in practice this initial domain is often not available. Instead, an approximate solution x * for a nominal parameter value a * ∈ [a] is available. In the sequel, x * is supposed to satisfy exactly f (a * , x * ) = 0, but the usage of an approximate solution has no incidence in practice. From the sensitivity analysis point of view, we need to prove that each parameter a ∈ [a] is mapped to an unique solution x and to enclose the set of these solutions.

Provided that the parameters domains are small enough, the iteration

Although this limit can be proved to contain the solution set, the inclusion [x k+1 ] ⊆ int[x k ] will never be satisfies because this iteration somehow translates and inflates the initial approximation x * . It is more practical to additionally inflate each iterate of a fixed ration δ so as to obtain the inclusion [x k+1 ] ⊆ int[x k ] after a finite number of steps, and hence

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