Geodesic
On the worst scenario method: Application to a quasilinear elliptic 2D-problem with uncertain coefficients
Applications of Mathematics, Tome 56 (2011) no. 5, pp. 459-480.

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We apply a theoretical framework for solving a class of worst scenario problems to a problem with a nonlinear partial differential equation. In contrast to the one-dimensional problem investigated by P. Harasim in Appl. Math. 53 (2008), No. 6, 583–598, the two-dimensional problem requires stronger assumptions restricting the admissible set to ensure the monotonicity of the nonlinear operator in the examined state problem, and, as a result, to show the existence and uniqueness of the state solution. The existence of the worst scenario is proved through the convergence of a sequence of approximate worst scenarios. Furthermore, it is shown that the Galerkin approximation of the state solution can be calculated by means of the Kachanov method as the limit of a sequence of solutions to linearized problems.
DOI : 10.1007/s10492-011-0026-z
Classification : 35D30, 35G30, 35J62, 47H05, 47J05, 49M25, 49N45, 65J15, 65N30
Keywords: worst scenario problem; nonlinear differential equation; uncertain input parameters; Galerkin approximation; Kachanov method 
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Harasim, Petr. On the worst scenario method: Application to a quasilinear elliptic 2D-problem with uncertain coefficients. Applications of Mathematics, Tome 56 (2011) no. 5, pp. 459-480. doi : 10.1007/s10492-011-0026-z. http://geodesic.mathdoc.fr/articles/10.1007/s10492-011-0026-z/

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