Abstract
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.
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The work was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z 30860518.
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Harasim, P. On the worst scenario method: Application to a quasilinear elliptic 2D-problem with uncertain coefficients. Appl Math 56, 459–480 (2011). https://doi.org/10.1007/s10492-011-0026-z
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DOI: https://doi.org/10.1007/s10492-011-0026-z
Keywords
- worst scenario problem
- nonlinear differential equation
- uncertain input parameters
- Galerkin approximation
- Kachanov method