Reliable Solution for a 1D Quasilinear Elliptic Equation with Uncertain Coefficients

Abstract

The solution of a quasilinear elliptic state equation depends on the coefficient function belonging to an admissible set. The solution is evaluated by a cost functional the value of which is to be maximized over the admissible set, i.e., the reliable (safe) solution is searched for. Due to the nature of the equation, the Kirchhoff transformation can be applied to obtain both the existence of the true state solution and a cost sensitivity formula. In many cases, the latter makes it possible to determine the reliable solution immediately. The problem is approximated by means of the finite element method, and some convergence results are proven. Numerical examples illustrate the theory which can be directly generalized to spatial problems.

On the worst scenario method: A modified convergence theorem and its application to an uncertain differential equation

“…Quasilinear elliptic boundary value problems with uncertain coefficients were studied in [2], [3], [6], [7], see also [4,Chapter III]. In these works the coefficient of the state equation is a u-dependent function.…”

On the worst scenario method: Application to a quasilinear elliptic 2D-problem with uncertain coefficients

“…Quasilinear elliptic boundary value problems with uncertain coefficients were studied in [6], [7], [1], [2], see also [9,Chapter III]. This paper, primarily, generalizes the one-dimensional problem examined in [5] to a two-dimensional uncertain partial differential equation.…”

On Fuzzy Input Data and the Worst Scenario Method

“…However, admissible sets are infinite-dimensional in advanced worst scenario problems, see [6], [10], for example. To apply the formula for µ MΦ , i.e., to extend the fuzziness from inputs to outputs, we need membership functions suitable for admissible sets common for complex worst scenario problems.…”

“…To guarantee U h ad ⊂ U ad , we could start with a sequence {U h ad } h→0 and define U ad as the closure of h→0 U h ad , see [6].…”