On the convergence of adaptive gPC for non-linear random difference equations: Theoretical analysis and some practical recommendations

In this paper, the application of adaptive generalized polynomial chaos (gPC) to quantify the uncertainty for non-linear random difference equations is analyzed. It is proved in detail that, under certain assumptions, the stochastic Galerkin projection technique converges algebraically in mean square to the solution process of the random recursive equation. The effect of the numerical errors on the convergence is also studied. A full numerical experiment illustrates our theoretical findings and gives useful insights to reduce the accumulation of numerical errors in practice.