For stochastic C0-semigroups on L1-spaces there is a wealth of results that show strong convergence to an equilibrium as t ? ?, given the semigroup contains partial integral operator. This has plenty applications transport equations and in mathematical biology. However, up now operators do not play prominent role theorems which yield uniform rather than only convergence. In this article we prov...

We study sample average approximations under adaptive importance sampling. Based on a Banach-space-valued martingale strong law of large numbers, we establish uniform convergence of the sample average approximation to the function being approximated. In the optimization context, we obtain convergence of the optimal value and optimal solutions of the sample average approximation.

The singularly perturbed parabolic boundary value problem is considered. Difference scheme is obtained by using cubic spline difference scheme on Shishkin’s mesh in space and classical discretization on uniform mesh in time. To obtain better stability and simpler matrix the fitting factor in polynomial form is used. The uniform convergence is achieved. Numerical results are presented. AMS Mathe...