| Popis: |
In this thesis, we propose, analyse and implement numerical methods for time-dependent non-linear parabolic and Schrödinger-type equations with uncertain parameters. The discretisation of the parameter space which incorporates the uncertainty of the problem is performed via single- and multi-level collocation strategies. To deal with the possibly large dimension of the parameter space, sparse grid collocation techniques are used to alleviate the curse of dimensionality to a certain extent. We prove that the multi-level method is capable of reducing the overall computational costs significantly. In the parabolic case, the time discretisation is performed via an implicit-explicit splitting strategy of order two which consists shortly speaking of a combination of an implicit trapezoidal rule for the stiff linear part and Heun's method for the non-linear part. In the Schrödinger case, time is discretised via the famous second-order Strang splitting method. For both problem classes we review known error bounds for both discretizations and prove new error bounds for the time discretisations which take the regularity in the parameter space into account. In the parabolic case, a new error bound for the "implicit-explicit trapezoidal method" (IMEXT) method is proved. To our knowledge, this error bound stating second-order convergence of the IMEXT method closes a current gap in the literature. Utilising the aforementioned new error bounds for both problem classes, we can rigorously prove convergence of the single- and multi-level methods. Additionally, cost savings of the multi-level methods compared to the single-level approach are predicted and verifed by numerical examples. The results mentioned above are novel contributions in two areas of mathematics. The first one is (analysis of) numerical methods for uncertainty quantification and the second one is numerical analysis of time-integration schemes for PDEs. |
