# Algebraic preconditioners, initial condition reconstruction, stochastic Galerkin finite element method

## Dr. Huidong Yang

**Nov. 3, 2020, 2:30 p.m. ZOOM**

The aim of this talk is to give a short summary on our recent ongoing work.

We will first present some numerical results for algebraic multigrid (AMG) and balancing domain decomposition by constraints (BDDC) preconditioners in solving singularly perturbed reaction-diffusion equations. These equations are arising from, e.g, elimination of co-state in the coupled state and co-state optimality system when using energy regularization [Neumüller, Steinbach 2020]. We consider given right hand sides which are smooth, discontinuous, or having boundary layers.

In the second part, we discuss inverse estimates of the initial conditions for the heat equation using the space-time finite element method [Steinbach 2015]. This kind of inverse problems are formulated as optimal control of parabolic equations in the space-time domain. The objective is a standard terminal observation functional including the Tikhonov regularization, in contrast to tracking type objectives considered in our recent work [Langer, Steinbach, Tröltzsch, Yang, 2020].

Finally, we provide some numerical experiments on an adaptive stochastic Galerkin finite element method for elliptic PDEs with random coefficients in the countably-parametric domain. This leads to elliptic PDEs in high dimension. Therefore, in order to reduce the number of the degrees of freedom and the computational cost, an adaptive method in both the spatial and stochastic refinement is very helpful [Eigel, etc., 2015].

This presentation is mainly based on joint work with U. Langer (RICAM), O. Steinbach (TU Graz), and F. Tröltzsch (TU Berlin).