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

Dr. Huidong Yang

Nov. 3, 2020, 3:30 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
[Neum\"{u}ller, 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, Tr\"{o}ltzsch, 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.~Tr\"{o}ltzsch (TU Berlin).