16 research outputs found
Accuracy controlled data assimilation for parabolic problems
This paper is concerned with the recovery of (approximate) solutions to
parabolic problems from incomplete and possibly inconsistent observational
data, given on a time-space cylinder that is a strict subset of the
computational domain under consideration. Unlike previous approaches to this
and related problems our starting point is a regularized least squares
formulation in a continuous infinite-dimensional setting that is based on
stable variational time-space formulations of the parabolic PDE. This allows us
to derive a priori as well as a posteriori error bounds for the recovered
states with respect to a certain reference solution. In these bounds the
regularization parameter is disentangled from the underlying discretization. An
important ingredient for the derivation of a posteriori bounds is the
construction of suitable Fortin operators which allow us to control oscillation
errors stemming from the discretization of dual norms. Moreover, the
variational framework allows us to contrive preconditioners for the discrete
problems whose application can be performed in linear time, and for which the
condition numbers of the preconditioned systems are uniformly proportional to
that of the regularized continuous problem.
In particular, we provide suitable stopping criteria for the iterative
solvers based on the a posteriori error bounds. The presented numerical
experiments quantify the theoretical findings and demonstrate the performance
of the numerical scheme in relation with the underlying discretization and
regularization
PACE solver description: tdULL
We describe tdULL, an algorithm for computing treedepth decompositions of minimal depth. An implementation was submitted to the exact track of PACE 2020. tdULL is a branch and bound algorithm branching on inclusion-minimal separators
PACE Solver Description: tdULL
We describe tdULL, an algorithm for computing treedepth decompositions of minimal depth. An implementation was submitted to the exact track of PACE 2020. tdULL is a branch and bound algorithm branching on inclusion-minimal separators
On p-robust saturation on quadrangulations
For the Poisson problem in two dimensions, posed on a domain partitioned into axis-aligned rectangles with up to one hanging node per edge, we envision an efficient error reduction step in an instance-optimal hp-adaptive finite element method. Central to this is the problem: Which increase in local polynomial degree ensures p-robust contraction of the error in energy norm? We reduce this problem to a small number of saturation problems on the reference square, and provide strong numerical evidence for their solution