981 research outputs found
Reliable a-posteriori error estimators for -adaptive finite element approximations of eigenvalue/eigenvector problems
We present reliable a-posteriori error estimates for -adaptive finite
element approximations of eigenvalue/eigenvector problems. Starting from our
earlier work on adaptive finite element approximations we show a way to
obtain reliable and efficient a-posteriori estimates in the -setting. At
the core of our analysis is the reduction of the problem on the analysis of the
associated boundary value problem. We start from the analysis of Wohlmuth and
Melenk and combine this with our a-posteriori estimation framework to obtain
eigenvalue/eigenvector approximation bounds.Comment: submitte
A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems
In this paper, we propose a method for the approximation of the solution of
high-dimensional weakly coercive problems formulated in tensor spaces using
low-rank approximation formats. The method can be seen as a perturbation of a
minimal residual method with residual norm corresponding to the error in a
specified solution norm. We introduce and analyze an iterative algorithm that
is able to provide a controlled approximation of the optimal approximation of
the solution in a given low-rank subset, without any a priori information on
this solution. We also introduce a weak greedy algorithm which uses this
perturbed minimal residual method for the computation of successive greedy
corrections in small tensor subsets. We prove its convergence under some
conditions on the parameters of the algorithm. The residual norm can be
designed such that the resulting low-rank approximations are quasi-optimal with
respect to particular norms of interest, thus yielding to goal-oriented order
reduction strategies for the approximation of high-dimensional problems. The
proposed numerical method is applied to the solution of a stochastic partial
differential equation which is discretized using standard Galerkin methods in
tensor product spaces
Block Iterative Eigensolvers for Sequences of Correlated Eigenvalue Problems
In Density Functional Theory simulations based on the LAPW method, each
self-consistent field cycle comprises dozens of large dense generalized
eigenproblems. In contrast to real-space methods, eigenpairs solving for
problems at distinct cycles have either been believed to be independent or at
most very loosely connected. In a recent study [7], it was demonstrated that,
contrary to belief, successive eigenproblems in a sequence are strongly
correlated with one another. In particular, by monitoring the subspace angles
between eigenvectors of successive eigenproblems, it was shown that these
angles decrease noticeably after the first few iterations and become close to
collinear. This last result suggests that we can manipulate the eigenvectors,
solving for a specific eigenproblem in a sequence, as an approximate solution
for the following eigenproblem. In this work we present results that are in
line with this intuition. We provide numerical examples where opportunely
selected block iterative eigensolvers benefit from the reuse of eigenvectors by
achieving a substantial speed-up. The results presented will eventually open
the way to a widespread use of block iterative eigensolvers in ab initio
electronic structure codes based on the LAPW approach.Comment: 12 Pages, 5 figures. Accepted for publication on Computer Physics
Communication
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