801 research outputs found
Quantitative Anderson localization of Schr\"odinger eigenstates under disorder potentials
This paper concerns spectral properties of linear Schr\"odinger operators
under oscillatory high-amplitude potentials on bounded domains. Depending on
the degree of disorder, we prove the existence of spectral gaps amongst the
lowermost eigenvalues and the emergence of exponentially localized states. We
quantify the rate of decay in terms of geometric parameters that characterize
the potential. The proofs are based on the convergence theory of iterative
solvers for eigenvalue problems and their optimal local preconditioning by
domain decomposition.Comment: accepted for publication in M3A
Computational multiscale methods for linear heterogeneous poroelasticity
We consider a strongly heterogeneous medium saturated by an incompressible
viscous fluid as it appears in geomechanical modeling. This poroelasticity
problem suffers from rapidly oscillating material parameters, which calls for a
thorough numerical treatment. In this paper, we propose a method based on the
local orthogonal decomposition technique and motivated by a similar approach
used for linear thermoelasticity. Therein, local corrector problems are
constructed in line with the static equations, whereas we propose to consider
the full system. This allows to benefit from the given saddle point structure
and results in two decoupled corrector problems for the displacement and the
pressure. We prove the optimal first-order convergence of this method and
verify the result by numerical experiments
Computational Multiscale Methods for Linear Poroelasticity with High Contrast
In this work, we employ the Constraint Energy Minimizing Generalized
Multiscale Finite Element Method (CEM-GMsFEM) to solve the problem of linear
heterogeneous poroelasticity with coefficients of high contrast. The proposed
method makes use of the idea of energy minimization with suitable constraints
in order to generate efficient basis functions for the displacement and the
pressure. These basis functions are constructed by solving a class of local
auxiliary optimization problems based on eigenfunctions containing local
information on the heterogeneity. Techniques of oversampling are adapted to
enhance the computational performance. Convergence of first order is shown and
illustrated by a number of numerical tests.Comment: 14 pages, 9 figure
Localization and delocalization of ground states of Bose-Einstein condensates under disorder
This paper studies the localization behaviour of Bose-Einstein condensates in
disorder potentials, modeled by a Gross-Pitaevskii eigenvalue problem on a
bounded interval. In the regime of weak particle interaction, we are able to
quantify exponential localization of the ground state, depending on statistical
parameters and the strength of the potential. Numerical studies further show
delocalization if we leave the identified parameter range, which is in
agreement with experimental data. These mathematical and numerical findings
allow the prediction of physically relevant regimes where localization of
ground states may be observed experimentally.Comment: accepted for publication in SIAM J. Appl. Mat
Energy-adaptive Riemannian optimization on the Stiefel manifold
This paper addresses the numerical solution of nonlinear eigenvector problems such as the Gross-Pitaevskii and Kohn-Sham equation arising in computational physics and chemistry. These problems characterize critical points of energy minimization problems on the infinite-dimensional Stiefel manifold. To efficiently compute minimizers, we propose a novel Riemannian gradient descent method induced by an energyadaptive metric. Quantified convergence of the methods is established under suitable assumptions on the underlying problem. A non-monotone line search and the inexact evaluation of Riemannian gradients substantially improve the overall efficiency of the method. Numerical experiments illustrate the performance of the method and demonstrates its competitiveness with well-established schemes
Energy-adaptive Riemannian optimization on the Stiefel manifold
This paper addresses the numerical solution of nonlinear eigenvector problems
such as the Gross-Pitaevskii and Kohn-Sham equation arising in computational
physics and chemistry. These problems characterize critical points of energy
minimization problems on the infinite-dimensional Stiefel manifold. To
efficiently compute minimizers, we propose a novel Riemannian gradient descent
method induced by an energy-adaptive metric. Quantified convergence of the
methods is established under suitable assumptions on the underlying problem. A
non-monotone line search and the inexact evaluation of Riemannian gradients
substantially improve the overall efficiency of the method. Numerical
experiments illustrate the performance of the method and demonstrates its
competitiveness with well-established schemes.Comment: accepted for publication in M2A
Re-Annotator: Annotation Pipeline for Microarray Probe Sequences
Microarray technologies are established approaches for high throughput gene expression, methylation and genotyping analysis. An accurate mapping of the array probes is essential to generate reliable biological findings. However, manufacturers of the microarray platforms typically provide incomplete and outdated annotation tables, which often rely on older genome and transcriptome versions that differ substantially from up-to-date sequence databases. Here, we present the Re-Annotator, a re-annotation pipeline for microarray probe sequences. It is primarily designed for gene expression microarrays but can also be adapted to other types of microarrays. The Re-Annotator uses a custom-built mRNA reference database to identify the positions of gene expression array probe sequences. We applied Re-Annotator to the Illumina Human-HT12 v4 microarray platform and found that about one quarter (25%) of the probes differed from the manufacturer's annotation. In further computational experiments on experimental gene expression data, we compared Re-Annotator to another probe re-annotation tool, ReMOAT, and found that Re-Annotator provided an improved re-annotation of microarray probes. A thorough re-annotation of probe information is crucial to any microarray analysis. The Re-Annotator pipeline is freely available at http://sourceforge.net/projects/reannotator along with re-annotated files for Illumina microarrays HumanHT-12 v3/v4 and MouseRef-8 v2
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