arXiv subjects

Zhaojun Bai

Publications and source records attributed to Zhaojun Bai.

23 records · Page 2Linked to original sources

arXiv2016

Convergence analysis of a locally accelerated preconditioned steepest descent method for Hermitian-definite generalized eigenvalue problems

By extending the classical analysis techniques due to Samokish, Faddeev and Faddeeva, and Longsine and McCormick among others, we prove the convergence of preconditioned steepest descent with implicit deflation (PSD-id) method for solving Hermitian-definite generalized eigenvalue problems. Furthermore, we derive a nonasymptotic estimate of the rate of convergence of the \psdid method. We show that with the proper choice of the shift, the indefinite shift-and-invert preconditioner is a locally accelerated preconditioner, and is asymptotically optimal that leads to superlinear convergence. Numerical examples are presented to verify the theoretical results on the convergence behavior of the \psdid method for solving ill-conditioned Hermitian-definite generalized eigenvalue problems arising from electronic structure calculations. While rigorous and full-scale convergence proofs of preconditioned block steepest descent methods in practical use still largely eludes us, we believe the theoretical results presented in this paper sheds light on an improved understanding of the convergence behavior of these block methods.

math.NA↗

arXiv2016

Impurity-Induced Antiferromagnetic Domains in the Periodic Anderson Model

A central feature of the Periodic Anderson Model is the competition between antiferromagnetism, mediated by the Ruderman-Kittel-Kasuya-Yosida interaction at small conduction electron-local electron hybridization $V$, and singlet formation at large $V$. At zero temperature, and in dimension $d>1$, these two phases are separated by a quantum critical point $V_c$. We use Quantum Monte Carlo simulations to explore the effect of impurities which have a local hybridization $V_{*} < V_c$ in the AF regime which are embedded in a bulk singlet phase with $V > V_c$. We measure the suppression of singlet correlations and the antiferromagnetic correlations which form around the impurity, as well as the size of the resulting domain. Our calculations agree qualitatively with NMR measurements in CeCoIn$_{5-x}$Cd$_x$.

cond-mat.str-el↗

arXiv2013

Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations

The iterative diagonalization of a sequence of large ill-conditioned generalized eigenvalue problems is a computational bottleneck in quantum mechanical methods employing a nonorthogonal basis for {\em ab initio} electronic structure calculations. We propose a hybrid preconditioning scheme to effectively combine global and locally accelerated preconditioners for rapid iterative diagonalization of such eigenvalue problems. In partition-of-unity finite-element (PUFE) pseudopotential density-functional calculations, employing a nonorthogonal basis, we show that the hybrid preconditioned block steepest descent method is a cost-effective eigensolver, outperforming current state-of-the-art global preconditioning schemes, and comparably efficient for the ill-conditioned generalized eigenvalue problems produced by PUFE as the locally optimal block preconditioned conjugate-gradient method for the well-conditioned standard eigenvalue problems produced by planewave methods.

cond-mat.mtrl-sci↗

arXiv2010

Error Control of Iterative Linear Solvers for Integrated Groundwater Models

An open problem that arises when using modern iterative linear solvers, such as the preconditioned conjugate gradient (PCG) method or Generalized Minimum RESidual method (GMRES) is how to choose the residual tolerance in the linear solver to be consistent with the tolerance on the solution error. This problem is especially acute for integrated groundwater models which are implicitly coupled to another model, such as surface water models, and resolve both multiple scales of flow and temporal interaction terms, giving rise to linear systems with variable scaling. This article uses the theory of 'forward error bound estimation' to show how rescaling the linear system affects the correspondence between the residual error in the preconditioned linear system and the solution error. Using examples of linear systems from models developed using the USGS GSFLOW package and the California State Department of Water Resources' Integrated Water Flow Model (IWFM), we observe that this error bound guides the choice of a practical measure for controlling the error in rescaled linear systems. It is found that forward error can be controlled in preconditioned GMRES by rescaling the linear system and normalizing the stopping tolerance. We implemented a preconditioned GMRES algorithm and benchmarked it against the Successive-Over-Relaxation (SOR) method. Improved error control reduces redundant iterations in the GMRES algorithm and results in overall simulation speedups as large as 7.7x. This research is expected to broadly impact groundwater modelers through the demonstration of a practical approach for setting the residual tolerance in line with the solution error tolerance.

math.NA↗

arXiv2007

CompostBin: A DNA composition-based algorithm for binning environmental shotgun reads

A major hindrance to studies of microbial diversity has been that the vast majority of microbes cannot be cultured in the laboratory and thus are not amenable to traditional methods of characterization. Environmental shotgun sequencing (ESS) overcomes this hurdle by sequencing the DNA from the organisms present in a microbial community. The interpretation of this metagenomic data can be greatly facilitated by associating every sequence read with its source organism. We report the development of CompostBin, a DNA composition-based algorithm for analyzing metagenomic sequence reads and distributing them into taxon-specific bins. Unlike previous methods that seek to bin assembled contigs and often require training on known reference genomes, CompostBin has the ability to accurately bin raw sequence reads without need for assembly or training. It applies principal component analysis to project the data into an informative lower-dimensional space, and then uses the normalized cut clustering algorithm on this filtered data set to classify sequences into taxon-specific bins. We demonstrate the algorithm's accuracy on a variety of simulated data sets and on one metagenomic data set with known species assignments. CompostBin is a work in progress, with several refinements of the algorithm planned for the future.

q-bio.GN↗