Structural Representations of Schur Complements in Sparse Matrices

1993 ◽  
pp. 85-100 ◽  
Author(s):  
Stanley C. Eisenstat ◽  
Joseph W. H. Liu
Keyword(s):  
Sparse Matrices ◽  
Schur Complements

scholarly journals Emerging issues in genomic selection

2021 ◽  
Author(s):  
I Misztal ◽  
I Aguilar ◽  
D Lourenco ◽  
L Ma ◽  
J Steibel ◽  
...  
Keyword(s):  
Genomic Selection ◽  
Sparse Matrices ◽  
Large Datasets ◽  
Single Step ◽  
Genomic Information ◽  
Long Term Effects ◽  
Additive Variance ◽  
Recent Developments ◽  
Emerging Issues ◽  
Using Data

Abstract Genomic selection is now practiced successfully across many species. However, many questions remain such as long-term effects, estimations of genomic parameters, robustness of GWAS with small and large datasets, and stability of genomic predictions. This study summarizes presentations from at the 2020 ASAS symposium. The focus of many studies until now is on linkage disequilibrium (LD) between two loci. Ignoring higher level equilibrium may lead to phantom dominance and epistasis. The Bulmer effect leads to a reduction of the additive variance; however, selection for increased recombination rate can release anew genetic variance. With genomic information, estimates of genetic parameters may be biased by genomic preselection, but costs of estimation can increase drastically due to the dense form of the genomic information. To make computation of estimates feasible, genotypes could be retained only for the most important animals, and methods of estimation should use algorithms that can recognize dense blocks in sparse matrices. GWAS studies using small genomic datasets frequently find many marker-trait associations whereas studies using much bigger datasets find only a few. Most current tools use very simple models for GWAS, possibly causing artifacts. These models are adequate for large datasets where pseudo-phenotypes such as deregressed proofs indirectly account for important effects for traits of interest. Artifacts arising in GWAS with small datasets can be minimized by using data from all animals (whether genotyped or not), realistic models, and methods that account for population structure. Recent developments permit computation of p-values from GBLUP, where models can be arbitrarily complex but restricted to genotyped animals only, and to single-step GBLUP that also uses phenotypes from ungenotyped animals. Stability was an important part of nongenomic evaluations, where genetic predictions were stable in the absence of new data even with low prediction accuracies. Unfortunately, genomic evaluations for such animals change because all animals with genotypes are connected. A top ranked animal can easily drop in the next evaluation, causing a crisis of confidence in genomic evaluations. While correlations between consecutive genomic evaluations are high, outliers can have differences as high as one SD. A solution to fluctuating genomic evaluations is to base selection decisions on groups of animals. While many issues in genomic selection have been solved, many new issues that require additional research continue to surface.

Schur complement spectral bounds for large hybrid FETI-DP clusters and huge three-dimensional scalar problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Zdeněk Dostál ◽  
Tomáš Brzobohatý ◽  
Oldřich Vlach
Keyword(s):  
Condition Number ◽  
Three Dimensional ◽  
Schur Complement ◽  
Decomposition Methods ◽  
Stiffness Matrices ◽  
Spectral Bounds ◽  
Schur Complements ◽  
Linear Problems ◽  
Primal Method ◽  
Large Clusters

Abstract Bounds on the spectrum of Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients of the convergence analysis of FETI (finite element tearing and interconnecting) based domain decomposition methods. Here we give bounds on the regular condition number of Schur complements of “floating” clusters arising from the discretization of 3D Laplacian on a cube decomposed into cube subdomains. The results show that the condition number of the cluster defined on a fixed domain decomposed into m × m × m cube subdomains connected by face and optionally edge averages increases proportionally to m. The estimates support scalability of unpreconditioned H-FETI-DP (hybrid FETI dual-primal) method. Though the research is most important for the solution of variational inequalities, the results of numerical experiments indicate that unpreconditioned H-FETI-DP with large clusters can be useful also for the solution of huge linear problems.

PARALLEL COMPUTING OF NUMERICAL SCHEMES AND BIG DATA ANALYTIC FOR SOLVING REAL LIFE APPLICATIONS

2016 ◽  
Vol 78 (8-2) ◽  
Author(s):  
Norma Alias ◽  
Nadia Nofri Yeni Suhari ◽  
Hafizah Farhah Saipan Saipol ◽  
Abdullah Aysh Dahawi ◽  
Masyitah Mohd Saidi ◽  
...  
Keyword(s):  
Big Data ◽  
Parallel Computing ◽  
Parallel Algorithm ◽  
Sparse Matrices ◽  
Real Life ◽  
Poor Performance ◽  
Equation System ◽  
Numerical Schemes ◽  
Linear Equation System ◽  
Data Analytic

This paper proposed the several real life applications for big data analytic using parallel computing software. Some parallel computing software under consideration are Parallel Virtual Machine, MATLAB Distributed Computing Server and Compute Unified Device Architecture to simulate the big data problems. The parallel computing is able to overcome the poor performance at the runtime, speedup and efficiency of programming in sequential computing. The mathematical models for the big data analytic are based on partial differential equations and obtained the large sparse matrices from discretization and development of the linear equation system. Iterative numerical schemes are used to solve the problems. Thus, the process of computational problems are summarized in parallel algorithm. Therefore, the parallel algorithm development is based on domain decomposition of problems and the architecture of difference parallel computing software. The parallel performance evaluations for distributed and shared memory architecture are investigated in terms of speedup, efficiency, effectiveness and temporal performance.

scholarly journals Sparse Recovery Using Sparse Matrices

2010 ◽  
Vol 98 (6) ◽  
pp. 937-947 ◽  
Author(s):  
Anna Gilbert ◽  
Piotr Indyk
Keyword(s):  
Sparse Matrices ◽  
Sparse Recovery

scholarly journals Amesos2 and Belos: Direct and Iterative Solvers for Large Sparse Linear Systems

2012 ◽  
Vol 20 (3) ◽  
pp. 241-255 ◽  
Author(s):  
Eric Bavier ◽  
Mark Hoemmen ◽  
Sivasankaran Rajamanickam ◽  
Heidi Thornquist
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Sparse Matrix ◽  
Sparse Matrices ◽  
Direct Methods ◽  
Iterative Solvers ◽  
Software Project ◽  
Sparse Linear Systems ◽  
Scalar Type ◽  
Mixed Precision

Solvers for large sparse linear systems come in two categories: direct and iterative. Amesos2, a package in the Trilinos software project, provides direct methods, and Belos, another Trilinos package, provides iterative methods. Amesos2 offers a common interface to many different sparse matrix factorization codes, and can handle any implementation of sparse matrices and vectors, via an easy-to-extend C++ traits interface. It can also factor matrices whose entries have arbitrary “Scalar” type, enabling extended-precision and mixed-precision algorithms. Belos includes many different iterative methods for solving large sparse linear systems and least-squares problems. Unlike competing iterative solver libraries, Belos completely decouples the algorithms from the implementations of the underlying linear algebra objects. This lets Belos exploit the latest hardware without changes to the code. Belos favors algorithms that solve higher-level problems, such as multiple simultaneous linear systems and sequences of related linear systems, faster than standard algorithms. The package also supports extended-precision and mixed-precision algorithms. Together, Amesos2 and Belos form a complete suite of sparse linear solvers.

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