Two generalizations of the singular value decomposition are given. These generalizations provided a unified way of regarding certain matrix problems and the numerical techniques which are used to s...

Generalizing the Singular Value Decomposition

Finite element method — the natural approach

We consider linear systems arising from the use of the finite element method for solving scalar linear elliptic problems. Our main result is that these linear systems, which are symmetric and positive semidefinite, are well approximated by symmetric diagonally dominant matrices. Our framework for defining matrix approximation is support theory. Significant graph theoretic work has already been developed in the support framework for preconditioners in the diagonally dominant case, and, in particular, it is known that such systems can be solved with iterative methods in nearly linear time. Thus, our approximation result implies that these graph theoretic techniques can also solve a class of finite element problems in nearly linear time. We show that the support number bounds, which control the number of iterations in the preconditioned iterative solver, depend on mesh quality measures but not on the problem size or shape of the domain.

pdf/solving-elliptic-finite-element-systems-in-near-linear-time-4bo3jv4tls.pdf

Solving Elliptic Finite Element Systems in Near-Linear Time with Support Preconditioners

1 The model.- 2 Simultaneous diagonalisation (Modal damping).- 3 Phase space.- 4 The singular mass case.- 5 "Indefinite metric".- 6 Matrices and indefinite scalar products.- 7 Oblique projections.- 8 J-orthogonal projections.- 9 Spectral properties and reduction of J-Hermitian matrices.- 10 Definite spectra.- 11 General Hermitian matrix pairs.- 12 Spectral decomposition of a general J-Hermitian matrix.- 13 The matrix exponential.- 14 The quadratic eigenvalue problem.- 15 Simple eigenvalue inclusions.- 16 Spectral shift.- 17 Resonances and resolvents.- 18 Well-posedness .- 19 Modal approximation.- 20 Modal approximation and overdampedness.- 21 Passive control.- 22 Perturbing matrix exponential.- 23 Notes and remarks.

http://library.um.ac.id/free-contents/downloadpdf.php/buku/damped-oscillations-of-linear-systems-a-mathematical-introduction-kresimir-veselic-45392.pdf

Damped oscillations of linear systems: a mathematical introduction / Kresimir Veselic

We describe the design and implementation of a parallel QR decomposition algorithm for a large sparse matrix A. The algorithm is based on the multifrontal approach and makes use of Householder transformations. The tasks are distributed among processors according to an assembly tree which is built from the symbolic factorization of the matrix ATA.

We first address uniprocessor issues and then discuss the multiprocessor implementation of the method. We consider the parallelization of both the factorization phase and the solve phase. We use relaxation of the sparsity structure of both the original matrix and the frontal matrices to improve the performance. We show that, in this case, the use of Level 3 BLAS can lead to very significant gains in performance. We use the eight processor Alliant˜FX/80 at CERFACS to illustrate our discussion.

Multifrontal QR Factorization in a Multiprocessor Environment

The natural factor formulation of the stiffness for the matrix displacement method

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

O.E. Brønlund

Author Tools

Chat about Author

Papers

The natural factor formulation of the stiffness for the matrix displacement method