FRINK Linked Data Fragments server

Matches in SemOpenAlex for { <https://semopenalex.org/work/W324598049> ?p ?o ?g. }

• W324598049 abstract "A BLOCK ORTHOGONALIZATION PROCEDURE WITH CONSTANT SYNCHRONIZATION REQUIREMENTS ANDREAS STATHOPOULOS AND KESHENG WU y Abstract. We propose an alternative orthonormalization method that computes the orthonor- mal basis from the right singular vectors of a matrix. Its advantage are: a all operations are matrix-matrix multiplications and thus cache-e cient, b only one synchronization point is required in parallel implementations, c could be more stable than Gram-Schmidt. In addition, we consider the problem of incremental orthonormalization where a block of vectors is orthonormalized against a previously orthonormal set of vectors and among itself. We solve this problem by alternating itera- tively between a phase of Gram-Schmidt and a phase of the new method. We provide error analysis and use it to derive bounds on how accurately the two successive orthonormalization phases should be performed to minimize total work performed. Our experiments con rm the favorable numerical behavior of the new method and its e ectiveness on modern parallel computers. Key words. Gram-Schmidt, orthogonalization, Householder, QR factorization, singular value decomposition, Poincare AMS Subject Classi cation. 65F15 1. Introduction. Computing an orthonormal basis from a given set of vectors is a basic computation, common in most scienti c applications. Often, it is also one of the most computationally demanding procedures because the vectors are of large dimension, and because the computation scales as the square of the number of vectors involved. Further, among several orthonormalization techniques the ones that ensure high accuracy are the more expensive ones. In many applications, orthonormalization occurs in an incremental fashion, where a new set of vectors we call this internal set is orthogonalized against a previously orthonormal set of vectors we call this external, and then among themselves. This computation is typical in block Krylov methods, where the Krylov basis is expanded by a block of vectors 12, 11 . It is also typical when certain external orthogonalization constraints have to be applied to the vectors of an iterative method. Locking of converged eigenvectors in eigenvalue iterative methods is such an example 19, 22 . This problem di ers from the classical QR factorization in that the external set of vectors should not be modi ed. Therefore, a two phase process is required; rst orthogonalizing the internal vectors against the external, and second the internal among themselves. Usually, the number of the internal vectors is much smaller than the external ones, and signi cantly smaller than their dimension. Another important di erence is that the accuracy of the R matrix of the QR factorization is not of pri- mary interest, but rather the orthonormality of the produced vectors Q . A variety of orthogonalization techniques exist for both phases. For the external phase, Gram- Schmidt GS and its modi ed version MGS are the most competitive choices. For the internal phase, QR factorization using Householder transformations is the most stable, albeit more expensive method 11 . When the number of vectors is signi - cantly smaller than their dimension, MGS or GS with reorthogonalization are usually preferred. Computationally, MGS, GS and Housholder transformations are based on level 1 or level 2 BLAS operations 15, 9, 8 . These basic kernel computations, dot prod- ucts, vector updates and sometimes matrix-vector operations, cannot fully utilize the Department of Computer Science, College of William and Mary, Williamsburg, Virginia 23187- 8795, andreas@cs.wm.edu . y NERSC, Lawrence Berkeley National Laboratory, Berkeley, California 94720, kwu@lbl.gov ." @default.
• W324598049 created "2016-06-24" @default.
• W324598049 creator A5043129695 @default.
• W324598049 creator A5079391957 @default.
• W324598049 date "2000-04-17" @default.
• W324598049 modified "2023-09-26" @default.
• W324598049 title "A block orthogonalization procedure with constant synchronization requirements" @default.
• W324598049 hasPublicationYear "2000" @default.
• W324598049 type Work @default.
• W324598049 sameAs 324598049 @default.
• W324598049 citedByCount "0" @default.
• W324598049 crossrefType "journal-article" @default.
• W324598049 hasAuthorship W324598049A5043129695 @default.
• W324598049 hasAuthorship W324598049A5079391957 @default.
• W324598049 hasConcept C106487976 @default.
• W324598049 hasConcept C11413529 @default.
• W324598049 hasConcept C121332964 @default.
• W324598049 hasConcept C158693339 @default.
• W324598049 hasConcept C159985019 @default.
• W324598049 hasConcept C187834632 @default.
• W324598049 hasConcept C188060507 @default.
• W324598049 hasConcept C192562407 @default.
• W324598049 hasConcept C199360897 @default.
• W324598049 hasConcept C22789450 @default.
• W324598049 hasConcept C2777027219 @default.
• W324598049 hasConcept C33923547 @default.
• W324598049 hasConcept C41008148 @default.
• W324598049 hasConcept C47559304 @default.
• W324598049 hasConcept C5806529 @default.
• W324598049 hasConcept C62520636 @default.
• W324598049 hasConcept C92951342 @default.
• W324598049 hasConceptScore W324598049C106487976 @default.
• W324598049 hasConceptScore W324598049C11413529 @default.
• W324598049 hasConceptScore W324598049C121332964 @default.
• W324598049 hasConceptScore W324598049C158693339 @default.
• W324598049 hasConceptScore W324598049C159985019 @default.
• W324598049 hasConceptScore W324598049C187834632 @default.
• W324598049 hasConceptScore W324598049C188060507 @default.
• W324598049 hasConceptScore W324598049C192562407 @default.
• W324598049 hasConceptScore W324598049C199360897 @default.
• W324598049 hasConceptScore W324598049C22789450 @default.
• W324598049 hasConceptScore W324598049C2777027219 @default.
• W324598049 hasConceptScore W324598049C33923547 @default.
• W324598049 hasConceptScore W324598049C41008148 @default.
• W324598049 hasConceptScore W324598049C47559304 @default.
• W324598049 hasConceptScore W324598049C5806529 @default.
• W324598049 hasConceptScore W324598049C62520636 @default.
• W324598049 hasConceptScore W324598049C92951342 @default.
• W324598049 hasLocation W3245980491 @default.
• W324598049 hasOpenAccess W324598049 @default.
• W324598049 hasPrimaryLocation W3245980491 @default.
• W324598049 hasRelatedWork W1409456526 @default.
• W324598049 hasRelatedWork W1605345756 @default.
• W324598049 hasRelatedWork W1629244413 @default.
• W324598049 hasRelatedWork W2001135345 @default.
• W324598049 hasRelatedWork W2014168697 @default.
• W324598049 hasRelatedWork W2016499269 @default.
• W324598049 hasRelatedWork W2079903960 @default.
• W324598049 hasRelatedWork W2087891564 @default.
• W324598049 hasRelatedWork W2102181290 @default.
• W324598049 hasRelatedWork W2120585954 @default.
• W324598049 hasRelatedWork W2128770615 @default.
• W324598049 hasRelatedWork W2131347209 @default.
• W324598049 hasRelatedWork W2133884101 @default.
• W324598049 hasRelatedWork W2479312855 @default.
• W324598049 hasRelatedWork W2493468169 @default.
• W324598049 hasRelatedWork W2617923577 @default.
• W324598049 hasRelatedWork W3135020728 @default.
• W324598049 hasRelatedWork W1567737801 @default.
• W324598049 hasRelatedWork W281564841 @default.
• W324598049 hasRelatedWork W3099348172 @default.
• W324598049 isParatext "false" @default.
• W324598049 isRetracted "false" @default.
• W324598049 magId "324598049" @default.
• W324598049 workType "article" @default.