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Conditioning analysis of nonlocal integral operators in fractional Sobolev spaces

Show simple item record Aksoylu, Burak Unlu, Zuhal 2019-07-04T14:19:40Z 2019-07-04T14:19:40Z 2014-03
dc.identifier.citation Aksoylu, B., & Unlu, Z. (2014). Conditioning analysis of nonlocal integral operators in fractional Sobolev spaces. SIAM Journal on Numerical Analysis, 52(2), 653-677. en_US
dc.identifier.issn 0036-1429
dc.description.abstract We study the conditioning of nonlocal integral operators with singular and integrable kernels in fractional Sobolev spaces. These operators are used, for instance, in peridynamics formulation and nonlocal diffusion. In one dimension (1D), we present sharp quantification of the extremal eigenvalues in all three parameters: size of nonlocality, mesh size, and regularity of the fractional Sobolev space. We accomplish sharpness both rigorously and numerically. For the minimal eigenvalue, we obtain sharpness analytically by using a nonlocal characterization of Sobolev spaces. We verify this estimate by exploiting the Toeplitz property of the stiffness matrix. However, the analytical approach fails to give sharp quantification of the maximal eigenvalue. Hence, in 1D, we take an algebraic approach by directly working with the stiffness matrix entries, which have complicated expressions due to all three parameters. We systematically characterize the nonzero entries and dramatically simplify their expressions by using convenient algebra. We establish the zero row sum property of the stiffness matrix and negativity of the off-diagonal entries. Eventually, we arrive at sharpness through the use of the Gerschgorin circle theorem. en_US
dc.language.iso eng en_US
dc.publisher Society for Industrial and Applied Mathematics Publications en_US
dc.rights info:eu-repo/semantics/closedAccess
dc.subject Condition number en_US
dc.subject  Nonlocal operators en_US
dc.subject  Peridynamics en_US
dc.subject  Nonlocal diffusion en_US
dc.subject  Toeplitz matrix en_US
dc.subject  The Gerschgorin circle theorem en_US
dc.subject  Preconditioning en_US
dc.title Conditioning analysis of nonlocal integral operators in fractional Sobolev spaces en_US
dc.type article en_US
dc.relation.journal SIAM Journal on Numerical Analysis en_US
dc.contributor.department TOBB ETU, Faculty of Science and Literature, Department of Mathematics en_US
dc.contributor.department TOBB ETÜ, Fen Edebiyat Fakültesi, Matematik Bölümü tr_TR
dc.identifier.volume 52
dc.identifier.issue 2
dc.identifier.startpage 653
dc.identifier.endpage 677
dc.relation.tubitak info:eu-repo/grantAgreement/TÜBİTAK/TBAG/112T240 en_US
dc.relation.tubitak info:eu-repo/grantAgreement/TÜBİTAK/MAG/112M891 en_US European Union Marie Curie Career Integration [293978] en_US
dc.identifier.wos WOS:000335818000004
dc.identifier.scopus 2-s2.0-84902581457
dc.contributor.tobbetuauthor Aksoylu, Burak
dc.contributor.YOKid 29230
dc.identifier.doi 10.1137/13092407X
dc.contributor.wosresearcherID C-4948-2016
dc.contributor.ScopusAuthorID 23979376500
dc.relation.publicationcategory Uluslararası yayın tr_TR National Science Foundation DMS [1016190] en_US

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