A scalable solver for a class of variable-order fractional diffusion problems.
Saved in:
| Title: | A scalable solver for a class of variable-order fractional diffusion problems. |
|---|---|
| Authors: | Lei, Wenyu1 (AUTHOR) wenyu.lei@uestc.edu.cn, Turkiyyah, George2 (AUTHOR), Knio, Omar2 (AUTHOR) |
| Source: | Computers & Mathematics with Applications. Oct2026, Vol. 219, p317-332. 16p. |
| Subjects: | Sparse matrices, Integral operators, Numerical analysis |
| Abstract: | We present a scalable method to solve multidimensional fractional diffusion problems with spatially varying fractional order and diffusivity. The nonlocal nature of the operator yields dense discrete systems that are expensive to solve at scale, making the development of scalable solvers for the variable-order case needed in many practical applications an important challenge. In this work, we propose a V-cycle multilevel preconditioner and demonstrate its scalability in general geometries and in the presence of discontinuities in the fractional order. At every level of the spatial hierarchy, the preconditioner represents the discrete system matrix as the sum of a sparse matrix related to the singular near-field interactions, a hierarchical low-rank approximation of the dense matrix representing the far-field effects of the nonlocal setting, and a sparse weighted mass matrix representing nonlocal boundary (volume) conditions. The overall storage requirements and construction costs of the preconditioner are linear in both storage and construction cost. We prove that the condition number of the preconditioned system is bounded independently of the mesh size, and numerical experiments on two-dimensional test problems confirm this finding, demonstrating that end-to-end linear-cost solution of variable-order fractional diffusion problems on general geometries is achievable. [ABSTRACT FROM AUTHOR] |
| Copyright of Computers & Mathematics with Applications is the property of Pergamon Press - An Imprint of Elsevier Science and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
|---|---|
| Header | DbId: egs DbLabel: Engineering Source An: 197133459 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
| IllustrationInfo | |
| Items | – Name: Title Label: Title Group: Ti Data: A scalable solver for a class of variable-order fractional diffusion problems. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Lei%2C+Wenyu%22">Lei, Wenyu</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> wenyu.lei@uestc.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Turkiyyah%2C+George%22">Turkiyyah, George</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Knio%2C+Omar%22">Knio, Omar</searchLink><relatesTo>2</relatesTo> (AUTHOR) – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Computers+%26+Mathematics+with+Applications%22">Computers & Mathematics with Applications</searchLink>. Oct2026, Vol. 219, p317-332. 16p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Sparse+matrices%22">Sparse matrices</searchLink><br /><searchLink fieldCode="DE" term="%22Integral+operators%22">Integral operators</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We present a scalable method to solve multidimensional fractional diffusion problems with spatially varying fractional order and diffusivity. The nonlocal nature of the operator yields dense discrete systems that are expensive to solve at scale, making the development of scalable solvers for the variable-order case needed in many practical applications an important challenge. In this work, we propose a V-cycle multilevel preconditioner and demonstrate its scalability in general geometries and in the presence of discontinuities in the fractional order. At every level of the spatial hierarchy, the preconditioner represents the discrete system matrix as the sum of a sparse matrix related to the singular near-field interactions, a hierarchical low-rank approximation of the dense matrix representing the far-field effects of the nonlocal setting, and a sparse weighted mass matrix representing nonlocal boundary (volume) conditions. The overall storage requirements and construction costs of the preconditioner are linear in both storage and construction cost. We prove that the condition number of the preconditioned system is bounded independently of the mesh size, and numerical experiments on two-dimensional test problems confirm this finding, demonstrating that end-to-end linear-cost solution of variable-order fractional diffusion problems on general geometries is achievable. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Computers & Mathematics with Applications is the property of Pergamon Press - An Imprint of Elsevier Science and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
| PLink | https://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=egs&AN=197133459 |
| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.camwa.2026.07.021 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 16 StartPage: 317 Subjects: – SubjectFull: Sparse matrices Type: general – SubjectFull: Integral operators Type: general – SubjectFull: Numerical analysis Type: general Titles: – TitleFull: A scalable solver for a class of variable-order fractional diffusion problems. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Lei, Wenyu – PersonEntity: Name: NameFull: Turkiyyah, George – PersonEntity: Name: NameFull: Knio, Omar IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 10 Text: Oct2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 08981221 Numbering: – Type: volume Value: 219 Titles: – TitleFull: Computers & Mathematics with Applications Type: main |
| ResultId | 1 |
