Radial Basis Function Collocation With L‐BFGS Optimization for Higher Order Initial Value Problems.
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| Title: | Radial Basis Function Collocation With L‐BFGS Optimization for Higher Order Initial Value Problems. |
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| Authors: | Al-Shimari, Nolf Shaker1 (AUTHOR) noflshaker8@gmail.com, Habib, Mohammad Rezwan1 (AUTHOR) mohabib@wiley.com |
| Source: | Journal of Applied Mathematics. 6/30/2026, Vol. 2026, p1-13. 13p. |
| Subjects: | Radial basis functions, Initial value problems, Optimization algorithms, Collocation methods, Numerical analysis, Ordinary differential equations, Meshfree methods |
| Abstract: | This paper describes a radial basis function collocation method that uses limited‐memory Broyden–Fletcher–Goldfarb–Shanno optimization to solve one‐dimensional ordinary differential initial value problems. Numerical tests include first‐order problems and manufactured fourth‐, sixth‐, and eighth‐order problems. The method constructs the trial solution from a polynomial part that satisfies the initial data exactly, with the remaining component expressed as a weighted Gaussian RBF expansion. The centers are located on the computational interval, and the selection of shape parameters happens prior to the optimization of coefficients. Therefore, the only trainable degrees of freedom are those represented by RBF coefficients. This formulation in the coefficient space is analytically differentiable, enables high‐order derivatives to be computed directly, and transforms the finite‐dimensional solution procedure into a simple residual minimization problem with explicit control over conditioning. The convergence interpretation is deliberately restricted to the applicable quasi‐Newton setting: L‐BFGS is used as a practical curvature‐based accelerator for the smooth coefficient‐space loss, whereas no global or unconditional superlinear‐convergence claim is made for nonlinear RBF parameterizations. Five benchmark problems are investigated: a first‐order linear IVP, a fourth‐order polynomial IVP, a sixth‐order exponential manufactured IVP, an eighth‐order trigonometric manufactured IVP, and a nonlinear Van der Pol oscillator. The additional sixth‐ and eighth‐order tests support the use of the weighted RBF trial form beyond fourth order, whereas the conditioning data confirm that accuracy must be interpreted together with shape‐parameter selection and regularization. The results indicate that RBF‐L‐BFGS collocation is a flexible meshless strategy for smooth left‐end IVPs when degree‐of‐freedom accounting, residual minimization, and numerical conditioning are treated explicitly. [ABSTRACT FROM AUTHOR] |
| Copyright of Journal of Applied Mathematics is the property of Wiley-Blackwell 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.) | |
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| Header | DbId: egs DbLabel: Engineering Source An: 194973509 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Radial Basis Function Collocation With L‐BFGS Optimization for Higher Order Initial Value Problems. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Al-Shimari%2C+Nolf+Shaker%22">Al-Shimari, Nolf Shaker</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> noflshaker8@gmail.com</i><br /><searchLink fieldCode="AR" term="%22Habib%2C+Mohammad+Rezwan%22">Habib, Mohammad Rezwan</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> mohabib@wiley.com</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Applied+Mathematics%22">Journal of Applied Mathematics</searchLink>. 6/30/2026, Vol. 2026, p1-13. 13p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Radial+basis+functions%22">Radial basis functions</searchLink><br /><searchLink fieldCode="DE" term="%22Initial+value+problems%22">Initial value problems</searchLink><br /><searchLink fieldCode="DE" term="%22Optimization+algorithms%22">Optimization algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Collocation+methods%22">Collocation methods</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink><br /><searchLink fieldCode="DE" term="%22Ordinary+differential+equations%22">Ordinary differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Meshfree+methods%22">Meshfree methods</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: This paper describes a radial basis function collocation method that uses limited‐memory Broyden–Fletcher–Goldfarb–Shanno optimization to solve one‐dimensional ordinary differential initial value problems. Numerical tests include first‐order problems and manufactured fourth‐, sixth‐, and eighth‐order problems. The method constructs the trial solution from a polynomial part that satisfies the initial data exactly, with the remaining component expressed as a weighted Gaussian RBF expansion. The centers are located on the computational interval, and the selection of shape parameters happens prior to the optimization of coefficients. Therefore, the only trainable degrees of freedom are those represented by RBF coefficients. This formulation in the coefficient space is analytically differentiable, enables high‐order derivatives to be computed directly, and transforms the finite‐dimensional solution procedure into a simple residual minimization problem with explicit control over conditioning. The convergence interpretation is deliberately restricted to the applicable quasi‐Newton setting: L‐BFGS is used as a practical curvature‐based accelerator for the smooth coefficient‐space loss, whereas no global or unconditional superlinear‐convergence claim is made for nonlinear RBF parameterizations. Five benchmark problems are investigated: a first‐order linear IVP, a fourth‐order polynomial IVP, a sixth‐order exponential manufactured IVP, an eighth‐order trigonometric manufactured IVP, and a nonlinear Van der Pol oscillator. The additional sixth‐ and eighth‐order tests support the use of the weighted RBF trial form beyond fourth order, whereas the conditioning data confirm that accuracy must be interpreted together with shape‐parameter selection and regularization. The results indicate that RBF‐L‐BFGS collocation is a flexible meshless strategy for smooth left‐end IVPs when degree‐of‐freedom accounting, residual minimization, and numerical conditioning are treated explicitly. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Journal of Applied Mathematics is the property of Wiley-Blackwell 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.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1155/jama/4733090 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 13 StartPage: 1 Subjects: – SubjectFull: Radial basis functions Type: general – SubjectFull: Initial value problems Type: general – SubjectFull: Optimization algorithms Type: general – SubjectFull: Collocation methods Type: general – SubjectFull: Numerical analysis Type: general – SubjectFull: Ordinary differential equations Type: general – SubjectFull: Meshfree methods Type: general Titles: – TitleFull: Radial Basis Function Collocation With L‐BFGS Optimization for Higher Order Initial Value Problems. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Al-Shimari, Nolf Shaker – PersonEntity: Name: NameFull: Habib, Mohammad Rezwan IsPartOfRelationships: – BibEntity: Dates: – D: 30 M: 06 Text: 6/30/2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 1110757X Numbering: – Type: volume Value: 2026 Titles: – TitleFull: Journal of Applied Mathematics Type: main |
| ResultId | 1 |
