Logo: to the web site of Uppsala University

uu.sePublications from Uppsala University
Change search
Link to record
Permanent link

Direct link
Publications (10 of 10) Show all publications
Mirzaei, D. & Soodbakhsh, N. (2024). A non-oscillatory finite volume scheme using a weighted smoothed reconstruction. Journal of Computational Physics, 508, Article ID 112981.
Open this publication in new window or tab >>A non-oscillatory finite volume scheme using a weighted smoothed reconstruction
2024 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 508, article id 112981Article in journal (Refereed) Published
Abstract [en]

In this research article, we introduce a high-order and non-oscillatory finite volume method in combination with radial basis function approximations and use it for the solution of scalar conservation laws on unstructured meshes. This novel approach departs from conventional non-oscillatory techniques, which often require the use of multiple stencils to achieve smooth reconstructions. Instead, the new method uses a single central stencil and hinges on an approximate interpolation methodology called the weighted smoothed reconstruction (WSR), with a foundation on polyharmonic spline interpolation. Through some numerical experiments, we demonstrate the efficiency and accuracy of the new approach. It reduces the computational cost and performs well in capturing shocks and sharp solution fronts.

Place, publisher, year, edition, pages
Elsevier, 2024
Keywords
Conservation laws, Finite volume method, Radial basis functions, Polyharmonic splines, Weighted smoothed reconstruction (WSR)
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-530459 (URN)10.1016/j.jcp.2024.112981 (DOI)001225394700001 ()
Available from: 2024-06-05 Created: 2024-06-05 Last updated: 2025-01-07Bibliographically approved
Mirzaei, D. & Soodbakhsh, N. (2023). A fault detection method based on partition of unity and kernel approximation. Numerical Algorithms, 93(4), 1759-1794
Open this publication in new window or tab >>A fault detection method based on partition of unity and kernel approximation
2023 (English)In: Numerical Algorithms, ISSN 1017-1398, E-ISSN 1572-9265, Vol. 93, no 4, p. 1759-1794Article in journal (Refereed) Published
Abstract [en]

In this paper, we present a scattered data approximation method for detecting and approximating the discontinuities of a bivariate function and its gradient. The new algorithm is based on partition of unity, polyharmonic kernel interpolation, and principal component analysis. Localized polyharmonic interpolation in partition of unity setting is applied for detecting a set of fault points on or close to discontinuity curves. Then a combination of partition of unity and principal component regression is used to thinning the detected points by moving them approximately on the fault curves. Finally, an ordered subset of these narrowed points is extracted and a parametric spline interpolation is applied to reconstruct the fault curves. A selection of numerical examples with different behaviors and an application for solving scalar conservation law equations illustrate the performance of the algorithm.

Place, publisher, year, edition, pages
Springer, 2023
Keywords
Partition of unity, Radial basis functions, Polyharmonic splines, Fault curves, Fault points, Principal component regression
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-495533 (URN)10.1007/s11075-022-01488-4 (DOI)000913250100002 ()
Funder
Uppsala University
Available from: 2023-01-30 Created: 2023-01-30 Last updated: 2024-09-26Bibliographically approved
Mir, R. & Mirzaei, D. (2023). The D-RBF-PU method for solving surface PDEs. Journal of Computational Physics, 479, Article ID 112001.
Open this publication in new window or tab >>The D-RBF-PU method for solving surface PDEs
2023 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 479, article id 112001Article in journal (Refereed) Published
Abstract [en]

In this paper a new direct RBF partition of unity (D-RBF-PU) method is developed for numerical solution of partial differential equations defined on smooth orientable surfaces which are discretized with sets of scattered nodes and with approximations to normal vectors at each of the nodes. The accuracy, stability and efficiency of the new method are studied through some theoretical and experimental results. This method is a localized RBF based technique, results in a perfectly sparse final linear system, uses only scattered nodes on the surface rather than a connected mesh, and is applicable for a large class of PDEs on manifolds. Applications to some biological and chemical reaction-diffusion models are also given. Results show that the new method outperforms other comparable techniques for surface PDEs.

Place, publisher, year, edition, pages
Elsevier, 2023
Keywords
Radial basis function, Partial differential equations, Surface PDEs, Partition of unity, D-RBF-PU method
National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-499924 (URN)10.1016/j.jcp.2023.112001 (DOI)000944332800001 ()
Available from: 2023-04-06 Created: 2023-04-06 Last updated: 2023-08-09Bibliographically approved
Arefian, S. & Mirzaei, D. (2022). A compact radial basis function partition of unity method. Computers and Mathematics with Applications, 127, 1-11
Open this publication in new window or tab >>A compact radial basis function partition of unity method
2022 (English)In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 127, p. 1-11Article in journal (Refereed) Published
Abstract [en]

In this work we develop the standard Hermite interpolation based RBF-generated finite difference (RBF-HFD) method into a new faster and more accurate technique based on partition of unity (PU) method. In the new approach, much fewer number of local linear systems needs to be solved for calculating the stencil weights. This reduces the computational cost of the method, remarkably. In addition, the method is flexible in using different types of PU weight functions, smooth or discontinuous, each results in a different scheme with additional nice properties. We also investigate the scaling property of polyharmonic spline (PHS) kernels to develop a simple and stable algorithm for computing local approximants in PU patches. Experimental results confirm the efficiency and applicability of the proposed method.

Place, publisher, year, edition, pages
Elsevier, 2022
Keywords
Radial basis function (RBF), Partial differential equations (PDEs), Partition of unity (PU), RBF-FD, Generalized interpolation, Hermit-Birkhoff interpolation, Compact formulas
National Category
Computational Mathematics Computer Sciences
Identifiers
urn:nbn:se:uu:diva-489263 (URN)10.1016/j.camwa.2022.09.029 (DOI)000875980200001 ()
Available from: 2022-11-29 Created: 2022-11-29 Last updated: 2023-03-03Bibliographically approved
Farazandeh, E. & Mirzaei, D. (2021). A rational RBF interpolation with conditionally positive definite kernels. Advances in Computational Mathematics, 47(5), Article ID 74.
Open this publication in new window or tab >>A rational RBF interpolation with conditionally positive definite kernels
2021 (English)In: Advances in Computational Mathematics, ISSN 1019-7168, E-ISSN 1572-9044, Vol. 47, no 5, article id 74Article in journal (Refereed) Published
Abstract [en]

In this paper, we present a rational RBF interpolation method to approximate multivariate functions with poles or other singularities on or near the domain of approximation. The method is based on scattered point layouts and is flexible with respect to the geometry of the problem’s domain. Despite the existing rational RBF-based techniques, the new method allows the use of conditionally positive definite kernels as basis functions. In particular, we use polyharmonic kernels and prove that the rational polyharmonic interpolation is scalable. The scaling property results in a stable algorithm provided that the method be implemented in a localized form. To this aim, we combine the rational polyharmonic interpolation with the partition of unity method. Sufficient number of numerical examples in one, two and three dimensions are given to show the efficiency and the accuracy of the method.

Place, publisher, year, edition, pages
Springer Nature, 2021
Keywords
Rational interpolation, Radial basis functions, Polyharmonic splines, Scalable approximations
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-490875 (URN)10.1007/s10444-021-09900-8 (DOI)000700388200001 ()
Available from: 2022-12-15 Created: 2022-12-15 Last updated: 2023-06-28Bibliographically approved
Seyednazari, S., Tatari, M. & Mirzaei, D. (2021). Error and stability estimates of a least-squares variational kernel-based method for second order elliptic PDEs. Computers and Mathematics with Applications, 103, 1-11
Open this publication in new window or tab >>Error and stability estimates of a least-squares variational kernel-based method for second order elliptic PDEs
2021 (English)In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 103, p. 1-11Article in journal (Refereed) Published
Abstract [en]

We consider a least-squares variational kernel-based method for numerical solution of second order elliptic partial differential equations on a multi-dimensional domain. In this setting it is not assumed that the differential operator is self-adjoint or positive definite as it should be in the Rayleigh-Ritz setting. However, the new scheme leads to a symmetric and positive definite algebraic system of equations. Moreover, the resulting method does not rely on certain subspaces satisfying the boundary conditions. The trial space for discretization is provided via standard kernels that reproduce the Sobolev spaces as their native spaces. The error analysis of the method is given, but it is partly subjected to an inverse inequality on the boundary which is still an open problem. The condition number of the final linear system is approximated in terms of the smoothness of the kernel and the discretization quality. Finally, the results of some computational experiments support the theoretical error bounds.

Place, publisher, year, edition, pages
Elsevier, 2021
Keywords
Meshfree methods, Least-squares principles, Radial basis functions, Inverse inequalities, Error estimates
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-495535 (URN)10.1016/j.camwa.2021.10.019 (DOI)000721358500001 ()
Available from: 2023-01-30 Created: 2023-01-30 Last updated: 2023-02-01Bibliographically approved
Mirzaei, D. (2021). The Direct Radial Basis Function Partition of Unity (D-RBF-PU) Method for Solving PDEs. SIAM Journal on Scientific Computing, 43(1), A54-A83
Open this publication in new window or tab >>The Direct Radial Basis Function Partition of Unity (D-RBF-PU) Method for Solving PDEs
2021 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 43, no 1, p. A54-A83Article in journal (Refereed) Published
Abstract [en]

In this paper, a new localized radial basis function (RBF) method based on partition of unity (PU) is proposed for solving boundary and initial-boundary value problems. The new method benefits from a direct discretization approach and is called the “direct RBF partition of unity (D-RBF-PU)” method. Thanks to avoiding all derivatives of PU weight functions as well as all lower derivatives of local approximants, the new method is faster and simpler than the standard RBF-PU method. Besides, the discontinuous PU weight functions can now be utilized to develop the method in a more efficient and less expensive way. Alternatively, the new method is an RBF-generated finite difference (RBF-FD) method in a PU setting which is much faster and in some situations more accurate than the original RBF-FD. The polyharmonic splines are used for local approximations, and the error and stability issues are considered. Some numerical experiments on irregular two- and three-dimensional domains, as well as cost comparison tests, are performed to support the theoretical analysis and to show the efficiency of the new method.

Place, publisher, year, edition, pages
Society for Industrial and Applied Mathematics, 2021
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-490880 (URN)10.1137/19m128911x (DOI)000623833100010 ()
Available from: 2022-12-15 Created: 2022-12-15 Last updated: 2023-06-29Bibliographically approved
Mirzaei, D. (2020). On analysis of kernel collocation methods for spherical PDEs. Applied Numerical Mathematics, 150, 222-232
Open this publication in new window or tab >>On analysis of kernel collocation methods for spherical PDEs
2020 (English)In: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 150, p. 222-232Article in journal (Refereed) Published
Abstract [en]

In this paper the error analysis of the kernel collocation method for partial differential equations on the unit sphere is presented. A simple analysis is given when the true solutions lie in arbitrary Sobolev spaces. This also extends the previous studies for true solutions outside the associated native spaces. Finally, some experimental results support the theoretical error bounds

Place, publisher, year, edition, pages
Elsevier, 2020
Keywords
Partial differential equations, Collocation method, Zonal kernels, Sobolev spaces, Error analysis
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-495536 (URN)10.1016/j.apnum.2019.10.001 (DOI)000513295400016 ()
Available from: 2023-01-30 Created: 2023-01-30 Last updated: 2023-02-01Bibliographically approved
Mirzaei, D. (2018). A Petrov-Galerkin Kernel Approximation on the Sphere. SIAM Journal on Numerical Analysis, 56(1), 274-295
Open this publication in new window or tab >>A Petrov-Galerkin Kernel Approximation on the Sphere
2018 (English)In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 56, no 1, p. 274-295Article in journal (Refereed) Published
Abstract [en]

In this paper, a numerical solution of partial differential equations on the unit sphere is given by using a kernel trial approximation in combination with a special Petrov--Galerkin test discretization. The solvability of the scheme is proved, and the error bounds are obtained for functions in appropriate Sobolev spaces. The condition number of the final system is estimated in terms of discretization parameters. The method is meshless because in the trial side the numerical solution parameterizes entirely in terms of scattered points and in the test side everything breaks down to simple numerical integrations over independent spherical caps. This means that no connected background mesh is required for either approximation or integration.

Place, publisher, year, edition, pages
Society for Industrial and Applied Mathematics, 2018
Keywords
radial basis functions, spherical basis functions, Petrov–Galerkin method, convo-lution data, error analysis, condition numbers
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-490882 (URN)10.1137/16m1106626 (DOI)000426741600012 ()
Available from: 2022-12-15 Created: 2022-12-15 Last updated: 2023-06-30Bibliographically approved
Mirzaei, D. (2015). Analysis of moving least squares approximation revisited. Journal of Computational and Applied Mathematics, 282, 237-250
Open this publication in new window or tab >>Analysis of moving least squares approximation revisited
2015 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 282, p. 237-250Article in journal (Refereed) Published
Place, publisher, year, edition, pages
Elsevier, 2015
Keywords
Moving least squares approximation, Error bounds, Sobolev spaces, Meshless methods
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-495543 (URN)10.1016/j.cam.2015.01.007 (DOI)000350923100018 ()
Available from: 2023-01-30 Created: 2023-01-30 Last updated: 2023-02-01Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-0166-4760

Search in DiVA

Show all publications