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Pooladi, F. & Larsson, E. (2024). Stabilized interpolation using radial basis functions augmented with selected radial polynomials. Journal of Computational and Applied Mathematics, 437, Article ID 115482.
Open this publication in new window or tab >>Stabilized interpolation using radial basis functions augmented with selected radial polynomials
2024 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 437, article id 115482Article in journal (Refereed) Published
Abstract [en]

Infinitely smooth radial basis functions (RBFs) have a shape parameter that controls their shapes. When using these RBFs (e.g., the Gaussian RBF) for interpolation problems, we have ill-conditioning when the shape parameter is very small, while in some cases small shape parameters lead to high accuracy. In this study, we are going to reduce the effect of the ill-conditioning of the infinitely smooth RBFs. We propose a new basis augmenting the infinitely smooth RBFs at different locations with radial polynomials of different even powers. Numerical experiments show that the new basis is stable for all values of the shape parameter.

Place, publisher, year, edition, pages
Elsevier, 2024
Keywords
Interpolation, Radial basis function, Radial polynomial, Flat limit, Augmented basis function
National Category
Computational Mathematics
Research subject
Scientific Computing
Identifiers
urn:nbn:se:uu:diva-509014 (URN)10.1016/j.cam.2023.115482 (DOI)001062173800001 ()
Funder
Swedish Research Council, 2020-03488
Available from: 2023-08-14 Created: 2023-08-14 Last updated: 2024-01-26Bibliographically approved
Larsson, E. & Schaback, R. (2023). Scaling of radial basis functions. IMA Journal of Numerical Analysis, 1-23
Open this publication in new window or tab >>Scaling of radial basis functions
2023 (English)In: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, p. 1-23Article in journal (Refereed) Published
Abstract [en]

This paper studies the influence of scaling on the behavior of radial basis function interpolation. It focuses on certain central aspects, but does not try to be exhaustive. The most important questions are: How does the error of a kernel-based interpolant vary with the scale of the kernel chosen? How does the standard error bound vary? And since fixed functions may be in spaces that allow scalings, like global Sobolev spaces, is there a scale of the space that matches the function best? The last question is answered in the affirmative for Sobolev spaces, but the required scale may be hard to estimate. Scalability of functions turns out to be restricted for spaces generated by analytic kernels, unless the functions are band-limited. In contrast to other papers, polynomials and polyharmonics are included as flat limits when checking scales experimentally, with an independent computation. The numerical results show that the hunt for near-flat scales is questionable, if users include the flat limit cases right from the start. When there are not enough data to evaluate errors directly, the scale of the standard error bound can be varied, up to replacing the norm of the unknown function by the norm of the interpolant. This follows the behavior of the actual error qualitatively well, but is only of limited value for estimating error-optimal scales. For kernels and functions with unlimited smoothness, the given interpolation data are proven to be insufficient for determining useful scales.

National Category
Computational Mathematics
Research subject
Scientific Computing with specialization in Numerical Analysis
Identifiers
urn:nbn:se:uu:diva-509012 (URN)10.1093/imanum/drad035 (DOI)
Funder
Swedish Research Council, 2020-03488
Available from: 2023-08-14 Created: 2023-08-14 Last updated: 2023-08-15Bibliographically approved
Larsson, E., Mavrič, B., Michael, A. & Pooladi, F. (2022). A numerical investigation of some RBF-FD error estimates. Dolomites Research Notes on Approximation, 15(5), 78-95
Open this publication in new window or tab >>A numerical investigation of some RBF-FD error estimates
2022 (English)In: Dolomites Research Notes on Approximation, ISSN 2035-6803, Vol. 15, no 5, p. 78-95Article in journal (Refereed) Published
Abstract [en]

In a recent paper by Tominec, Larsson and Heryudono a convergence proof for an oversampled version of the RBF-FD method, using polyharmonic spline basis functions augmented with polynomials, was derived. In this paper, we take a closer look at the individual estimates involved in this proof. We investigate how large the bounds are and how they depend on the node layout, the stencil size, and the polynomial degree. We find that a moderate amount of oversampling is sufficient for the method to be stable when Halton nodes are used for the stencil approximations, while a random node layout may require a very high oversampling factor. From a practical perspective, this indicates the importance of having a locally quasi uniform node layout for the method to be stable and give reliable results. We see an overall growth of the error constant with the polynomial degree and with the stencil size.

Place, publisher, year, edition, pages
Padova University Press, 2022
Keywords
Radial Basis Functions, RBF-FD, error estimates, numerical analysis, least squares, oversampling
National Category
Computational Mathematics
Research subject
Scientific Computing
Identifiers
urn:nbn:se:uu:diva-492668 (URN)10.14658/pupj-drna-2022-5-8 (DOI)000979620400009 ()
Projects
eSSENCE - An eScience Collaboration
Funder
Swedish Research Council, 2020-03488eSSENCE - An eScience CollaborationSwedish National Infrastructure for Computing (SNIC), 2021/22-457Swedish Research Council, 2018-05973
Available from: 2023-01-09 Created: 2023-01-09 Last updated: 2023-10-31Bibliographically approved
Tominec, I., Villard, P.-F., Larsson, E., Bayona, V. & Cacciani, N. (2022). An unfitted radial basis function generated finite difference method applied to thoracic diaphragm simulations. Journal of Computational Physics, 469, 111496
Open this publication in new window or tab >>An unfitted radial basis function generated finite difference method applied to thoracic diaphragm simulations
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2022 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 469, p. 111496-Article in journal (Refereed) Published
Abstract [en]

The thoracic diaphragm is the muscle that drives the respiratory cycle of a human being. Using a system of partial differential equations (PDEs) that models linear elasticity we compute displacements and stresses in a two-dimensional cross section of the diaphragm in its contracted state. The boundary data consists of a mix of displacement and traction conditions. If these are imposed as they are, and the conditions are not compatible, this leads to reduced smoothness of the solution. Therefore, the boundary data is first smoothed using the least-squares radial basis function generated finite difference (RBF-FD) framework. Then the boundary conditions are reformulated as a Robin boundary condition with smooth coefficients. The same framework is also used to approximate the boundary curve of the diaphragm cross section based on data obtained from a slice of a computed tomography (CT) scan. To solve the PDE we employ the unfitted least-squares RBF-FD method. This makes it easier to handle the geometry of the diaphragm, which is thin and non-convex. We show numerically that our solution converges with high-order towards a finite element solution evaluated on a fine grid. Through this simplified numerical model we also gain an insight into the challenges associated with the diaphragm geometry and the boundary conditions before approaching a more complex three-dimensional model. 

Place, publisher, year, edition, pages
ElsevierElsevier BV, 2022
Keywords
unfitted, RBF-FD, least-squares, elasticity, diaphragm, mixed boundary condition
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-470981 (URN)10.1016/j.jcp.2022.111496 (DOI)000884362300004 ()
Available from: 2022-03-31 Created: 2022-03-31 Last updated: 2024-01-15Bibliographically approved
Tominec, I., Larsson, E. & Heryudono, A. (2021). A Least Squares Radial Basis Function Finite Difference Method with Improved Stability Properties. SIAM Journal on Scientific Computing, 43(2), A1441-A1471
Open this publication in new window or tab >>A Least Squares Radial Basis Function Finite Difference Method with Improved Stability Properties
2021 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 43, no 2, p. A1441-A1471Article in journal (Refereed) Published
Abstract [en]

Localized collocation methods based on radial basis functions (RBFs) for elliptic problems appear to be nonrobust in the presence of Neumann boundary conditions. In this paper, we overcome this issue by formulating the RBF-generated finite difference method in a discrete least squares setting instead. This allows us to prove high-order convergence under node refinement and to numerically verify that the least squares formulation is more accurate and robust than the collocation formulation. The implementation effort for the modified algorithm is comparable to that for the collocation method.

Place, publisher, year, edition, pages
Society for Industrial and Applied MathematicsSIAM PUBLICATIONS, 2021
Keywords
radial basis function, least squares, partial differential equation, elliptic problem, Neumann condition, RBF-FD
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-444700 (URN)10.1137/20M1320079 (DOI)000646026400018 ()
Funder
Swedish Research Council, 2016-04849eSSENCE - An eScience Collaboration
Available from: 2021-06-09 Created: 2021-06-09 Last updated: 2024-01-15Bibliographically approved
Cacciani, N., Larsson, E., Lauro, A., Meggiolaro, M., Scatto, A., Tominec, I. & Villard, P.-F. (2020). A first meshless approach to simulation of the elastic behaviour of the diaphragm. In: Spectral and High Order Methods for Partial Differential Equations: ICOSAHOM 2018. Paper presented at ICOSAHOM 2018, July 9–13, London, UK (pp. 501-512). Springer
Open this publication in new window or tab >>A first meshless approach to simulation of the elastic behaviour of the diaphragm
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2020 (English)In: Spectral and High Order Methods for Partial Differential Equations: ICOSAHOM 2018, Springer, 2020, p. 501-512Conference paper, Published paper (Refereed)
Place, publisher, year, edition, pages
Springer, 2020
Series
Lecture Notes in Computational Science and Engineering ; 134
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-409858 (URN)10.1007/978-3-030-39647-3_40 (DOI)978-3-030-39646-6 (ISBN)
Conference
ICOSAHOM 2018, July 9–13, London, UK
Projects
eSSENCE
Available from: 2020-08-12 Created: 2020-04-30 Last updated: 2020-08-27Bibliographically approved
Larsson, E. & Sundin, U. (2020). An investigation of global radial basis function collocation methods applied to Helmholtz problems. Dolomites Research Notes on Approximation, 13, 65-85
Open this publication in new window or tab >>An investigation of global radial basis function collocation methods applied to Helmholtz problems
2020 (English)In: Dolomites Research Notes on Approximation, ISSN 2035-6803, Vol. 13, p. 28p. 65-85Article in journal (Refereed) Published
Abstract [en]

Global radial basis function (RBF) collocation methods with inifinitely smooth basis functions for partial differential equations (PDEs) work in general geometries, and can have exponential convergence properties for smooth solution functions. At the same time, the linear systems that arise are dense and severly ill-conditioned for large numbers of unknowns and small values of the shape parameter that determines how flat the basis functions are. We use Helmholtz equation as an application problem for the theoretical analysis and numerical experiments. We analyse and characterise the convergence properties as a function of the number of unknowns and for different shape parameter ranges. We provide theoretical results for the flat limit of the PDE solutions and investigate when the non-symmetric collocation matrices become singular. We also provide practical strategies for choosing the method parameters and evaluate the results on Helmholtz problems in acurved waveguide geometry

Place, publisher, year, edition, pages
Padova University Press, 2020. p. 28
Keywords
Radial basis function, Helmholtz equation, shape parameter, flat limit, error estimate
National Category
Computational Mathematics
Research subject
Scientific Computing
Identifiers
urn:nbn:se:uu:diva-404563 (URN)10.14658/PUPJ-DRNA-2020-1-8 (DOI)000604606300001 ()
Funder
Swedish Research CouncileSSENCE - An eScience Collaboration
Available from: 2020-02-24 Created: 2020-02-24 Last updated: 2022-01-14Bibliographically approved
Ahmad, M., Islam, S.-u. & Larsson, E. (2020). Local meshless methods for second order elliptic interface problems with sharp corners. Journal of Computational Physics, 416, Article ID 109500.
Open this publication in new window or tab >>Local meshless methods for second order elliptic interface problems with sharp corners
2020 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 416, article id 109500Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-414049 (URN)10.1016/j.jcp.2020.109500 (DOI)000539999800008 ()
Projects
eSSENCE
Available from: 2020-05-04 Created: 2020-06-22 Last updated: 2022-01-10Bibliographically approved
von Sydow, L., Milovanović, S., Larsson, E., In't Hout, K., Wiktorsson, M., Oosterlee, C. W., . . . Waldén, J. (2019). BENCHOP–SLV: The BENCHmarking project in Option Pricing – Stochastic and local volatility problems. International Journal of Computer Mathematics, 96, 1910-1923
Open this publication in new window or tab >>BENCHOP–SLV: The BENCHmarking project in Option Pricing – Stochastic and local volatility problems
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2019 (English)In: International Journal of Computer Mathematics, ISSN 0020-7160, E-ISSN 1029-0265, Vol. 96, p. 1910-1923Article in journal (Refereed) Published
National Category
Computational Mathematics
Identifiers
urn:nbn:se:uu:diva-357218 (URN)10.1080/00207160.2018.1544368 (DOI)000475440700002 ()
Projects
eSSENCE
Available from: 2018-11-07 Created: 2018-08-14 Last updated: 2019-08-29Bibliographically approved
Zafari, A., Larsson, E. & Tillenius, M. (2019). DuctTeip: An efficient programming model for distributed task-based parallel computing. Parallel Computing, 90, Article ID 102582.
Open this publication in new window or tab >>DuctTeip: An efficient programming model for distributed task-based parallel computing
2019 (English)In: Parallel Computing, ISSN 0167-8191, E-ISSN 1872-7336, Vol. 90, article id 102582Article in journal (Refereed) Published
National Category
Software Engineering
Identifiers
urn:nbn:se:uu:diva-338832 (URN)10.1016/j.parco.2019.102582 (DOI)000501649400002 ()
Projects
UPMARCeSSENCE
Available from: 2019-11-04 Created: 2018-01-14 Last updated: 2020-01-24Bibliographically approved
Projects
INVIVE: The Individual Virtual Ventilator [2016-04849_VR]; Uppsala UniversityREVIVE: Reliable and efficient virtual ventilator experiments [2020-03488_VR]; Uppsala University
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0003-1154-9587

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