Energy-stable global radial basis function methods on summation-by-parts form

Glaubitz, Jan; Nordström, Jan; Öffner, Philipp

doi:10.48550/arxiv.2204.03291

Cited by 3 publications

(6 citation statements)

References 38 publications

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“…Our findings imply that the concept of MSBP operators can be applied to a significantly larger class of methods than is currently known. Another aspect of introducing the SBP framework in an existing "old" method is to gain stability, as done for the FV method in [55,56] and for RBF methods in [33], when combined with weakly enforced boundary data [31,32]. Here, we demonstrated the advantage of MFSBP operators for different linear problems.…”

mentioning

confidence: 85%

“…Although non-polygonal elements are rarely used in the context of SE-like methods for hyperbolic conservation laws, they are crucial in mesh-free RBF methods [20,23]. In the recent work [33], we established a stability theory for global RBF methods using FSBP operators. The MFSBP operators introduced here will allow us to extend this theory to the genuine multi-dimensional case, which will be addressed in future work, together with the extension to local RBF methods.…”

Section: Examples Of Mfsbp Operatorsmentioning

confidence: 99%

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Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction

Glaubitz¹,

Simon-Christian²,

Nordström³

et al. 2023

Preprint

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Summation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that polynomials can accurately approximate the solution, and SBP operators should thus be exact for them. However, polynomials do not provide the best approximation for some problems, with other approximation spaces being more appropriate. We recently addressed this issue and developed a theory for one-dimensional SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the theory of FSBP operators to multiple dimensions. We focus on their existence, connection to quadratures, construction, and mimetic properties. A more exhaustive numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their application will be provided in future works. Similar to the one-dimensional case, we demonstrate that most of the established results for polynomial-based multidimensional SBP (MSBP) operators carry over to the more general class of MFSBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently done. This can increase the accuracy of the numerical solutions and/or provide stability to the methods.

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confidence: 85%

Section: Examples Of Mfsbp Operatorsmentioning

confidence: 99%

Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction

Glaubitz¹,

Simon-Christian²,

Nordström³

et al. 2023

Preprint

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show abstract

“…where κ = 1 and cond(B j ) = ||B j || F ||B −1 j || F , B j is the interpolation matrix built using the interpolation nodes x j and the predicted ε. The loss function is as in (17) to ensure that it is continuous, and to enforce that the condition number B j is between 10 10 and 10 12 .…”

Section: Data Driven Modelmentioning

confidence: 99%

“…• The RBF interpolants becomes exact for polynomials up to a certain degree. Including constants is necessary for the RBF method to be conservative [17,18].…”

mentioning

confidence: 99%

A new variable shape parameter strategy for RBF approximation using neural networks

Mojarrad¹,

Veiga²,

Hesthaven³

et al. 2022

Preprint

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The choice of the shape parameter highly effects the behaviour of radial basis function (RBF) approximations, as it needs to be selected to balance between ill-condition of the interpolation matrix and high accuracy. In this paper, we demonstrate how to use neural networks to determine the shape parameters in RBFs. In particular, we construct a multilayer perceptron trained using an unsupervised learning strategy, and use it to predict shape parameters for inverse multiquadric and Gaussian kernels. We test the neural network approach in RBF interpolation tasks and in a RBFfinite difference method in one and two-space dimensions, demonstrating promising results.

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“…The possible advantage of stable LSRBF-QFs compared to (potentially unstable) interpolatory RBF-QFs is demonstrated in §7.2. Finally, we point out the potential application of stable LSRBF-QFs to the construction of stable RBF methods for time-dependent hyperbolic partial differential equations [90,41]. A crucial part of these methods is replacing exact integrals involving the approximate solution-in this case, an (local) RBF function-with a quadrature that should be as accurate as possible for functions from the approximation space.…”

mentioning

confidence: 99%

Towards stability results for global radial basis function based quadrature formulas

Glaubitz,

Reeger

2023

Preprint

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Quadrature formulas (QFs) based on radial basis functions (RBFs) have become an essential tool for multivariate numerical integration of scattered data. Although numerous works have been published on RBF-QFs, their stability theory can still be considered as underdeveloped. Here, we strive to pave the way towards a more mature stability theory for global and function-independent RBF-QFs. In particular, we prove stability of these for compactly supported RBFs under certain conditions on the shape parameter and the data points. As an alternative to changing the shape parameter, we demonstrate how the least-squares approach can be used to construct stable RBF-QFs by allowing the number of data points used for numerical integration to be larger than the number of centers used to generate the RBF approximation space. Moreover, it is shown that asymptotic stability of many global RBF-QFs is independent of polynomial terms, which are often included in RBF approximations. While our findings provide some novel conditions for stability of global RBF-QFs, the present work also demonstrates that there are still many gaps to fill in future investigations.

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Energy-stable global radial basis function methods on summation-by-parts form

Cited by 3 publications

References 38 publications

Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction

Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction

A new variable shape parameter strategy for RBF approximation using neural networks

Towards stability results for global radial basis function based quadrature formulas

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