On the Construction of Kernel-Based Adaptive Particle Methods in Numerical Flow Simulation

Iske, Armin

doi:10.1007/978-3-642-33221-0_12

Cited by 8 publications

(5 citation statements)

References 24 publications

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“…As discussed before, the approximation values u in and u out in (21) should be reconstructed from the cell average values {u T (t)} T ∈T in each time step. An efficient particle reconstruction scheme based on polyharmonic spline interpolation is described [32]. For some other reconstruction methods, see, for example, [1,42].…”

Section: Reconstruction Stepmentioning

confidence: 99%

“…The weights are chosen in such way that the oscillations are minimized. We use a WENO reconstruction with an oscillation indicator parameter based on native space norm of the underlying polyharmonic kernel which is fully described in [32]. Details are left and the reader is referred to original sources.…”

Section: Reconstruction Stepmentioning

confidence: 99%

“…In the RBF context and on scattered points, this approach is applicable only for PHS and excludes all other well-known kernels. For more details, see [20,31,32]. In a more general case, assume that X is a set of points in a local domain D with fill distance h. Assume further that the polyharmonic approximation of Lu(x) for a fixed x ∈ D is sought.…”

Section: Introductionmentioning

confidence: 99%

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A fault detection method based on partition of unity and kernel approximation

Mirzaei

Soodbakhsh

2023

Numer Algor

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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.

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Section: Reconstruction Stepmentioning

confidence: 99%

Section: Reconstruction Stepmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A fault detection method based on partition of unity and kernel approximation

Mirzaei

Soodbakhsh

2023

Numer Algor

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“…This works without serious instabilities for local approximations where h decreases while the number of points in X is fixed; the situation in RBF-FD, RBF-PU, D-RBF-PU and all other local RBF-based methods. For proofs and more details about the scaling property of polyharmonic kernels, see [5,22,23].…”

Section: Polyharmonic Kernels and Scalabilitymentioning

confidence: 99%

The Direct Radial Basis Function Partition of Unity (D-RBF-PU) Method for Solving PDEs

Mirzaei¹

2020

Preprint

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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 is benefited 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, we are allowed to use discontinuous weight functions for developing other efficient numerical algorithms. 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 2D and 3D domains, as well as cost comparison tests, are performed to support the theoretical analysis and to show the efficiency of the new method.

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“…These occur in many papers in Science and Engineering, e.g. [1,2,4,8,15,14,16,17,18,22,26,32,33,34,35,36,37,38,41,42], and several authors have analyzed the construction of nodal approximations mathematically, e.g. [9,10,11,19,20,24,40], but without considering optimal convergence rates.…”

Section: Introductionmentioning

confidence: 99%

Optimal stencils in Sobolev spaces

Davydov

Schaback

2017

IMA Journal of Numerical Analysis

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This paper proves that the approximation of pointwise derivatives of order s of functions in Sobolev space W m 2 (R d ) by linear combinations of function values cannot have a convergence rate better than m − s − d/2, no matter how many nodes are used for approximation and where they are placed. These convergence rates are attained by scalable approximations that are exact on polynomials of order at least ⌊m − d/2⌋ + 1, proving that the rates are optimal for given m, s, and d. And, for a fixed node set X ⊂ R d , the convergence rate in any Sobolev space W m 2 (Ω) cannot be better than q − s where q is the maximal possible order of polynomial exactness of approximations based on X , no matter how large m is. In particular, scalable stencil constructions via polyharmonic kernels are shown to realize the optimal convergence rates, and good approximations of their error in Sobolev space can be calculated via their error in Beppo-Levi spaces. This allows to construct near-optimal stencils in Sobolev spaces stably and efficiently, for use in meshless methods to solve partial differential equations via generalized finite differences (RBF-FD). Numerical examples are included for illustration. 1 Univ. Gießen, Oleg.Davydov@math.uni-giessen.de https://www.staff.uni-giessen.de/odavydov/ 2 Univ. Göttingen, schaback@math.uni-goettingen.de

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On the Construction of Kernel-Based Adaptive Particle Methods in Numerical Flow Simulation

Cited by 8 publications

References 24 publications

A fault detection method based on partition of unity and kernel approximation

A fault detection method based on partition of unity and kernel approximation

The Direct Radial Basis Function Partition of Unity (D-RBF-PU) Method for Solving PDEs

Optimal stencils in Sobolev spaces

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