The overlapped radial basis function-finite difference (RBF-FD) method: A generalization of RBF-FD

Shankar, Varun

doi:10.1016/j.jcp.2017.04.037

Cited by 60 publications

(84 citation statements)

References 41 publications

(62 reference statements)

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“…We first present a mathematical description of the overlapped RBF-FD method, recently developed by the first author [24]. Let X = {x k } N k=1 be a global set of nodes on the domain.…”

Section: Mathematical Descriptionmentioning

confidence: 99%

“…Thus, this technique results in larger stencils than if one were to use RBFs or polynomials alone. Fortunately, the cost associated with the increase in stencil sizes for this new augmented RBF-FD method can be dramatically reduced by using a variant called the overlapped RBF-FD method [24], developed by the first author and used in this article. Interestingly, there is also evidence that the inclusion of polynomials improves the global stability of RBF-FD methods, at least in the context of elliptic PDEs [23].…”

Section: Introductionmentioning

confidence: 99%

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Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection–diffusion equations

Shankar

Fogelson

2018

Journal of Computational Physics

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We present a novel hyperviscosity formulation for stabilizing RBF-FD discretizations of the advectiondiffusion equation. The amount of hyperviscosity is determined quasi-analytically for commonly-used explicit, implicit, and implicit-explicit (IMEX) time integrators by using a simple 1D semi-discrete Von Neumann analysis. The analysis is applied to an analytical model of spurious growth in RBF-FD solutions that uses auxiliary differential operators mimicking the undesirable properties of RBF-FD differentiation matrices. The resulting hyperviscosity formulation is a generalization of existing ones in the literature, but is free of any tuning parameters and can be computed efficiently. To further improve robustness, we introduce a simple new scaling law for polynomial-augmented RBF-FD that relates the degree of polyharmonic spline (PHS) RBFs to the degree of the appended polynomial. When used in a novel ghost node formulation in conjunction with the recently-developed overlapped RBF-FD method, the resulting method is robust and free of stagnation errors. We validate the high-order convergence rates of our method on 2D and 3D test cases over a wide range of Peclet numbers (1-1000). We then use our method to solve a 3D coupled problem motivated by models of platelet aggregation and coagulation, again demonstrating high-order convergence rates.

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“…We first present a mathematical description of the overlapped RBF-FD method, recently developed by the first author [24]. Let X = {x k } N k=1 be a global set of nodes on the domain.…”

Section: Mathematical Descriptionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection–diffusion equations

Shankar

Fogelson

2018

Journal of Computational Physics

Self Cite

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“…However, if δ < 1, then N δ << N , giving a significant speedup over the standard RBF-FD method. For a detailed complexity analysis, see [35].…”

Section: Descriptionmentioning

confidence: 99%

“…The degree m of the PHS RBF can either be fixed [3,15,[31][32][33], or increased with respect to [6,37]. In Euclidean domains, it appears beneficial to fix m [3,14,15,35]. On the other hand, the traditional scaling law m = 2 + 1, appears to give the greatest accuracy and stability on manifolds [37].…”

Section: Parameter Selectionmentioning

confidence: 99%

RBF-LOI: Augmenting Radial Basis Functions (RBFs) with Least Orthogonal Interpolation (LOI) for solving PDEs on surfaces

Shankar

Narayan

Kirby

2018

Journal of Computational Physics

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We present a new method for the solution of PDEs on manifolds M ⊂ R d of co-dimension one using stable scale-free radial basis function (RBF) interpolation. Our method involves augmenting polyharmonic spline (PHS) RBFs with polynomials to generate RBF-finite difference (RBF-FD) formulas. These polynomial basis elements are obtained using the recently-developed least orthogonal interpolation technique (LOI) on each RBF-FD stencil to obtain local restrictions of polynomials in R 3 to stencils on M. The resulting RBF-LOI method uses Cartesian coordinates, does not require any intrinsic coordinate systems or projections of points onto tangent planes, and our tests illustrate robustness to stagnation errors. We show that our method produces high orders of convergence for PDEs on the sphere and torus, and present some applications to reaction-diffusion PDEs motivated by biology.

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“…In the application of neural network, back propagation (BP) neural network "as stated in [10]" is a widely used model. Because RBF is more suitable for warehouse environment monitoring information processing, the RBF network model is adopted.…”

Section: Heterogeneous Sensor Information Fusion Based On Rbf Neural mentioning

confidence: 99%

An Algorithm of Wireless Sensor Monitoring System

Hongri

2018

Int. J. Onl. Eng.

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Abstract-In order to realize more intelligent storage monitoring system, the information fusion model of wireless sensor network for storage environment monitoring is studied on the basis of analyzing information fusion technology. By analyzing the structure of storage monitoring system based on wireless sensor network, a two-layer information fusion method is established. The information fusion of homogeneous sensor based on adaptive weighting and the fusion method of heterogeneous sensor based on radial basis function neural network are designed and verified. The experimental results show that the design method can fuse the storage environment information and realize the accurate identification of the environmental state. Therefore, the algorithm can effectively improve the speed of network training, and the classification effect is good. To a certain extent, it can help enterprises to establish a safe and efficient storage system, to enhance the efficiency of enterprise warehousing operations.Keywords-Wireless Sensor Network, Warehouse Monitoring System, Information Fusion IntroductionIn order to ensure safe and efficient logistics operation, and further improve the overall efficiency and safety of the logistics system, "intelligent logistics" has become an inevitable trend of the development of the logistics industry. As an important part of logistics, storage is bound to move in the direction of intelligence. Wireless sensor network adopts a new mode of information acquisition and processing, which can realize the integration of information acquisition, transmission and processing. Therefore, the warehouse monitoring system based on wireless sensor network can effectively solve the problems such as numerous equipment, complex wiring, poor scalability, high management and maintenance cost. However, the system still cannot meet the requirements of "intelligent logistics" for intelligent warehousing monitoring system "as stated in [1]". In order to realize the intelligence of the warehouse monitoring system, it is necessary to process the information of the whole process of the system operation. It is mainly reflected in the recognition and utilization of perception information, and the information processing of whole process "as stated in [2]".Taking the warehouse monitoring system based on wireless sensor network as the research object, the theory of information fusion and its application in warehouse 52

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The overlapped radial basis function-finite difference (RBF-FD) method: A generalization of RBF-FD

Cited by 60 publications

References 41 publications

Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection–diffusion equations

Hyperviscosity-based stabilization for radial basis function-finite difference (RBF-FD) discretizations of advection–diffusion equations

RBF-LOI: Augmenting Radial Basis Functions (RBFs) with Least Orthogonal Interpolation (LOI) for solving PDEs on surfaces

An Algorithm of Wireless Sensor Monitoring System

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