Α Study on Radial Basis Function and Quasi-Monte Carlo Methods

Chen, W.; He, Jianyu

doi:10.1515/ijnsns.2000.1.4.337

Cited by 5 publications

(5 citation statements)

References 11 publications

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“…Moreover, the numerical experiments showed that the BKM could avoid the curse of dimensionality in the solution of the 3D problem. This attractive advantage is endorsed by the theoretical analysis and experimental ÿndings that the use of higher order smooth radial basis functions can o set the dimensional a ect [16,25,26].…”

Section: Discussionmentioning

confidence: 99%

“…It can be observed from the tables that the BKM worked equally well for this 3D problem as in the previous 2D cases. Based on some numerical experiments and theoretical analysis concerning the dimensional e ect on the error bounds of the RBF interpolation, Chen and He [25] conjectured that the RBF-based numerical scheme may circumvent the curse of dimensionality like the Monte Carlo method. To be more precise, the computational e ort in using the RBF on solving higher-dimensional problems only grows linearly instead of exponentially.…”

Section: A 3d Homogeneous Helmholtz Problemmentioning

confidence: 99%

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Boundary knot method for 2D and 3D Helmholtz and convection–diffusion problems under complicated geometry

Hon

Chen

2003

Numerical Meth Engineering

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SUMMARYThe boundary knot method (BKM) of very recent origin is an inherently meshless, integration-free, boundary-type, radial basis function collocation technique for the numerical discretization of general partial di erential equation systems. Unlike the method of fundamental solutions, the use of non-singular general solution in the BKM avoids the unnecessary requirement of constructing a controversial artiÿcial boundary outside the physical domain. The purpose of this paper is to extend the BKM to solve 2D Helmholtz and convection-di usion problems under rather complicated irregular geometry. The method is also ÿrst applied to 3D problems. Numerical experiments validate that the BKM can produce highly accurate solutions using a relatively small number of knots. For inhomogeneous cases, some inner knots are found necessary to guarantee accuracy and stability. The stability and convergence of the BKM are numerically illustrated and the completeness issue is also discussed.

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Section: Discussionmentioning

confidence: 99%

Section: A 3d Homogeneous Helmholtz Problemmentioning

confidence: 99%

Boundary knot method for 2D and 3D Helmholtz and convection–diffusion problems under complicated geometry

Hon

Chen

2003

Numerical Meth Engineering

Self Cite

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“…The first method adopts cubic splines [17][18][19][20][21][22][23]. The second one interpolates with radial basis functions [24][25][26][27][28][29][30]. Finally, the third method is an ad hoc neuronal network: the multilayer perceptron (MLP) [31][32][33][34][35][36][37][38][39][40], expecting in this case that fitting weighed and biased algorithms would be compatible with a simple calculation based on metaheuristic techniques such as simulated annealing (SA) or particle swarm optimization (PSO), [41].…”

Section: Introductionmentioning

confidence: 99%

Assessing the Best Gap-Filling Technique for River Stage Data Suitable for Low Capacity Processors and Real-Time Application Using IoT

Luna

Lineros

Estévez

et al. 2020

Sensors

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Hydrometeorological data sets are usually incomplete due to different reasons (malfunctioning sensors, collected data storage problems, etc.). Missing data do not only affect the resulting decision-making process, but also the choice of a particular analysis method. Given the increase of extreme events due to climate change, it is necessary to improve the management of water resources. Due to the solution of this problem requires the development of accurate estimations and its application in real time, this work present two contributions. Firstly, different gap-filling techniques have been evaluated in order to select the most adequate one for river stage series: (i) cubic splines (CS), (ii) radial basis function (RBF) and (iii) multilayer perceptron (MLP) suitable for small processors like Arduino or Raspberry Pi. The results obtained confirmed that splines and monolayer perceptrons had the best performances. Secondly, a pre-validating Internet of Things (IoT) device was developed using a dynamic seed non-linear autoregressive neural network (NARNN). This automatic pre-validation in real time was tested satisfactorily, sending the data to the catchment basin process center (CPC) by using remote communication based on 4G technology.

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“…If one views a PDE from a numerical Green's function formulation, low-discrepancy sequences are one approach to numerically computing a solution in complex domains. These methods have been used in the solution of integral equations [6,7], and more recently, in the solution of PDE's [7]. The basic premise is that a good estimate of the solution may be obtained with relatively few samples, i.e., a sparse mesh.…”

Section: Introductionmentioning

confidence: 99%

Scandalously Parallelizable Mesh Generation

Bortz,

Christlieb

2011

Preprint

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We propose a novel approach which employs random sampling to generate an accurate non-uniform mesh for numerically solving Partial Differential Equation Boundary Value Problems (PDE-BVP's). From a uniform probability distribution U over a 1D domain, we sample M discretizations of size N where M ≫ N . The statistical moments of the solutions to a given BVP on each of the M ultra-sparse meshes provide insight into identifying highly accurate non-uniform meshes. Essentially, we use the pointwise mean and variance of the coarse-grid solutions to construct a mapping Q(x) from uniformly to non-uniformly spaced mesh-points. The error convergence properties of the approximate solution to the PDE-BVP on the non-uniform mesh are superior to a uniform mesh for a certain class of BVP's. In particular, the method works well for BVP's with locally non-smooth solutions. We present a framework for studying the sampled sparse-mesh solutions and provide numerical evidence for the utility of this approach as applied to a set of example BVP's. We conclude with a discussion of how the near-perfect paralellizability of our approach suggests that these strategies have the potential for highly efficient utilization of massively parallel multi-core technologies such as General Purpose Graphics Processing Units (GPGPU's). We believe that the proposed algorithm is beyond embarrassingly parallel; implementing it on anything but a massively multi-core architecture would be scandalous.

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Α Study on Radial Basis Function and Quasi-Monte Carlo Methods

Cited by 5 publications

References 11 publications

Boundary knot method for 2D and 3D Helmholtz and convection–diffusion problems under complicated geometry

Boundary knot method for 2D and 3D Helmholtz and convection–diffusion problems under complicated geometry

Assessing the Best Gap-Filling Technique for River Stage Data Suitable for Low Capacity Processors and Real-Time Application Using IoT

Scandalously Parallelizable Mesh Generation

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