A Multiscale RBF Collocation Method for the Numerical Solution of Partial Differential Equations

Li, Yong; Xu, Qiuyan

doi:10.3390/math7100964

Cited by 12 publications

(4 citation statements)

References 23 publications

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“…The multilevel scattered approximation was implemented first in [8] and then studied by a number of other researchers [9][10][11][12][13]. In the multilevel algorithm, the residual can be formed on the coarsest level first and then be approximated on the later finer level by the compactly supported radial basis functions with gradually smaller support.…”

Section: Multilevel Approximationmentioning

confidence: 99%

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Scattered Data Interpolation and Approximation with Truncated Exponential Radial Basis Function

Xu¹

2019

Mathematics

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Surface modeling is closely related to interpolation and approximation by using level set methods, radial basis functions methods, and moving least squares methods. Although radial basis functions with global support have a very good approximation effect, this is often accompanied by an ill-conditioned algebraic system. The exceedingly large condition number of the discrete matrix makes the numerical calculation time consuming. The paper introduces a truncated exponential function, which is radial on arbitrary n-dimensional space R n and has compact support. The truncated exponential radial function is proven strictly positive definite on R n while internal parameter l satisfies l ≥ ⌊ n 2 ⌋ + 1 . The error estimates for scattered data interpolation are obtained via the native space approach. To confirm the efficiency of the truncated exponential radial function approximation, the single level interpolation and multilevel interpolation are used for surface modeling, respectively.

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Section: Multilevel Approximationmentioning

confidence: 99%

“…Similar experiments and observations are reported in detail in Fasshauer's book [3], where Wendland's function C 4 has been used for approximation. However, to improve the allocation memory needs of the multilevel algorithm, we can make use of the hierarchical collocation method developed in [13].…”

Section: Multilevel Approximationmentioning

confidence: 99%

Scattered Data Interpolation and Approximation with Truncated Exponential Radial Basis Function

Xu¹

2019

Mathematics

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“…Approximation using a sparse linear combination of elements from a fixed redundant family is actively used because of its concise representations and increased computational efficiency. It has been applied widely to signal processing, image compression, machine learning and PDE solvers (see [1][2][3][4][5][6][7][8][9][10]). Among others, simultaneous sparse approximation has been utilized in signal vector processing and multi-task learning (see [11][12][13][14]).…”

Section: Introductionmentioning

confidence: 99%

Approximation Properties of the Vector Weak Rescaled Pure Greedy Algorithm

Guo

et al. 2023

Mathematics

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We first study the error performances of the Vector Weak Rescaled Pure Greedy Algorithm for simultaneous approximation with respect to a dictionary D in a Hilbert space. We show that the convergence rate of the Vector Weak Rescaled Pure Greedy Algorithm on A1(D) and the closure of the convex hull of the dictionary D is optimal. The Vector Weak Rescaled Pure Greedy Algorithm has some advantages. It has a weaker convergence condition and a better convergence rate than the Vector Weak Pure Greedy Algorithm and is simpler than the Vector Weak Orthogonal Greedy Algorithm. Then, we design a Vector Weak Rescaled Pure Greedy Algorithm in a uniformly smooth Banach space setting. We obtain the convergence properties and error bound of the Vector Weak Rescaled Pure Greedy Algorithm in this case. The results show that the convergence rate of the VWRPGA on A1(D) is sharp. Similarly, the Vector Weak Rescaled Pure Greedy Algorithm is simpler than the Vector Weak Chebyshev Greedy Algorithm and the Vector Weak Relaxed Greedy Algorithm.

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“…There are finite difference methods [5], finite element methods [6], spectral methods [7], boundary element methods [8], finite volume methods [9], Monte Carlo methods [10], and adaptive mesh refinement. With the development of the research, many new numerical methods appear (e.g., the fast curvilinear finite difference method [11], multiscale RBF collocation method [12], Haar wavelet collocation method [13], and sinccollocation method [14]). However, we often need to analyze and design PDE solution algorithms via numerical methods.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Method for Solving Two-Dimensional Partial Differential Equations with the Deep Operator Network

Zhang,

Wang,

Peng

et al. 2023

Axioms

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Partial differential equations (PDEs) usually apply for modeling complex physical phenomena in the real world, and the corresponding solution is the key to interpreting these problems. Generally, traditional solving methods suffer from inefficiency and time consumption. At the same time, the current rise in machine learning algorithms, represented by the Deep Operator Network (DeepONet), could compensate for these shortcomings and effectively predict the solutions of PDEs by learning the operators from the data. The current deep learning-based methods focus on solving one-dimensional PDEs, but the research on higher-dimensional problems is still in development. Therefore, this paper proposes an efficient scheme to predict the solution of two-dimensional PDEs with improved DeepONet. In order to construct the data needed for training, the functions are sampled from a classical function space and produce the corresponding two-dimensional data. The difference method is used to obtain the numerical solutions of the PDEs and form a point-value data set. For training the network, the matrix representing two-dimensional functions is processed to form vectors and adapt the DeepONet model perfectly. In addition, we theoretically prove that the discrete point division of the data ensures that the model loss is guaranteed to be in a small range. This method is verified for predicting the two-dimensional Poisson equation and heat conduction equation solutions through experiments. Compared with other methods, the proposed scheme is simple and effective.

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A Multiscale RBF Collocation Method for the Numerical Solution of Partial Differential Equations

Cited by 12 publications

References 23 publications

Scattered Data Interpolation and Approximation with Truncated Exponential Radial Basis Function

Scattered Data Interpolation and Approximation with Truncated Exponential Radial Basis Function

Approximation Properties of the Vector Weak Rescaled Pure Greedy Algorithm

An Efficient Method for Solving Two-Dimensional Partial Differential Equations with the Deep Operator Network

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