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Solving moving-boundary problems with the wavelet adaptive radial basis functions method

COMPUTERS & FLUIDS 2013年 86卷 37-44页

作者： Vrankar, Leopold Libre, Nicolas Ali Ling, Leevan Turk, Goran Runovc, Franc Univ Ljubljana Fac Nat Sci & Engn Ljubljana Slovenia Hong Kong Baptist Univ Dept Math Hong Kong Hong Kong Peoples R China Univ Ljubljana Fac Civil & Geodet Engn Ljubljana Slovenia

Moving boundaries are associated with the time-dependent problems where the momentary position of boundaries needs to be determined as a function of time. The level set method has become an effective tool for tracking, modeling and simulating the motion of free boundaries in fluid mechanics, computer animation and image processing. This work extends our earlier work on solving moving boundary problems with adaptive meshless methods. In particular, the objective of this paper is to investigate numerical performance the radial basis functions (RBFs) methods, with compactly supported basis and with global basis, coupled with a wavelet node refinement technique and a greedy trial space selection technique. Numerical simulations are provided to verify the effectiveness and robustness of RBFs methods with different adaptive techniques. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Moving-boundary problems Wavelet method Level set method Global RBFs Compactly supported RBFs Partial differential equations adaptive greedy algorithm

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adaptive Sparse Channel Estimator for OFDM Systems

Adaptive Sparse Channel Estimator for OFDM Systems

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International Conference on Wireless Communications and Signal Processing (WCSP)

作者： Wu, Yun Zhou, Xiaolin Chen, Wen Liao, Xiaofei Donghua Univ Col Informat Sci & Technol Shanghai Peoples R China Fudan Univ Dept Commun Sci & Engn Shanghai 200433 Peoples R China

ISBN: (纸本)9781479903085

An adaptive sparse channel estimator based on compressive sensing for OFDM systems is proposed. Unlike conventional compressive channel estimator with a prior knowledge of the channel spaisity, the proposed channel estimator can extract information on sparsity of the channel adaptively by introducing the ASMP algorithm. Moreover, the proposed channel estimator has a superior estimate performance due to the backtracking refinement process in the ASMP algorithm. The estimation performance for the sparse channel is analysis in terms of mean square error (MSE). Simulation results demonstrate substantial performance gains of the proposed estimator over conventional compressive channel estimator.

关键词： channel estimation adaptive greedy algorithm compressive sensing OFDM

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An adaptive-hybrid meshfree approximation method

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2012年第5期89卷 637-657页

作者： Ling, Leevan Hong Kong Baptist Univ Dept Math Kowloon Tong Hong Kong Peoples R China

It is now commonly agreed that the global radial basis functions (GRBF) method is an attractive approach for approximating smooth functions. This superiority does not come free;one must find ways to circumvent the associated problem of ill-conditioning and the high computational cost for solving dense matrix systems. We previously proposed different variants of adaptive methods for selecting proper trial subspaces so that the instability caused by inappropriately shaped parameters were minimized. In contrast, the compactly supported radial basis functions (CSRBF) are more relaxing on the smoothness requirements. By settling with the algebraic order of convergence only, the CSRBF method, provided the support radii are properly chosen, can approximate functions with less smoothness. The reality is that end users must know the functions to be approximated a priori to decide which method to be used;this is not practical if one is solving a time-evolving partial differential equation. The solution could be smooth at the beginning but the formation of shocks may come later. In this paper, we propose a hybrid algorithm making use of both GRBF and CSRBF with other developed techniques for meshfree approximation with minimal fine tuning. The first contribution here is an adaptive node refinement scheme. Second, we apply the GRBFs (with adaptive subspace selection) on the adaptively generated data sites, and lastly, the CSRBF (with adaptive support selection) that can be used as a blackbox algorithm for robust approximations to a wider class of functions and for solving PDEs. Copyright (C) 2011 John Wiley & Sons, Ltd.

关键词： radial basis function adaptive greedy algorithm data refinement convergence proof Burgers' equation meshfree methods adaptivity partial differential equations parabolic

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adaptive Sparsity Matching Pursuit algorithm for Sparse Reconstruction

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IEEE SIGNAL PROCESSING LETTERS 2012年第8期19卷 471-474页

作者： Wu, Honglin Wang, Shu Huazhong Univ Sci & Technol Dept Elect & Informat Engn Wuhan 430074 Peoples R China

This letter presents a new greedy method, called adaptive Sparsity Matching Pursuit (ASMP), for sparse solutions of underdetermined systems with a typical/random projection matrix. Unlike anterior greedy algorithms, ASMP can extract information on sparsity of the target signal adaptively with a well-designed stagewise approach. Moreover, it takes advantage of backtracking to refine the chosen supports and the current approximation in the process. With these improvements, ASMP provides even more attractive results than the state-of-the-art greedy algorithm CoSaMP without prior knowledge of the sparsity level. Experiments validate the proposed algorithm works well for both noiseless signals and noisy signals, with the recovery quality often outperforming that of l(1)-minimization and other greedy algorithms.

关键词： adaptive greedy algorithm blind sparse reconstruction compressive sensing

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An adaptive greedy technique for inverse boundary determination problem

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JOURNAL OF COMPUTATIONAL PHYSICS 2010年第22期229卷 8484-8496页

作者： Yang, F. L. Ling, L. Wei, T. Lanzhou Univ Sch Math & Stat Lanzhou 730000 Peoples R China Hong Kong Baptist Univ Dept Math Hong Kong Hong Kong Peoples R China

In this paper, the method of fundamental solutions (MFS) is employed for determining an unknown portion of the boundary from the Cauchy data specified on parts of the boundary. We propose a new numerical method with adaptive placement of source points in the MFS to solve the inverse boundary determination problem. Since the MFS source points placement here is not trivial due to the unknown boundary, we employ an adaptive technique to choose a sub-optimal arrangement of source points on various fictitious boundaries. Afterwards, the standard Tikhonov regularization method is used to solve ill-conditional matrix equation, while the regularization parameter is chosen by the L-curve criterion. The numerical studies of both open and closed fictitious boundaries are considered. It is shown that the proposed method is effective and stable even for data with relatively high noise levels. (C) 2010 Elsevier Inc. All rights reserved.

关键词： Inverse problem Method of fundamental solutions Source points placement adaptive greedy algorithm

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Numerical simulations of 2D fractional subdiffusion problems

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JOURNAL OF COMPUTATIONAL PHYSICS 2010年第18期229卷 6613-6622页

作者： Brunner, Hermann Ling, Leevan Yamamoto, Masahiro Hong Kong Baptist Univ Dept Math Kowloon Hong Kong Peoples R China Univ Tokyo Grad Sch Math Sci Tokyo Japan

The growing number of applications of fractional derivatives in various fields of science and engineering indicates that there is a significant demand for better mathematical algorithms for models with real objects and processes. Currently, most algorithms are designed for 1D problems due to the memory effect in fractional derivatives. In this work, the 2D fractional subdiffusion problems are solved by an algorithm that couples an adaptive time stepping and adaptive spatial basis selection approach. The proposed algorithm is also used to simulate a subdiffusion-convection equation. (C) 2010 Elsevier Inc. All rights reserved.

关键词： Fractional differential equations Kansas method Radial basis functions Collocation adaptive greedy algorithm Geometric time grids

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adaptive METHOD OF PARTICULAR SOLUTION FOR SOLVING 3D INHOMOGENEOUS ELLIPTIC EQUATIONS

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INTERNATIONAL JOURNAL OF COMPUTATIONAL METHODS 2010年第3期7卷 499-511页

作者： Chen, C. S. Kwok, T. O. Ling, L. Hong Kong Baptist Univ Dept Math Kowloon Tong Hong Kong Peoples R China Univ So Mississippi Dept Math Hattiesburg MS 39406 USA

Recently, particular solutions using radial basis functions have been used as a basis for solving inhomogeneous partial differential equations as a one-stage approach without the need of finding homogeneous solutions. In this paper, we adopt a newly developed adaptive greedy algorithm to enhance the performance of the one-stage method and alleviate the difficulty of ill-conditioning of the resultant matrix. To demonstrate the effectiveness of coupling these two methods, we give two 3D examples with excellent numerical results.

关键词： Radial basis functions adaptive greedy algorithm meshless method

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Dimension-splitting data points redistribution for meshless approximation

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2010年第3期235卷 736-746页

作者： Kwok, Ting-On Ling, Leevan Hong Kong Baptist Univ Dept Math Hong Kong Hong Kong Peoples R China

To better approximate nearly singular functions with meshless methods, we propose a data points redistribution method extended from the well-known one-dimensional equidistribution principle. With properly distributed data points, nearly singular functions can be well approximated by linear combinations of global radial basis functions. The proposed method is coupled with an adaptive trial subspace selection algorithm in order to reduce computational cost. In our numerical examples, clear exponential convergence (with respect to the numbers of data points) can be observed. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Radial basis function Meshless interpolation adaptive greedy algorithm Exponential convergence Dual reciprocity method

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Moving-boundary problems solved by adaptive radial basis functions

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COMPUTERS & FLUIDS 2010年第9期39卷 1480-1490页

作者： Vrankar, L. Kansa, E. J. Ling, L. Runovc, F. Turk, G. Slovenian Nucl Safety Adm Ljubljana Slovenia Hong Kong Baptist Univ Dept Math Hong Kong Hong Kong Peoples R China Univ Ljubljana Fac Nat Sci & Engn Ljubljana Slovenia Univ Ljubljana Fac Civil & Geodet Engn Ljubljana Slovenia

The objective of this paper is to present an alternative approach to the conventional level set methods for solving two-dimensional moving-boundary problems known as the passive transport. Moving boundaries are associated with time-dependent problems and the position of the boundaries need to be determined as a function of time and space. The level set method has become an attractive design tool for tracking, modeling and simulating the motion of free boundaries in fluid mechanics, combustion, computer animation and image processing. Recent research on the numerical method has focused on the idea of using a meshless methodology for the numerical solution of partial differential equations. In the present approach, the moving interface is captured by the level set method at all time with the zero contour of a smooth function known as the level set function. A new approach is used to solve a convective transport equation for advancing the level set function in time. This new approach is based on the asymmetric meshless collocation method and the adaptive greedy algorithm for trial subspaces selection. Numerical simulations are performed to verify the accuracy and stability of the new numerical scheme which is then applied to simulate a bubble that is moving, stretching and circulating in an ambient flow to demonstrate the performance of the new meshless approach. (C) 2010 Elsevier Ltd. All rights reserved.

关键词： Moving-boundary problems Stefan problems Level set method Multiquadric Partial differential equations adaptive greedy algorithm

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An improved subspace selection algorithm for meshless collocation methods

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2009年第13期80卷 1623-1639页

作者： Ling, Leevan Schaback, Robert Hong Kong Baptist Univ Dept Math Kowloon Tong Hong Kong Peoples R China Univ Gottingen Inst Numer & Angew Math D-37083 Gottingen Germany

Choosing data points is a common problem for researchers who employ various meshless methods for solving partial differential equations. On the one hand, high accuracy is always desired: oil the other, ill-conditioning problems of the resultant matrices, which may lead to unstable algorithms, prevent some researchers from using meshless methods. For example, the optimal placements of source points in the method of fundamental solutions or of the centers in the radial basis functions method are always unclear. Intuitively, Such optimal locations will depend on many factors: the partial differential equations, the domain, the trial basis used (i.e. the employed method itself), the computational precisions, some user-defined parameters, and so on. Such complexity makes the hope of having an optimal centers placement unpromising. In this paper, we provide a data-dependent algorithm that adaptively selects centers based on all the other variables. Copyright (C) 2009 John Wiley & Sons, Ltd.

关键词： radial basis function Kansa's method collocation adaptive greedy algorithm

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