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Solving Differential Equations with Radial Basis Functions: Multilevel Methods and Smoothing
 Advances in Comp. Math
"... . Some of the meshless radial basis function methods used for the numerical solution of partial differential equations are reviewed. In particular, the differences between globally and locally supported methods are discussed, and for locally supported methods the important role of smoothing within a ..."
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. Some of the meshless radial basis function methods used for the numerical solution of partial differential equations are reviewed. In particular, the differences between globally and locally supported methods are discussed, and for locally supported methods the important role of smoothing within a multilevel framework is demonstrated. A possible connection between multigrid finite elements and multilevel radial basis function methods with smoothing is explored. Various numerical examples are also provided throughout the paper. 1. Introduction During the past few years the idea of using socalled meshless methods for the numerical solution of partial differential equations (PDEs) has received much attention throughout the scientific community. As a few representative examples we mention Belytschko and coworker's results [3] using the socalled elementfree Galerkin method, Duarte and Oden's work [11] using hp clouds, Babuska and Melenk 's work [1] on the partition of unity method, ...
General Projective Splitting Methods for Sums of Maximal Monotone Operators
, 2007
"... R u t c o r ..."
Algorithms Defined by Nash Iteration: Some Implementations via Multilevel Collocation and Smoothing
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Meshfree methods
 Handbook of Theoretical and Computational Nanotechnology. American Scientific Publishers
, 2005
"... Meshfree methods are the topic of recent research in many areas of computational science and approximation theory. These methods come in various flavors, most of which can be explained either by what is known in the literature as radial basis functions (RBFs), or in terms of the moving least squares ..."
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Cited by 2 (0 self)
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Meshfree methods are the topic of recent research in many areas of computational science and approximation theory. These methods come in various flavors, most of which can be explained either by what is known in the literature as radial basis functions (RBFs), or in terms of the moving least squares (MLS) method. Over the past several years meshfree approximation methods have found their way into many different application areas ranging from artificial intelligence, computer graphics, image processing and optimization to the numerical solution of all kinds of (partial) differential equations problems. Applications in computational nanotechnology are still somewhat rare, but do exist in the literature. In this chapter we will focus on the mathematical foundation of meshfree methods, and the discussion of various computational techniques presently available for a successful implementation of meshfree methods. At the end of this review we mention some initial applications of meshfree methods to problems in computational nanotechnology, and hope that this introduction will serve as a motivation for others to apply meshfree methods to many other challenging problems in computational nanotechnology.
Nash Iteration as a Computational Tool for Differential Equations
"... We describe Nash iteration in numerical analysis, as applied to the solution of linear differential equations. We employ an adaptation involving a splitting of the inversion and the smoothing into two separate steps. We had earlier shown how these ideas apply to scattered data approximation. In this ..."
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Cited by 1 (0 self)
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We describe Nash iteration in numerical analysis, as applied to the solution of linear differential equations. We employ an adaptation involving a splitting of the inversion and the smoothing into two separate steps. We had earlier shown how these ideas apply to scattered data approximation. In this work, we review the ideas in the context of the general nonlinear problem, before proceeding to a detailed set of simulation examples for the linear problem, primarily involving collocation and radial basis functions. We make use of approximate smoothers, involving the solution of evolution equations with calibrated time steps. Acknowledgments: The second author is supported by the National Science Foundation under grant DMS9870420. The third author is supported by the National Science Foundation under grant DMS9704458. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616. email:fass@amadeus.csam.iit.edu y Department of Mathematics and Comp. Sc. , Ke...
Adaptive Search of Autoregression Model Order
"... A new adaptive procedure for search of autoregression model order in real time is proposed. This procedure is threestage adaptive algorithm. Where at the first stage the unknown parameter of autoregression model are adjusted. At the second stage the probabilities that assign to order of autoregress ..."
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A new adaptive procedure for search of autoregression model order in real time is proposed. This procedure is threestage adaptive algorithm. Where at the first stage the unknown parameter of autoregression model are adjusted. At the second stage the probabilities that assign to order of autoregression model are calculated. And at the third stage the autoregression models set entropy are calculated and search of true autoregression model order is made. 1.PROBLEM FORMULATION The search of autoregression model order problem can be considered as a particular case of the structural identification problem. Many tests exist for solution of the problem [1]. But the tests are not adaptive and can not be used in real time statement. In the case which we discuss in the paper we assumed that effect of nonstationarity of stochastic sequences lead to parametrical or structural changes in real time. And it is clear that structural changes reduced to changes of model order because of autoregressive m...
1.1 History and Outline
"... Originally, the motivation for the basic meshfree approximation methods (radial basis functions and moving least squares methods) came from applications in geodesy, geophysics, mapping, or meteorology. Later, applications were found in many areas such ..."
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Originally, the motivation for the basic meshfree approximation methods (radial basis functions and moving least squares methods) came from applications in geodesy, geophysics, mapping, or meteorology. Later, applications were found in many areas such
On SteepestDescentKaczmarz Methods for Regularizing Systems of Nonlinear Illposed Equations
, 2008
"... We investigate modified steepest descent methods coupled with a loping Kaczmarz strategy for obtaining stable solutions of nonlinear systems of illposed operator equations. We show that the proposed method is a convergent regularization method. Numerical tests are presented for a linear problem rel ..."
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We investigate modified steepest descent methods coupled with a loping Kaczmarz strategy for obtaining stable solutions of nonlinear systems of illposed operator equations. We show that the proposed method is a convergent regularization method. Numerical tests are presented for a linear problem related to photoacoustic tomography and a nonlinear problem related to the testing of semiconductor devices.
ADAPTIVE NOISE CANCELLATION FOR MULTISENSORY SIGNALS
"... Accepted (accepted date) This paper describes a fast adaptive algorithm for noise cancellation using multisensory signal recordings of the same noisy source. It is shown that the performance of the new procedure for noise cancellation for multisensory signals is improved when compared to previousl ..."
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Accepted (accepted date) This paper describes a fast adaptive algorithm for noise cancellation using multisensory signal recordings of the same noisy source. It is shown that the performance of the new procedure for noise cancellation for multisensory signals is improved when compared to previously proposed methods. A short overview of the previously proposed methods is given. Optimality of the algorithm is discussed and numerical simulation is included to show the validity and effectiveness of the algorithm.
KACZMARZ ALGORITHM IN HILBERT SPACE AND TIGHT FRAMES
, 2005
"... Abstract. We prove that any tight frame in Hilbert space can be obtained by the Kaczmarz algorithm. An explicit way of constructing this correspondence is given. The uniqueness of the correspondence is determined. Let {en} ∞ n=0 space H. Define ..."
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Abstract. We prove that any tight frame in Hilbert space can be obtained by the Kaczmarz algorithm. An explicit way of constructing this correspondence is given. The uniqueness of the correspondence is determined. Let {en} ∞ n=0 space H. Define