Singularly perturbed boundary value problems often have solutions with very thinlayers in which t... more Singularly perturbed boundary value problems often have solutions with very thinlayers in which the solution changes rapidly. This paper concentrates on the case where theses layers occur near the boundary, although our method can be applied to problems with interior layers. One technique to deal with the increased resolution requirements in these layers is the use of domain transformations. A
In this paper, we propose a multilevel univariate quasi-interpolation scheme usingmultiquadric ba... more In this paper, we propose a multilevel univariate quasi-interpolation scheme usingmultiquadric basis. It is practical as it does not require derivative values of the function being interpolated. It has a higher degree of smoothness than the original level-0 formula as it allows a shape parameter c=O(h). Our level-1 quasi-interpolation costs O(nlogn) flops to set up. It preserves strict convexity and
ABSTRACT In this paper, a new general scheme, based on the method of fundamental solutions, is pr... more ABSTRACT In this paper, a new general scheme, based on the method of fundamental solutions, is presented for inverse source identification problems. This is fulfilled by coupling a linear combination of fundamental solutions and radial basis functions associated with particular solutions. Under this scheme, we can determine harmonic and nonharmonic source terms from partially accessible boundary measurements. Numerical results for several general inverse source identification problems show that the proposed numerical algorithm is simple, accurate, stable and computationally efficient.
The Effective-Condition-Number (ECN) is a sensitivity measure for a linear system; it differs fro... more The Effective-Condition-Number (ECN) is a sensitivity measure for a linear system; it differs from the traditional condition-number in the sense that the ECN is also right-hand side vector dependent. The first part of this work, in [EABE 33(5): 637-43], revealed the close connection between the ECN and the accuracy of the Method of Fundamental Solutions (MFS) for each given problem. In this paper, we show how the ECN can help achieve the problem-dependent quasi-optimal settings for MFS calculations—that is, determining the position and density of the source points. A series of examples on Dirichlet and mixed boundary conditions shows the reliability of the proposed scheme; whenever the MFS fails, the corresponding value of the ECN strongly indicates to the user to switch to other numerical methods.
Detecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in... more Detecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the pipe. The interior surface of the pipe is inaccessible and the nondestructive detection is solely based on measurements from the outer layer. The Cauchy problem for an elliptic equation is a typical ill-posed problem whose solution does not depend continuously on the boundary data. In this work, we assume that the measurements are available on the whole outer boundary on an annulus domain. By imposing reasonable assumptions, the theoretical goal here is to derive the stabilities of the Cauchy solutions and an energy regularization method. Relationship between the proposed energy regularization method and the Tikhonov regularization with Morozov principle is also given. A novel numerical algorithm is proposed and numerical examples are given.
SUMMARY It is now commonly agreed that the global radial basis functions method is an attractive ... more SUMMARY It is now commonly agreed that the global radial basis functions 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 inappropriate shape parameters were minimized. In contrast, the compactly supported radial basis functions are more relaxing on the smoothness requirements. By settling with algebraic order of convergence only, compactly supported radial basis functions method, provided the support radius are properly chosen, can approximate functions with less smoothness. The reality is that end-users must know the functions to be approximated a priori in order 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 time. In this paper, we propose a hybrid algorithm making use of both global and compactly supported radial basis functions with other developed techniques for meshfree approximation with minimal fine tuning. The first contribution here is an adaptive node refinement scheme. Secondly, we apply the global radial basis functions (with adaptive subspace selection) on the adaptively generated data sites and lastly, the compactly supported radial basis functions (with adaptive support selection) that can be used as a blackbox algorithm for robust approximation to a wider class of functions and for solving partial differential equations.
Partial differential equations (PDEs) on surfaces arise in a variety of application areas includi... more Partial differential equations (PDEs) on surfaces arise in a variety of application areas including biological systems, medical imaging, fluid dynamics, mathematical physics, image processing and computer graphics. In this paper, we propose a radial basis function (RBF) discretization of the closest point method. The corresponding localized meshless method may be used to approximate diffusion on smooth or folded surfaces. Our method has the benefit of having an a priori error bound in terms of percentage of the norm of the solution. A stable solver is used to avoid the ill-conditioning that arises when the radial basis functions (RBFs) become flat.
In this paper, we give the uniqueness on the identification of unknown source locations in two-di... more In this paper, we give the uniqueness on the identification of unknown source locations in two-dimensional heat equations from scattered measurements. Based on the assumption that the unknown source function is a sum of some known functions, we prove that one measurement point is sufficient to identify the number of sources and three measurement points are sufficient to determine all unknown source locations. For verification, we propose a numerical reconstruction scheme for recovering the number of unknown sources and all source locations.
The recently developed multiscale kernel of R. Opfer is applied to approximate numerical derivati... more The recently developed multiscale kernel of R. Opfer is applied to approximate numerical derivatives. The proposed method is truly mesh-free and can handle unstructured data with noise in any dimension. The method of Tikhonov and the method of L-curve are employed for regularization; no information about the noise level is required. An error analysis is provided in a general setting for all dimensions. Numerical comparisons are given in two dimensions which show competitive results with recently published thin plate spline methods.
Over the last decade, there has been a considerable amount of new numerical methods being develop... more Over the last decade, there has been a considerable amount of new numerical methods being developed for solving the Cauchy problems of elliptic operators. In this paper, with some new classes of numerical experiments, we re-verify the conclusions in the review article [EABE,31(4):373-385,2007] concerning the effectiveness of solving Cauchy problems with the method of fundamental solutions.
The condition number of a matrix is commonly used for investigating the stability of solutions to... more The condition number of a matrix is commonly used for investigating the stability of solutions to linear algebraic systems. Recent meshless techniques for solving partial differential equations have been known to give rise to ill-conditioned matrices, yet are still able to produce results that are close to machine accuracy. In this work, we consider the method of fundamental solutions (MFS), which is known to solve, with extremely high accuracy, certain partial differential equations, namely those for which a fundamental solution is known. To investigate the applicability of the MFS, either when the boundary is not analytic or when the boundary data is not harmonic, we examine the relationship between its accuracy and the effective condition number. Three numerical examples are presented in which various boundary value problems for the Laplace equation are solved. We show that the effective condition number, which estimates system stability with the right-hand side vector taken into account, is roughly inversely proportional to the maximum error in the numerical approximation. Unlike the proven theories in literature, we focus on cases when the boundary and the data are not analytic. The effective condition number numerically provides an estimate of the quality of the MFS solution without any knowledge of the exact solution and allows the user to decide whether the MFS is, in fact, an appropriate method for a given problem, or what is the appropriate formulation of the given problem.
We construct several explicit asymptotic two-sided confidence intervals (CIs) for the difference ... more We construct several explicit asymptotic two-sided confidence intervals (CIs) for the difference between two correlated proportions using the method of variance of estimates recovery (MOVER). The basic idea is to recover variance estimates required for the proportion difference from the confidence limits for single proportions. The CI estimators for a single proportion, which are incorporated with the MOVER, include the Agresti-Coull, the Wilson, and the Jeffreys CIs. Our simulation results show that the MOVER-type CIs based on the continuity corrected Phi coefficient and the Tango score CI perform satisfactory in small sample designs and spare data structures. We illustrate the proposed CIs with several real examples.
In the theoretical part of this paper, we introduce a simplified proof technique for error bounds... more In the theoretical part of this paper, we introduce a simplified proof technique for error bounds and convergence of a variation of E. Kansa's well-known unsymmetric meshless collocation method. For a numerical implementation of the convergent variation, a previously proposed greedy technique is coupled with linear optimization. This algorithm allows a fully adaptive on-the-fly data-dependent meshless selection of test and
Abstract. We analyze,a least-squares asymmetric,radial basis function collocation method for solv... more Abstract. We analyze,a least-squares asymmetric,radial basis function collocation method for solving the modified Helmholtz equations. In the theoretical part, we proved,the convergence,of the proposed,method,providing,that the collocation points are sufficiently dense. For numerical verification, direct solver and a sub- space selection process for the trial space (the so-called adaptive,greedy,algorithm) is employed, respectively, for small and large scale problems. AMS subject classifications: 35J25, 65N12, 65N15, 65N35 Key words: Radial basis function, adaptive greedy algorithm, asymmetric collocation, Kansa’s
ABSTRACT Previously, based on the method of (radial powers) radial basis functions, we proposed a... more ABSTRACT Previously, based on the method of (radial powers) radial basis functions, we proposed a procedure for approximating derivative values from one-dimensional scattered noisy data. In this work, we show that the same approach also allows us to approximate the values of (Caputo) fractional derivatives (for orders between 0 and 1). With either an a priori or a posteriori strategy of choosing the regularization parameter, our convergence analysis shows that the approximated fractional derivative values converge at the same rate as in the case of integer order 1.
Singularly perturbed boundary value problems often have solutions with very thinlayers in which t... more Singularly perturbed boundary value problems often have solutions with very thinlayers in which the solution changes rapidly. This paper concentrates on the case where theses layers occur near the boundary, although our method can be applied to problems with interior layers. One technique to deal with the increased resolution requirements in these layers is the use of domain transformations. A
In this paper, we propose a multilevel univariate quasi-interpolation scheme usingmultiquadric ba... more In this paper, we propose a multilevel univariate quasi-interpolation scheme usingmultiquadric basis. It is practical as it does not require derivative values of the function being interpolated. It has a higher degree of smoothness than the original level-0 formula as it allows a shape parameter c=O(h). Our level-1 quasi-interpolation costs O(nlogn) flops to set up. It preserves strict convexity and
ABSTRACT In this paper, a new general scheme, based on the method of fundamental solutions, is pr... more ABSTRACT In this paper, a new general scheme, based on the method of fundamental solutions, is presented for inverse source identification problems. This is fulfilled by coupling a linear combination of fundamental solutions and radial basis functions associated with particular solutions. Under this scheme, we can determine harmonic and nonharmonic source terms from partially accessible boundary measurements. Numerical results for several general inverse source identification problems show that the proposed numerical algorithm is simple, accurate, stable and computationally efficient.
The Effective-Condition-Number (ECN) is a sensitivity measure for a linear system; it differs fro... more The Effective-Condition-Number (ECN) is a sensitivity measure for a linear system; it differs from the traditional condition-number in the sense that the ECN is also right-hand side vector dependent. The first part of this work, in [EABE 33(5): 637-43], revealed the close connection between the ECN and the accuracy of the Method of Fundamental Solutions (MFS) for each given problem. In this paper, we show how the ECN can help achieve the problem-dependent quasi-optimal settings for MFS calculations—that is, determining the position and density of the source points. A series of examples on Dirichlet and mixed boundary conditions shows the reliability of the proposed scheme; whenever the MFS fails, the corresponding value of the ECN strongly indicates to the user to switch to other numerical methods.
Detecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in... more Detecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the pipe. The interior surface of the pipe is inaccessible and the nondestructive detection is solely based on measurements from the outer layer. The Cauchy problem for an elliptic equation is a typical ill-posed problem whose solution does not depend continuously on the boundary data. In this work, we assume that the measurements are available on the whole outer boundary on an annulus domain. By imposing reasonable assumptions, the theoretical goal here is to derive the stabilities of the Cauchy solutions and an energy regularization method. Relationship between the proposed energy regularization method and the Tikhonov regularization with Morozov principle is also given. A novel numerical algorithm is proposed and numerical examples are given.
SUMMARY It is now commonly agreed that the global radial basis functions method is an attractive ... more SUMMARY It is now commonly agreed that the global radial basis functions 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 inappropriate shape parameters were minimized. In contrast, the compactly supported radial basis functions are more relaxing on the smoothness requirements. By settling with algebraic order of convergence only, compactly supported radial basis functions method, provided the support radius are properly chosen, can approximate functions with less smoothness. The reality is that end-users must know the functions to be approximated a priori in order 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 time. In this paper, we propose a hybrid algorithm making use of both global and compactly supported radial basis functions with other developed techniques for meshfree approximation with minimal fine tuning. The first contribution here is an adaptive node refinement scheme. Secondly, we apply the global radial basis functions (with adaptive subspace selection) on the adaptively generated data sites and lastly, the compactly supported radial basis functions (with adaptive support selection) that can be used as a blackbox algorithm for robust approximation to a wider class of functions and for solving partial differential equations.
Partial differential equations (PDEs) on surfaces arise in a variety of application areas includi... more Partial differential equations (PDEs) on surfaces arise in a variety of application areas including biological systems, medical imaging, fluid dynamics, mathematical physics, image processing and computer graphics. In this paper, we propose a radial basis function (RBF) discretization of the closest point method. The corresponding localized meshless method may be used to approximate diffusion on smooth or folded surfaces. Our method has the benefit of having an a priori error bound in terms of percentage of the norm of the solution. A stable solver is used to avoid the ill-conditioning that arises when the radial basis functions (RBFs) become flat.
In this paper, we give the uniqueness on the identification of unknown source locations in two-di... more In this paper, we give the uniqueness on the identification of unknown source locations in two-dimensional heat equations from scattered measurements. Based on the assumption that the unknown source function is a sum of some known functions, we prove that one measurement point is sufficient to identify the number of sources and three measurement points are sufficient to determine all unknown source locations. For verification, we propose a numerical reconstruction scheme for recovering the number of unknown sources and all source locations.
The recently developed multiscale kernel of R. Opfer is applied to approximate numerical derivati... more The recently developed multiscale kernel of R. Opfer is applied to approximate numerical derivatives. The proposed method is truly mesh-free and can handle unstructured data with noise in any dimension. The method of Tikhonov and the method of L-curve are employed for regularization; no information about the noise level is required. An error analysis is provided in a general setting for all dimensions. Numerical comparisons are given in two dimensions which show competitive results with recently published thin plate spline methods.
Over the last decade, there has been a considerable amount of new numerical methods being develop... more Over the last decade, there has been a considerable amount of new numerical methods being developed for solving the Cauchy problems of elliptic operators. In this paper, with some new classes of numerical experiments, we re-verify the conclusions in the review article [EABE,31(4):373-385,2007] concerning the effectiveness of solving Cauchy problems with the method of fundamental solutions.
The condition number of a matrix is commonly used for investigating the stability of solutions to... more The condition number of a matrix is commonly used for investigating the stability of solutions to linear algebraic systems. Recent meshless techniques for solving partial differential equations have been known to give rise to ill-conditioned matrices, yet are still able to produce results that are close to machine accuracy. In this work, we consider the method of fundamental solutions (MFS), which is known to solve, with extremely high accuracy, certain partial differential equations, namely those for which a fundamental solution is known. To investigate the applicability of the MFS, either when the boundary is not analytic or when the boundary data is not harmonic, we examine the relationship between its accuracy and the effective condition number. Three numerical examples are presented in which various boundary value problems for the Laplace equation are solved. We show that the effective condition number, which estimates system stability with the right-hand side vector taken into account, is roughly inversely proportional to the maximum error in the numerical approximation. Unlike the proven theories in literature, we focus on cases when the boundary and the data are not analytic. The effective condition number numerically provides an estimate of the quality of the MFS solution without any knowledge of the exact solution and allows the user to decide whether the MFS is, in fact, an appropriate method for a given problem, or what is the appropriate formulation of the given problem.
We construct several explicit asymptotic two-sided confidence intervals (CIs) for the difference ... more We construct several explicit asymptotic two-sided confidence intervals (CIs) for the difference between two correlated proportions using the method of variance of estimates recovery (MOVER). The basic idea is to recover variance estimates required for the proportion difference from the confidence limits for single proportions. The CI estimators for a single proportion, which are incorporated with the MOVER, include the Agresti-Coull, the Wilson, and the Jeffreys CIs. Our simulation results show that the MOVER-type CIs based on the continuity corrected Phi coefficient and the Tango score CI perform satisfactory in small sample designs and spare data structures. We illustrate the proposed CIs with several real examples.
In the theoretical part of this paper, we introduce a simplified proof technique for error bounds... more In the theoretical part of this paper, we introduce a simplified proof technique for error bounds and convergence of a variation of E. Kansa's well-known unsymmetric meshless collocation method. For a numerical implementation of the convergent variation, a previously proposed greedy technique is coupled with linear optimization. This algorithm allows a fully adaptive on-the-fly data-dependent meshless selection of test and
Abstract. We analyze,a least-squares asymmetric,radial basis function collocation method for solv... more Abstract. We analyze,a least-squares asymmetric,radial basis function collocation method for solving the modified Helmholtz equations. In the theoretical part, we proved,the convergence,of the proposed,method,providing,that the collocation points are sufficiently dense. For numerical verification, direct solver and a sub- space selection process for the trial space (the so-called adaptive,greedy,algorithm) is employed, respectively, for small and large scale problems. AMS subject classifications: 35J25, 65N12, 65N15, 65N35 Key words: Radial basis function, adaptive greedy algorithm, asymmetric collocation, Kansa’s
ABSTRACT Previously, based on the method of (radial powers) radial basis functions, we proposed a... more ABSTRACT Previously, based on the method of (radial powers) radial basis functions, we proposed a procedure for approximating derivative values from one-dimensional scattered noisy data. In this work, we show that the same approach also allows us to approximate the values of (Caputo) fractional derivatives (for orders between 0 and 1). With either an a priori or a posteriori strategy of choosing the regularization parameter, our convergence analysis shows that the approximated fractional derivative values converge at the same rate as in the case of integer order 1.
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Papers by Leevan Ling