This paper presents a fourth-order Cartesian grid based boundary integral method(BIM)for heterogeneous interface problems in two and three dimensional space,where the problem interfaces are irregular and can be explic...This paper presents a fourth-order Cartesian grid based boundary integral method(BIM)for heterogeneous interface problems in two and three dimensional space,where the problem interfaces are irregular and can be explicitly given by parametric curves or implicitly defined by level set functions.The method reformulates the governing equation with interface conditions into boundary integral equations(BIEs)and reinterprets the involved integrals as solutions to some simple interface problems in an extended regular region.Solution of the simple equivalent interface problems for integral evaluation relies on a fourth-order finite difference method with an FFT-based fast elliptic solver.The structure of the coefficient matrix is preserved even with the existence of the interface.In the whole calculation process,analytical expressions of Green’s functions are never determined,formulated or computed.This is the novelty of the proposed kernel-free boundary integral(KFBI)method.Numerical experiments in both two and three dimensions are shown to demonstrate the algorithm efficiency and solution accuracy even for problems with a large diffusion coefficient ratio.展开更多
In this paper,a new quadratic kernel-free least square twin support vector machine(QLSTSVM)is proposed for binary classification problems.The advantage of QLSTSVM is that there is no need to select the kernel function...In this paper,a new quadratic kernel-free least square twin support vector machine(QLSTSVM)is proposed for binary classification problems.The advantage of QLSTSVM is that there is no need to select the kernel function and related parameters for nonlinear classification problems.After using consensus technique,we adopt alternating direction method of multipliers to solve the reformulated consensus QLSTSVM directly.To reduce CPU time,the Karush-Kuhn-Tucker(KKT)conditions is also used to solve the QLSTSVM.The performance of QLSTSVM is tested on two artificial datasets and several University of California Irvine(UCI)benchmark datasets.Numerical results indicate that the QLSTSVM may outperform several existing methods for solving twin support vector machine with Gaussian kernel in terms of the classification accuracy and operation time.展开更多
This work proposes a generalized boundary integral method for variable coefficients elliptic partial differential equations(PDEs),including both boundary value and interface problems.The method is kernel-free in the s...This work proposes a generalized boundary integral method for variable coefficients elliptic partial differential equations(PDEs),including both boundary value and interface problems.The method is kernel-free in the sense that there is no need to know analytical expressions for kernels of the boundary and volume integrals in the solution of boundary integral equations.Evaluation of a boundary or volume integral is replaced with interpolation of a Cartesian grid based solution,which satisfies an equivalent discrete interface problem,while the interface problem is solved by a fast solver in the Cartesian grid.The computational work involved with the generalized boundary integral method is essentially linearly proportional to the number of grid nodes in the domain.This paper gives implementation details for a secondorder version of the kernel-free boundary integral method in two space dimensions and presents numerical experiments to demonstrate the efficiency and accuracy of the method for both boundary value and interface problems.The interface problems demonstrated include those with piecewise constant and large-ratio coefficients and the heterogeneous interface problem,where the elliptic PDEs on two sides of the interface are of different types.展开更多
基金the National Natural Science Foundation of China(Grant No.DMS-12101553,Grant No.DMS-11771290)the Natural Science Foundation of Zhejiang Province(Grant No.LQ22A010017)+4 种基金the National Key Research and Development Program of China(Project No.2020YFA0712000)the Science Challenge Project of China(Grant No.TZ2016002)the Strategic Priority Research Program of Chinese Academy of Sciences(Grant No.XDA25000400)the National Science Foundation of America(Grant No.ECCS-1927432)also partially supported by the National Science Foundation of America(Grant No.DMS-1720420).
文摘This paper presents a fourth-order Cartesian grid based boundary integral method(BIM)for heterogeneous interface problems in two and three dimensional space,where the problem interfaces are irregular and can be explicitly given by parametric curves or implicitly defined by level set functions.The method reformulates the governing equation with interface conditions into boundary integral equations(BIEs)and reinterprets the involved integrals as solutions to some simple interface problems in an extended regular region.Solution of the simple equivalent interface problems for integral evaluation relies on a fourth-order finite difference method with an FFT-based fast elliptic solver.The structure of the coefficient matrix is preserved even with the existence of the interface.In the whole calculation process,analytical expressions of Green’s functions are never determined,formulated or computed.This is the novelty of the proposed kernel-free boundary integral(KFBI)method.Numerical experiments in both two and three dimensions are shown to demonstrate the algorithm efficiency and solution accuracy even for problems with a large diffusion coefficient ratio.
基金This research was supported by the National Natural Science Foundation of China(No.11771275).
文摘In this paper,a new quadratic kernel-free least square twin support vector machine(QLSTSVM)is proposed for binary classification problems.The advantage of QLSTSVM is that there is no need to select the kernel function and related parameters for nonlinear classification problems.After using consensus technique,we adopt alternating direction method of multipliers to solve the reformulated consensus QLSTSVM directly.To reduce CPU time,the Karush-Kuhn-Tucker(KKT)conditions is also used to solve the QLSTSVM.The performance of QLSTSVM is tested on two artificial datasets and several University of California Irvine(UCI)benchmark datasets.Numerical results indicate that the QLSTSVM may outperform several existing methods for solving twin support vector machine with Gaussian kernel in terms of the classification accuracy and operation time.
基金supported in part by the National Science Foundation of the USA under Grant DMS-0915023is supported by the National Natural Science Foundation of China under Grants DMS-11101278 and DMS-91130012+2 种基金supported by the Young Thousand Talents Program of Chinasupported in part by National Science Committee of Taiwan under Grant 99-2115-M-007-002-MY2supported in part by National Center for Theoretical Sciences of Taiwan,too.
文摘This work proposes a generalized boundary integral method for variable coefficients elliptic partial differential equations(PDEs),including both boundary value and interface problems.The method is kernel-free in the sense that there is no need to know analytical expressions for kernels of the boundary and volume integrals in the solution of boundary integral equations.Evaluation of a boundary or volume integral is replaced with interpolation of a Cartesian grid based solution,which satisfies an equivalent discrete interface problem,while the interface problem is solved by a fast solver in the Cartesian grid.The computational work involved with the generalized boundary integral method is essentially linearly proportional to the number of grid nodes in the domain.This paper gives implementation details for a secondorder version of the kernel-free boundary integral method in two space dimensions and presents numerical experiments to demonstrate the efficiency and accuracy of the method for both boundary value and interface problems.The interface problems demonstrated include those with piecewise constant and large-ratio coefficients and the heterogeneous interface problem,where the elliptic PDEs on two sides of the interface are of different types.
