We consider the interior inverse scattering problem for recovering the shape of a penetrable partially coated cavity with external obstacles from the knowledge of measured scattered waves due to point sources.In the f...We consider the interior inverse scattering problem for recovering the shape of a penetrable partially coated cavity with external obstacles from the knowledge of measured scattered waves due to point sources.In the first part,we obtain the well-posedness of the direct scattering problem by the variational method.In the second part,we establish the mathematical basis of the linear sampling method to recover both the shape of the cavity,and the shape of the external obstacle,however the exterior transmission eigenvalue problem also plays a key role in the discussion of this paper.展开更多
To reduce high computational cost of existing Direction-Of-Arrival(DOA) estimation techniques within a sparse representation framework,a novel method with low computational com-plexity is proposed.Firstly,a sparse lin...To reduce high computational cost of existing Direction-Of-Arrival(DOA) estimation techniques within a sparse representation framework,a novel method with low computational com-plexity is proposed.Firstly,a sparse linear model constructed from the eigenvectors of covariance matrix of array received signals is built.Then based on the FOCal Underdetermined System Solver(FOCUSS) algorithm,a sparse solution finding algorithm to solve the model is developed.Compared with other state-of-the-art methods using a sparse representation,our approach also can resolve closely and highly correlated sources without a priori knowledge of the number of sources.However,our method has lower computational complexity and performs better in low Signal-to-Noise Ratio(SNR).Lastly,the performance of the proposed method is illustrated by computer simulations.展开更多
By using the super-symmetric quantum mechanics (SUSYQM) method, this paper obtains the analytical solutions for the spin-weighted spheroidal wave equation in the case of s = 2. Based on the derived W0 to W4 the gene...By using the super-symmetric quantum mechanics (SUSYQM) method, this paper obtains the analytical solutions for the spin-weighted spheroidal wave equation in the case of s = 2. Based on the derived W0 to W4 the general form for the n-th-order super-potential is summarized and is proved correct by mathematical induction. Hence the ground eigenvalue problem is completely solved. Particularly, the novel solutions of the excited state are investigated according to the shape-invariance property.展开更多
This paper introduces the principle method and simulation of an asymmetric TE (transverse electric) mode absorption in a lossy artificial metamaterial (LHM (left-handed material)). LHM is sandwiched between a lo...This paper introduces the principle method and simulation of an asymmetric TE (transverse electric) mode absorption in a lossy artificial metamaterial (LHM (left-handed material)). LHM is sandwiched between a lossy substrate and covered by a lossless dielectric cladding. The asymmetry solutions of the eigenvalue equation describe lossy-guided modes with complex-valued propagation constants. The dispersion relations, normalized field and the longitudinal attenuation were numerically solved for a given set of parameters: frequency range; film's thicknesses; and TE mode order. We found that high order modes, which are guided in thinner films, generally have more loss of power than low-order modes since the mode attenuation along z-axis Ofz increases to negative values as the mode's number increases, and the film thickness decreases. Moreover, for LHM, at incident wavelength = 1.9 /an, refractive index = -3.74+i2 and at thickness = 0.3μm, the modes of order (4, 5, 6) attain high positive attenuation which means these modes have larger absorption lengths and they are better absorber than the others. This LHM is appropriate for solar cell applications. For arbitrary LHM, at frequency band of wavelengt (600, 700 to 900 nm), the best absorption is attained at longer wavelengths and for lower order modes at wider films. The obtained results could be useful for the design of future light absorbers.展开更多
Optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering, in particular in problems of the design. The solution of such class of PDE-constrained op...Optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering, in particular in problems of the design. The solution of such class of PDE-constrained optimization problems is usually a major computational task. Because of the complexion for directly seeking the solution of PDE-constrained op- timization problem, we transform it into a system of linear equations of the saddle-point form by using the Galerkin finite-element discretization. For the discretized linear system, in this paper we construct a block-symmetric and a block-lower-triangular preconditioner, for solving the PDE-constrained optimization problem. Both preconditioners exploit the structure of the coefficient matrix. The explicit expressions for the eigenvalues and eigen- vectors of the corresponding preconditioned matrices are derived. Numerical implementa- tions show that these block preconditioners can lead to satisfactory experimental results for the preconditioned GMRES methods when the regularization parameter is suitably small.展开更多
In this paper, we present a posteriori error estimator for the nonconforming finite element approximation, including using Crouzeix–Raviart element and extended Crouzeix–Raviart element, of the Stokes eigenvalue pro...In this paper, we present a posteriori error estimator for the nonconforming finite element approximation, including using Crouzeix–Raviart element and extended Crouzeix–Raviart element, of the Stokes eigenvalue problem. With the technique of Helmholtz decomposition, we first give out a posteriori error estimator and prove it as the global upper bound and local lower bound of the approximation error. Then, by deleting a jump term in the indicator, another simpler but equivalent indicator is obtained. Some numerical experiments are provided to verify our analysis.展开更多
We present a computational framework for isolating spatial patterns arising in the steady states of reaction-diffusion systems. Such systems have been used to model many nat- ural phenomena in areas such as developmen...We present a computational framework for isolating spatial patterns arising in the steady states of reaction-diffusion systems. Such systems have been used to model many nat- ural phenomena in areas such as developmental and cancer biology, cell motility and material science. In many of these applications, often one is interested in identifying parameters which will lead to a particular pattern for a given reaction-diffusion model. To attempt to answer this, we compute eigenpairs of the Laplacian on a variety of domains and use linear stability analysis to determine parameter values for the system that will lead to spatially inhomogeneous steady states whose patterns correspond to particular eigenfunctions. This method has previously been used on domains and surfaces where the eigenvalues and eigenfunctions are found analytically in closed form. Our contribution to this methodology is that we numerically compute eigenpairs on arbitrary domains and surfaces. Here we present examples and demonstrate that mode isolation is straightforward especially for low eigenvalues. Additionally, we show that in some cases the inhomogeneous steady state can be a linear combination of eigenfunctions. Finally,we show an example suggesting that pattern formation is robust on similar surfaces in cases that the surface either has or does not have a boundary.展开更多
We present in this paper a numerical method for hypersingular boundary integral equations. This method was developed for planar crack problems: additional edge singularities are known to develop in that case. This pa...We present in this paper a numerical method for hypersingular boundary integral equations. This method was developed for planar crack problems: additional edge singularities are known to develop in that case. This paper includes a rigorous error analysis proving the convergence of our numerical scheme. Three types of examples are covered: the Laplace equation in free space, the linear elasticity equation in free space, and in half space.展开更多
基金supported by the Natural Science Foundation of Xinjiang Uygur Autonomous Region of China(2019D01A05)supported by the NSFC(11571132)。
文摘We consider the interior inverse scattering problem for recovering the shape of a penetrable partially coated cavity with external obstacles from the knowledge of measured scattered waves due to point sources.In the first part,we obtain the well-posedness of the direct scattering problem by the variational method.In the second part,we establish the mathematical basis of the linear sampling method to recover both the shape of the cavity,and the shape of the external obstacle,however the exterior transmission eigenvalue problem also plays a key role in the discussion of this paper.
基金Supported by the National Natural Science Foundation of China (No. 60502040)the Innovation Foundation for Outstanding Postgraduates in the Electronic Engineering Institute of PLA (No. 2009YB005)
文摘To reduce high computational cost of existing Direction-Of-Arrival(DOA) estimation techniques within a sparse representation framework,a novel method with low computational com-plexity is proposed.Firstly,a sparse linear model constructed from the eigenvectors of covariance matrix of array received signals is built.Then based on the FOCal Underdetermined System Solver(FOCUSS) algorithm,a sparse solution finding algorithm to solve the model is developed.Compared with other state-of-the-art methods using a sparse representation,our approach also can resolve closely and highly correlated sources without a priori knowledge of the number of sources.However,our method has lower computational complexity and performs better in low Signal-to-Noise Ratio(SNR).Lastly,the performance of the proposed method is illustrated by computer simulations.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10875018 and 10773002)
文摘By using the super-symmetric quantum mechanics (SUSYQM) method, this paper obtains the analytical solutions for the spin-weighted spheroidal wave equation in the case of s = 2. Based on the derived W0 to W4 the general form for the n-th-order super-potential is summarized and is proved correct by mathematical induction. Hence the ground eigenvalue problem is completely solved. Particularly, the novel solutions of the excited state are investigated according to the shape-invariance property.
文摘This paper introduces the principle method and simulation of an asymmetric TE (transverse electric) mode absorption in a lossy artificial metamaterial (LHM (left-handed material)). LHM is sandwiched between a lossy substrate and covered by a lossless dielectric cladding. The asymmetry solutions of the eigenvalue equation describe lossy-guided modes with complex-valued propagation constants. The dispersion relations, normalized field and the longitudinal attenuation were numerically solved for a given set of parameters: frequency range; film's thicknesses; and TE mode order. We found that high order modes, which are guided in thinner films, generally have more loss of power than low-order modes since the mode attenuation along z-axis Ofz increases to negative values as the mode's number increases, and the film thickness decreases. Moreover, for LHM, at incident wavelength = 1.9 /an, refractive index = -3.74+i2 and at thickness = 0.3μm, the modes of order (4, 5, 6) attain high positive attenuation which means these modes have larger absorption lengths and they are better absorber than the others. This LHM is appropriate for solar cell applications. For arbitrary LHM, at frequency band of wavelengt (600, 700 to 900 nm), the best absorption is attained at longer wavelengths and for lower order modes at wider films. The obtained results could be useful for the design of future light absorbers.
基金Acknowledgments. This work was supported by the National Natural Science Foundation of China(l1271174). The authors are very much indebted to the referees for providing very valuable suggestions and comments, which greatly improved the original manuscript of this paper. The authors would also like to thank Dr. Zeng-Qi Wang for helping on forming the MATLAB data of the matrices.
文摘Optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering, in particular in problems of the design. The solution of such class of PDE-constrained optimization problems is usually a major computational task. Because of the complexion for directly seeking the solution of PDE-constrained op- timization problem, we transform it into a system of linear equations of the saddle-point form by using the Galerkin finite-element discretization. For the discretized linear system, in this paper we construct a block-symmetric and a block-lower-triangular preconditioner, for solving the PDE-constrained optimization problem. Both preconditioners exploit the structure of the coefficient matrix. The explicit expressions for the eigenvalues and eigen- vectors of the corresponding preconditioned matrices are derived. Numerical implementa- tions show that these block preconditioners can lead to satisfactory experimental results for the preconditioned GMRES methods when the regularization parameter is suitably small.
基金Supported by National Science Foundation of China(NSFC 91330202,11001259,11371026,11201501,11031006,11071265,2011CB309703,2010DFR00700)the National Center for Mathematics and Interdisciplinary Science,CAS+1 种基金the President Foundation of AMSS-CASthe Program for Innovation Research in Central University of Finance and Economics
文摘In this paper, we present a posteriori error estimator for the nonconforming finite element approximation, including using Crouzeix–Raviart element and extended Crouzeix–Raviart element, of the Stokes eigenvalue problem. With the technique of Helmholtz decomposition, we first give out a posteriori error estimator and prove it as the global upper bound and local lower bound of the approximation error. Then, by deleting a jump term in the indicator, another simpler but equivalent indicator is obtained. Some numerical experiments are provided to verify our analysis.
文摘We present a computational framework for isolating spatial patterns arising in the steady states of reaction-diffusion systems. Such systems have been used to model many nat- ural phenomena in areas such as developmental and cancer biology, cell motility and material science. In many of these applications, often one is interested in identifying parameters which will lead to a particular pattern for a given reaction-diffusion model. To attempt to answer this, we compute eigenpairs of the Laplacian on a variety of domains and use linear stability analysis to determine parameter values for the system that will lead to spatially inhomogeneous steady states whose patterns correspond to particular eigenfunctions. This method has previously been used on domains and surfaces where the eigenvalues and eigenfunctions are found analytically in closed form. Our contribution to this methodology is that we numerically compute eigenpairs on arbitrary domains and surfaces. Here we present examples and demonstrate that mode isolation is straightforward especially for low eigenvalues. Additionally, we show that in some cases the inhomogeneous steady state can be a linear combination of eigenfunctions. Finally,we show an example suggesting that pattern formation is robust on similar surfaces in cases that the surface either has or does not have a boundary.
文摘We present in this paper a numerical method for hypersingular boundary integral equations. This method was developed for planar crack problems: additional edge singularities are known to develop in that case. This paper includes a rigorous error analysis proving the convergence of our numerical scheme. Three types of examples are covered: the Laplace equation in free space, the linear elasticity equation in free space, and in half space.