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We use higher dimensional B-splines as basis functions to find the approximations for the Dirichlet problem of the Poisson equation in dimension two and three. We utilize the boundary data to remove unnecessary bases. Our method is applicable to more general linear partial differential equations. We provide new basis functions which do not require as many B-splines. The number of new bases coincides with that of the necessary knots. The reducing process uses the boundary conditions to redefine a basis without extra artificial assumptions on knots which are outside the domain. Therefore, more accuracy would be expected from our method. The approximation solutions satisfy the Poisson equation at each mesh point and are solved explicitly using tensor product of matrices. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 393-405, 2014 [ABSTRACT FROM AUTHOR]

Abstract: We aim at developing methods to track minimal energy solutions of time-independent m-component coupled discrete nonlinear Schrödinger (DNLS) equations. We first propose a method to find energy minimizers of the 1-component DNLS equation and use it as the initial point of the m-component DNLS equations in a continuation scheme. We then show that the change of local optimality occurs only at the bifurcation points. The fact leads to a minimal energy tracking method that guides the choice of bifurcation branch corresponding to the minimal energy solution curve. By combining all these techniques with a parameter-switching scheme, we successfully compute a non-radially symmetric energy minimizer that can not be computed by existing numerical schemes straightforwardly. [Copyright &y& Elsevier]

We present a simple and efficient numerical method for the simulation of the three-dimensional pyramid quantum dot heterostructure. The pyramid-shaped quantum dot is placed in a computational box with uniform mesh in Cartesian coordinates. The corresponding Schro¨dinger equation is discretized using the finite volume method and the interface conditions are incorporated into the discretization scheme without explicitly enforcing them. The resulting matrix eigenvalue problem is then solved using a Jacobi–Davidson based method. Both linear and non-linear eigenvalue problems are simulated. The scheme is 2nd order accurate and converges extremely fast. The superior performance is a combined effect of the uniform spacing of the grids and the nice structure of the resulting matrices. We have successfully simulated a variety of test problems, including a quintic polynomial eigenvalue problem with more than 32 million variables. [Copyright &y& Elsevier]

Two very general, fast and simple iterative methods were proposed by Bosma and de Rooij (Bosma, P. B., de Rooij, W. A. (1983). Efficient methods to calculate Chandrasekhar's H functions. Astron. Astrophys. 126:283-292.) to determine Chandrasekhar's H-functions. The methods are based on the use of the equation h =&Ftilde;(h), where &Ftilde = (&ftilde;[sub1],&ftilde;[sub2],...,&ftilde;[subn]supT] is a nonlinear map from R[supn] to R[supn]. Here &ftilde; =1/√1-c+Σ[supn][subk]=1(c[sub;]μ[subk]h[subk/μ[subi]+μ[subk])), 0 < c ≤ 1, i = 1, 2,...,n. One such method is essentially a nonlinear Gauss-Seidel iteration with respect to F. The other ingenious approach is to normalize each iterate after a nonlinear Gauss-Jacobi iteration with respect to P is taken. The purpose of this article is two-fold. First, we prove that both methods converge locally. Moreover, the convergence rate of the second iterative method is shown to be strictly less than (√3-2)/2. Second, we show that both the Gauss-Jacobi method and Gauss-Seidel method with respect to some other known alternative forms of the Chandrasekhar's H-functions either do not converge or essentially stall for c = 1. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient ΓQR algorithm for solving the linear response eigenvalue problem H x = λ x , where H is Π − -symmetric with respect to Γ 0 = diag ( I n , − I n ) . Based on newly introduced Γ -orthogonal transformations, the ΓQR algorithm preserves the Π − -symmetric structure of H throughout the whole process, and thus guarantees the computed eigenvalues to appear pairwise ( λ , − λ ) as they should. With the help of a newly established implicit Γ -orthogonality theorem, we incorporate the implicit multi-shift technique to accelerate the convergence of the ΓQR algorithm. Numerical experiments are given to show the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

Abstract: Three dimensional photonic crystals can be modeled by the Maxwell equations as a generalized eigenvalue problem (GEVP). Simulations based on the numerical solutions of the GEVP are used to reveal physical properties and boost innovative applications of photonic crystals. However, to solve these GEVP remains a computational challenge in both timing and accuracy. The GEVP corresponding to the photonic crystals with face centered cubic (FCC) lattice is one of the challenging eigenvalue problems. From a viewpoint of matrix computation, we demonstrate how such obstacles can be overcome. Our main contribution is an explicit matrix representation of the double-curl operator associated with the photonic crystal with FCC lattice. This particular matrix represents the degenerate coefficient matrix of the discrete GEVP obtained by Yee’s scheme. The explicit matrix leads to an eigendecomposition of the degenerate coefficient matrix and then a fast eigenvalue solver. Promising numerical results in terms of timing and accuracy are reported for solving the discrete GEVP arising in three dimensional photonic crystals with various geometric parameters. [Copyright &y& Elsevier]

Abstract: We consider the solution of large-scale algebraic Riccati equations with numerically low-ranked solutions. For the discrete-time case, the structure-preserving doubling algorithm has been adapted, with the iterates for not explicitly computed but in the recursive form , with and being low-ranked and being small in dimension. For the continuous-time case, the algebraic Riccati equation will be first treated with the Cayley transform before doubling is applied. With being the dimension of the algebraic equations, the resulting algorithms are of an efficient computational complexity per iteration, without the need for any inner iterations, and essentially converge quadratically. Some numerical results will be presented. For instance in Section 5.2, Example 3, of dimension with 204 million variables in the solution , was solved using MATLAB on a MacBook Pro within 45 s to a machine accuracy of . [Copyright &y& Elsevier]

Abstract: We consider the solution of the ★-Sylvester equations AX ± X ★ B ★ = C, for ★= T, H and , and the related linear matrix equations AXB ★ ± X ★ = C, AXB ★ ± CX ★ D ★ = E and AX ± X ★ A ★ = C. Solvability conditions and numerical methods are considered, in terms of the (generalized and periodic) Schur and QR decompositions. We emphasize the square cases where m = n but the rectangular cases will be considered. [Copyright &y& Elsevier]

This article presents numerical methods for computing bound state energies and associated wave functions of three-dimensional semiconductor heterostructures with special interest in the numerical treatment of the effect of band nonparabolicity. A nonuniform finite difference method is presented to approximate a model of a cylindrical-shaped semiconductor quantum dot embedded in another semiconductor matrix. A matrix reduction method is then proposed to dramatically reduce huge eigenvalue systems to relatively very small subsystems. Moreover, the nonparabolic band structure results in a cubic type of nonlinear eigenvalue problems for which a cubic Jacobi–Davidson method with an explicit nonequivalence deflation method are proposed to compute all the desired eigenpairs. Numerical results are given to illustrate the spectrum of energy levels and the corresponding wave functions in rather detail. [Copyright &y& Elsevier]

The problem of bisymmetric damping and stiffness matrices calibration with test data of vibration systems was discussed. Based on the eigen equation as well as bisymmetry of the damping and stiffness matrices, existence and uniqueness of the solution to the problem was studied by means of the theory and method for the inverse algebraic quadratic eigenvalue problem. A new method for the calibration of damping and stiffness matrices was presented. According to the properties of bisymmetric matrices, the bisymmetric solution to the matrix equation was studied. The general expression of the bisymmetric solution was obtained. Moreover, the related optimal approximation problem of any related matrix was addressed and the solution given. The damping and stiffness matrices calibrated with the method not only satisfy the quadratic eigen equation, but also are the unique bisymmetric matrix solution. A numerical example proves efficiency of the present method. [ABSTRACT FROM AUTHOR]

Abstract: We extend the Rayleigh–Ritz method to the eigen-problem of periodic matrix pairs. Assuming that the deviations of the desired periodic eigenvectors from the corresponding periodic subspaces tend to zero, we show that there exist periodic Ritz values that converge to the desired periodic eigenvalues unconditionally, yet the periodic Ritz vectors may fail to converge. To overcome this potential problem, we minimize residuals formed with periodic Ritz values to produce the refined periodic Ritz vectors, which converge under the same assumption. These results generalize the corresponding well-known ones for Rayleigh–Ritz approximations and their refinement for non-periodic eigen-problems. In addition, we consider a periodic Arnoldi process which is particularly efficient when coupled with the Rayleigh–Ritz method with refinement. The numerical results illustrate that the refinement procedure produces excellent approximations to the original periodic eigenvectors. [ABSTRACT FROM AUTHOR]

Abstract: To explore band structures of three-dimensional photonic crystals numerically, we need to solve the eigenvalue problems derived from the governing Maxwell equations. The solutions of these eigenvalue problems cannot be computed effectively unless a suitable combination of eigenvalue solver and preconditioner is chosen. Taking eigenvalue problems due to Yee’s scheme as examples, we propose using Krylov–Schur method and Jacobi–Davidson method to solve the resulting eigenvalue problems. For preconditioning, we derive several novel preconditioning schemes based on various preconditioners, including a preconditioner that can be solved by Fast Fourier Transform efficiently. We then conduct intensive numerical experiments for various combinations of eigenvalue solvers and preconditioning schemes. We find that the Krylov–Schur method associated with the Fast Fourier Transform based preconditioner is very efficient. It remarkably outperforms all other eigenvalue solvers with common preconditioners like Jacobi, Symmetric Successive Over Relaxation, and incomplete factorizations. This promising solver can benefit applications like photonic crystal structure optimization. [ABSTRACT FROM AUTHOR]