A new method for determining excited states of quantum systems

Summary A new method for computing excited states of a given operatorH is here presented. Our procedure is of particular value when its representation requires an orthonormal basis set of large dimension. In order to obtain the excited state ofH nearest in energy to any chosen trial energyE _t, we consider the auxiliary operatorA=(H−E _t)². We show that a reasonable number of relaxations on appropriately generated low-order Krylov subspaces forA is sufficient to produce better and better approximations of its ground state; a high-accuracy final refinement of the ground state ofA is then possible through the standard Lanczos procedure. An important feature of our method is that storage memory limitations, encountered in the conventional determination of all eigenvalues of large systems, are here overcome. As an illustration of the method two significant examples are discussed. In honour of Prof. Fausto Fumi on the occasion of his retirement from teaching.     

Discriminating resonant targets from clutter using Lanczos iterated single-channel time reversal

Power iterated single-channel time-reversal is extended to employ Lanczos iterations. The properties of these algorithms are studied in the presence of varying levels of noise and broadband clutter. It is shown the Lanczos iterated method possesses superior convergence properties in comparison to the standard power iterated technique. Results demonstrate that such algorithms provide an efficient means through which to isolate and extract the properties of resonant scatterers in the presence of noise and coherent interference.     

Spectral distribution calculations of the level density of20Ne

A recent paper compared moment method and Lanczos method calculations of the level density of²⁰Ne. The present work extends this comparison to include a new type of Lanczos expansion. This expansion procedure shares the advantage of the traditional Lanczos method of a positive definite level density, but unlike the latter, may be used in an essentially infinite basis. Calculations utilizing dsdf and ppdsd bases have been compared with experimental values for the level density, parity ratio and spin cutoff parameters to determine the appropriateness of the input parameters and the model, while calculations in a dsd basis are compared with exact calculated results to test the accuracy of representation provided by the expansion technique.     

Monoslope and multislope MUSCL methods for unstructured meshes

We present new MUSCL techniques associated with cell-centered finite volume method on triangular meshes. The first reconstruction consists in calculating one vectorial slope per control volume by a minimization procedure with respect to a prescribed stability condition. The second technique we propose is based on the computation of three scalar slopes per triangle (one per edge) still respecting some stability condition. The resulting algorithm provides a very simple scheme which is extensible to higher dimensional problems. Numerical approximations have been performed to obtain the convergence order for the advection scalar problem whereas we treat a nonlinear vectorial example, namely the Euler system, to show the capacity of the new MUSCL technique to deal with more complex situations.     

Systematic improvement of variational Monte Carlo using Lanczos iterations

We present a new numerical technique which combines the variational Monte Carlo and the Lanczos methods without suffering from the fermion sign problem. Lanczos iterations allow systematic improvement of trial wavefunctions while Monte Carlo sampling permits treatment of large lattices. As in the usual Lanczos method we find it useful to symmetrize the starting wavefunction in order to accelerate convergence. We apply our method to the 2D AFM Heisenberg model in the fermionic electron representation, which allows us to compare with results from the equivalent bosonic spin representation. Using d-wave RVB states as starting wavefunctions shows that after only one iteration between 70 and 80% of the difference between the variational energy and the ground state energy (as determined by GFMC) is recovered, and a similar improvement is observed in the second iteration. Leaving the spin-singlet sector by introducing antiferromagnetic correlations reduces the symmetry and the relative improvement in energy drops below 50% for one iteration. Our method allows us also to see trends in observables. Relative to the d-wave RVB states we find an enhancement in the spinspin correlations, consistent with the expectation that the true ground state has long-range order.     

A new real-space algorithm for realistic density functional calculations

We present the implementation of a fast real-space algorithm for density functional calculations for atomic nanoclusters. The numerical method is based on a fourth-order operator splitting technique for the solution of the Kohn-Sham equation [1]. The convergence of the procedure is about one order of magnitude better than that of previously used second-order operator factorizations. The method has now been extended to deal with non-local pseudopotentials of the Kleinman-Bylander [2] type, permitting calculations for realistic systems, without significantly degrading the convergence rate. We demonstrate the convergence of the method for the examples C and C₆₀ and present examples of structure calculations of Na and Mg clusters.     

On the relation between the connected-moments expansion and the Lanczos variational scheme

Summary The recently introduced Connected-Moments Expansion (CMX) is compared to a variational Lanczos scheme for estimating ground-state energies of many-body systems. A systematic approach is given for three quantum-mechanical systems: harmonic oscillator, anharmonic oscillator and the Kondo model. A second-order analysis is given in terms of particular ratios of moments of the Hamiltonian.     

A comparative study of iterative solutions to linear systems arising in quantum mechanics

This study is mainly focused on iterative solutions with simple diagonal preconditioning to two complex-valued nonsymmetric systems of linear equations arising from a computational chemistry model problem proposed by Sherry Li of NERSC. Numerical experiments show the feasibility of iterative methods to some extent when applied to the problems and reveal the competitiveness of our recently proposed Lanczos biconjugate A-orthonormalization methods to other classic and popular iterative methods. By the way, experiment results also indicate that application specific preconditioners may be mandatory and required for accelerating convergence.     

一种改进的用于合成孔径雷达图像相干斑抑制的双边滤波参数配置方法

双边滤波能够有效光滑合成孔径雷达图像, 同时保持边缘信息. 最优配置双边滤波参数一直非常困难. 本文作者曾提出了一种迭代参数配置方法, 具有高精度、高效率的特点, 但是该方法会出现迭代错误终止的情况. 本文提出了一种改进的参数配置方法, 能够确保获得最优折衷配置的参数, 并且给出了该方法收敛性的证明. 对真实合成孔径雷达图像的实验结果显示, 在迭代精度相同的情况下, 改进方法不仅能够获得与迭代方法相当的可视效果, 而且灰度值近似方差参数具有更高的精度; 随着迭代精度的提高, 改进方法比迭代方法有更快的收敛速度.     

Photoionization of atoms in parallel electric and magnetic fields: Cross-section calculations using the Lanczos algorithm and the complex coordinate method in a discrete variable representation

A combination of the complex-coordinate method and the Lanczos recursion scheme is implemented in the discrete variable representation (DVR) to obtain total photoionization cross-sections using an iterative procedure. Applications to photoionization of hydrogen atoms in electric fields and sodium atoms in electric and parallel electric and magnetic fields are presented and discussed. Received 15 May 2000 and Received in final form 4 October 2000     

Improved Lanczos algorithms for blackbox MRS data quantitation

Magnetic resonance spectroscopy (MRS) has been shown to be a potentially important medical diagnostic tool. The success of MRS depends on the quantitative data analysis, i.e., the interpretation of the signal in terms of relevant physical parameters, such as frequencies, decay constants, and amplitudes. A variety of time-domain algorithms to extract parameters have been developed. On the one hand, there are so-called blackbox methods. Minimal user interaction and limited incorporation of prior knowledge are inherent to this type of method. On the other hand, interactive methods exist that are iterative, require user involvement, and allow inclusion of prior knowledge. We focus on blackbox methods. The computationally most intensive part of these blackbox methods is the computation of the singular value decomposition (SVD) of a Hankel matrix. Our goal is to reduce the needed computational time without affecting the accuracy of the parameters of interest. To this end, algorithms based on the Lanczos method are suitable because the main computation at each step, a matrix-vector product, can be efficiently performed by means of the fast Fourier transform exploiting the structure of the involved matrix. We compare the performance in terms of accuracy and efficiency of four algorithms: the classical SVD algorithm based on the QR decomposition, the Lanczos algorithm, the Lanczos algorithm with partial reorthogonalization, and the implicitly restarted Lanczos algorithm. Extensive simulation studies show that the latter two algorithms perform best.     

基于预条件LANCZOS算法快速实现三维地电场正演计算

针对三维地电场正演计算过程中形成的超大规模稀疏线性方程组,采用不完全Cholesky分解方法进行预条件处理,经过条件数改善后形成的新线性方程组的系数矩阵变为一个近似的单位矩阵,再应用Lanczos算法将会提高数值计算的稳定性,加快迭代收敛的速度,通常在迭代次数远小于系数矩阵阶数时就能得到较好精确解的近似值,为下一步的电阻率三维反演计算打下了非常好的基础.     

Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics

Non-overlapping domain decomposition (DD) methods provide efficient algorithms for solving time-harmonic Maxwell equations. It has been shown that the convergence of DD algorithms can be improved significantly by using high order transmission conditions. In this paper, we extend a newly developed second-order transmission condition (SOTC), which involves two second-order transverse derivatives, to facilitate fast convergence in the non-conformal DD algorithms. However, the non-conformal nature of the DD methods introduces an additional technical difficulty, which results in poor convergence in many real-life applications. To mitigate the difficulty, a corner-edge penalty method is proposed and implemented in conjunction with the SOTC to obtain truly robust solver performance. Numerical results verify the analysis and demonstrate the effectiveness of the proposed methods on a few model problems. Finally, drastically improved convergence, compared to the conventional Robin transmission condition, was observed for an electrically large problem of practical interest.     

Ground-state energy of two-dimensional interacting electrons in strong magnetic fields

Summary The ground-state energy of two-dimensional interacting electrons in a strong magnetic field in the angular-momentum representation is studied using a modified Lanczos diagnonalization method. In the presence of a positive background density the ground-state energy increases with the total angular momentum, while a cusp-type behaviour appears. The problem of the electron-impurities interaction is formulated as well. The results are discussed in connection with the observed fractional quantum Hall states. Due to the relevance of its scientific content, this paper has been given priority by the Journal Direction. The author of this paper has agreed to not receive the proofs for correction.     

An eigen-based high-order expansion basis for structured spectral elements

We present an eigen-based high-order expansion basis for the spectral element approach with structured elements. The new basis exhibits a numerical efficiency significantly superior, in terms of the conditioning of coefficient matrices and the number of iterations to convergence for the conjugate gradient solver, to the commonly-used Jacobi polynomial-based expansion basis. This basis results in extremely sparse mass matrices, and it is very amenable to the diagonal preconditioning. Ample numerical experiments demonstrate that with the new basis and a simple diagonal preconditioner the number of conjugate gradient iterations to convergence has essentially no dependence or only a very weak dependence on the element order. The expansion bases are constructed by a tensor product of a set of special one-dimensional (1D) basis functions. The 1D interior modes are constructed such that the interior mass and stiffness matrices are simultaneously diagonal and have identical condition numbers. The 1D vertex modes are constructed to be orthogonal to all the interior modes. The performance of the new basis has been investigated and compared with other expansion bases.     

Optimal homotopy asymptotic method with application to thin film flow

A new approximate analytical technique to address for non-linear problems, namely Optimal Homotopy Asymptotic Method (OHAM) is proposed and has been applied to thin film flow of a fourth grade fluid down a vertical cylinder. This approach however, does not depend upon any small/large parameters in comparison to other perturbation method. This method provides a convenient way to control the convergence of approximation series and allows adjustment of convergence regions where necessary. The series solution has been developed and the recurrence relations are given explicitly. The results reveal that the proposed method is very accurate, effective and easy to use.      

二维非结构自适应多重网格的Euler方程解

将原来在结构网格中提出的AUSM⁺(Advection Upstream Splitting Method)格式,推广应用于二维非结构网格中。利用格林-高斯公式对控制体内的变量进行线性重构,获得空间二阶精度,并采用Barth型限制器以抑制数值解的振荡。为提高计算效率,采用了自适应多重网格法。在生成非结构网格时,采用了一种新颖的数据结构——数组链接表。最后给出了NACA0012翼型和带10%鼓包的管道流动的几个Euler方程的算例,并进行了分析比较。     

A Spectral Time-Domain Method for Computational Electrodynamics

Ever since its introduction by Kane Yee over forty years ago, the finite-difference time-domain (FDTD) method has been a widely-used technique for solving the time-dependent Maxwell's equations that has also inspired many other methods. This paper presents an alternative approach to these equations in the case of spatially-varying electric permittivity and/or magnetic permeability, based on Krylov subspace spectral (KSS) methods. These methods have previously been applied to the variable-coefficient heat equation and wave equation, and have demonstrated high-order accuracy, as well as stability characteristic of implicit time-stepping schemes, even though KSS methods are explicit. KSS methods for scalar equations compute each Fourier coefficient of the solution using techniques developed by Golub and Meurant for approximating elements of functions of matrices by Gaussian quadrature in the spectral, rather than physical, domain. We show how they can be generalized to coupled systems of equations, such as Maxwell's equations, by choosing appropriate basis functions that, while induced by this coupling, still allow efficient and robust computation of the Fourier coefficients of each spatial component of the electric and magnetic fields. We also discuss the application of block KSS methods to problems involving non-self-adjoint spatial differential operators, which requires a generalization of the block Lanczos algorithm of Golub and Underwood to unsymmetric matrices.     

Theoretical Calculations of Rydberg Stark Effect of Hydrogen Atom

The Stark shifts and widths of the highly excited states near the classical ionization threshold of a hydrogen atom are calculated by the B spline technique plus complex scaling method. The Lanczos method has been used in our calculations and is proved to be powerful. Our results are in agreement with the experimental results and theoretical ones obtained by other methods. The method can also be used to calculate the same problem for atoms in parallel and cross electric and magnetic fields.