Improvement of the recursive projection method for linear iterative scheme stabilization based on an approximate eigenvalue problem

An algorithm for stabilizing linear iterative schemes is developed in this study. The recursive projection method is applied in order to stabilize divergent numerical algorithms. A criterion for selecting the divergent subspace of the iteration matrix with an approximate eigenvalue problem is introduced. The performance of the present algorithm is investigated in terms of storage requirements and CPU costs and is compared to the original Krylov criterion. Theoretical results on the divergent subspace selection accuracy are established. The method is then applied to the resolution of the linear advection–diffusion equation and to a sensitivity analysis for a turbulent transonic flow in the context of aerodynamic shape optimization. Numerical experiments demonstrate better robustness and faster convergence properties of the stabilization algorithm with the new criterion based on the approximate eigenvalue problem. This criterion requires only slight additional operations and memory which vanish in the limit of large linear systems.     

An optimal Galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere

The kinematic dynamo approximation describes the generation of magnetic field in a prescribed flow of electrically-conducting liquid. One of its main uses is as a proof-of-concept tool to test hypotheses about self-exciting dynamo action. Indeed, it provided the very first quantitative evidence for the possibility of the geodynamo. Despite its utility, due to the requirement of resolving fine structures, historically, numerical work has proven difficult and reported solutions were often plagued by poor convergence. In this paper, we demonstrate the numerical superiority of a Galerkin scheme in solving the kinematic dynamo eigenvalue problem in a full sphere. After adopting a poloidal–toroidal decomposition and expanding in spherical harmonics, we express the radial dependence in terms of a basis of exponentially convergent orthogonal polynomials. Each basis function is constructed from a terse sum of one-sided Jacobi polynomials that not only satisfies the boundary conditions of matching to an electrically insulating exterior, but is everywhere infinitely differentiable, including at the origin. This Galerkin method exhibits more rapid convergence, for a given problem size, than any other scheme hitherto reported, as demonstrated by a benchmark of the magnetic diffusion problem and by comparison to numerous kinematic dynamos from the literature. In the axisymmetric flows we consider in this paper, at a magnetic Reynolds number of O(100), a convergence of 9 significant figures in the most unstable eigenvalue requires only 40 radial basis functions; alternatively, 4 significant figures requires 20 radial functions. The terse radial discretization becomes particularly advantageous when considering flows whose associated numerical solution requires a large number of coupled spherical harmonics. We exploit this new method to confirm the tentatively proposed positive growth rate of the planar flow of Bachtiar et al. [4], thereby verifying a counter-example to the Zel’dovich anti-dynamo theorem in a spherical geometry.     

An iterative solution of the three-colour problem on a random lattice

We study the generalisation of Baxter's three-colour problem to a random lattice. Rephrasing the problem as a matrix model problem we discuss the analyticity structure and critical behaviour of the resulting matrix model. Based on a set of loop equations we develop an algorithm which enables us to solve the three-colour problem recursively.     

A numerical study of the solution of the partial eigenvalue problem

The partial eigenvalue problem that arises from the application of the finite element method is considered. A number of iterative methods are examined which consist in seeking the stationary points of the Rayleigh quotient and thus avoiding the physical assembling of the matrices involved. The computational efficiencies of the steepest descent, the conjugate gradient and the co-ordinate overrelaxation methods are compared. Several other modifications to the original conjugate gradient algorithm as well as an orthogonalization process are studied. The dependence of the convergence of the methods on the initial estimate of the eigenvectors and on different values of the relaxation parameter, in the case of the co-ordinate overrelaxation, are also examined.     

Linear prediction based on the solution of an eigenvalue problem for the autocorrelation matrix

To solve the widespread data-processing problem of predicting the signal or process behavior, we propose a unconventional method for determining the prediction coefficients, which is based on the solution of the problem of finding the eigenvalues of the autocorrelation matrix of the process. The method has some advantages. In particular, the roots of the characteristic polynomial based on the prediction coefficients are located on the unit circle. Physical and Technical Research Institute, N. I. Lobachevsky State University, Nizhny Novgorod, Russia Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 43, No. 1, pp. 66–70, September 2000.     

Consistent second-order approximate solution of the n-fermion problem

A simplified approximation method for the n-fermion problem is developed and tested within the exactly solvable model of Lipkin, Meshkov and Glick. Results are compared with the exact solutions and those of another approximation, which we have discussed in a previous paper, and those of the RPA.     

厄米本征值问题的探究

给出了用探索性方法进行数学物理方法教学的一个案例.从厄米本征值问题出发,经过合情推理,归纳出厄米多项式的递推公式,并猜想出通项公式.该方法可以在传授知识的同时,培养学生的探索意识与创新能力.     

A mixed problem for the Einstein equations and an approximate method of solution

The present paper poses a mixed problem for the Einstein equations. A combined method for solving the problem is introduced. The method consists of a combination of the finite-difference method for the time coordinate, and Galerkin's method for solving the system of equations so obtained. Existence and uniqueness conditions are found for the mixed problem in an appropriately introduced functional space. The convergence conditions for the method are found.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 113–118, May, 1976.     

An algebraic approach to the planar coloring problem

We point out a general relationship between the planar coloring problem withQ colors and the Temperley-Lieb algebra with parameter . This allows us to give a complete algebraic reformulation of the four color result, and to give algebraic interpretations of various other aspects of planar colorings.Work supported in part by NSF Grant #DMS-882602, the program for Mathematics and Molecular Biology, UC Berkeley, and a visiting fellowship of the Japan Society for the promotion of science at Kyoto University, Kyoto, JapanWork supported in part by DOE Contact #DE-AC02-76ERO3075 and by a Packard Fellowship for Science and Engineering     

An improved approximate solution of Altarelli-Parisi equations

We have presented here an improved solution to Altarelli-Parisi equations, and it is found to be in good agreement with SLAC-MIT data up tox⩾0.1.     

A lagrangian scheme for the solution of the optimal mass transfer problem

A lagrangian method to numerically solve the L² optimal mass transfer problem is presented. The initial and final density distributions are approximated by finite mass particles having a gaussian kernel. Mass conservation and the Hamilton–Jacobi equation for the potential are identically satisfied by constant mass transport along straight lines. The scheme is described in the context of existing methods to solve the problem and a set of numerical examples including applications to medical imagery are presented.     

Commutator eigenvalue problem for finite groups

The commutator eigenproblem for an arbitrary finite group is defined by analogy to that of the Lie group. After considering possible arrangements of the group basis of a Frobenius algebra, two convenient matrix forms of commutator eigenproblem are found. They enabled us to find an analytical formula for the eigenvalues and allowed to propose a method for finding the eigenvectors for this problem. The results are demonstrated on the point group T_d of tetrahedron symmetry.     

Generalized eigenvalue problem for non-compact generators

An expansion in generalized eigenvectors of non-compact generators, based on the asymptotic behaviour of the corresponding one-parameter operator groups, is constructed. In particular, such expansion exists for the infinitesimal translations of the Poincaré group.     

An adaptive inverse iteration for Maxwell eigenvalue problem based on edge elements

We propose and analyze an adaptive inverse iterative method for solving the Maxwell eigenvalue problem with discontinuous physical parameters in three dimensions. The adaptive method updates the eigenvalue and eigenfunction based on an a posteriori error estimate of the edge element discretization. At each iteration, the involved saddle-point Maxwell system is transformed into an equivalent system consisting of a singular Maxwell equation and two Poisson equations, for both of which preconditioned iterative solvers are available with optimal convergence rate in terms of the total degrees of freedom. Numerical results are presented, which confirms the quasi-optimal convergence of the adaptive edge element method in terms of the numerical accuracy and the total degrees of freedom.     

二维耦合非线性谐振子的近似求解

利用自洽平均值方法计算二维耦合非线性谐振子的能量本征值,然后将基态能量本征值的计算结果与微扰法的计算结果进行比较,其结果是非常接近的.     

An algebraic treatment of the MIC-Kepler problem on S 3 sphere

The quantum-mechanical problem of motion in a dual charged Coulomb field modified by a centrifugal term (MIC-Kepler problem) is considered in a three-dimensional space of constant positive curvature, S ³. Conserved operators are found, and their commutation relations are derived. It is shown that, in the MIC-Kepler problem in S ³ space, conserved operators form a cubic algebra similar to that of the Kepler problem in the same space. This symmetry algebra is used to obtain the energy spectrum of the problem.     

An eigenvalue problem with limitations in a finite movable basis

An asymptotic method for taking into account the constraints of the orthogonality type when solving eigenvalue problems is developed. The features of the variational determination of eigenvalues are considered, with various finite-dimensional Hilbert subspaces being used to calculate different eigenvalues. A hydrogen molecular ion is used as an example to study the influence of the finite-dimensional approximation and basis optimization on the accuracy of determining the states of identical symmetry. Calculations have shown that the method proposed makes it possible to construct a flexible, consistent scheme for determining the energies of the ground and excited states. The total energy values obtained in a basis of 29 primitive Gaussian functions differ from exact values by 0.228 μH for the ground 1σ_g state and 0.413 μH (microhartree) for the excited 2σ_g state.     

An approximate analytic solution of the heat conduction equation at nanoscale

The Adomian decomposition method (ADM) and the Adomian double decomposition method (ADDM) for solving the 3D non-Fourier heat conduction equation at nanoscale based on the dual-phase-lag framework are proposed. We show that the noise terms that appear in ADM solution can be removed, if the ADDM is employed.