An efficient method for evaluating the natural frequencies of structures with uncertain-but-bounded parameters

Using interval theory and the second-order Taylor series, the eigenvalue problems of structures with multi-parameter can be transformed into those with single parameter. The epsilon-algorithm is used to accelerate the convergence of the Neumann series to obtain the bounds of eigenvalues of structures with single interval parameter, thus increasing the computing accuracy and reducing the computational effort. Finally, the effect of uncertain parameters on natural frequencies is evaluated. Two engineering examples show that the proposed method can give better results than those obtained by the first-order approximation, even if the uncertainties of parameters are fairly large.     

Application of the Feynman-Kac path integral method in finding excited states of quantum systems

Group theory considerations and properties of a continuous path are used to define a failure tree procedure for finding eigenvalues of the Schrödinger equation using stochastic methods. The procedure is used to calculate the lowest excited state eigenvalues of eigenfunctions possessing anti-symmetric nodal regions in configuration space using the Feynman-Kac path integral method. Within this method the solution of the imaginary time Schrödinger equation is approximated by random walk simulations on a discrete grid constrained only by symmetry considerations of the Hamiltonian. The required symmetry constraints on random walk simulations are associated with a given irreducible representation and are found by identifying the eigenvalues for the irreducible representation corresponding to symmetric or antisymmetric eigenfunctions for each group operator. The method provides exact eigenvalues of excited states in the limit of infinitesimal step size and infinite time. The numerical method is applied to compute the eigenvalues of the lowest excited states of the hydrogenic atom that transform as Γ² and Γ⁴ irreducible representations. Numerical results are compared with exact analytical results.     

Nonclassical Asymptotic Formulas and Approximation with an Arbitrary Order of Accuracy of Eigenvalues of the Sturm-Liouville Problem with Bitsadze-Samarskii Conditions

The Sturm-Liouville problem with Bitsadze-Samarskii conditions is analyzed in the paper. The study is performed in two directions: (i) construction and substantiation of a numerical-analytical method used for deriving eigenfunctions and eigenvalues and (ii) deriving sufficient conditions that are based on the FD-method, that has to provide real eigenvalues, and simplicity and convergence of the generalized nonclassical analytical series.     

A New Nonsymmetric Discontinuous Galerkin Method for Time Dependent Convection Diffusion Equations

We propose a discontinuous Galerkin finite element method for convection diffusion equations that involves a new methodology handling the diffusion term. Test function derivative numerical flux term is introduced in the scheme formulation to balance the solution derivative numerical flux term. The scheme has a nonsymmetric structure. For general nonlinear diffusion equations, nonlinear stability of the numerical solution is obtained. Optimal kth order error estimate under energy norm is proved for linear diffusion problems with piecewise P ^k polynomial approximations. Numerical examples under one-dimensional and two-dimensional settings are carried out. Optimal (k+1)th order of accuracy with P ^k polynomial approximations is obtained on uniform and nonuniform meshes. Compared to the Baumann-Oden method and the NIPG method, the optimal convergence is recovered for even order P ^k polynomial approximations.     

Two-dimensional approximations of three-dimensional eigenvalue problems in plate theory

The eigenvalues and eigenfunctions corresponding to the three-dimensional equations for the linear elastic equilibrium of a clamped plate of thickness 2?, are shown to converge (in a specific sense) to the eigenvalues and eigenfunctions of the well-known two-dimensional biharmonic operator of plate theory, as ? approaches zero. In the process, it is found in particular that the displacements and stresses are indeed of the specific forms usually assumed a priori in the literature. It is also shown that the limit eigenvalues and eigenfunctions can be equivalently characterized as the leading terms in an asymptotic expansion of the three-dimensional solutions, in terms of powers of ?. The method presented here applies equally well to the stationary problem of linear plate theory, as shown elsewhere by P. Destuynder.     

Difetti ed eccessi degli autovalori del classico problema di «buckling»

The classical buckling eigenvalue problem for a square plate clamped along its boundary is here considered. By using the Rayleigh-Ritz method and the method of orthogonal invariants, we obtain upper and lower bounds for the first 60 eigenvalues. Numerical tables are given. The multiplicity of the first eigenvalues and the symmetries of the corresponding eigenfunctions are also studied.      

Control of 1D parabolic PDEs with Volterra nonlinearities, Part II: Analysis

For a class of stabilizing boundary controllers for nonlinear 1D parabolic PDEs introduced in a companion paper, we derive bounds for the gain kernels of our nonlinear Volterra controllers, prove the convergence of the series in the feedback laws, and establish the stability properties of the closed-loop system. We show that the state transformation is at least locally invertible and include an explicit construction for computing the inverse of the transformation. Using the inverse, we show L² and H¹ exponential stability and explicitly construct the exponentially decaying closed-loop solutions. We then illustrate the theoretical results on an analytically tractable example.     

Anelastic Stokes Eigenmodes in Infinite Channel

The anelastic Stokes eigenmodes are computed for a fluid confined, in presence of gravity, between two horizontally infinite plates. These eigenmodes are described by one horizontal wave number k. The eigenvalues λ(k ²) are proved to be all negative. They depend monotonically upon k, behaving like k ² for very large k. Two particular values of k are considered, i.e., k=2?π and k=0, and the stratification parameter of the equilibrium state is taken between 0 (incompressible approximation) and 10 (upper limit of the anelastic configuration). The k=2?π eigenvalue problem is solved numerically while the k=0 is solved both numerically and analytically. Two physical configurations are analyzed, one with no-slip boundary conditions imposed on both horizontal walls, and one with no-stress, while imposing no flow through these boundaries in both cases. The main results are: (i) the smaller the stratification, the larger the decay rate, (ii) the eigenmodes are localized in the lower part of the channel, their vertical extension increasing with the eigenmode spatial frequency, (iii) the Neumann eigenmode decay rates are smaller than their Dirichlet counterparts, except for k=0, where it is just the reverse, (iv) a general trend seems to emerge from the present study, regarding the way the numerical eigenvalues of an elliptic operator compare with the analytical ones, viz., the numerical spectrum overestimates (in absolute value) the analytical spectrum, slightly in the low frequency part of the spectrum and more and more strongly in the upper part.     

Numerical solution of singular ODE eigenvalue problems in electronic structure computations

We put forward a new method for the solution of eigenvalue problems for (systems of) ordinary differential equations, where our main focus is on eigenvalue problems for singular Schrödinger equations arising for example in electronic structure computations. In most established standard methods, the generation of the starting values for the computation of eigenvalues of higher index is a critical issue. Our approach comprises two stages: First we generate rough approximations by a matrix method, which yields several eigenvalues and associated eigenfunctions simultaneously, albeit with moderate accuracy. In a second stage, these approximations are used as starting values for a collocation method which yields approximations of high accuracy efficiently due to an adaptive mesh selection strategy, and additionally provides reliable error estimates. We successfully apply our method to the solution of the quantum mechanical Kepler, Yukawa and the coupled ODE Stark problems.     

Extended backward differentiation formulae in the numerical solution of general Volterra integro-differential equations

In this paper nonlinear Volterra integro-differential equations are considered with kernels of the formP(x,s,y(s)) andK(x,s,y(x),y(s)) and extended backward differentiation methods are applied as extended from their introduction for the solution of ordinary differential equations by Cash [4]. An error bound is obtained and a rate of convergence is found and validated by testing the method on some examples. The numerical results are compared with those obtained by applying standard backward differentiation and collocation methods.     

Reanalysis of linear and nonlinear structures using iterated Shanks transformation

Reanalysis of structures is a common task in two important fields of structural engineering, namely optimization and reliability. It implies the solution of the kinematic equations for each new set of data. In order to simplify this task several approximate methods have been developed. In this paper a simple method for reanalysis, applicable to either linear or nonlinear structures, is proposed. It is based on the improvement of the Neumann resolvent series, which diverges beyond a certain radius, by means of the iterated Shanks transformation. The method is compared with two relevant proposals intended to improve the convergence of the Neumann series, namely the Padé series and the scaled approximation methods. The accuracy and simplicity of the proposed method is demonstrated through some examples concerning linear and nonlinear structures.     

Non-polynomial cubic spline methods for the solution of parabolic equations

Second-order parabolic partial differential equations are solved by using a new three level method based on non-polynomial cubic spline in the space direction and finite difference in the time direction. Stability analysis of the method has been carried out and we have shown that our method is unconditionally stable. It has been shown that by suitably choosing the parameters most of the previous known methods for homogeneous and non-homogeneous cases can be obtained from our method. We also obtain a new high accuracy scheme of O(k ⁴+h ⁴). Numerical examples are given to illustrate the applicability and efficiency of the new method.     

Lower bounds in spectral tests for vectors of nonsuccessive values produced by multiple recursive generator with some zero multipliers

L'Ecuyer develops lower bounds for the maximum distance of the parallel hyperplanes generated by the vectors of nonsuccessive random numbers (RNs) obtained from multiple recursive generators (MRGs). In this paper, a stronger bound, twice that obtained by L'Ecuyer, is derived. For k^th-order MRGs with fewer than k terms in the recursive relationship, much stronger bounds are found. Large distance implies bad lattice structure. In designing RN generators, this factor should be taken into consideration.     

Learning Rates for Regularized Classifiers Using Trigonometric Polynomial Kernels

Regularized classifiers are known to be a kind of kernel-based classification methods generated from Tikhonov regularization schemes, and the trigonometric polynomial kernels are ones of the most important kernels and play key roles in signal processing. The main target of this paper is to provide convergence rates of classification algorithms generated by regularization schemes with trigonometric polynomial kernels. As a special case, an error analysis for the support vector machines (SVMs) soft margin classifier is presented. The norms of Fejér operator in reproducing kernel Hilbert space and properties of approximation of the operator in L ¹ space with periodic function play key roles in the analysis of regularization error. Some new bounds on the learning rate of regularization algorithms based on the measure of covering number for normalized loss functions are established. Together with the analysis of sample error, the explicit learning rates for SVM are also derived.     

Truncations of infinite matrices and algebraic series associated with some of grammars

We consider an algebraic system over $\text{R}$[x] of the form X = a₀(x)X^k+ a_k1(x)X+a_k(x), where a₀(x) and a_k(x) are in x$\text{R}$[x] and a_k?1(x) is in x$\text{R}$. Let A be the infinite incidence matrix associated with the algebraic system. Then we prove that the eigenvalues of northwest corner truncations of A are dense in some algebraic curves.Using this we get a result on positive algebraic series. We consider the case that the coefficients of a₁(x)(i = 0,…,k?1, k) are positive. The algebraic series generated by the algebraic system may be viewed as a function in the complex variable x. Then by the above fact we prove that the radius of convergence of the function equals the least positive zero of the modified discriminant of the system.As an application to context free languages we show a procedure for calculating the entropy of some one counter languages. Other applications to Dyck languages and the Lukasiewicz language are also described.     

Finite difference discretization of the extended Fisher-Kolmogorov equation in two dimensions

Approximate solutions are considered for the extended Fisher-Kolmogorov (EFK) equation in two space dimension with Dirichlet boundary conditions by a Crank-Nicolson type finite difference scheme. A priori bounds are proved using Lyapunov functional. Further, existence, uniqueness and convergence of difference solutions with order O(h²+k²) in the L^∞-norm are proved. Numerical results are also given in order to check the properties of analytical solutions.     

Block splittings for the conjugate gradient method

We study the generalized conjugate gradient scheme, based on the k-line and k × k block Jacobi splittings A = M ? N, for solving two-dimensional parabolic and elliptic difference equations $\text{AU =}\text{F}\text{?}$. Here A represents the difference operator ch^α ? h²Δ_h. Computational experiments suggest that the eigenvalues of K: = I ? M^?1N cluster, and that the cluster radii decrease as k or ch^α increases. As is well known, clustering improves convergence of the conjugate gradient iterates. We discuss computational results for k = 4, 8, 16, 32 and for ch^α = 0, h, 2. Moreover, we establish the number and size of eigenvalue clusters for the model problem.     

On Spectral Bounds for the k-Partitioning of Graphs

When executing processes on parallel computer systems a major bottle-neck is interprocessor communication. One way to address this problem is to minimize the communication between processes that are mapped to different processors. This translates to the k-partitioning problem of the corresponding process graph, where k is the number of processors. The classical spectral lower bound of (|V|/2k)\sum ^k _i=1λ _i for the k-section width of a graph is well known. We show new relations between the structure and the eigenvalues of a graph and present a new method to get tighter lower bounds on the k-section width. This method makes use of the level structure defined by the k-section. We define a global expansion property and prove that for graphs with the same k-section width the spectral lower bound increases with this global expansion. We also present examples of graphs for which our new bounds are tight up to a constant factor.     

On the use of parallel processors for implicit Runge-Kutta methods

An iteration scheme, for solving the non-linear equations arising in the implementation of implicit Runge-Kutta methods, is proposed. This scheme is particularly suitable for parallel computation and can be applied to any method which has a coefficient matrixA with all eigenvalues real (and positive). For such methods, the efficiency of a modified Newton scheme may often be improved by the use of a similarity transformation ofA but, even when this is the case, the proposed scheme can have advantages for parallel computation. Numerical results illustrate this. The new scheme converges in a finite number of iterations when applied to linear systems of differential equations, achieving this by using the nilpotency of a strictly lower triangular matrixS ^?1 AS — Λ, with Λ a diagonal matrix. The scheme reduces to the modified Newton scheme whenS ^?1 AS is diagonal.A convergence result is obtained which is applicable to nonlinear stiff systems.