Multimodel inference based on smoothed information criteria

The multimodel inference makes statistical inferences from a set of plausible models rather than from a single model. In this paper, we focus on the multimodel inference based on smoothed information criteria proposed by seminal monographs(see Buckland et al.(1997) and Burnham and Anderson(2003)), which are termed as smoothed Akaike information criterion(SAIC) and smoothed Bayesian information criterion(SBIC)methods. Due to their simplicity and applicability, these methods are very widely used in many fields. By using an illustrative example and deriving limiting properties for the weights in the linear regression, we find that the existing variance estimation for SAIC is not applicable because of a restrictive condition, but for SBIC it is applicable. Especially, we propose a simulation-based inference for SAIC based on the limiting properties. Both the simulation study and the real data example show the promising performance of the proposed simulationbased inference.     

On large-scale diagonalization techniques for the Anderson model of localization

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multi-level incomplete LDL^T factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. Our numerical examples reveal that recent algebraic multi-level preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Open-loop control of parameter-dependent discrete-time systems

We consider parameter-dependent linear time-invariant discrete-time single input systems, where the system matrix and the input vector are assumed to depend continuously on a parameter varying over a compact interval. We face the problem of steering the zero state simultaneously arbitrarily close towards a given continuous family of desired terminal states with a finite parameter independent open-loop input sequence. Starting from existing sufficient conditions, which include simplicity of the eigenvalues of the system matrices, we examine the case of multiple eigenvalues. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Poisson-Type Limit Theorems for Eigenvalues of Finite-Volume Anderson Hamiltonians

We consider the spectral problem for the random Schrödinger operator on the multidimensional lattice torus increasing to the whole of lattice, with an i.i.d. potential (Anderson Hamiltonian). We prove complete Poisson-type limit theorems for the (normalized) eigenvalues and their locations, provided that the upper tails of the distribution of potential decay at infinity slower than the double exponential tails. For the fractional-exponential tails, the strong influence of the parameters of the model on a specification of the normalizing constants is described.     

The eigenvalues of Hermite and rational spectral differentiation matrices

Summary We derive expressions for the eigenvalues of spectral differentiation matrices for unbounded domains. In particular, we consider Galerkin and collocation methods based on Hermite functions as well as rational functions (a Fourier series combined with a cotangent mapping). We show that (i) first derivative matrices have purely imaginary eigenvalues and second derivative matrices have real and negative eigenvalues, (ii) for the Hermite method the eigenvalues are determined by the roots of the Hermite polynomials and for the rational method they are determined by the Laguerre polynomials, and (iii) the Hermite method has attractive stability properties in the sense of small condition numbers and spectral radii.     

Krylov-Bogoliubov-Mitropolsky averaging used to construct effective Hamiltonians in the theory of strongly correlated electron systems

We show that the Krylov-Bogoliubov-Mitropolsky averaging in the canonical formulation can be used as a method for constructing effective Hamiltonians in the theory of strongly correlated electron systems. As an example, we consider the transition from the Hamiltonians of the Hubbard and Anderson models to the respective Hamiltonians of the t-J and Kondo models. This is a very general method, has several advantages over other methods, and can be used to solve a wide range of problems in the physics of correlated systems.     

Uniform Anderson localization,unimodal eigenstates and simple spectra in a class of “haarsh” deterministic potentials

We consider a particular class of families of multi-dimensional lattice Schrödinger operators with deterministic (e.g., quasi-periodic) potentials generated by the “hull” given by an orthogonal series over the Haar wavelet basis on the torus of arbitrary dimension, with expansion coefficients considered as independent parameters. In the strong disorder regime, we prove Anderson localization for generic operator families and show that all localized eigenfunctions are unimodal and feature uniform exponential decay away from their respective localization centers. We also prove a variant of the Minami estimate for deterministic potentials and simplicity of the spectrum.     

Persistence of Anderson localization in Schrödinger operators with decaying random potentials

We show persistence of both Anderson and dynamical localization in Schrödinger operators with non-positive (attractive) random decaying potential. We consider an Anderson-type Schrödinger operator with a non-positive ergodic random potential, and multiply the random potential by a decaying envelope function. If the envelope function decays slower than |x|^-2 at infinity, we prove that the operator has infinitely many eigenvalues below zero. For envelopes decaying as |x|^-α at infinity, we determine the number of bound states below a given energy E<0, asymptotically as α↓0. To show that bound states located at the bottom of the spectrum are related to the phenomenon of Anderson localization in the corresponding ergodic model, we prove: (a) these states are exponentially localized with a localization length that is uniform in the decay exponent α; (b) dynamical localization holds uniformly in α.     

Stabilization of nonlinear systems in compound critical cases

In this paper, we study the stabilization of nonlinear systems in critical cases by using the center manifold reduction technique. Three degenerate cases are considered, wherein the linearized model of the system has two zero eigenvalues, one zero eigenvalue and a pair of nonzero pure imaginary eigenvalues, or two distinct pairs of nonzero pure imaginary eigenvalues; while the remaining eigenvalues are stable. Using a local nonlinear mapping (normal form reduction) and Liapunov stability criteria, one can obtain the stability conditions for the degenerate reduced models in terms of the original system dynamics. The stabilizing control laws, in linear and/or nonlinear feedback forms, are then designed for both linearly controllable and linearly uncontrollable cases. The normal form transformations obtained in this paper have been verified by using code MACSYMA.     

Spectral dependence of the localization degree in the one-dimensional disordered Lloyd model

We calculate the Anderson criterion and the spectral dependence of the degree of localization in the first nonvanishing approximation with respect to disorder for one-dimensional diagonally disordered models with a site energy distribution function that has no finite even moments higher than the zeroth. For this class of models (for which the usual perturbation theory is inapplicable), we show that the perturbation theory can be consistently constructed for the joint statistics of advanced and retarded Green’s functions. Calculations for the Lloyd model show that the Anderson criterion in this case is a linear (not quadratic as usual) function of the disorder degree. We illustrate the calculations with computer experiments.     

Eigenproblem for p-Laplacian and nonlinear elliptic equation with nonlinear boundary conditions

In this article, we study the solvability of nonlinear problem for p-Laplacian with nonlinear boundary conditions. We give some characterization of the first eigenvalue of an intermediary eigenvalue problem as simplicity, isolation and its strict monotonicity. Afterward, we character also the second eigenvalue and its strictly partial monotony. On the other hand, in some sense, we establish the non-resonance below the first and furthermore between the first and second eigenvalues of nonlinear Steklov–Robin.     

A FEAST algorithm with oblique projection for generalized eigenvalue problems

The contour integral‐based eigensolvers are the recent efforts for computing the eigenvalues inside a given region in the complex plane. The best‐known members are the Sakurai–Sugiura method, its stable version CIRR, and the FEAST algorithm. An attractive computational advantage of these methods is that they are easily parallelizable. The FEAST algorithm was developed for the generalized Hermitian eigenvalue problems. It is stable and accurate. However, it may fail when applied to non‐Hermitian problems. Recently, a dual subspace FEAST algorithm was proposed to extend the FEAST algorithm to non‐Hermitian problems. In this paper, we instead use the oblique projection technique to extend FEAST to the non‐Hermitian problems. Our approach can be summarized as follows: (a) construct a particular contour integral to form a search subspace containing the desired eigenspace and (b) use the oblique projection technique to extract desired eigenpairs with appropriately chosen test subspace. The related mathematical framework is established. Comparing to the dual subspace FEAST algorithm, we can save the computational cost roughly by a half if only the eigenvalues or the eigenvalues together with their right eigenvectors are needed. We also address some implementation issues such as how to choose a suitable starting matrix and design‐efficient stopping criteria. Numerical experiments are provided to illustrate that our method is stable and efficient.     

Diagram theory for the twofold-degenerate Anderson impurity model

We develop a diagram technique for investigating the twofold-degenerate Anderson impurity model in the normal state with the strong electronic correlations of d electrons of the impurity ion taken into account. We discuss the properties of the Slater-Kanamori model of d electrons. After finding the eigenfunctions and eigenvalues of all 16 local states, we determine the local one-particle propagator. We construct the perturbation theory around the atomic limit of the impurity ion and obtain a Dyson-type equation establishing the relation between the impurity electron propagator and the normal correlation function. As a result of summing infinite series of ladder diagrams, we obtain an approximation for the correlation function.     

Nordsieck methods with computationally verified algebraic stability

We describe the search for algebraically stable Nordsieck methods of order p = s and stage order q = p, where s is the number of stages. This search is based on the theoretical criteria for algebraic stability proposed recently by Hill, and Hewitt and Hill, for general linear methods for ordinary differential equations. These criteria, which are expressed in terms of the non-negativity of the eigenvalues of a Hermitian matrix on the unit circle, are then verified computationally for the derived Nordsieck methods of order p ? 2.     

Estimating a largest eigenvector by Lanczos and polynomial algorithms with a random start

We study Lanczos and polynomial algorithms with random start for estimating an eigenvector corresponding to the largest eigenvalue of an n × n large symmetric positive definite matrix. We analyze the two error criteria: the randomized error and the randomized residual error. For the randomized error, we prove that it is not possible to get distribution-free bounds, i.e., the bounds must depend on the distribution of eigenvalues of the matrix. We supply such bounds and show that they depend on the ratio of the two largest eigenvalues. For the randomized residual error, distribution-free bounds exist and are provided in the paper. We also provide asymptotic bounds, as well as bounds depending on the ratio of the two largest eigenvalues. The bounds for the Lanczos algorithm may be helpful in a practical implementation and termination of this algorithm. © 1998 John Wiley & Sons, Ltd.     

On computation of real eigenvalues of matrices via the Adomian decomposition

The problem of matrix eigenvalues is encountered in various fields of engineering endeavor. In this paper, a new approach based on the Adomian decomposition method and the Faddeev-Leverrier’s algorithm is presented for finding real eigenvalues of any desired real matrices. The method features accuracy and simplicity. In contrast to many previous techniques which merely afford one specific eigenvalue of a matrix, the method has the potential to provide all real eigenvalues. Also, the method does not require any initial guesses in its starting point unlike most of iterative techniques. For the sake of illustration, several numerical examples are included.     

Performance evaluation of nonlinear optimization methods via pairwise comparison and fuzzy numbers

In this paper we extend the deterministic performance evaluation of nonlinear optimization methods: we carry out a pairwise comparison using fuzzy estimates of the performance ratios to obtain fuzzy final scores of the methods under consideration. The key instrument is the concept of fuzzy numbers with triangular membership functions. The algebraic operations on them are simple extensions of the operations on real numbers; they are exact in the parameters (lower, modal, and upper values), not necessarily exact in the shape of the membership function. We illustrate the fuzzy performance evaluation by the ranking and rating of five methods (geometric programming and four general methods) for solving geometric-programming problems, using the results of recent computational studies. Some general methods appear to be leading, an outcome which is not only due to their performance under subjective criteria like domain of applications and conceptual simplicity of use; they also score higher under more objective criteria like robustness and efficiency.     

Tensor eigenvalue complementarity problems

This paper studies tensor eigenvalue complementarity problems. Basic properties of standard and complementarity tensor eigenvalues are discussed. We formulate tensor eigenvalue complementarity problems as constrained polynomial optimization. When one tensor is strictly copositive, the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity. When no tensor is strictly copositive, we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process. The complementarity eigenvalues can be computed sequentially. The formulated polynomial optimization can be solved by Lasserre’s hierarchy of semidefinite relaxations. We show that it has finite convergence for generic tensors. Numerical experiments are presented to show the efficiency of proposed methods.     

A hybrid iterative method for symmetric positive definite linear systems

A hybrid iterative scheme that combines the Conjugate Gradient (CG) method with Richardson iteration is presented. This scheme is designed for the solution of linear systems of equations with a large sparse symmetric positive definite matrix. The purpose of the CG iterations is to improve an available approximate solution, as well as to determine an interval that contains all, or at least most, of the eigenvalues of the matrix. This interval is used to compute iteration parameters for Richardson iteration. The attraction of the hybrid scheme is that most of the iterations are carried out by the Richardson method, the simplicity of which makes efficient implementation on modern computers possible. Moreover, the hybrid scheme yields, at no additional computational cost, accurate estimates of the extreme eigenvalues of the matrix. Knowledge of these eigenvalues is essential in some applications.Research supported in part by NSF grant DMS-9409422.Research supported in part by NSF grant DMS-9205531.