Non-periodic Problems Research Articles

Block Circulant and Toeplitz Structures in the Linearized Hartree–Fock Equation on Finite Lattices: Tensor Approach

Abstract This paper introduces and analyzes the new grid-based tensor approach to approximate solutions of the elliptic eigenvalue problem for the 3D lattice-structured systems. We consider the linearized Hartree–Fock equation over a spatial L 1 × L 2 × L 3 {L_{1}\times L_{2}\times L_{3}} lattice for both periodic and non-periodic problem setting, discretized in the localized Gaussian-type orbitals basis. In the periodic case, the Galerkin system matrix obeys a three-level block-circulant structure that allows the FFT-based diagonalization, while for the finite extended systems in a box (Dirichlet boundary conditions) we arrive at the perturbed block-Toeplitz representation providing fast matrix-vector multiplication and low storage size. The proposed grid-based tensor techniques manifest the twofold benefits: (a) the entries of the Fock matrix are computed by 1D operations using low-rank tensors represented on a 3D grid, (b) in the periodic case the low-rank tensor structure in the diagonal blocks of the Fock matrix in the Fourier space reduces the conventional 3D FFT to the product of 1D FFTs. Lattice type systems in a box with Dirichlet boundary conditions are treated numerically by our previous tensor solver for single molecules, which makes possible calculations on rather large L 1 × L 2 × L 3 {L_{1}\times L_{2}\times L_{3}} lattices due to reduced numerical cost for 3D problems. The numerical simulations for both box-type and periodic L × 1 × 1 {L\times 1\times 1} lattice chain in a 3D rectangular “tube” with L up to several hundred confirm the theoretical complexity bounds for the block-structured eigenvalue solvers in the limit of large L.

Does the complex Langevin method give unbiased results?

We investigate whether the stationary solution of the Fokker-Planck equation of the complex Langevin algorithm reproduces the correct expectation values. When the complex Langevin algorithm for an action $S(x)$ is convergent, it produces an equivalent complex probability distribution $P(x)$ which ideally would coincide with $e^{-S(x)}$. We show that the projected Fokker-Planck equation fulfilled by $P(x)$ may contain an anomalous term whose form is made explicit. Such term spoils the relation $P(x)=e^{-S(x)}$, introducing a bias in the expectation values. Through the analysis of several periodic and non-periodic one-dimensional problems, using either exact or numerical solutions of the Fokker-Planck equation on the complex plane, it is shown that the anomaly is present quite generally. In fact, an anomaly is expected whenever the Langevin walker needs only a finite time to go to infinity and come back, and this is the case for typical actions. We conjecture that the anomaly is the rule rather than the exception in the one-dimensional case, however, this could change as the number of variables involved increases.

Hybrid Compact-WENO Finite Difference Scheme with Conjugate Fourier Shock Detection Algorithm for Hyperbolic Conservation Laws

For discontinuous solutions of hyperbolic conservation laws, a hybrid scheme, based on the high order nonlinear characteristicwise weighted essentially nonoscillatory (WENO) conservative finite difference scheme and the high resolution spectral-like linear compact finite difference (Compact) scheme, is developed for capturing shocks and strong gradients accurately and resolving smooth scale structures efficiently. The key issue in any hybrid scheme is the design of an accurate, robust, and efficient high order shock detection algorithm that is capable of determining the smoothness of the solution at any given grid point. The conjugate Fourier (cF) partial sum and its derivative are investigated for its applicability as a shock detector due to its unique property, namely, the cF partial sum converges to the location and strength of an isolated jump. For a nonperiodic problem, the data are first evenly extended before the derivative of the cF partial sum and its mean are computed. The mean allows one to par...

Hidden Markov model for municipal waste generation forecasting under uncertainties

Waste generation forecasting is a complex process that is found to be influenced by some latent influencing parameters and their uncertainties, such as economic growth, demography, individual behaviors, activities and events, and management policies. These hidden features play an important role in forecasting the fluctuations of waste generation. We therefore focus on revealing the trend of waste generation in megacities which face significant influences of social and economic changes to achieve urban sustainable development. To dynamically trace fluctuations caused by these uncertainties, we propose a probability model-driven statistical learning approach which hybridizes a wavelet de-noising, a Gaussian mixture model, and a hidden Markov model. First, to gain the actual underlying trend, wavelet de-noising is used to eliminate the noise of data. Next, the Expectation–Maximization and the Viterbi algorithms are employed for learning parameters and discerning the most probable sequence of hidden states, respectively. Subsequently, the state transition matrix is updated by fractional predictable changes of influencing parameters to perform non-periodic fluctuation problem forecasting, and the forward algorithm is utilized to search the most similar data pattern for the current pattern from historical data in order to forecast the future trend of the periodic fluctuation problem. Finally, we apply the approaches into two kinds of case studies that test both a small dataset and a large dataset. How uncertainty factors influence forecasted results is analyzed in the subsection of results and discussion. The computational results demonstrate that the proposed approaches are effective in solving the municipal waste generation forecasting problem.

Single-line rail rapid transit timetabling under dynamic passenger demand

Railway planning is a complex activity which is usually decomposed into several stages, traditionally network design, line design, timetabling, rolling stock, and staffing. In this paper, we study the design and optimization of train timetables for a rail rapid transit (RRT) line adapted to a dynamic demand environment, which focuses on creating convenient timetables for passengers. The objective is to minimize the average passenger waiting time at the stations, thus focusing on passenger welfare. We first propose two mathematical programming formulations which generalize the non-periodic train timetabling problem on a single line under a dynamic demand pattern. We then analyze the properties of the problem before introducing a fast adaptive large neighborhood search (ALNS) metaheuristic in order to solve large instances of the problem within short computation times. The algorithm yields timetables that may not be regular or periodic, but are adjusted to a dynamic demand behavior. Through extensive computational experiments on artificial and real-world based instances, we demonstrate the computational superiority of our ALNS compared with a truncated branch-and-cut algorithm. The average reduction in passenger waiting times is 26%, while the computational time of our metaheuristic is less than 1% of that required by the alternative CPLEX-based algorithm. Out of 120 open instances, we obtain 84 new best known solutions and we reach the optimum on 10 out of 14 instances with known optimal solutions.

Non-uniform dependence on initial data for the modified Camassa-Holm equation on the line

In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt+umx+2uxm=0, m=(1-∂x2)2u,u(x,0)=u0(x)∈Hs(ℝ), x∈ℝ, t>0,and show that the solution map is not uniformly continuous in Sobolev spaces HS(ℝ) for s>7/2. Compared with the periodic problem, the non-periodic problem is more difficult, e.g., it depends on the conservation law. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part.

Analysis and approximation of a strain-limiting nonlinear elastic model

Elastic solids with strain-limiting response to external loading represent an interesting class of material models, capable of describing stress concentration at strains with small magnitude. A theoretical justification of this class of models comes naturally from implicit constitutive theory. We investigate mathematical properties of static deformations for such strain-limiting nonlinear models. Focusing on the spatially periodic setting, we obtain results concerning existence, uniqueness and regularity of weak solutions, and existence of renormalized solutions for the full range of the positive scalar parameter featuring in the model. These solutions are constructed via a Fourier spectral method. We formulate a sufficient condition for ensuring that a renormalized solution is in fact a weak solution, and we comment on the extension of the analysis to nonperiodic boundary-value problems.

Ground state homoclinic orbits of damped vibration problems

In this paper, we consider a class of non-periodic damped vibration problems with superquadratic nonlinearities. We study the existence of nontrivial ground state homoclinic orbits for this class of damped vibration problems under some conditions weaker than those previously assumed. To the best of our knowledge, there has been no work focused on this case. MSC: 49J40, 70H05.

Chebyshev-Fourier Spectral Methods for Nonperiodic Boundary Value Problems

A new class of spectral methods for solving two-point boundary value problems for linear ordinary differential equations is presented in the paper. Although these methods are based on trigonometric functions, they can be used for solving periodic as well as nonperiodic problems. Instead of using basis functions periodic on a given interval−1,1, we use functions periodic on a wider interval. The numerical solution of the given problem is sought in terms of the half-range Chebyshev-Fourier (HCF) series, a reorganization of the classical Fourier series using half-range Chebyshev polynomials of the first and second kind which were first introduced by Huybrechs (2010) and further analyzed by Orel and Perne (2012). The numerical solution is constructed as a HCF series via differentiation and multiplication matrices. Moreover, the construction of the method, error analysis, convergence results, and some numerical examples are presented in the paper. The decay of the maximal absolute error according to the truncation numberNfor the new class of Chebyshev-Fourier-collocation (CFC) methods is compared to the decay of the error for the standard class of Chebyshev-collocation (CC) methods.

Ill-posedness results for generalized Boussinesq equations

In this article, we investigate both the periodic and non-periodic initial-value problems for generalized Boussinesq equations. We show that the associated flow map is not smooth for a range of Sobolev indices, thus providing a threshold for the regularity needed to perform a Picard iteration for these problems.

A Stability Criterion for Semi-Discrete Difference Schemes of Hyperbolic Conservation Laws on Uniform Grids

In the present study, the stability condition for semi-discrete difference schemes of hyperbolic conservation laws obtained from Fourier analysis is simplified. This stability condition can be applied only to linear difference schemes with constant coefficients implemented with periodic boundary treatment. It could often give useful results for other cases, such as schemes with variable coefficients, schemes for nonperiodic problem and nonlinear problem. However, this condition usually leads to a trigonometric inequality, which makes it not convenient to use. For explicit difference schemes on uniform grids, this trigonometric inequality can be converted to polynomial form. Furthermore, if the scheme is a high-order one, the polynomial can be factorized into a simple form. Thus, it is much easier to solve than the inequality obtained directly from Fourier analysis. For compact difference schemes and conservative schemes, similar results are obtained. Some applications of this new stability criterion are shown, including judging the stability of two schemes, proving the upstream central schemes to be stable, constructing a stable upwind dissipation relation preserving (DRP) scheme and constructing an optimized weighted essentially non-oscillatory (WENO) scheme. Since WENO schemes are nonlinear schemes, the stability analysis in the present study is performed on their underlying linear schemes. According to the numerical tests, the underlying linear scheme should be stable, otherwise the corresponding WENO scheme may display instability. These applications demonstrate that this criterion is convenient and efficient for judging the linear stability of semi-discrete difference schemes and constructing stable upwind difference schemes.

Modal method based on subsectional Gegenbauer polynomial expansion for nonperiodic structures: complex coordinates implementation

In this paper we present an extension of the modal method by Gegenbauer expansion (MMGE) [J. Opt. Soc. Am. A28, 2006 (2011)], [Progress Electromagn. Res.133, 17 (2013)] to the study of nonperiodic problems. The nonperiodicity is introduced through the perfectly matched layers (PMLs) concept, which can be introduced in an equivalent way either by a change of coordinates or by the use of a uniaxial anisotropic medium. These PMLs can generate strong irregularities of the electromagnetic fields that can significantly alter the convergence and stability of the numerical scheme. This is the case, e.g., for the famous Fourier modal method, especially when using complex stretching coordinates. In this work, it will be shown that the MMGE equipped with PMLs is a robust approach because of its natural immunity against spurious modes.

Long-time stability estimates for the non-periodic Littlewood boundedness problem

We consider the nonlinear oscillator equation x ¨ + | x | α − 1 x = p ( t ) for α⩾3 and non-periodic forcing p. For any solution x which is unbounded in the ( x , x ˙ ) -phase plane, it is shown that, for ε>0 small enough, there is a solution xε which is bounded in the phase plane and such that the actions of xε and x remain close on a time interval of length ε−2; further precise information is available on the location of the zeros of xε. In addition, it is possible to take the bounded solution xε from a fixed countable family of such solutions.

Analysis of Two Parareal Algorithms for Time-Periodic Problems

The parareal algorithm, which permits us to solve evolution problems in a time parallel fashion, has created a lot of attention over the past decade. The algorithm has its roots in the multiple shooting method for boundary value problems, which in the parareal algorithm is applied to initial value problems, with a particular coarse approximation of the Jacobian matrix. It is therefore of interest to formulate parareal-type algorithms for time-periodic problems, which also couple the end of the time interval with the beginning, and to analyze their performance in this context. We present and analyze two parareal algorithms for time-periodic problems: one with a periodic coarse problem and one with a nonperiodic coarse problem. An interesting advantage of the algorithm with the nonperiodic coarse problem is that no time-periodic problems need to be solved during the iteration, since on the time subdomains, the problems are not time-periodic either. We prove for both linear and nonlinear problems convergence...

On non-periodic homogenization of time-dependent equations

The recently introduced needle problem approach for the homogenization of non-periodic problems was originally designed for the homogenization of elliptic problems. After a short review of the needle problem approach we demonstrate in this note how the stationary results can be transferred to time-dependent problems. The standard parabolic problem of the corresponding heat equation in a heterogeneous material is considered. Furthermore, we include an application to a hysteresis problem which appears in the theory of porous media.

Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems

In this work, we are concerned with the convergence of the multiscale finite element method (MsFEM) for elliptic homogenization problems, where we do not assume a certain periodic or stochastic structure, but an averaging assumption which in particular covers periodic and ergodic stochastic coefficients. We also give a result on the convergence in the case of an arbitrary coupling between grid size $H$ and a parameter $\epsilon$. $\epsilon$ is an indicator for the size of the fine scale which converges to zero. The findings of this work are based on the homogenization results obtained in [B. Schweizer and M. Veneroni, The needle problem approach to non-periodic homogenization, Netw. Heterog. Media, 6 (4), 2011].

Peakon solutions of the Novikov equation and properties of the data-to-solution map

We consider both the periodic and the non-periodic Cauchy problem for the Novikov equation and discuss continuity results for the data-to-solution map in Sobolev spaces. In particular, we show that the data-to-solution map is not (globally) uniformly continuous in Sobolev spaces with exponent less than 3/2. To accomplish this, we construct sequences of peakon solutions whose distance initially goes to zero but later becomes large.

Existence of Multiple Positive Solutions of a Type of Impulsive Functional Differential Equations

In this paper, we consider the non-periodic boundary value problem for a type of first order impulsive functional differential equation in Banach spaces. The existence of pulse in differential equations makes them an important area of investigation. We make use of fixed point index theory on the cone to prove existence of positive solutions. The conditions for existence of two and three positive solutions are given.

Solution of linearized rotating shallow water equations by compact schemes with different grid-staggering strategies

High accuracy solution of PDEs requires proper error analysis. Previous analysis for a non-dispersive system [T.K. Sengupta, A. Dipankar, P. Sagaut, Error dynamics: beyond von Neumann analysis, J. Comput. Phys. 226 (2007) 1211–1218] identified sources of error correctly. Here, the aim is to extend the spectral analysis for the model linearized rotating shallow water equations (LRSWE), as an example of dispersive system. We perform the analysis when high accuracy compact schemes are used to solve the LRSWE relevant to geophysical fluid dynamics, using different grid arrangements proposed in Mesinger and Arakawa [F. Mesinger, A. Arakawa, Numerical Methods Used in Atmospheric Models, GARP Publ. Ser. No. 17, vol. 1, WMO, Geneva, 1976, pp. 43–64] and Randall [D.A. Randall, Geostrophic adjustment and the finite-difference shallow-water equations, Mon. Wea. Rev. 122 (1994) 1371–1377]. Compact schemes are used for fluid dynamical problem, as these afford near-spectral accuracy in solving non-periodic problems. However, higher accuracy methods also suffer from errors, those are often filtered by low order methods. For example, dispersion error is present in all numerical methods and extreme form of it leads to q-waves, which appear at higher wavenumbers for compact schemes as compared to lower order method. We also evaluate a compact scheme specifically designed for use with staggered grids. Here, two and four time-level temporal discretization methods have been compared for solving LRSWE by considering classical fourth-order, four-stage Runge–Kutta ( RK 4), two time-level forward–backward (FB) and four time-level generalized FB temporal integration schemes.

The EPS method: A new method for constructing pseudospectral derivative operators

We develop a new type of derivative matrix for pseudospectral methods. The norm of these matrices grows at the optimal rate O( N 2) for N-by- N matrices, in contrast to standard pseudospectral constructions that result in O( N 4) growth of the norm. The smaller norm has a big advantage when using the derivative matrix for solving time dependent problems such as wave propagation. The construction is based on representing the derivative operator as an integral kernel, and does not rely on the interpolating polynomials. In particular, we construct second derivative matrices that incorporate Dirichlet or Neumann boundary conditions on an interval and on the disk, but the method can be used to construct a wide variety of commonly used operators for solving PDEs and integral equations. The construction can be used with any quadrature, including traditional Gauss–Legendre quadratures, but we have found that by using quadratures based on prolate spheroidal wave functions, we can achieve a near optimal sampling rate close to two points per wavelength, even for non-periodic problems. We provide numerical results for the new construction and demonstrate that the construction achieves similar or better accuracy than traditional pseudospectral derivative matrices, while resulting in a norm that is orders of magnitude smaller than the standard construction. To demonstrate the advantage of the new construction, we apply the method for solving the wave equation in constant and discontinuous media and for solving PDEs on the unit disk. We also present two compression algorithms for applying the derivative matrices in O( N log N) operations.