Polynomial Projection Research Articles

A power moment reformulation of the Nikiforov–Uvarov method for exactly solvable systems

Exactly solvable (ES) systems are those for which the full, discrete spectrum can be solved in closed form. In this work, we argue that a moment’s representation analysis can generate these closed-form expressions for the energy in a more direct and transparent manner than the popular Nikiforov–Uvarov (NU) procedure. NU analysis strips the asymptotic form of the physical states. We retain these to generate appropriate moment equations. We show how the form of these moment equations leads to closed-form energy expressions. The wave functions can then be generated as well. Our analysis is extendable to quasi-exactly solvable systems (QES; those for which a subset of the discrete spectrum can be generated in closed form). Two formulations are presented. One of these affirms that a previously developed, general, moment quantization procedure is exact for ES and QES states. This method is referred to as the orthogonal polynomial projection quantization method. It combines moment equation representations for physical states with weighted polynomial expansions (Handy and Vrinceanu. J. Phys. A: Math. Theor. 46, 135202 (2013). doi:10.1088/1751-8113/46/13/135202 ). We also show that in implementing any numerical search procedure to determine the quantum parameter regimes corresponding to ES or QES states, our procedure is more reliable (i.e., numerically stable) than using a Hill determinant formulation. We develop our formalism, demonstrate its effectiveness, and prove its equivalence to the NU approach for ES systems.

Momentum space orthogonal polynomial projection quantization

The orthogonal polynomial projection quantization (OPPQ) is an algebraic method for solving Schrödinger’s equation by representing the wave function as an expansion in terms of polynomials orthogonal with respect to a suitable reference function R(x), which decays asymptotically not faster than the bound state wave function. The expansion coefficients are obtained as linear combinations of power moments . In turn, the 's are generated by a linear recursion relation derived from Schrödinger’s equation from an initial set of low order moments. It can be readily argued that for square integrable wave functions representing physical states . Rapidly converging discrete energies are obtained by setting Ω coefficients to zero at arbitrarily high order. This paper introduces an extention of OPPQ in momentum space by using the representation , where Qn(k) are polynomials orthogonal with respect to a suitable reference function T(k). The advantage of this new representation is that it can help solving problems for which there is no coordinate space moment equation. This is because the power moments in momentum space are the Taylor expansion coefficients, which are recursively calculated via Schrödinger’s equation. We show the convergence of this new method for the sextic anharmonic oscillator and an algebraic treatment of Gross–Pitaevskii nonlinear equation.

Pointwise Reconstruction of Wave Functions from Their Moments through Weighted Polynomial Expansions: An Alternative Global-Local Quantization Procedure

Many quantum systems admit an explicit analytic Fourier space expansion, besides the usual analytic Schrödinger configuration space representation. We argue that the use of weighted orthonormal polynomial expansions for the physical states (generated through the power moments) can define an L2 convergent, non-orthonormal, basis expansion with sufficient pointwise convergent behaviors, enabling the direct coupling of the global (power moments) and local (Taylor series) expansions in configuration space. Our formulation is elaborated within the orthogonal polynomial projection quantization (OPPQ) configuration space representation previously developed The quantization approach pursued here defines an alternative strategy emphasizing the relevance of OPPQ to the reconstruction of the local structure of the physical states.

A High-Order Compact Limiter Based on Spatially Weighted Projections for the Spectral Volume and the Spectral Differences Method

This paper exposes the theoretical developments needed to design a class of spatially weighted polynomial projections used in the definition of a compact limiter dedicated to high-order methods. The spectral volume framework and its integral representation of the solution is used to introduce the degree reduction of the polynomial interpolation. The degree reduction is conducted through a linear projection onto a smaller polynomial space. A particular care is taken regarding the conservativity property and results in a parametric framework where projections can be monitored with spatial weights. These projections are used to define a simple and compact high-order limiting procedure, the SWeP limiter. Then, numerical evaluations are performed using the spectral differences method for the mono-dimensional Euler equations and demonstrate the high-order behavior of the SWeP limiter.

On justification of general polynomial projection method for solving periodic fractional integral equations

In this paper we subtantiate a general polynomial projection method for solving periodic integral equations with Holder coefficients and fractional integral Weyl operator in the main part.

Alignment-free row-co-occurrence cancelable palmprint Fuzzy Vault

Texture coding approach of palmprint biometric enjoys several merits; however, palmprint texture code suffers from alignment problem during matching, which hinders it from being directly adopted in biometric cryptosystems, such as Fuzzy Vault. On the other hand, Fuzzy Vault itself is plagued by several security and privacy issues if biometric template is left unprotected. In this paper, we proposed two hybrid cancelable palmprint cryptosystems based on palmprint texture code, dubbed row-alone and row-co-occurrence Fuzzy Vaults. To be specific, we blend 2DPalmHash Code (2DPHC), a cancelable biometric scheme, and Fuzzy Vault primitive to jointly protect palmprint templates. In row-alone Fuzzy Vault, we leverage the horizontal displacement tolerance trait of 2DPHC to extract an alignment-free feature set for polynomial projection. We further generalize row-alone Fuzzy Vault to row-co-occurrence Fuzzy Vault to improve the key recovery rate. Besides that, we demonstrate that 2DPHC-based Fuzzy Vault enhances the robustness against several security attacks.

A Family of $H(div)$ Finite Element Approximations on Polygonal Meshes

In this paper, we present a family of $H(\div)$-compatible finite element spaces on strictly convex $n$-gons, whose construction makes use of generalized barycentric coordinates. In particular, for integers $0\leq k\leq2$, we define finite element spaces with edge degrees of freedom that include polynomial vector fields of order $k$ and whose vector fields have piecewise $k$th-order polynomial normal traces along the element boundary. These spaces suffer from the shortcoming that the image of the divergence operator includes nonpolynomial functions and, as such, their direct use in a mixed setting along with polynomial scalar fields may lead to unstable discretizations exhibiting degraded or no convergence. We present a general remedy for restoration of optimal convergence that involves “polynomial corrections” of vector fields and their divergence at the element level. These corrections are consistent with one another and require computation of suitable polynomial projection maps. In addition to the theoretical discussion, the performance of the discretization schemes based on the proposed spaces and the accompanying correction maps is numerically evaluated through their application to $H(\div)$ eigenvalue problems, mixed and least-squares approximations of linear and nonlinear porous media flow problems.

High-Spatial-Diversity Imaging Receiver Using Fisheye Lens for Indoor MIMO VLCs

This letter proposes an imaging receiver scheme for an indoor multiple-input-multiple-output (MIMO) visible light communication (VLC), in which a fisheye lens with ultrawide field-of-view is used for high-quality imaging, so it can realize omnidirectional receiving and provide high-spatial diversity for decoding of the MIMO signals. In addition, the fisheye lens projects planar and small-sized images, which means that the integration with compact planar receiver array is feasible. Using the polynomial projection model, the optical intensity on the receiving plane is obtained, which is in accordance with the experimental result and shows that the images of the light-emitting diodes are clearly separated. The simulation results indicate that low correlations of the channel matrix are achieved, so high spectral efficiency is realized with various receiver positions under indoor circumstance. Consequently, this fisheye-lens-based imaging receiver is a potential candidate for high-performance indoor MIMO VLC applications.

A polynomial projection algorithm for linear feasibility problems

We propose a polynomial algorithm for linear feasibility problems. The algorithm represents a linear problem in the form of a system of linear equations and non-negativity constraints. Then it uses a procedure which either finds a solution for the respective homogeneous system or provides the information based on which the algorithm rescales the homogeneous system so that its feasible solutions in the unit cube get closer to the vector of all ones. In a polynomial number of calls to the procedure the algorithm either proves that the original system is infeasible or finds a solution in the relative interior of the feasible set.

A Projection Bias in Frequent Subgraph Mining Can Make a Difference

The aim of the frequent subgraph mining task is to find frequently occurring subgraphs in a large graph database. However, this task is a thriving challenge, as graph and subgraph isomorphisms play a key role throughout the computations. Since subgraph isomorphism testing is a hard problem, subgraph miners are exponential in runtime. To alleviate the complexity issue, we propose to introduce a bias in the projection operator and instead of using the costly subgraph isomorphism projection, one can use a polynomial projection having a semantically-valid structural interpretation. This paper presents a new projection operator for graphs named AC-projection, which exhibits nice theoretical complexity properties. We study the size of the search space as well as some practical properties of the projection operator. We also introduce a novel breadth-first algorithm for frequent AC-reduced subgraphs mining. Then, we prove experimentally that we can achieve an important performance gain (polynomial complexity projection) without or with non-significant loss of discovered patterns in terms of quality.

LC-mine: a framework for frequent subgraph mining with local consistency techniques

Developing algorithms that discover all frequently occurring subgraphs in a large graph database is computationally extensive, as graph and subgraph isomorphisms play a key role throughout the computations. Since subgraph isomorphism testing is a hard problem, fragment miners are exponential in runtime. To alleviate the complexity issue, we propose to introduce a bias in the projection operator and instead of using the costly subgraph isomorphism projection, one can use a polynomial projection having a semantically valid structural interpretation. In this paper, our purpose is to present LC-mine, a generic and efficient framework to mine frequent subgraphs by the means of local consistency techniques used in the constraint programming field. Two instances of the framework based on the arc consistency technique are developed and presented in this paper. The first instance follows a breadth-first order, while the second is a pattern-growth approach that follows a depth-first search space exploration strategy. Then, we prove experimentally that we can achieve an important performance gain without or with nonsignificant loss of discovered patterns in terms of quality.

A moments' analysis of quasi-exactly solvable systems: a new perspective on the sextic potential

A quantum system is quasi-exactly solvable (QES) when a subset of its eigenvalues can be obtained in closed form, or they are the roots of closed form expressions. In one dimension, QES states assume the form , involving a known positive reference function and a polynomial P(x), so that the Taylor expansion of truncates at a finite order. For non-QES states this truncation procedure, for suitable reference functions, can provide approximate values for the eigenvalues. It corresponds to the Hill determinant method, which is known to be unstable when assumes the controlling asymptotic form of the physical state. For this reason, it cannot be used to simultaneously generate exact QES states while approximating non-QES states. To address these limitations, the orthogonal polynomial projection quantization (OPPQ) method was developed (Handy and Vrinceanu 2013 J. Phys. A: Math. Theor. 46 135202; 2013 J. Phys. B: Atom. Mol. Opt. Phys. 46 115002). It is demonstrated in this paper that OPPQ can directly and transparently provide, at the same time, both algebraic solutions for QES states and convergent, numerically stable, approximations for non-QES states. Within the OPPQ analysis for the wavefunction representation , the Bender–Dunne energy orthogonal polynomials correspond, exactly (up to a numerical factor), to the energy dependent power moments ν(p) = ∫dx xpΦ(x). Within this perspective, the existence of QES states is associated with an anomalous kink behavior in the order of the finite difference moment equation corresponding to the ν's, suggesting a change in the number of degrees of freedom. This was first noted through the implementation of the eigenvalue moment method, the first application of semidefinite programming analysis to quantum operators (Handy and Bessis 1985 Phys. Rev. Lett. 55 931). This moments' perspective also reveals additional properties for the non-QES states of the same symmetry as the QES states: their lower order ν-moments must be zero. We demonstrate our results in the context of the two sextic potentials: Vsa(x) = x6 + mx2 + bx4 and Vss(x) = x6 + mx2 + b/x2.

Addressing integration error for polygonal finite elements through polynomial projections: A patch test connection

Polygonal finite elements generally do not pass the patch test as a result of quadrature error in the evaluation of weak form integrals. In this work, we examine the consequences of lack of polynomial consistency and show that it can lead to a deterioration of convergence of the finite element solutions. We propose a general remedy, inspired by techniques in the recent literature of mimetic finite differences, for restoring consistency and thereby ensuring the satisfaction of the patch test and recovering optimal rates of convergence. The proposed approach, based on polynomial projections of the basis functions, allows for the use of moderate number of integration points and brings the computational cost of polygonal finite elements closer to that of the commonly used linear triangles and bilinear quadrilaterals. Numerical studies of a two-dimensional scalar diffusion problem accompany the theoretical considerations.

Projection Pursuit Regression Based on Hermite Polynomial for Atmospheric Corrosion Data Application

A Projection Pursuit Regression method by using Hermite Polynomial is put forward to make modeling and forcasting of corrosion data, because of small sample of acumulation data of metal material corrosion in atmosphere, Multi-dimensional Properties and Non-orthogonality of influence factors. Analyses and prediction of atmospheric corrosion data of a metal are made by using this method. Compared with PCA+SVM method, this method improves significantly the accuration of prediction and correctness of corrosion vehavior trend. The result proves that the Hermite Polynomial Projection Pursuit Regression method has great huge advantage in data analysis of steel corrosion in atmosphere.

Rayleigh-Ritz variational method with suitable asymptotic behaviour

AbstractThis paper considers the Rayleigh-Ritz variational calculations with non-orthogonal basis sets that exhibit the correct asymptotic behaviour. This approach is illustrated by constructing suitable basis sets for one-dimensional models such as the two double-well oscillators recently considered by other authors. The rate of convergence of the variational method proves to be considerably greater than the one exhibited by the recently developed orthogonal polynomial projection quantization.

Local polynomial and penalized trigonometric series regression

We investigate the connections between local polynomial regression, mixed models, and penalized trigonometric series regression. Expressing local polynomial regression in a projection framework, we derive equivalent kernels for both the interior and boundary points. For interior points, it is shown that the asymptotic bias decreases as the order of polynomial increases. Then we show that, under some conditions, the local polynomial projection approach admits an equivalent mixed model formulation where the fixed effects part includes the polynomial functions. The random effects part in the representation is shown to be the trigonometric series asymptotically. The connections are extended to partial linear models and additive models. These results suggest a new smoothing approach using a combination of unpenalized polynomials and penalized trigonometric functions. We illustrate the potential usefulness of the new approach with real data analysis.

Reconstructing Distributions of Multivariate Random Functions under Orthogonal Polynomial Projection Approximation*

This paper deals with the problem of reconstructing probabilistic distributions of random functions, which describe images dynamically formed in tomographic experiments. For a certain class of random functions, a reconstruction method is developed. Stability properties of the proposed method are studied.

Saturation estimates for hp-finite element methods

In this paper we will prove saturation estimates for the adaptive $$hp$$hp-finite element method for linear, second order partial differential equations. More specifically we will consider a sequence of nested finite element discretizations where we allow for both, local mesh refinement and locally increasing the polynomial order. We will prove that the energy norm of the error on the finer level can be estimated by the sum of a contraction of the old error and data oscillations. We will derive estimates of the contraction factor which are explicit with respect to the local mesh width and the local polynomial degree. In order to cover $$p$$p-refinement of finite element spaces new polynomial projection operators will be introduced and new polynomial inverse estimates will be derived.

Rapidly converging bound state eigenenergies for the two dimensional quantum dipole

We examine the effectiveness of a new spectral method in solving the two dimensional dipole problem (DP), as originally formulated by Dasbiswas et al (2010 Phys. Rev. B: At. Mol. Opt. Phys. 81 064516), and recently analysed by Amore and Fernandez (AF, 2012 Phys. Rev. B: At. Mol. Opt. Phys. 45 235004), through a large, non-orthogonal basis, Rayleigh–Ritz (RR) analysis. This deceptively simple problem has a long history of poorly approximated energy values, particularly for the ground state, until the recent work by AF. In contrast to their approach, we implement an orthogonal polynomial projection quantization (OPPQ) analysis (Handy and Vrinceanu 2013 J. Phys. A: Math. Theor. 46 135202), involving expanding the wavefunction in terms of a complete basis, , where are the orthogonal polynomials relative to the weight . For systems transformable into a moment equation, such as DP, the projection coefficients are determinable in closed form, yielding an efficient quantization procedure, particularly when the weight assumes the asymptotic form of the physical solutions. There are several theoretical reasons why the OPPQ should be more effective than the above RR approach. Indeed, comparable results are achieved with significantly fewer OPPQ variational parameters as compared to RR-variational parameters. For instance, with regards to the delicate ground state energy, 130 OPPQ variables are required to achieve Egr = −0.137 7614 (Egr = −0.137 7514 after a Shanks transform) as opposed to the 821 required within the RR formulation: Egr = −0.137 7478. Despite this, the relative slow convergence for low lying even parity states, within both the OPPQ and RR formulations, suggests that significant logarithmic contributions to the wavefunction, at the origin, have been ignored by all previous investigators. Modifying the RR variational analysis to include log-dependent basis, affirms this through an accelerated RR convergence for the ground state: Egr = −0.137 748 utilizing 68 variational parameters. This result suggests further improvements within either the OPPQ or RR formulation.

Orthogonal polynomial projection quantization: a new Hill determinant method

Accurate energy eigenvalues are obtained by simply projecting the unknown bound state wave function on, essentially, arbitrary sets of orthogonal polynomials, and setting a subset of these to zero. The projection integrals are represented in terms of the power moments of the wave function, obtained recursively by transforming Schrödinger’s equation into a moment equation. Because unbounded wave functions do not have power moments, all solutions are guaranteed to be L2, resulting in a more robust, rapidly converging and stable method when compared with configuration space Hill determinant methods. More importantly, our approach permits the use of arbitrary, nonanalytic, positive reference functions, including those that manifest the true asymptotic behavior of the discrete states. These advantages are not usually possible with the standard Hill approach. The formulation presented here can be applied to any problem for which Schrödinger’s equation can be transformed into a moment equation. In this regard it is related to the L2 quantization prescription developed by Tymczak et al (1998 Phys. Rev. Lett. 80 3674; 1998 Phys. Rev. A 58 2708) corresponding to the Hill determinant method in momentum space, and to the eigenvalue moment method (Handy and Bessis 1985 Phys. Rev. Lett. 55 931; Handy et al 1988 Phys. Rev. Lett. 60 253), the first use of semidefinite programming related analysis in quantum physics. The latter method can be used to generate the ground state, whose orthogonal polynomials can serve to generate even more rapidly converging estimates for the excited discrete state energies. We demonstrate the power of this new approach on the sextic and quartic anharmonic oscillators, as well as on recently studied one- and two-dimensional pseudo-Hermitian Hamiltonians.