Multidimensional Polynomial Research Articles

Notes on factor prime factorizations for n-D polynomial matrices

Multidimensional (n?D) polynomial matrix factorizations have been widely investigated during the past years due to the fundamental importance in the areas of multidimensional systems and signal processing. This note is a continuation of the papers in Wang (Circuits Syst I 54: 1398---1405, 2007 and Circuits Syst II 55: 61---64, 2008). A complete constructive criterion is given for a matrix of 2 × m full row rank to be FLP. We also present a reduction method for dealing with the factorization problems with cases of non-regular divisors.

Geotechnical probabilistic analysis by collocation-based stochastic response surface method: An Excel add-in implementation

A general probabilistic method called collocation-based stochastic response surface method (CSRSM) was previously developed. It involves the propagation of input uncertainties through a computation model to arrive at a random output vector. It is assumed that the unknown random output can be expanded using a polynomial chaos basis with corresponding unknown coefficients. The unknown coefficients are evaluated using a collocation method because it has the important practical advantage of allowing existing deterministic numerical codes to be used as ‘black boxes’. The roots of the Hermite polynomial provide efficient collocation points to evaluate the coefficients in the stochastic response surface. An Excel add-in is developed to produce the basis functions (multi-dimensional Hermite polynomials) without resorting to symbolic algebra practitioners. This is a major practical advantage that would bring realistic probabilistic analyses within reach of the practitioners. Full Excel implementation details are illustrated using a simple slope problem involving six input random variables. A second problem (sum of exponential random variables) is studied to examine CSRSM over a wider range of conditions. It also provides further validation because the solution is available in closed-form. The results show the ease and successful implementation of the proposed Excel-based CSRSM. However, the add-in is unable to handle correlated input parameters thus far. Future development work is needed.

Operational methods and solutions of boundary-value problems with periodic data

We introduce suitable exponential operators and use the multi-dimensional polynomials considered by Hermite and subsequently studied by P. Appell and J. Kampe de Feriet, H. W. Gould and A. T. Hopper, and G. Dattoli et al. in order to obtain explicit solutions of classical boundary-value problems with periodic data.

Polynomial modeling and optimization for colorimetric characterization of scanners

We present different computational strategies for colorimetric characterization of scanners using multidimensional polynomials. The designed strategies allow us to determine the coefficients of an a priori fixed polynomial, taking into account different color error statistics. Moreover, since there is no clear relationship between the polynomial chosen for the characterization and the intrinsic characteristics of the scanner, we show how genetic programming could be used to generate the best polynomial. Experimental results on different devices are reported to confirm the effectiveness of our methods with respect to others in the state of the art.

The multidimensional moment-constrained maximum entropy problem: A BFGS algorithm with constraint scaling

In a recent paper we developed a new algorithm for the moment-constrained maximum entropy problem in a multidimensional setting, using a multidimensional orthogonal polynomial basis in the dual space of Lagrange multipliers to achieve numerical stability and rapid convergence of the Newton iterations. Here we introduce two new improvements for the existing algorithm, adding significant computational speedup in situations with many moment constraints, where the original algorithm is known to converge slowly. The first improvement is the use of the BFGS iterations to progress between successive polynomial reorthogonalizations rather than single Newton steps, typically reducing the total number of computationally expensive polynomial reorthogonalizations for the same maximum entropy problem. The second improvement is a constraint rescaling, aimed to reduce relative difference in the order of magnitude between different moment constraints, improving numerical stability of iterations due to reduced sensitivity of different constraints to changes in Lagrange multipliers. We observe that these two improvements can yield an average wall clock time speedup of 5–6 times compared to the original algorithm.

Whittaker Limits of Difference Spherical Functions

We introduce the (global) q-Whittaker function as the limit at t=0 of the q,t-spherical function extending the symmetric Macdonald polynomials to arbitrary eigenvalues. The construction heavily depends on the technique of the q-Gaussians developed by the author (and Stokman in the non-reduced case). In this approach, the q-Whittaker function is given by a series convergent everywhere, a kind of generating function for multi-dimensional q-Hermite polynomials (closely related to the level 1 Demazure characters). One of the applications is a q-version of the Shintani- Casselman- Shalika formula, which appeared directly connected with q-Mehta- Macdonald identities in terms of the Jackson integrals. This formula generalizes that of type A due to Gerasimov et al. to arbitrary reduced root systems. At the end of the paper, we obtain a q,t-counterpart of the Harish-Chandra asymptotic formula for the spherical functions, including the Whittaker limit.

Remarks on $n$-D Polynomial Matrix Factorization Problems

Multidimensional (n-D) polynomial matrix factorizations are intimately linked to many problems of multidimensional systems and signal processing. This paper gives a new result for a n-D polynomial matrix to have an minor prime factorization using methods from computer algebra. This result may be regarded as a generalization of a previous criterion under a special restriction [IEEE Trans. Circuits Syst. II: Exp Briefs, vol. 52, no. 9 (2005)]. Examples are given to illustrate results using computer algebra software system Singular.

Two-index Clifford–Hermite polynomials with applications in wavelet analysis

Clifford analysis may be regarded as a higher-dimensional analogue of the theory of holomorphic functions in the complex plane. It has proven to be an appropriate framework for higher-dimensional continuous wavelet transforms, based on specific types of multi-dimensional orthogonal polynomials, such as the Clifford–Hermite polynomials, which form the building blocks for so-called Clifford–Hermite wavelets, offering a refinement of the traditional Marr wavelets. In this paper, a generalization of the Clifford–Hermite polynomials to a two-parameter family is obtained by taking the double monogenic extension of a modulated Gaussian, i.e. the classical Morlet wavelet. The eventual goal being the construction of new Clifford wavelets refining the Morlet wavelet, we first investigate the properties of the underlying polynomials.

Operational methods: an extension from ordinary monomials to multi-dimensional Hermite polynomials

The operational point of view combined with the principle of monomiality provides efficient tools in the theory of ordinary and generalized special functions. In this paper, we extend this approach for the purpose of deriving a wealth of relations involving multi-dimensional Hermite polynomials.

Monomiality and multi-index multi-variable special polynomials

In this paper, we will use the operational recipes from monomiality and umbral calculus to deal with multi-dimensional polynomials.

An improved algorithm for the multidimensional moment-constrained maximum entropy problem

The maximum entropy principle is a versatile tool for evaluating smooth approximations of probability density functions with the least bias beyond specified constraints. In the recent paper we introduced new computational framework for the moment-constrained maximum entropy problem in a multidimensional domain, and developed a simple numerical algorithm capable of computing maximum entropy problem in a two-dimensional domain with moment constraints of order up to 4. Here we design an improved numerical algorithm for computing the maximum entropy problem in a two- and higher-dimensional domain with higher order moment constraints. The algorithm features multidimensional orthogonal polynomial basis in the dual space of Lagrange multipliers to achieve numerical stability and rapid convergence of Newton iterations. The new algorithm is found to be capable of solving the maximum entropy problem in the two-dimensional domain with moment constraints of order up to 8, in the three-dimensional domain with moment constraints of order up to 6, and in the four-dimensional domain with moment constraints of order up to 4, corresponding to the total number of moment constraints of 44, 83 and 69, respectively. The two- and higher-dimensional maximum entropy test problems in the current work are based upon long-term statistics of numerical simulation of the real-world geophysical model for wind stress driven oceanic currents such as the Gulf Stream and the Kuroshio.

Assessment of Predictive Ability of Artificial Neural Networks Using Holographic Mapping

In this study, artificial neural networks (ANNs) were used to reveal a quantitative relationship between catalytic composition and catalytic activity. This relationship was predefined using a hypothetical experimental space described by a multidimensional polynomial. The predictive ability of ANNs was investigated, i.e. an attempt was done to evaluate how ANNs can envisage a given hypothetical experimental space. Data sets for training, validation and testing of ANNs were obtained from the hypothetical experimental space using two different ways of sampling. Data were selected, (i) by means of our optimization algorithm called Holographic Research Strategy (HRS); and (ii) randomly. In order to model real experimentation, data were also generated with error. The relationship between the complexity of different network topologies and their predictive ability was investigated. It was shown that when data used for training have been perturbed with a given level of noise, less complex network architectures give acceptable accuracy. Additionally, estimated experimental spaces were visualized in a 2D layout by means of Holographic Mappings (HMs). Analysis of HMs revealed that ANNs trained by data sets obtained upon an optimization procedure provides better description of the experimental space in the vicinity of the optimum than ANNs trained by randomly selected data sets. This fact indicates again the importance of the optimization in combinatorial catalyst library design.

High-order limit state functions in the response surface method for structural reliability analysis

The stochastic response surface method (SRSM) is a technique for the reliability analysis of complex systems with low failure probabilities, for which Monte Carlo simulation (MCS) is too computationally intensive and for which approximate methods are inaccurate. Typically, the SRSM approximates a limit state function with a multi-dimensional quadratic polynomial by fitting the polynomial to a number of sampling points from the limit state function. This method can give biased approximations of the failure probability for cases in which the quadratic response surface can not conform to the true limit state function’s nonlinearities. In contrast to recently proposed algorithms which focus on the positions of sample points to improve the accuracy of the quadratic SRSM, this paper describes the use of higher order polynomials in order to approximate the true limit state more accurately. The use of higher order polynomials has received relatively little attention to date because of problems associated with ill-conditioned systems of equations and an approximated limit state which is very inaccurate outside the domain of the sample points. To address these problems, an algorithm using orthogonal polynomials is proposed to determine the necessary polynomial orders. Four numerical examples compare the proposed algorithm with the conventional quadratic polynomial SRSM and a detailed MCS.

A new approach to obtain the non-Condon factors in closed form for two one-dimensional harmonic oscillators

A simple algebraic approach to calculate general Franck-Condon overlaps is extended to evaluate non-Condon factors for two one-dimensional harmonic oscillators. The method is based on the use of eigenstates of the harmonic oscillator annihilation operator which allows to obtain in terms of a multi-dimensional Hermite polynomial the overlap of harmonic oscillator functions associated with different Born-Oppenheimer potentials. The presented approach is self-contained, only basic concepts of quantum mechanics associated with the harmonic oscillator system are needed. The obtained expression for the Franck-Condon overlaps is similar to the Ansbacher’s formula and equivalent to the one calculated by Malkin and Man’ko. However our final expression has the advantages that only real numbers are involved and it is straightforward to get the limit case of equal frequencies. Concerning the non-Condon factors two approaches leading to different formulas are considered, both of which reduce to triple sums of products of three Hermite polynomials.

HERMITE FUNCTION EXPANSIONS VERSUS HERMITE POLYNOMIAL EXPANSIONS

We consider expansions with respect to the multi-dimensional Hermite functions and to the multi-dimensional Hermite polynomials. They are respectively eigenfunctions of the Harmonic oscillator The corresponding heat semigroups and Riesz transforms are considered and results on both aspects (polynomials and functions) are obtained.

Hybridization of stochastic reduced basis methods with polynomial chaos expansions

We propose a hybrid formulation combining stochastic reduced basis methods with polynomial chaos expansions for solving linear random algebraic equations arising from discretization of stochastic partial differential equations. Our objective is to generalize stochastic reduced basis projection schemes to non-Gaussian uncertainty models and simplify the implementation of higher-order approximations. We employ basis vectors spanning the preconditioned stochastic Krylov subspace to represent the solution process. In the present formulation, the polynomial chaos decomposition technique is used to represent the stochastic basis vectors in terms of multidimensional Hermite polynomials. The Galerkin projection scheme is then employed to compute the undetermined coefficients in the reduced basis approximation. We present numerical studies on a linear structural problem where the Youngs modulus is represented using Gaussian as well as lognormal models to illustrate the performance of the hybrid stochastic reduced basis projection scheme. Comparison studies with the spectral stochastic finite element method suggest that the proposed hybrid formulation gives results of comparable accuracy at a lower computational cost.

Multidimensional bell polynomials of higher order

We develop some extensions of the classical Bell polynomials, previously obtained, by introducing a further class of these polynomials called multidimensional Bell polynomials of higher order. They arise considering the derivatives of functions f in several variables φ ( i) , ( i = 1, 2, …, m), where φ ( i) are composite functions of different orders, i.e. φ ( i) ( t) = ( i,1) ( ( i,2) (… ( ( i,r i ) ( t))), ( i = 1, 2, …, m). We show that these new polynomials are always expressible in terms of the ordinary Bell polynomials, by means of suitable recurrence relations or formal multinomial expansions. Moreover, we give a recurrence relation for their computation.

New method for calculating the moments of thermally averaged densities of distribution of internuclear distances in polyatomic molecules using the Bloch integral equation

A convenient method is suggested for calculating thermally averaged powers of the normal vibrational coordinates Q i by iteratively solving the Bloch integral equation with an anharmonic function of potential energy using multidimensional Hermite polynomials. Analytical formulas of the first approximation regarding anharmonicity constant have been obtained for the following moments of thermally averaged density: 〈Q 1〉, 〈 Q 1 2 〉, 〈Q 1 Q 2〉, 〈Q 1 3 〉 〈Q 1 3 〉, 〈Q 1 Q 2 Q 3〉, 〈Q 1 4 〉, 〈Q 1 2 Q 2 2 〉, 〈Q 1 Q 2/3〉, 〈Q 1 Q 2 Q 3 2 〉, 〈 Q 1 Q 2 Q 3 Q 4〉.

Design of complex-coefficient variable digital filters using successive vector-array decomposition

Singular-value decomposition (SVD) can be efficiently utilized to obtain the optimal vector-array decomposition (VAD) for simplifying real-coefficient variable digital filter design problem, but the SVD-based VAD methods are not applicable to the design of complex-coefficient variable filters. This paper proposes a successive algorithm for decomposing arbitrary multidimensional complex array into the VAD form, and thus, a complex-coefficient variable digital filter with arbitrary variable frequency response can be easily obtained through constant complex-coefficient filter design and multidimensional polynomial fitting. The new VAD algorithm successively decomposes the complex array and its residual arrays into the vector-array pairs stage by stage, and each stage contains an iterative optimization that can be easily solved in a closed-form. Our computer simulations have demonstrated that the successive VAD converges very fast to the optimal solution.

Design of arbitrary-phase variable digital filters using SVD-based vector-array decomposition

This paper proposes a straightforward method for designing variable digital filters with arbitrary variable magnitude as well as arbitrary fixed-phase or variable fractional delay (VFD) responses. The basic idea is to avoid the complicated direct design of one-dimensional (1-D) variable digital filters by decomposing the original variable filter design problem into easier subproblems that only require constant 1-D filter designs and multidimensional polynomial approximations. Through constant 1-D filter designs and multidimensional polynomial fits, we can easily obtain a variable digital filter satisfying the given variable design specifications. To decompose the original variable filter design into constant 1-D filter designs and multidimensional polynomial fits, a new multidimensional complex array decomposition called vector array decomposition (VAD) is proposed, which is based on two new theorems using the singular value decomposition (SVD). Once the VAD is obtained, the subproblems can be easily solved. Furthermore, we show that the VAD can also be generalized to the weighted least squares (WLS) case (WLS-VAD) for the WLS variable filter design. Three design examples are given to illustrate that the WLS-VAD and VAD-based techniques are considerably efficient for designing variable digital filters with arbitrary variable magnitude and arbitrary fixed-phase or VFD responses.