Multivariate Orthogonal Polynomials Research Articles

Mini-Workshop: Multivariate Orthogonal Polynomials: New synergies with Numerical Analysis

Multivariate polynomials and, in particular, multivariate orthogonal polynomials (MOPs) are research areas within the fields of special functions, Lie groups, quantum groups, computer algebra to name only some of them. However, there are many important areas in the field of numerical analysis where multivariate polynomials (of high order) play a crucial role: approximation by spectral methods and finite elements, discrete calculus, polynomial trace liftings, exact sequence properties, sparsity, efficient and stable recursions, analysis of the geometry of the zeros. The miniworkshop brought together experts from the fields of MOPs and numerical analysis of partial differential equations.

A Stieltjes Algorithm for Generating Multivariate Orthogonal Polynomials

Orthogonal polynomials of several variables have a vector-valued three-term recurrence relation, much like the corresponding one-dimensional relation. This relation requires only knowledge of certain recurrence matrices, and allows simple and stable evaluation of multivariate orthogonal polynomials. In the univariate case, various algorithms can stably and accurately evaluate the recurrence coefficients given the ability to compute polynomial moments, but it is difficult to identify analogous procedures in multiple dimensions. We present a new Multivariate Stieltjes (MS) algorithm that fills this gap in the multivariate case, allowing computation of recurrence matrices assuming moments are available. The algorithm is essentially explicit in two and three dimensions, but requires the numerical solution to a nonconvex problem in more than three dimensions. Compared to direct Gram–Schmidt-type orthogonalization, we demonstrate on several examples in up to three dimensions that the MS algorithm is far more stable, and allows accurate computation of orthogonal bases in the multivariate setting, in contrast to direct orthogonalization approaches.

A unified approach to multivariate polynomial sequences with real stability

We give new sufficient conditions for a sequence of multivariate polynomials to be real stable. As applications, we obtain several known results, such as the real stability of multivariate Eulerian polynomials, multivariate Bell polynomials and multivariate polynomials over Stirling permutations, in a unified manner. And we also show some new results, such as the real stability of multivariate Lah polynomials and multivariate polynomials over Jacobi-Stirling permutations, and the proper position property of multivariate orthogonal polynomials and multivariate matching polynomials. Finally, we obtain a multivariate generalization of Fisk's result by presenting a 2×2 matrix, which preserves the proper position property.

On multivariate orthogonal polynomials and elementary symmetric functions

We study families of multivariate orthogonal polynomials with respect to the symmetric weight function in d variables Bγ(x)=∏i=1dω(xi)∏i<j|xi-xj|2γ+1,x∈(a,b)d,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} B_{\\gamma }(\\mathtt {x}) = \\prod \\limits _{i=1}^{d} \\omega (x_{i}) \\prod \\limits _{i<j} |x_{i}-x_{j}|^{2\\gamma +1}, \\quad \\mathtt {x}\\in (a,b)^{d}, \\end{aligned}$$\\end{document}for gamma >-1, where omega (t) is an univariate weight function in t in (a,b) and mathtt {x} = (x_{1},x_{2}, ldots , x_{d}) with x_{i} in (a,b). Applying the change of variables x_{i},i=1,2,ldots ,d, into u_{r},r=1,2,ldots ,d, where u_{r} is the r-th elementary symmetric function, we obtain the domain region in terms of the discriminant of the polynomials having x_{i},i=1,2,ldots ,d, as its zeros and in terms of the corresponding Sturm sequence. Choosing the univariate weight function as the Hermite, Laguerre, and Jacobi weight functions, we obtain the representation in terms of the variables u_{r} for the partial differential operators such that the respective Hermite, Laguerre, and Jacobi generalized multivariate orthogonal polynomials are the eigenfunctions. Finally, we present explicitly the partial differential operators for Hermite, Laguerre, and Jacobi generalized polynomials, for d=2 and d=3 variables.

Fourier Transform of the Orthogonal Polynomials on the Unit Ball and Continuous Hahn Polynomials

Some systems of univariate orthogonal polynomials can be mapped into other families by the Fourier transform. The most-studied example is related to the Hermite functions, which are eigenfunctions of the Fourier transform. For the multivariate case, by using the Fourier transform and Parseval’s identity, very recently, some examples of orthogonal systems of this type have been introduced and orthogonality relations have been discussed. In the present paper, this method is applied for multivariate orthogonal polynomials on the unit ball. The Fourier transform of these orthogonal polynomials on the unit ball is obtained. By Parseval’s identity, a new family of multivariate orthogonal functions is introduced. The results are expressed in terms of the continuous Hahn polynomials.

Three term relations for multivariate Uvarov orthogonal polynomials

Three term relations for orthogonal polynomials in several variables associated to a moment linear functional obtained by a Uvarov modification of a given moment functional are studied. Existence of Uvarov orthogonal polynomials is analyzed, stating conditions to ensure it. The matrices of the three term relations of the Uvarov orthogonal polynomials are explicitly given in terms of the matrices of the three term relations satisfied by the original family. Two examples are presented in order to show that the results are valid for positive definite linear functionals and also for some quasi definite linear functionals which are not positive definite.

Fourier Transforms of Some Special Functions in Terms of Orthogonal Polynomials on the Simplex and Continuous Hahn Polynomials

In this paper, Fourier transform of multivariate orthogonal polynomials on the simplex is presented. A new family of multivariate orthogonal functions is obtained using the Parseval’s identity and several recurrence relations are derived.

Multivariate Orthogonal Laurent Polynomials and Integrable Systems

An ordering for Laurent polynomials in the algebraic torus (\mathbb{C}^*)^D , inspired by the Cantero–Moral–Vel´azquez approach to orthogonal Laurent polynomials in the unit circle, leads to the construction of a moment matrix for a given Borel measure in the unit torus \mathbb{T}^D . The Gauss–Borel factorization of this moment matrix allows for the construction of multivariate biorthogonal Laurent polynomials in the unit torus, which can be expressed as last quasi-determinants of bordered truncations of the moment matrix. The associated second-kind functions are expressed in terms of the Fourier series of the given measure. Persymmetries and partial persymmetries of the moment matrix are studied and Cauchy integral representations of the second-kind functions are found, as well as Plemelj-type formulae. Spectral matrices give string equations for the moment matrix, which model the three-term relations as well as the Christoffel–Darboux formulae. Christoffel-type perturbations of the measure given by the multiplication by Laurent polynomials are studied. Sample matrices on poised sets of nodes, which belong to the algebraic hypersurface of the perturbing Laurent polynomial, are used to find a Christoffel formula that expresses the perturbed orthogonal Laurent polynomials in terms of a last quasi-determinant of a bordered sample matrix constructed in terms of the original orthogonal Laurent polynomials. Poised sets exist only for prepared Laurent polynomials, which are analyzed from the perspective of Newton polytopes and tropical geometry. Then, an algebraic geometrical characterization of prepared Laurent polynomial perturbation and poised sets is given; full-column-rankness of the corresponding multivariate Laurent–Vandermonde matrices and a product of different prime prepared Laurent polynomials leads to such sets. Some examples are constructed in terms of perturbations of the Lebesgue–Haar measure. Discrete and continuous deformations of the measure lead to a Toda-type integrable hierarchy, being the corresponding flows described through Lax and Zakharov–Shabat equations; bilinear equations and vertex operators are found. Varying size matrix nonlinear partial difference and differential equations of two-dimensional Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows are connected with a Gauss–Borel factorization of the Jacobi-type matrices and its quasi-determinants allow for expressions for the multivariate orthogonal polynomials in terms of shifted quasi-tau matrices, which generalize those that relate the Baker functions with ratios of Miwa shifted \tau -functions in the one-dimensional scenario. It is shown that the discrete and continuous flows are deeply connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomtsev–Petviashvili-type system.

Quasi-birth-and-death processes and multivariate orthogonal polynomials

The aim of this paper is to study some models of quasi-birth-and-death (QBD) processes arising from the theory of bivariate orthogonal polynomials. First we will see how to perform the spectral analysis in the general setting as well as to obtain results about recurrence and the invariant measure of these processes in terms of the spectral measure supported on some domain Ω⊂Rd. Afterwards, we will apply our results to several examples of bivariate orthogonal polynomials, namely product orthogonal polynomials, orthogonal polynomials on a parabolic domain and orthogonal polynomials on the triangle. We will focus on linear combinations of the Jacobi matrices generated by these polynomials and produce families of either continuous or discrete-time QBD processes. Finally, we show some urn models associated with these QBD processes.

Sparse spectral methods for partial differential equations on spherical caps

Abstract In recent years, sparse spectral methods for solving partial differential equations have been derived using hierarchies of classical orthogonal polynomials (OPs) on intervals, disks, disk-slices and triangles. In this work, we extend the methodology to a hierarchy of non-classical multivariate OPs on spherical caps. The entries of discretizations of partial differential operators can be effectively computed using formulae in terms of (non-classical) univariate OPs. We demonstrate the results on partial differential equations involving the spherical Laplacian and biharmonic operators, showing spectral convergence with discretizations that can be made well conditioned using a simple preconditioner.

Analyzing genomic data using tensor-based orthogonal polynomials with application to synthetic RNAs.

An important goal in molecular biology is to quantify both the patterns across a genomic sequence and the relationship between phenotype and underlying sequence. We propose a multivariate tensor-based orthogonal polynomial approach to characterize nucleotides or amino acids in a given sequence and map corresponding phenotypes onto the sequence space. We have applied this method to a previously published case of small transcription activating RNAs. Covariance patterns along the sequence showcased strong correlations between nucleotides at the ends of the sequence. However, when the phenotype is projected onto the sequence space, this pattern does not emerge. When doing second order analysis and quantifying the functional relationship between the phenotype and pairs of sites along the sequence, we identified sites with high regressions spread across the sequence, indicating potential intramolecular binding. In addition to quantifying interactions between different parts of a sequence, the method quantifies sequence–phenotype interactions at first and higher order levels. We discuss the strengths and constraints of the method and compare it to computational methods such as machine learning approaches. An accompanying command line tool to compute these polynomials is provided. We show proof of concept of this approach and demonstrate its potential application to other biological systems.

Bivariate continuous q-Hermite polynomials and deformed quantum Serre relations

To Nicolás Andruskiewitsch on his 60th birthday, with admiration We introduce bivariate versions of the continuous [Formula: see text]-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence relations) and analytic properties (determining the orthogonality measure). We find a direct link between bivariate continuous [Formula: see text]-Hermite polynomials and the star product method of [S. Kolb and M. Yakimov, Symmetric pairs for Nichols algebras of diagonal type via star products, Adv. Math. 365 (2020), Article ID: 107042, 69 pp.] for quantum symmetric pairs to establish deformed quantum Serre relations for quasi-split quantum symmetric pairs of Kac–Moody type. We prove that these defining relations are obtained from the usual quantum Serre relations by replacing all monomials by multivariate orthogonal polynomials.

A data-driven polynomial chaos method considering correlated random variables

Variable correlation commonly exists in practical engineering applications. However, most of the existing polynomial chaos (PC) approaches for uncertainty propagation (UP) assume that the input random variables are independent. To address variable correlation, an intrusive PC method has been developed for dynamic system, which however is not applicable to problems with black-box-type functions. Therefore, based on the existing data-driven PC method, a new non-intrusive data-driven polynomial chaos approach that can directly consider variable correlation for UP of black-box computationally expensive problems is developed in this paper. With the proposed method, the multivariate orthogonal polynomial basis corresponding to the correlated input random variables is conveniently constructed by solving the moment-matching equations based on the correlation statistical moments to consider the variable correlation. A comprehensive comparative study on several numerical examples of UP and design optimization under uncertainty with correlated input random variables is conducted to verify the effectiveness and advantage of the proposed method. The results show that the proposed method is more accurate than the existing data-driven PC method with Nataf transformation when the variable distribution is known, and it can produce accurate results with unknown variable distribution, demonstrating its effectiveness.

Quantitative study on SLIP model parameters based on multi-rigid-body dynamics

The spring-loaded inverted pendulum (SLIP) model is often used to describe the interaction between hindlimbs and the ground during the locomotion of animals or humans. This model is frequently adopted to qualitatively explain the flexible deformation of legs and feet and the trajectory change of center of mass (COM), caused by the impact between hindlimbs and the ground. However, such research cannot provide precise reference on the structural parameters, spring and damper selections and their collocations for the design of robotic legs and feet. In this study, an SLIP model was established on the multi-rigid-body dynamics software Adams. The main influence factors on the SLIP model were determined by targeting at the touchdown-phase duration of animal or human locomotion. Then the main factors were quantitatively studied by combining a multivariate orthogonal polynomial regression design. Simulation showed that spring stiffness coefficient (Z1), damping coefficient (Z2), swing angular velocity (Z3) and initial swing position (Z4) were the main factors affecting the trajectory of COM. Multivariate orthogonal polynomial regression analyses showed the relationship between the COM fluctuation (y) and the main factors satisfied the following equation: y = 217.33 - 16.25 Z1 + 0.3975 Z12 - 194.6 Z2 - 0.953 Z3 + 0.755 Z4 + 8.1 Z1 Z2 + 0.043 Z1 Z3 + 0.6 Z2 Z3 - 0.0055 Z3 Z4.

Image recognition using new set of separable three-dimensional discrete orthogonal moment invariants

In this paper, we propose new sets of 3D separable discrete orthogonal moment invariants, named Racah-Tchebichef-Krawtchouk Moment Invariants (RTKMI), Racah-Krawtchouk-Krawtchouk Moment Invariants (RKKMI) and Racah-Racah-Kr-awtchouk Moment Invariants (RRKMI), for 3D image recognition. The basis functions of these new sets of moment invariants are represented by multivariate discrete orthogonal polynomials. We also present theoretical framework to derive their Rotation, Scaling and Translation (RST) invariants based on the 3D geometric moment invariants. Accordingly, the performance of these proposed separable moment invariants is evaluated under heterogeneous databases and through several appropriate experiments, including 3D image invariance against geometric deformations, local feature extraction, computation time and recognition accuracy, in comparison with the traditional moment invariants. The obtained results showed that our proposed separable moment invariant are very efficient in terms of object recognition, numerical stability and local feature extraction, and can be highly useful for computer vision applications.

Multivariate orthogonal polynomials: Quantum decomposition, deficiency rank and support of measure

In this paper we investigate the multivariate orthogonal polynomials based on the theory of interacting Fock spaces. Our framework is on the same stream line of the recent paper by Accardi, Barhoumi, and Dhahri [1]. The (classical) coordinate variables are decomposed into non-commuting (quantum) operators called creation, annihilation, and preservation operators, in the interacting Fock spaces. Getting the commutation relations, which follow from the commuting property of the coordinate variables between themselves, we can develop the reconstruction theory of the measure, namely the Favard's theorem. We then further develop some related problems including the marginal distributions and the rank theory of the Jacobi operators. We will see that the deficiency rank of the Jacobi operator implies that the underlying measure is supported on some algebraic surface and vice versa. We will provide with some examples.

Dimensionwise multivariate orthogonal polynomials in general probability spaces

This paper puts forward a new generalized polynomial dimensional decomposition (PDD), referred to as GPDD, encompassing hierarchically organized, measure-consistent multivariate orthogonal polynomials in dependent random variables. In contrast to the existing PDD, which is valid strictly for independent random variables, no tensor-product structure or product-type probability measure is imposed or necessary. Fundamental mathematical properties of GPDD are examined by creating dimensionwise decomposition of polynomial spaces, deriving statistical properties of random orthogonal polynomials, demonstrating completeness of orthogonal polynomials for requisite assumptions, and upholding mean-square convergence to the correct limit. The GPDD approximation proposed should be effective in solving high-dimensional stochastic problems subject to dependent variables with a large class of non-product-type probability measures.

Multivariate orthogonal polynomial-based positioning error modeling and active compensation of dual-driven feed system

The nonsynchronous error of the dual-driven feed system seriously affects the positioning and machining accuracy of CNC machine tools. Improving the positioning error is an extremely important task as it determines the geometric accuracy of the manufactured parts. In this paper, a Multivariate Orthogonal Polynomial Regression model has been developed to predict the positioning error in terms of motion parameters such as position and speed of the X1 and X2 axes of the dual-driven worktable. Orthogonal Experimental Design has been engaged in conducting experiments. The positioning errors are measured by a dual-frequency laser interferometer. In addition, a real-time error compensation system is developed based on the proposed active compensation control strategy in the dual-driven feed system controlled by Beckhoff motion control card. Experimental results show that the proposed positioning error model and error compensation strategy can be utilized as an effective manner to improve the accuracy of dual-driven CNC machine tools.

Asymptotic Behaviour of the Christoffel Functions on the Unit Ball in the Presence of a Mass on the Sphere

We present a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which includes a mass uniformly distributed on the sphere. First, connection formulas relating these multivariate orthogonal polynomials and the classical ball polynomials are obtained. Then, using the representation formula for these polynomials in terms of spherical harmonics analytic properties will be deduced. Finally, we analyse the asymptotic behaviour of the Christoffel functions.

Uncertainty quantification under dependent random variables by a generalized polynomial dimensional decomposition

This paper is concerned with uncertainty quantification analysis of complex systems subject to dependent input random variables. The analysis focuses on a new, generalized version of polynomial dimensional decomposition (PDD), referred to as GPDD, entailing hierarchically ordered measure-consistent multivariate orthogonal polynomials in dependent variables. Under a few prescribed assumptions, GPDD exists for any square-integrable output random variable and converges in mean-square to the correct limit. New analytical formulae are proposed to calculate the mean and variance of a GPDD approximation of a general output variable in terms of the expansion coefficients and second-moment properties of multivariate orthogonal polynomials. However, unlike in PDD, calculating the coefficients of GPDD requires solving a coupled system of linear equations. Besides, the variance formula of GPDD contains extra terms due to statistical dependence among input variables. The extra terms disappear when the input variables are statistically independent, reverting GPDD to PDD. Two numerical examples, the one derived from a stochastic boundary-value problem and the other entailing a random eigenvalue problem, illustrate second-moment error analysis and estimation of the probabilistic characteristics of eigensolutions.