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Mathematics
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New Trends in Applications of Orthogonal Polynomials and Special Functions
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New Trends in Applications of Orthogonal Polynomials and Special Functions

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 January 2016) | Viewed by 21870

Special Issue Editors

Dr. Zhongqiang Zhang

E-Mail Website
Guest Editor
Department of Mathematical Sciences - Worcester Polytechnic Institute, Worcester, MA 01609, USA
Interests: numerical analysis; scientific computing; spectral methods
Dr. Mohsen Zayernouri

E-Mail Website
Guest Editor
Department of Mechanical Engineering and Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USA
Interests: numerical analysis; uncertainty quantification and statistical learning; fractional LES turbulence modeling; anomalous transport; multiscale material failure modeling

Special Issue Information

Dear Colleagues,

Orthogonal polynomials and special functions play an important role in developing numerical and analytical methods in mathematics, physics, and engineering. Over the past decades, this area of research has received an ever-increasing attention and has gained a growing momentum in modern topics, such as computational probability, numerical analysis, computational fluid dynamics, data assimilation, statistics, image and signal processing etc.

Orthogonal polynomials are crucial to the stability of high-order numerical methods, such as hp/spectral-element methods for ordinary and partial differential equations and fast Fourier or wavelet transformations in signal processing. These high-order numerical methods, originally formulated for partial differential equations, have been extended to integral, integro-differential equations, stochastic differential equations, and yet, these methods have not been well understood in various fields. The study of orthogonal polynomials and corresponding numerical methods helps us deepen our understanding of  these more general mathematical models that can capture non-Gaussian, non-Markovian, and non-Newtonian phenomenon.

The purpose of this special issue is to report and review the recent developments in applications of orthogonal polynomials and special functions as numerical and analytical methods. This special issue of Mathematics will contain contributions from leading experts in areas ranging from mathematical modeling, high-order numerical methods for differential, integral and integro-differential equations, stochastic differential equations, statistics, information and communication sciences and beyond.

The guest editors aim that the papers in this special issue will help understand the state-of-art high-order methods for models of complex systems and boost in-depth insights and discussion in a wide research community of related topics.

Dr. Zhongqiang Zhang
Dr. Mohsen Zayernouri
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • orthogonal functions,
  • nonlocal problems,
  • integral transforms,
  • fractional differential equations,
  • high-order numerical methods,
  • stochastic dynamics

Published Papers (5 papers)

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Research

5596 KiB  
Article
An Adaptive WENO Collocation Method for Differential Equations with Random Coefficients
by Wei Guo, Guang Lin, Andrew J. Christlieb and Jingmei Qiu
Mathematics 2016, 4(2), 29; https://doi.org/10.3390/math4020029 - 03 May 2016
Cited by 3 | Viewed by 4435
Abstract
The stochastic collocation method for solving differential equations with random inputs has gained lots of popularity in many applications, since such a scheme exhibits exponential convergence with smooth solutions in the random space. However, in some circumstance the solutions do not fulfill the [...] Read more.
The stochastic collocation method for solving differential equations with random inputs has gained lots of popularity in many applications, since such a scheme exhibits exponential convergence with smooth solutions in the random space. However, in some circumstance the solutions do not fulfill the smoothness requirement; thus a direct application of the method will cause poor performance and slow convergence rate due to the well known Gibbs phenomenon. To address the issue, we propose an adaptive high-order multi-element stochastic collocation scheme by incorporating a WENO (Weighted Essentially non-oscillatory) interpolation procedure and an adaptive mesh refinement (AMR) strategy. The proposed multi-element stochastic collocation scheme requires only repetitive runs of an existing deterministic solver at each interpolation point, similar to the Monte Carlo method. Furthermore, the scheme takes advantage of robustness and the high-order nature of the WENO interpolation procedure, and efficacy and efficiency of the AMR strategy. When the proposed scheme is applied to stochastic problems with non-smooth solutions, the Gibbs phenomenon is mitigated by the WENO methodology in the random space, and the errors around discontinuities in the stochastic space are significantly reduced by the AMR strategy. The numerical experiments for some benchmark stochastic problems, such as the Kraichnan-Orszag problem and Burgers’ equation with random initial conditions, demonstrate the reliability, efficiency and efficacy of the proposed scheme. Full article
(This article belongs to the Special Issue New Trends in Applications of Orthogonal Polynomials and Special Functions)
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11129 KiB  
Article
POD-Based Constrained Sensor Placement and Field Reconstruction from Noisy Wind Measurements: A Perturbation Study
by Zhongqiang Zhang, Xiu Yang and Guang Lin
Mathematics 2016, 4(2), 26; https://doi.org/10.3390/math4020026 - 14 Apr 2016
Cited by 12 | Viewed by 4414
Abstract
It is shown in literature that sensor placement at the extrema of Proper Orthogonal Decomposition (POD) modes is efficient and leads to accurate reconstruction of the field of quantity of interest (velocity, pressure, salinity, etc.) from a limited number of measurements in [...] Read more.
It is shown in literature that sensor placement at the extrema of Proper Orthogonal Decomposition (POD) modes is efficient and leads to accurate reconstruction of the field of quantity of interest (velocity, pressure, salinity, etc.) from a limited number of measurements in the oceanography study. In this paper, we extend this approach of sensor placement and take into account measurement errors and detect possible malfunctioning sensors. We use the 24 hourly spatial wind field simulation data sets simulated using the Weather Research and Forecasting (WRF) model applied to the Maine Bay to evaluate the performances of our methods. Specifically, we use an exclusion disk strategy to distribute sensors when the extrema of POD modes are close. We demonstrate that this strategy can improve the accuracy of the reconstruction of the velocity field. It is also capable of reducing the standard deviation of the reconstruction from noisy measurements. Moreover, by a cross-validation technique, we successfully locate the malfunctioning sensors. Full article
(This article belongs to the Special Issue New Trends in Applications of Orthogonal Polynomials and Special Functions)
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225 KiB  
Article
Recurrence Relations for Orthogonal Polynomials on Triangular Domains
by Abedallah Rababah
Mathematics 2016, 4(2), 25; https://doi.org/10.3390/math4020025 - 12 Apr 2016
Viewed by 3716
Abstract
In Farouki et al, 2003, Legendre-weighted orthogonal polynomials P n , r ( u , v , w ) , r = 0 , 1 , … , n , n ≥ 0 on the triangular domain [...] Read more.
In Farouki et al, 2003, Legendre-weighted orthogonal polynomials P n , r ( u , v , w ) , r = 0 , 1 , … , n , n ≥ 0 on the triangular domain T = { ( u , v , w ) : u , v , w ≥ 0 , u + v + w = 1 } are constructed, where u , v , w are the barycentric coordinates. Unfortunately, evaluating the explicit formulas requires many operations and is not very practical from an algorithmic point of view. Hence, there is a need for a more efficient alternative. A very convenient method for computing orthogonal polynomials is based on recurrence relations. Such recurrence relations are described in this paper for the triangular orthogonal polynomials, providing a simple and fast algorithm for their evaluation. Full article
(This article belongs to the Special Issue New Trends in Applications of Orthogonal Polynomials and Special Functions)
592 KiB  
Article
Pointwise Reconstruction of Wave Functions from Their Moments through Weighted Polynomial Expansions: An Alternative Global-Local Quantization Procedure
by Carlos R. Handy, Daniel Vrinceanu, Carl B. Marth and Harold A. Brooks
Mathematics 2015, 3(4), 1045-1068; https://doi.org/10.3390/math3041045 - 05 Nov 2015
Cited by 2 | Viewed by 3779
Abstract
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, [...] Read more.
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. Full article
(This article belongs to the Special Issue New Trends in Applications of Orthogonal Polynomials and Special Functions)
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216 KiB  
Article
The Spectral Connection Matrix for Any Change of Basis within the Classical Real Orthogonal Polynomials
by Tom Bella and Jenna Reis
Mathematics 2015, 3(2), 382-397; https://doi.org/10.3390/math3020382 - 14 May 2015
Cited by 3 | Viewed by 4001
Abstract
The connection problem for orthogonal polynomials is, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. Expansions in terms of orthogonal polynomials are very common in many [...] Read more.
The connection problem for orthogonal polynomials is, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. Expansions in terms of orthogonal polynomials are very common in many applications. While the connection problem may be solved by directly computing the change–of–basis matrix, this approach is computationally expensive. A recent approach to solving the connection problem involves the use of the spectral connection matrix, which is a matrix whose eigenvector matrix is the desired change–of–basis matrix. In Bella and Reis (2014), it is shown that for the connection problem between any two different classical real orthogonal polynomials of the Hermite, Laguerre, and Gegenbauer families, the related spectral connection matrix has quasiseparable structure. This result is limited to the case where both the source and target families are one of the Hermite, Laguerre, or Gegenbauer families, which are each defined by at most a single parameter. In particular, this excludes the large and common class of Jacobi polynomials, defined by two parameters, both as a source and as a target family. In this paper, we continue the study of the spectral connection matrix for connections between real orthogonal polynomial families. In particular, for the connection problem between any two families of the Hermite, Laguerre, or Jacobi type (including Chebyshev, Legendre, and Gegenbauer), we prove that the spectral connection matrix has quasiseparable structure. In addition, our results also show the quasiseparable structure of the spectral connection matrix from the Bessel polynomials, which are orthogonal on the unit circle, to any of the Hermite, Laguerre, and Jacobi types. Additionally, the generators of the spectral connection matrix are provided explicitly for each of these cases, allowing a fast algorithm to be implemented following that in Bella and Reis (2014). Full article
(This article belongs to the Special Issue New Trends in Applications of Orthogonal Polynomials and Special Functions)
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