Search Results (19)

We develop a semi-parametric framework for representing discrete probability mass functions through orthogonal polynomial representations. Classical count models, such as the Poisson and negative binomial distributions, impose restrictive structural assumptions that often fail to accommodate empirical features including heavy overdispersion, multimodality, and nonstandard tail behavior. To address these limitations, we introduce a linear-tilt model constructed from orthonormal polynomial systems associated with Poisson and negative binomial baselines, namely the Charlier and Meixner families. The proposed representation improves the baseline distribution using additional information from empirical moments. This allows the distribution to flexibly adjust its shape, capturing differences in skewness and kurtosis. We establish theoretical properties of the expansion within a weighted Hilbert space formulation, where the coefficients arise as orthogonal projections that can be expressed as expectations of the corresponding polynomial basis functions. In addition, we analyze approximation behavior and provide numerical bounds on the resulting numerical error and convergence properties of truncated approximations. The practical relevance of the proposed methodology is illustrated through applications to several empirical datasets, demonstrating its ability to capture complex distributional structures while preserving a tractable semi-parametric form. Full article

The development of efficient strategies for optimizing and controlling nonlinear bioprocesses remains a significant challenge due to their complex dynamics and sensitivity to operating conditions. This work addresses the problem by proposing a two-step methodology applied to a laboratory-scale fed-batch bioethanol process. The first step employs a dynamic optimization approach based on Fourier parameterization and orthonormal polynomials, which generates smooth and continuous substrate-feed profiles using only three parameters instead of the ten required by piecewise approaches. The second step introduces a controller formulated through basic linear algebra operations, which ensures accurate trajectory tracking of the optimized state variables. Simulation results demonstrate a $3.65\%$ increase in ethanol concentration at the end of the process, together with an accumulated tracking error of only $0.0189$ under nominal conditions. In addition, the closed-loop strategy outperforms open-loop implementation when the initial conditions deviate from their nominal values. These findings highlight that the proposed methodology reduces mathematical complexity and computational effort while producing continuous control profiles suitable for practical application. The combination of optimization and algebraic control thus provides a promising alternative for improving the efficiency of bioethanol-production processes. Full article

This paper presents an improved methodology for optimizing the fed-batch fermentation process of xylitol production, aiming to maximize the final concentration in a bioreactor co-fed with xylose and glucose. Xylitol is a valuable sugar alcohol widely used in the food and pharmaceutical industries, and its microbial production requires precise control over substrate feeding strategies. The proposed technique employs Legendre polynomials to parameterize two control actions (the feeding rates of glucose and xylose), and it uses a hybrid optimization algorithm combining Monte Carlo sampling with genetic algorithms for coefficient selection. Unlike traditional optimization approaches based on piecewise parameterization, which produce discontinuous control profiles and require post-processing, this method generates smooth profiles directly applicable to real systems. Additionally, it significantly reduces mathematical complexity compared to strategies that combine Fourier series with orthonormal polynomials while maintaining similar optimization results. The methodology achieves good results in xylitol production using only eight parameters, compared to at least twenty in other approaches. This dimensionality reduction improves the robustness of the optimization by decreasing the likelihood of convergence to local optima while also reducing the computational cost and enhancing feasibility for implementation. The results highlight the potential of this strategy as a practical and efficient tool for optimizing nonlinear multivariable bioprocesses. Full article

The Gram–Schmidt process (GSP) plays an important role in algebra. It provides a theoretical and practical approach for generating an orthonormal basis, QR decomposition, unitary matrices, etc. It also facilitates some applications in the fields of communication, machine learning, feature extraction, etc. The typical GSP is self-referential, while the non-self-referential GSP is based on the Gram determinant, which has exponential complexity. The motivation for this article is to find a way that could convert a set of linearly independent vectors ${\left\{{\overrightarrow{u}}_i\right\}}_{j=1}^n$ into a set of orthogonal vectors ${\left\{\overrightarrow{v}\right\}}_{j=1}^n$ via a non-self-referential GSP (NsrGSP). The approach we use is to derive a method that utilizes the recursive property of the standard GSP to retrieve a NsrGSP. The individual orthogonal vector form we obtain is ${\displaystyle {\overrightarrow{v}}_k=\sum _{j=1}^k{\beta }_{[k\to j]}{\overrightarrow{u}}_j}$, and the collective orthogonal vectors, in a matrix form, are $V_k=U_k\left(B_{{\Delta }_k}^{+}\right)$. This approach could reduce the exponential computational complexity to a polynomial one. It also has a neat representation. To this end, we also apply our approach on a classification problem based on real data. Our method shows the experimental results are much more persuasive than other familiar methods. Full article

This paper gives a new representation for the fractional Brownian motion that can be applied to simulate this self-similar random process in continuous time. Such a representation is based on the spectral form of mathematical description and the spectral method. The Legendre polynomials are used as the orthonormal basis. The paper contains all the necessary algorithms and their theoretical foundation, as well as the results of numerical experiments. Full article

In spectral/finite element methods, a robust and stable high-order polynomial approximation method for the solution can significantly reduce the required number of degrees of freedom (DOFs) to achieve a certain level of accuracy. In this work, a closed-form relation is proposed to approximate the Fekete points (AFPs) on arbitrary shape domains based on the singular value decomposition (SVD) of the Vandermonde matrix. In addition, a novel method is derived to compute the moments on highly complex domains, which may include discontinuities. Then, AFPs are used to generate compatible basis functions using SVD. Equations are derived and presented to determine orthogonal/orthonormal modal basis functions, as well as the Lagrange basis. Furthermore, theorems are proved to show the convergence and accuracy of the proposed method, together with an explicit form of the Weierstrass theorem for polynomial approximation. The method was implemented and some classical cases were analyzed. The results show the superior performance of the proposed method in terms of convergence and accuracy using many fewer DOFs and, thus, a much lower computational cost. It was shown that the orthogonal modal basis is the best choice to decrease the DOFs while maintaining a small Lebesgue constant when very high degree of polynomial is employed. Full article

The determination of the distributions of the eigenvalues associated with matrix-variate gamma and beta random variables of either type proves to be a challenging problem. Several of the approaches utilized so far yield unwieldy representations that, for instance, are expressed in terms of multiple integrals, functions of skew symmetric matrices, ratios of determinants, solutions of differential equations, zonal polynomials, and products of incomplete gamma or beta functions. In the present paper, representations of the density functions of the smallest, largest and $j^{\mathrm{th}}$ largest eigenvalues of matrix-variate gamma and each type of beta random variables are explicitly provided as finite sums when certain parameters are integers and, as explicit series, in the general situations. In each instance, both the real and complex cases are considered. The derivations initially involve an orthonormal or unitary transformation whereby the wedge products of the differential elements of the eigenvalues can be worked out from those of the original matrix-variate random variables. Some of these results also address the distribution of the eigenvalues of a central Wishart matrix as well as eigenvalue problems arising in connection with the analysis of variance procedure and certain tests of hypotheses in multivariate analysis. Additionally, three numerical examples are provided for illustration purposes. Full article

The purpose of this paper is to leverage the advantages of physics-informed neural network (PINN) and convolutional neural network (CNN) by using Legendre multiwavelets (LMWs) as basis functions to approximate partial differential equations (PDEs). We call this method Physics-Informed Legendre Multiwavelets CNN (PiLMWs-CNN), which can continuously approximate a grid-based state representation that can be handled by a CNN. PiLMWs-CNN enable us to train our models using only physics-informed loss functions without any precomputed training data, simultaneously providing fast and continuous solutions that generalize to previously unknown domains. In particular, the LMWs can simultaneously possess compact support, orthogonality, symmetry, high smoothness, and high approximation order. Compared to orthonormal polynomial (OP) bases, the approximation accuracy can be greatly increased and computation costs can be significantly reduced by using LMWs. We applied PiLMWs-CNN to approximate the damped wave equation, the incompressible Navier–Stokes (N-S) equation, and the two-dimensional heat conduction equation. The experimental results show that this method provides more accurate, efficient, and fast convergence with better stability when approximating the solution of PDEs. Full article

The 3D Zernike polynomials form an orthonormal basis of the unit ball. The associated 3D Zernike moments have been successfully applied for 3D shape recognition; they are popular in structural biology for comparing protein structures and properties. Many algorithms have been proposed for computing those moments, starting from a voxel-based representation or from a surface based geometric mesh of the shape. As the order of the 3D Zernike moments increases, however, those algorithms suffer from decrease in computational efficiency and more importantly from numerical accuracy. In this paper, new algorithms are proposed to compute the 3D Zernike moments of a homogeneous shape defined by an unstructured triangulation of its surface that remove those numerical inaccuracies. These algorithms rely on the analytical integration of the moments on tetrahedra defined by the surface triangles and a central point and on a set of novel recurrent relationships between the corresponding integrals. The mathematical basis and implementation details of the algorithms are presented and their numerical stability is evaluated. Full article

Orthogonal matching pursuit (OMP for short) is a classical method for sparse signal recovery in compressed sensing. In this paper, we consider the application of OMP to reconstruct sparse polynomials generated by uniformly bounded orthonormal systems, which is an extension of the work on OMP to reconstruct sparse trigonometric polynomials. Firstly, in both cases of sampled data with and without noise, sufficient conditions for OMP to recover the coefficient vector of a sparse polynomial are given, which are more loose than the existing results. Then, based on a more accurate estimation of the mutual coherence of a structured random matrix, the recovery guarantees and success probabilities for OMP to reconstruct sparse polynomials are obtained with the help of those sufficient conditions. In addition, the error estimation for the recovered coefficient vector is gained when the sampled data contain noise. Finally, the validity and correctness of the theoretical conclusions are verified by numerical experiments. Full article

Generating discrete orthogonal polynomials from the recurrence or difference equation is error-prone, as it is sensitive to error propagation and dependent on highly accurate initial values. Strategies to handle this, involving control over the deviation of norm and orthogonality, have already been developed for the discrete Chebyshev and Krawtchouk functions, i.e., the orthonormal basis in ${\ell }_2$ derived from the polynomials. Since these functions are limiting cases of the discrete Hahn functions, it suggests that the strategy could also be successful there. We outline the algorithmic strategies including the specific method of generating the initial values, and show that the orthonormal basis can indeed be generated for large supports and polynomial degrees with controlled numerical error. Special attention is devoted to symmetries, as the symmetric windows are most commonly used in signal processing, allowing for simplification of the algorithm due to this prior knowledge, and leading to savings in the required computational power. Full article

Deployment of the recurrence relation or difference equation to generate discrete classical orthogonal polynomials is vulnerable to error propagation. This issue is addressed for the case of Krawtchouk functions, i.e., the orthonormal basis derived from the Krawtchouk polynomials. An algorithm is proposed for stable determination of these functions. This is achieved by defining proper initial points for the start of the recursions, balancing the order of the direction in which recursions are executed and adaptively restricting the range over which equations are applied. The adaptation is controlled by a user-specified deviation from unit norm. The theoretical background is given, the algorithmic concept is explained and the effect of controlled accuracy is demonstrated by examples. Full article

Herein, we developed and analyzed a new fractal–fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials. We used this matrix to handle the FF Riccati differential equation with the new generalized Caputo FF derivative. Based on the developed operational matrix and the spectral Tau method, the nonlinear differential problem was reduced to a system of algebraic equations in the unknown expansion coefficients. Accordingly, the resulting system was solved by Newton’s solver with a small initial guess. The efficiency, accuracy, and applicability of the developed numerical method were checked by exhibiting various test problems. The obtained results were also compared with other recent methods, based on the available literature. Full article

In this paper, we consider a discrete Sobolev inner product involving the Jacobi weight with a twofold objective. On the one hand, since the orthonormal polynomials with respect to this inner product are eigenfunctions of a certain differential operator, we are interested in the corresponding eigenvalues, more exactly, in their asymptotic behavior. Thus, we can determine a limit value which links this asymptotic behavior and the uniform norm of the orthonormal polynomials in a logarithmic scale. This value appears in the theory of reproducing kernel Hilbert spaces. On the other hand, we tackle a more general case than the one considered in the literature previously. Full article

This paper discusses the applications of numerical inversion of the Laplace transform method based on the Bernstein operational matrix to find the solution to a class of fractional differential equations. By the use of Laplace transform, fractional differential equations are firstly converted to system of algebraic equations then the numerical inverse of a Laplace transform is adopted to find the unknown function in the equation by expanding it in a Bernstein series. The advantages and computational implications of the proposed technique are discussed and verified in some numerical examples by comparing the results with some existing methods. We have also combined our technique to the standard Laplace Adomian decomposition method for solving nonlinear fractional order differential equations. The method is given with error estimation and convergence criterion that exclude the validity of our method. Full article