Gram-Schmidt Orthogonalization Process Research Articles

A new approach for Solving a nonlinear system of second-order BVPs

In this paper, we introduce a new approach based on the Reproducing Kernel Method (RKM) for solving a nonlinear system of second-order Boundary Value Problems (BVPs) without the Gram-Schmidt orthogonalization process. What motivates us to use the RKM without the Gram-Schmidt orthogonalization process is its easy implementation, elimination of the Gram-Schmidt process, fewer calculations, and high accuracy. Finally, the compatibility of numerical results and theorems demonstrates that the Present method is effective.

Analytical solutions for the free vibration of cross-ply composite laminated plates with arbitrary biaxial symmetric geometry

This paper presents an analytical model for the free vibration of composite laminated plates with arbitrary biaxial symmetric geometry under elastic boundary conditions, applicable to cross-ply fiber-reinforced composites. In this model, the domain segmentation integral method proposed by the authors is extended to the first-order shear deformation theory (FSDT) and the laminated plate theory. The energy integrals pertaining to the FSDT are analytically simplified by segmenting the integration domains and synthesizing orthogonal polynomial displacement functions over the plate domain coordinate intervals. This synthesis is achieved through the amalgamation of the penalty function method and the Gram-Schmidt orthogonalization process. Then, using the Rayleigh-Ritz procedure, a series solution for the free vibration problem of cross-ply fiber-reinforced composite laminated plates is obtained. The model provides an analytical mathematical relationship between the profile curve of composite laminated plates with arbitrary biaxial symmetric geometry and the strain energy, kinetic energy, and boundary potential energy during vibrational processes. The correctness and applicability of the method are validated by comparing the obtained results with those from the literatures. New results for laminated plates, such as hexagonal and astroid-shaped plates, were presented as references for future research. Taking a track shaped plate as an example, the effect of the number of layers on the vibration characteristics of cross-ply fiber-reinforced composite laminated plates with a certain thickness is studied.

Intuitionistic Fuzzy Gram-Schmidt Orthogonalized Artificial Neural Network for Solving MAGDM Problems

Objectives: To propose a suitable decision-making model based on Intuitionistic Fuzzy sets (IFSs) and Gram-Schmidt orthogonalization process for Artificial Neural Network (ANN). Methods: The IFS data sets appearing in the form of matrices are aggregated using the available aggregation operators in the literature and then the collective aggregated information is processed through Gram-Schmidt orthogonalization for the revised input vectors which is then fed into the ANN algorithm following Delta Learning Rule for the next phase. The weight updation is performed through the ANN and the output is improvised. Findings: The proposed Gram-Scmidt Orthogonalization process is utilized in Intuitionistic Fuzzy Artificial Neural Network model. The Delta learning rule is utilized in the process of the Neural Network, where the Intuitionistic Fuzzy nature of the input data is transformed into a fuzzy data and then the ranking of the alternatives is done based on the weights updation through the learning phase of the ANN. Once the vector is trained out of the learning phase, it is then processed through the activation function for the final selection of the best alternative required of the Multiple Attribute Group Decision Making (MAGDM) problem posed in this work. To demonstrate the usefulness and applicability of this new method with the Gram-Schmidt process, the numerical example also adds more insight to the proposed methodology of ANN with the application of some Linear Space techniques. Novelty: Most of the research done on Intuitionistic Fuzzy Artificial Neural Network model are based on learning rules or using some other calculations. The proposed Gram-Schmidt Orthogonalization process is used to find the orthogonal basis that are used as input training vectors in the Delta learning rule for ANN. Keywords: MAGDM, ANN, Aggregation operators, Learning Rules, Intuitionistic Fuzzy sets, Gram-Scmidt Orthogonalization

An anisotropic hypoplastic model for clays considering the non-coaxial behavior

Previous experimental tests on clays have confirmed the non-coincidence of principal strain increments with principal stress axes even though the stress rate is colinear with the current stress. To simulate the non-coaxial behavior of saturated clay subjected to monotonic proportional or non-proportional loading with constant principal stress rate directions, an anisotropic hypoplastic model is presented in this paper. The model is proposed by incorporating an improved anisotropic asymptotic state boundary surface (ASBS) and a non-coaxial asymptotic strain rate direction into the original explicit-ASBS hypoplastic model. The non-coaxial flow results from a non-coaxial stress rate defined by a Gram-Schmidt orthogonalization process based on a reference stress tensor. The capability of the proposed model is demonstrated by simulating the test data performed in hollow cylindrical apparatus (HCA) on Wenzhou clay and Shanghai clay with different drainage conditions and initial consolidation states. In addition, a series of numerical stress probing tests have been conducted to gain further insights into the properties of the proposed model.

Failure Probability Prediction for Offshore Floating Structures Using Machine Learning

Summary Accurately estimating the failure probability is crucial in designing civil infrastructure systems, such as floating offshore platforms for oil and gas processing/production, to ensure their safe operation throughout their service periods. However, as a system becomes complex, the evaluation of a limit state function may involve the use of an external computer solver, resulting in a significant computational burden to perform Monte Carlo simulations (MCS). Moreover, the high-dimensionality of the limit state function may limit efficient sampling of input variables due to the “curse of dimensionality.” To address these issues, an efficient machine learning framework is proposed, combining polynomial chaos expansion (PCE) and active subspace. This will enable the accurate and efficient evaluation of the failure probability of an offshore structure, which typically involves a large number of uncertain parameters. Unlike conventional PCE schemes that use the original random variable space or the auxiliary variable space for building a surrogate model, the proposed method utilizes a reduced-dimension space to circumvent the “curse of dimensionality.” An appropriate coordinate transformation is first sought so that most of the variability of a limit state function can be accounted for. Next, a PCE surrogate limit state function is constructed on the derived low-dimensional “active subspace.” The Gram-Schmidt orthogonalization process is used for making basis polynomial functions, which is particularly effective when input random parameters do not follow the Askey scheme and/or when a dependence structure between the input parameters exists. Therefore, a nonlinear iso-probabilistic transformation, which makes the convergence of a surrogate to the true model difficult, is not required, unlike traditional PCE. Numerical examples, including limit state functions related to structural dynamics problems, are presented to illustrate the advantages of the proposed method in estimating failure probabilities for complex structural systems. Specifically, the method exhibits significantly improved efficiency in estimating the failure probability of an offshore floating structure without compromising accuracy as compared to traditional PCE and MCS.

The Impact of Compilation Flags and Choosing Single- or Double-Precision Variables in Linear Systems Solvers

This paper intends to show the impact of compiler optimization flags and the variable's precision on direct methods to solve linear systems. The chosen six methods are simple direct methods, so our work could be a study for new researchers in this field. The methods are LU decomposition, LDU decomposition, Gaussian Elimination, Gauss-Jordan Elimination, Cholesky decomposition, and QR decomposition using the Gram-Schmidt orthogonalization process. Our study showed a huge difference in time between single- and double-precision in all methods, but the error encountered in the single-precision was not so high. Also, the best flags to these methods were the `-O3' and the `-Ofast' ones.

An analytical method for vibration analysis of arbitrarily shaped non-homogeneous orthotropic plates of variable thickness resting on Winkler-Pasternak foundation

A highly applicable analytical approach is proposed for the dynamic analysis of composite plate structures with complex variation of geometric characteristics (shape/boundary and thickness) and material characteristics (Young's modulus, shear modulus and density) resting on Winkler-Pasternak foundation with general elastic boundary conditions. Boundary conditions of the plates are treated by the penalty method, so that the admissible functions can use any complete set of orthogonal polynomials, which is obtained by Gram-Schmidt orthogonalization process. The interval where the in-plane coordinate variables of the plate domain are located can be selected as the orthogonalization intervals, so as to avoid the normalization procedure of coordinates and simplify the operation process. In order to calculate the energy integral of the arbitrarily shaped plate, the plate domain segmentation strategy is introduced. To verify the effectiveness and accuracy of the proposed method, convergence study and numerical verifications for different orthogonalization intervals, weight functions, penalty spring stiffness and the truncation number of orthogonal polynomials are carried out. Taking regular hexagonal plates as examples, the effects of the parameters of thickness, nonhomogeneity and foundations on the vibration characteristics of the plates are studied. These results can be used as reference data for future research.

Towards a universal and privacy preserving EEG-based authentication system

EEG-based authentication has gained much interest in recent years. However, despite its growing appeal, there are still various challenges to their practical use, such as lack of universality, lack of privacy-preserving, and lack of ease of use. In this paper, we have tried to provide a model for EEG-based authentication by focusing on these three challenges. The proposed method, employing deep learning methods, can capture the fingerprint of the users’ EEG signals for authentication aim. It is capable of verifying any claimed identity just by having a genuine EEG fingerprint and taking a new EEG sample of the user who has claimed the identity, even those who were not observed during the training. The role of the fingerprint function is similar to the hash functions in password-based authentication and it helps preserve the user’s privacy by storing the fingerprint, rather than the raw EEG signals. Moreover, for targeting the lack of ease of use challenge, Gram-Schmidt orthogonalization process reduces the required number of channels to just three ones. The experiments show that the proposed method can reach around 98% accuracy in the authentication of completely new users with only three channels of Oz, T7, and Cz.

Combining the reproducing kernel method with a practical technique to solve the system of nonlinear singularly perturbed boundary value problems

In this paper, a reliable new scheme is presented based on combining Reproducing Kernel Method (RKM) with a practical technique for the nonlinear problem to solve the System of Singularly Perturbed Boundary Value Problems (SSPBVP). The Gram-Schmidt orthogonalization process is removed in the present RKM. However, we provide error estimation for the approximate solution and its derivative. Based on the present algorithm in this paper, can also solve linear problem. Several numerical examples demonstrate that the present algorithm does have higher precision.

A Signal Processing Framework for Agile RF Beamforming: From RF-Chain-Free to Hybrid Beamformers

In conventional hybrid beamforming approaches, the number of radio-frequency (RF) chains is the bottleneck on the achievable spatial multiplexing gain. Recent studies have overcome this limitation by increasing the update-rate of the RF beamformer. This paper presents a framework to design and evaluate such approaches, which we refer to as agile RF beamforming, from theoretical and practical points of view. In this context, we consider the impact of the number of RF-chains, phase shifters' speed, and resolution to design agile RF beamformers. Our analysis and simulations indicate that even an RF-chain-free transmitter, which its beamformer has no RF-chains, can provide a promising performance compared with fully-digital systems and significantly outperform the conventional hybrid beamformers. Then, we show that the phase shifter's limited switching speed can result in signal aliasing, in-band distortion, and out-of-band emissions. We introduce performance metrics and approaches to measure such effects and compare the performance of the proposed agile beamformers using the Gram-Schmidt orthogonalization process. Although this paper aims to present a generic framework for deploying agile RF beamformers, it also presents extensive performance evaluations in communication systems in terms of adjacent channel leakage ratio, sum-rate, power efficiency, error vector magnitude, and bit-error rates.

Reproducing kernel method to solve fractional delay differential equations

This paper is devoted to the numerical scheme for Fractional Delay Differential Equations (FDDEs). We use a semi-analytical method as Reproducing kernel Method (RKM) to solve FDDE such that the obtained approximate results are much better than other methods in comparison. The main obstacle to solve this problem is the existence of a Gram-Schmidt orthogonalization process in the general form of reproducing kernel method, that is very time consuming. So, we introduce a different implementation for the general form of the reproducing kernel method. In this method, the Gram-Schmidt orthogonalization process is eliminated to significantly reduce the CPU-time. Also, this new method, increases the accuracy of approximate solutions. Due to the increasing accuracy of approximate solutions, we will be able to provide a valid error analysis for this technique. The accuracy of the theoretical results are also illustrated by solving two numerical examples.

ON ORTHOGONALIZATION OF BOUBAKER POLYNOMIALS

In this work, we explore some unknown properties of the Boubaker polynomials. The orthogonalization of the Boubaker polynomials has not been discussed in the literature. Since most of the application areas of such polynomial sequences demand orthogonal polynomials, the orthogonality of the Boubaker polynomials will help extend its theareas of application. We investigate orthogonality of classical Boubaker polynomials using Sturm-Liouville form and then apply the Gram-Schmidt orthogonalization process to develop modified Boubaker polynomials which are also orthogonal. Some classical properties, like orthogonality and orthonormality relation and zeros, of the modified Boubaker polynomials, have been proved. The contributions from this study have an impact on the further application of modified Boubaker polynomials to not only the cases where classical polynomials could be used but also in cases where the classical ones could not be used due to orthogonality issue.

An efficient numerical technique for a biological population model of fractional order

In the present paper, a biological population model of fractional order (FBPM) with one carrying capacity has been examined with the help of reproducing kernel Hilbert space method (RKHSM). This important fractional model arises in many applications in computational biology. It is worth noting that, the considered FBPM is used to provide the changes that is made on the densities of the predator and prey populations by the fractional derivative . The technique employed to construct new numerical solutions for the FBPM which is considered of a system of two nonlinear fractional ordinary differential equations (FODEs). In the proposed investigation, the utilised fractional derivative is the Caputo derivative. The most valuable advantages of the RKHSM is that it is easily and fast implemented method. The solution methodology is based on the use of two important Hilbert spaces, as well as on the construction of a normal basis through the use of Gram-Schmidt orthogonalization process. We illustrate the high competency and capacity of the suggested approach through the convergence analysis. The computational results, which are compared with the homotopy perturbation Sumudu transform method (HPSTM), clearly show: On the one hand, the effect of the fractional derivative in the obtained outcomes, and on the other hand, the great agreement between the mentioned methods, also the superior performance of the RKHSM. The numerical computational are presented in illustrated graphically to show the variations of the predator and prey populations for various fractional order derivatives and with respect to time.

Optimal orthogonalization processes

Two optimal orthogonalization processes are devised to orthogonalize, possibly approximately, the columns of a very large and possibly sparse matrix A ∈ ℂn×k. Algorithmically the aim is, at each step, to optimally decrease nonorthogonality of all the columns of A. One process relies on using translated small rank corrections. Another is a polynomial orthogonalization process for performing the Lowdin orthogonalization. The steps rely on using iterative methods combined, preferably, with preconditioning which can have a dramatic effect on how fast nonorthogonality decreases. The speed of orthogonalization depends on how bunched the singular values of A are, modulo the number of steps taken. These methods put the steps of the Gram-Schmidt orthogonalization process into perspective regarding their (lack of) optimality. The constructions are entirely operator theoretic and can be extended to infinite dimensional Hilbert spaces.

Numerical solutions of hybrid fuzzy differential equations in a Hilbert space

The main goal of this work is to study a numerical method for certain hybrid fuzzy differential equations with an application of a reproducing kernel Hilbert space technique for fuzzy differential equations. Meanwhile, we construct a system of orthogonal functions of the space W22[a,b]⊕W22[a,b] depending on a Gram-Schmidt orthogonalization process to get approximate-analytical solutions of a hybrid fuzzy differential equation. A proof of convergence of this method is also discussed in detail. The exact as well as the approximate solutions are displayed by a series in terms of their α-cut representation form in the Hilbert space W22[a,b]⊕W22[a,b]. To demonstrate behavior, efficiency, and appropriateness of the present technique, two different numerical experiments are solved numerically in this paper.

Solving system of second-order BVPs using a new algorithm based on reproducing kernel Hilbert space

In this study, a new and reliable technique based on Reproducing Kernel Method (RKM) is introduced to solve the linear and nonlinear system of second-order boundary value problem. We propose that RKM without Gram-Schmidt orthogonalization process significantly reduces the CPU time and increases the accuracy of the approximate solutions. The convergence theorem and error analysis are also verified. An algorithm is presented to indicate the steps of implementing the proposed scheme. Since the system of second-order boundary value problem is solvable with the numerical commands in Mathematica software, we examined these commands for numerical examples, compared the results with the present method, and demonstrated the higher precision of the present algorithm.

Soft computing technique for a system of fuzzy Volterra integro-differential equations in a Hilbert space

In this article, we implement a relatively new computational technique, a reproducing kernel Hilbert space method, for solving a system of fuzzy Volterra integro-differential equations in the Hilbert space ⊕j=1n(W22[a,b]⊕W22[a,b]). Based on the concept of the reproducing kernel function combined with Gram-Schmidt orthogonalization process, we represent an exact solution in a form of Fourier series in the reproducing kernel Hilbert space ⊕j=1n(W22[a,b]⊕W22[a,b]). Accordingly, the approximate solution of the system of fuzzy Volterra integro-differential equations is obtained by the n-term intercept of the exact solution and proved to converge to the exact solution. Finally, two numerical examples are presented to illustrate the reliability, appropriateness and efficiency of the method.

A nonlinear analytical formula for forced vibration analysis of the hard-coating cylindrical shell based on the strain energy density principle

In this paper, a nonlinear analytical formulation of the forced vibration of the hard-coating cylindrical shell is developed to investigate the nonlinear resonant characteristics of the shell. The strain dependence of hard coating and the elastic constraint with continuous variable stiffness are considered in the formulation. In order to fully introduce the effects of the strain dependence of hard coating on the resonant characteristics with base excitation, a novel analytical method is presented to determine the equivalent strain of hard coating according to the principle of equal strain energy density. The nonlinear governing equations of motion and the admissible displacement function are derived based on the Love's first approximation theory and the Gram-Schmidt orthogonalization process. A unified Newton-Raphson iterative solution method is employed to solve the nonlinear resonant frequency and response of the shell. As an example to demonstrate the feasibility of the developed analytical model, the forced vibration of the cylindrical shell coated with NiCoCrAlY + YSZ hard coating is implemented numerically and experimentally. Moreover, the influences of the storage modulus, loss modulus and thickness of hard coating on the forced vibration characteristics of the hard-coating cylindrical shell are analyzed in detail.

Vibration and Buckling of Skew Plates Under Linearly Varying Edge Compression

Pre-buckling vibration and buckling behaviour of composite skew plates subjected to linearly varying in-plane edge loading with different boundary conditions are studied. The total energy functional of the skew plate mapped from physical domain to computational domain over which a set of orthonormal polynomials satisfying the essential boundary conditions is generated by Gram-Schmidt orthogonalization process. Using Rayleigh-Ritz method in conjunction with Boundary Characteristics Orthonormal Polynomials, the total energy functional is converted into sets of algebraic equations for static stability problems and ordinary differential equation for free vibration problem. Pre-buckling vibration frequencies of the stressed skew plate are obtained by solving associated linear eigen value problem for free vibration and solution of the eigen value problem for static case results critical buckling load. From different parametric study, it is observed that the pre-buckling vibration frequency and critical buckling load increase with the increase of skew angle and edge restraint.

An iterative polynomial chaos approach for solution of structural mechanics problem with Gaussian material property

A new approach for solution of Stochastic structural Mechanics problem with random coefficient in the framework of Polynomial Chaos (PC) expansion is proposed. The basic strategy of the proposed method is to iteratively construct a PC expansion with reduced numbers of unknown coefficients. The method is based on orthogonal expansion of stochastic responses and generation of an iterative PC based on the responses of the previous iteration. The polynomials are evaluated using Gram-Schmidt orthogonalization process. The number of random variables in PC expansion is reduced by considering only the dominant components of the response characteristics, which is evaluated using Karhunen-Loève (KL) expansion. In case of random material field problem, KL expansion is used to discretize the random field. The usefulness of the proposed method in terms of accuracy and computational efficiency is examined. The results of linear static structural mechanics problems are compared with MCS and PC method. The proposed method is observed to be computationally more efficient and accurate than PC method.