Explore the vast range of titles available.

To improve the computational efficiency of global sensitivity analysis (GSA) for complex structures, this study proposed a new importance analysis method (IE) based on the low deviation sequences and orthogonal polynomials to study the influence of parameters’ uncertainty on three structural seismic demands. A comparative investigation utilizing nonlinear time history analysis for these seismic demands was conducted using OpenSEES. The variance-based importance analysis method and the Tornado graphic sensitivity analysis method were employed to validate the accuracy of the proposed approach. The results regarding the order of importance are nearly consistent across methods, demonstrating the effectiveness of our proposed method. Notably, the sample size required by this new method is only 1024 to achieve reliable results, which is significantly lower than existing sampling methods that necessitate thousands of samples for effective importance analysis; thus, enhancing overall efficiency. Furthermore, the findings indicate that the influence of representative value of gravity load ( M s ) on seismic demands is relatively substantial, whereas the influence of modulus of elasticity of concrete ( E c ) is comparatively minor.

In parameter uncertainty analysis, traditional Monte Carlo simulation (MC) requires double sampling to calculate the importance index based on the moment-independent method, which takes a long time, especially when the failure probability is small. The Latin hypercube sampling method is used to simulate the uncertainty of the random variables in this study, which can consider the correlation between random variables and guarantee the accuracy in the small sample size. The moment-independent method combined with orthogonal polynomial estimation and kernel density estimation is developed to estimate the unconditional and conditional probability density functions of the output response to calculate the importance index. The results of the proposed method are in good agreement with those of the verification example and MC, and the efficiency is higher than that of the MC. Subsequently, the influence of the correlation between the shear strength parameters is investigated, and the differences in the influence of the random variables on the safety factor (Fs) and failure probability (Pf) are compared. The results show that the correlation between the shear strength parameters will greatly affect the importance indices, and the cumulative impact of each random variable on the Fs and Pf is different. This difference indicates that the influence of the parameter uncertainty is different when different dependent variables are taken as the output responses. Moreover, the correlation should not be ignored.

We study the estimation of a regression function by two classes of estimators, the Nadaraya-Watson Kernel type estimators and the orthogonal polynomial estimators. We obtain sharp pointwise rates of strong consistency by establishing laws of the iterated logarithm for the two classes of estimators. These results parallel those of Hall (1981) on density estimation and extend those of Noda (1976) on strong consistency of kernel regression estimators.

This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

We propose a flexible framework for modeling financial returns using orthogonal expansions of the standardized Student-t density. The Skew-Kurtotic t Expansion (SKTE) captures skewness and excess kurtosis while preserving heavy tails. Orthogonal polynomials are constructed with the t kernel, while a fourth-order truncation provides a parsimonious and interpretable parameterization. Squaring and normalization ensure a valid density. Truncation accuracy analysis shows that the fourth-order approximation balances computational efficiency and tail fidelity, with higher-order terms offering minimal improvement. Empirical analysis of daily returns for the Hang Seng Index, S&P 500, front-month natural gas futures, and EUR/USD exchange rate shows that SKTE outperforms the classical Gram–Charlier expansion, particularly in tail modeling. Maximum likelihood estimation confirms stable skewness and kurtosis adjustments, while location and scale parameters align the distribution to each asset. These results indicate that SKTE provides a robust and flexible framework for modeling asymmetric and heavy-tailed stock return distributions, which is highly relevant for understanding extreme market behavior.

This paper investigates a class of Jacobsthal-type polynomials (JTPs) that involves one parameter. We present several new formulas for these polynomials, including expressions for their derivatives, moments, and linearization formulas. The key idea behind the derivation of these formulas is based on developing a new connection formula that expresses the shifted Chebyshev polynomials of the third kind in terms of the JTPs. This connection formula is used to deduce a new inversion formula of the JTPs. Therefore, by utilizing the power form representation of these polynomials and their corresponding inversion formula, we can derive additional expressions for them. Additionally, we compute some definite integrals based on some formulas of these polynomials.

To speed-up the solution of parametrized differential problems, reduced order models (ROMs) have been developed over the years, including projection-based ROMs such as the reduced-basis (RB) method, deep learning-based ROMs, as well as surrogate models obtained through machine learning techniques. Thanks to its physics-based structure, ensured by the use of a Galerkin projection of the full order model (FOM) onto a linear low-dimensional subspace, the Galerkin-RB method yields approximations that fulfill the differential problem at hand. However, to make the assembling of the ROM independent of the FOM dimension, intrusive and expensive hyper-reduction techniques, such as the discrete empirical interpolation method (DEIM), are usually required, thus making this strategy less feasible for problems characterized by (high-order polynomial or nonpolynomial) nonlinearities. To overcome this bottleneck, we propose a novel strategy for learning nonlinear ROM operators using deep neural networks (DNNs). The resulting hyper-reduced order model enhanced by DNNs, to which we refer to as Deep-HyROMnet, is then a physics-based model, still relying on the RB method approach, however employing a DNN architecture to approximate reduced residual vectors and Jacobian matrices once a Galerkin projection has been performed. Numerical results dealing with fast simulations in nonlinear structural mechanics show that Deep-HyROMnets are orders of magnitude faster than POD-Galerkin-DEIM ROMs, still ensuring the same level of accuracy.

The classical curve-fitting problem to relate two variables, x and y, deals with polynomials. Generally, this problem is solved by the least squares method (LS), where the minimization function considers the vertical errors from the data points to the fitting curve. Another curve-fitting method is total least squares (TLS), which takes into account errors in both x and y variables. A further method is the orthogonal distances method (OD), which minimizes the sum of the squares of orthogonal distances from the data points to the fitting curve. In this work, we develop the OD method for the polynomial fitting of degree n and compare the TLS and OD methods. The results show that TLS and OD methods are not equivalent in general; however, both methods get the same estimates when a polynomial of degree 1 without an independent coefficient is considered. As examples, we consider the calibration curve-fitting problem of a R-type thermocouple by polynomials of degrees 1 to 4, with and without an independent coefficient, using the LS, TLS and OD methods.

Propagating uncertainty through a crash problem is very difficult due to non-linear and non-smooth behavior. The required number of model evaluations is often high, and therefore the computational cost is prohibitive. To deal with such problems, an adaptive meta-model is developed using a polynomial chaos expansion (PCE) and a proper orthogonal decomposition (POD). The adaptive meta-model is used for uncertainty quantification and for global sensitivity analysis of a crash box under impact loading. The time-dependent uncertain response quantities are expressed with the reduced POD modes. The predicted stochastic contact force and impactor velocity by the adaptive meta-model are quite close to the actual simulations. The time-dependent mean and standard deviation for all responses are predicted quite well with low number of model evaluations. Furthermore, it is found that the material property and the crash box thickness are the most influential parameters for the contact force, and the impactor mass is the most influential parameter for the total dissipated energy.

Uncertainty is a common feature in first-principles models that are widely used in various engineering problems. Uncertainty quantification (UQ) has become an essential procedure to improve the accuracy and reliability of model predictions. Polynomial chaos expansion (PCE) has been used as an efficient approach for UQ by approximating uncertainty with orthogonal polynomial basis functions of standard distributions (e.g., normal) chosen from the Askey scheme. However, uncertainty in practice may not be represented well by standard distributions. In this case, the convergence rate and accuracy of the PCE-based UQ cannot be guaranteed. Further, when models involve non-polynomial forms, the PCE-based UQ can be computationally impractical in the presence of many parametric uncertainties. To address these issues, the Gram–Schmidt (GS) orthogonalization and generalized dimension reduction method (gDRM) are integrated with the PCE in this work to deal with many parametric uncertainties that follow arbitrary distributions. The performance of the proposed method is demonstrated with three benchmark cases including two chemical engineering problems in terms of UQ accuracy and computational efficiency by comparison with available algorithms (e.g., non-intrusive PCE).