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The Taylor series expansion- and least squares-based lattice Boltzmann method (TLLBM) was used in this paper to extend the current thermal model to an arbitrary geometry so that it can be used to solve practical thermo-hydrodynamics in the incompressible limit. The new explicit method is based on the standard lattice Boltzmann method (LBM), Taylor series expansion and the least squares approach. The final formulation is an algebraic form and essentially has no limitation on the mesh structure and lattice model. Numerical simulations of natural convection in a square cavity on both uniform and nonuniform grids have been carried out. Favorable results were obtained and compared well with the benchmark data. It was found that, to get the same order of accuracy, the number of mesh points used on the nonuniform grid is much less than that used on the uniform grid.

The two-dimensional form of the Taylor series expansion- and least square-based lattice Boltzmann method (TLLBM) was recently presented by Shu et al.⁸ TLLBM is based on the standard lattice Boltzmann method (LBM), Taylor series expansion and the least square optimization. The final formulation is an algebraic form and essentially has no limitation on the mesh structure and lattice model. In this paper, TLLBM is extended to the three-dimensional case. The resultant form keeps the same features as the two-dimensional one. The present form is validated by its application to simulate the three-dimensional lid-driven cavity flow at Re=100, 400 and 1000. Very good agreement was achieved between the present results and those of Navier–Stokes solvers.

Lidar is an active remote sensing instrument, but its effective range is often limited by signal-to-noise (SNR) ratio. The reason is that noises or fluctuations always strongly affect the measured results. To resolve this problem, a novel approach of using least-squares support vector machine (LS-SVM) to reconstruct the Lidar signal is proposed in this paper. LS-SVM has been proven as robust to noisy data; the Lidar signal, which is strongly corrupted by noises or fluctuations, can be thought as a function of distance. So detecting Lidar signals from high noisy regime can be regarded as a robust regression procedure which involves estimating the underlying relationship from detected signal data set. To apply the LS-SVM on Lidar signal regression, firstly the noises in Lidar signal is analyzed and then the traditional LS-SVM algorithm is modified to incorporate the a priori knowledge of the Lidar signal in the training of LS-SVM. The experimental results demonstrate the effectiveness and efficiency of our approach.

This paper deals with the prediction of primary parameters of CMOS transistor for upcoming process using the robust Bisquare Weights method which is able to provide solutions to the challenges of some parameters of Nanoscale CMOS. Predicted parameters for 45 nm to 22 nm process nodes are obtained in order to solve design challenges generated by Nanoscale process. These predicted primary parameters are helpful to estimate the performance of a basic element circuit having a key role in the design of upcoming analog systems. Comparisons between predictive technology model data and predicted parameters are used to check the validity of the used method. As a study case, we will detail the behavior of optimized telescopic operational transconductance amplifier performance with process scaling.

We prove the $L^1$-norm and bounded variation norm convergence of a piecewise linear least squares method for the computation of an invariant density of the Foias operator associated with a random map with position dependent probabilities. Then we estimate the convergence rate of this least squares method in the $L^1$-norm and the bounded variation norm, respectively. The numerical results, which demonstrate a higher order accuracy than the linear spline Markov method, support the theoretical analysis.

Fuzzy c-regression models are known to be useful in real applications, but there are two drawbacks: strong dependency on the predefined number of clusters and sensitiveness against outliers or noises. To avoid these drawbacks, we propose sequential fuzzy regression models based on least absolute deviations which we call SFCRMLAD. This algorithm sequentially extracts one cluster at a time using a method of noise-detection, enabling the automatic determination of clusters and having robustness to noises. We compare this method with the ordinary fuzzy c-regression models based on least squares, fuzzy c-regression models based on least absolute deviations, and moreover sequential fuzzy regression models based on least squares. For this purpose we use a two-dimensional illustrative example whereby characteristics of the four methods are made clear. Moreover a simpler and more efficient algorithm of SFCRMLAD can be used for scalar input and output variables, while a general algorithm of SFCRMLAD uses linear programming solutions for multivariable input. By using the above example, we compare efficiency of different algorithms.

We adapt data analytic techniques to the software reliability setting. We develop an evaluation procedure based on scatterplots of transformed data, crossvalidation using the predicted residual sum of squares (PRESS) criterion, residual plots, and normal plots. We analyze a software failure data set collected at Storage Technology Corporation utilizing this evaluation technique. We identify a new model which, for this data set, outperforms several established software reliability models, including the delayed S-shaped, exponential, inverse linear, logarithmic, power, and log power models. The failure intensity, and hence the reliability, for this model at any point in time is a function of the time per failure, that is, the ratio of cumulative time divided by cumulative failures, a quantity that agrees with the mean time between failures for time points at which failures occur.

Elsewhere we have introduced a construction to produce biorthogonal multiresolutions from given subdivisions. This construction was formulated in matrix terms, which is appropriate for curves and tensor-product surfaces. For mesh surfaces of non-tensor connectivity, however, matrix notation is inconvenient. This work presents the construction for regular meshes using diagrams (stencils, masks) and interactions between diagrams to replace matrices and matrix multiplication. Regular triangular meshes with butterfly subdivision and a variant of Loop subdivision due to Litke, et al. are used as examples.

Objective: The main objective of this study was to assess the accuracy of bottom-up solution for three-dimensional (3D) inverse dynamics analysis of squat lifting using a 3D full body linked segment model. Least squares solution was used in this study as reference for assessment of the accuracy of bottom-up solution. Findings of this study may clarify how much the bottom-up solution can be reliable for calculating the joint kinetics in 3D inverse dynamics problems. Methods: Ten healthy males volunteered to perform squat lifting of a box with a load of one-tenth of their body weights. The joint moments were calculated using 110 reflective passive markers (46 anatomical markers and 64 tracking markers) and a 3D full body linked segment model. Ground reaction forces and kinematics data were recorded using a Vicon system with two parallel Kistler force plates. Three-dimensional Newton–Euler equations of motion with bottom-up and least squares solutions were applied to calculate joint moments. The peak and mean values of the joint moments were determined to check the quantitative differences as well as the time-to-peak value of the moment curves was determined to check the temporal differences between the two inverse dynamics solutions. Results: Significant differences (all $P$-$\mathrm{\text{values}}<0.05$) between the two inverse dynamics solutions were detected for the peak values of the hip (right and left sides) and L5–S1 joint moments in the lateral anatomical direction as well significant differences (all $P$-$\mathrm{\text{values}}<0.05$) were detected for the peak and mean values of the L5–S1 joint moment in all anatomical directions. Moreover, small differences (all $\mathrm{\text{RMSEs}}<0.01\%$) were detected between the two inverse dynamic solutions for the calculated lower body joint moments. Conclusions: The findings of this study clarified the disadvantages of the straightforward solutions and demonstrated that the bottom-up solution may not be accurate for more distal measures from the force plate (for hip and S1–L5) but it may be accurate for more proximal joints (ankle and knee) in 3D inverse dynamics analysis.

Objective: The main objective of this study was to assess the accuracy of bottom-up solution for three-dimensional (3D) inverse dynamics analysis of squat lifting using a 3D full body linked segment model. Least squares solution was used in this study as reference for assessment of the accuracy of bottom-up solution. Findings of this study may clarify how much the bottom-up solution can be reliable for calculating the joint kinetics in 3D inverse dynamics problems. Methods: Ten healthy males volunteered to perform squat lifting of a box with a load of one-tenth of their body weights. The joint moments were calculated using 110 reflective passive markers (46 anatomical markers and 64 tracking markers) and a 3D full body linked segment model. Ground reaction forces and kinematics data were recorded using a Vicon system with two parallel Kistler force plates. Three-dimensional Newton–Euler equations of motion with bottom-up and least squares solutions were applied to calculate joint moments. The peak and mean values of the joint moments were determined to check the quantitative differences as well as the time-to-peak value of the moment curves was determined to check the temporal differences between the two inverse dynamics solutions. Results: Significant differences (all $P$-values $<0.05$) between the two inverse dynamics solutions were detected for the peak values of the hip (right and left sides) and L5–S1 joint moments in the lateral anatomical direction as well significant differences (all $P$-values $<0.05$) were detected for the peak and mean values of the L5–S1 joint moment in all anatomical directions. Moreover, small differences (all RMSEs $<0.01$%) were detected between the two inverse dynamic solutions for the calculated lower body joint moments. Conclusions: The findings of this study clarified the disadvantages of the straightforward solutions and demonstrated that the bottom-up solution may not be accurate for more distal measures from the force plate (for hip and S1–L5) but it may be accurate for more proximal joints (ankle and knee) in 3D inverse dynamics analysis.

An unsupervised feature selection method is proposed for analysis of datasets of high dimensionality. The least square error (LSE) of approximating the complete dataset via a reduced feature subset is proposed as the quality measure for feature selection. Guided by the minimization of the LSE, a kernel least squares forward selection algorithm (KLS-FS) is developed that is capable of both linear and non-linear feature selection. An incremental LSE computation is designed to accelerate the selection process and, therefore, enhances the scalability of KLS-FS to high-dimensional datasets. The superiority of the proposed feature selection algorithm, in terms of keeping principal data structures, learning performances in classification and clustering applications, and robustness, is demonstrated using various real-life datasets of different sizes and dimensions.

We present a dimensionless fit index for phylogenetic trees that have been constructed from distance matrices. It is designed to measure the quality of the fit of the data to a tree in absolute terms, independent of linear transformations on the distance matrix. The index can be used as an absolute measure to evaluate how well a set of data fits to a tree, or as a relative measure to compare different methods that are expected to produce the same tree. The usefulness of the index is demonstrated in three examples.

Linear discrimination, from the point of view of numerical linear algebra, can be treated as solving an ill-posed system of linear equations. In order to generate a solution that is robust in the presence of noise, these problems require regularization. Here, we examine the ill-posedness involved in the linear discrimination of cancer gene expression data with respect to outcome and tumor subclasses. We show that a filter factor representation, based upon Singular Value Decomposition, yields insight into the numerical ill-posedness of the hyperplane-based separation when applied to gene expression data. We also show that this representation yields useful diagnostic tools for guiding the selection of classifier parameters, thus leading to improved performance.

A novel method for load bounds identification for uncertain structures is proposed in the frequency domain. The uncertain parameters are assumed to locate in their intervals and only their bounds rather than their precise information are needed. To quantitatively describe the effect of the interval uncertainty on the load identification in the frequency ranges, the interval extension is then introduced in the frequency response function (FRF)-based least squares approach. Therefore, the load bounds are determined through the summation of the two separate parts including the midpoint part and the perturbed part of the load. The midpoint part is computed by using the Moore–Penrose pseudo-inversion and the perturbed part is transformed into the first derivatives of the midpoint load with respect to the uncertain parameters by applying the truncated total least squares (TTLS). Two numerical examples are investigated to validate that the proposed method is very effective to predict the load bounds for the uncertain structure in frequency domain.

This paper presents an investigation of the numerical computations of nonweak/strong solutions of linear and nonlinear hyperbolic and parabolic differential and partial differential equations resulting from a single conservation law using C¹p-version least squares finite element formulation (LSFEF) and C¹¹p-version space time least squares finite element formulation (STLSFEF) for stationary and time dependent processes. It is demonstrated that with this approach it is possible to compute nonweak/strong solutions in the sense that the computed solutions possess the same orders of continuity in space and time as the strong solutions and satisfy nonweak (or residual) form of the governing differential equations (GDE's). Other benefits of this approach over weak solutions are also discussed and demonstrated. Stationary and time dependent convection-diffusion and Burgers equations are used as model problems.

In the present study, a statistical strength model is proposed, which aims at describing how the strength of geometrically irregular honeycomb material is affected by the scale. Hence, the samples are designed based on the selected geometrical irregularity and the number of the cells/scale. Simulation experiments are conducted on these samples under different loading combinations. The experiment results are linked to possible failure mechanisms in order to obtain the critical loads which are expressed in terms of cumulative distribution functions. The discrete distribution data of the critical loads are then fitted to analyze the effect of scale on different strength percentiles by virtue of the least squares function and closed quadric surface fitting. Eventually, the outcome is expressed in terms of ellipsoid surface representing the honeycomb material strength in three-dimensional stress space.

This paper presents a technique for deriving least-squares-based polynomial bubble functions to enrich the standard linear finite elements, employed in the formulation of Galerkin weighted-residual statements. The element-level linear shape functions are enhanced using supplementary polynomial bubble functions with undetermined coefficients. The enhanced shape functions are inserted into the model equation and the residual functional is constructed and minimized by using the method of the least squares, resulting in an algebraic system of equations which can be solved to determine the unknown polynomial coefficients in terms of element-level nodal values. The stiffness matrices are subsequently formed with the standard finite elements assembly procedures followed by using these enriched elements which require no additional nodes to be introduced and no extra degree of freedom incurred. Furthermore, the proposed technique is tested on a number of benchmark linear transport equations where the quadratic and cubic bubble functions are derived and the numerical results are compared against the exact and standard linear element solutions. It is demonstrated that low order bubble enriched elements provide more accurate approximations for the exact analytical solutions than the standard linear elements at no extra computational cost in spite of using relatively crude meshes. On the other hand, it is observed that a satisfactory solution of the strongly convection-dominated transport problems may require element enrichment by using significantly higher order polynomial bubble functions in addition to the use of extremely fine computational meshes.

A function with the form y = (a+ex)/(1+cx) is monotonic for all values of the parameters and can be inverted easily. This function describes a rectangular hyperbola, a segment of which might be used as a calibration curve for reading from x to y and, more importantly, from y to x. Methods are given for finding the hyperbolic segment that minimizes the sum of squared residuals to a set of points {(x, y)} and for finding a spline of hyperbolic segments that minimizes an objective function, e.g. the sum of squared residuals. A method is also given for constructing a spline of hyperbolic segments that interpolates a set of points. An analysis of model error is given.

This paper describes recent research by the authors on the development of improved computational methods and algorithms for physically correct solution of linear and nonlinear boundary and initial value problems of structural dynamics. The difficulties and anomalies commonly encountered in the numerical simulation of problems with localized high solution gradients are critically examined and new computational methodologies are proposed and demonstrated through some well-known problems that defied physically meaningful solutions by the currently available computational formulations. The root cause of the failure of the current computational methodology when applied to these problems is the lack of desired global differentiability of the computed solution. Two essential elements of the proposed mathematical and computational framework are (1) the variational consistency (VC) of the mathematical approach used in obtaining algebraic systems from the differential systems via an integral approach, and (2) the global differentiability (or smoothness) requiring the use of higher-order spaces for local approximation. The variational consistency is a fundamentally important aspect of the proposed approach. A violation of this leads to variational inconsistent (VIC) formulation that is shown to produce degenerate computational models. The degree of global differentiability is of critical importance for three reasons, (i) If the solutions of the algebraic system to be admissible in the weak form; (ii) if the solutions from the algebraic system are to yield the same global smoothness as the analytical solutions; and (iii) to permit correct incorporation of the physics of the processes in the numerical computations. The proposed computational technology provides mathematically sound, most general, comprehensive, and complete computational platform in which all problems can be treated with the same rigor and without any bias to their form or field of application. The important issues of concern in the numerical simulation of boundary value problems (BVP) and initial value problems (IVP) arising in solid and structural mechanics (e.g., fracture mechanics), as well as in coupled problems (e.g., fluid-structure interaction) are addressed using the proposed k-version finite element methodology. Theoretical basis and supporting numerical studies will be discussed to illustrate various features of the developed methodology. It will be demonstrated that the present approach is mathematically consistent, incorporates the physics correctly, requires no ‘ad-hoc’ approaches or semi-empirical adjustments, and it is rather a natural way to simulate correct physical behavior of such problems.

Support Vector Machines (SVMs) methods have become a popular tool for predictive data mining problems and novelty detection. They show good generalization performance on many real-life datasets and they are motivated theoretically through convex programming formulations. There are relatively few free parameters to adjust using cross validation and the architecture of the SVM learning machine does not need to be found by experimentation as in the case of Artificial Neural Networks (ANNs). We discuss the fundamentals of SVMs with emphasis to multiclass classification problems and applications in science, business and engineering.