Least-squares Formulation Research Articles

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Using Heteroscedasticity to Identify and Estimate Mismeasured and Endogenous Regressor Models

This article proposes a new method of obtaining identification in mismeasured regressor models, triangular systems, and simultaneous equation systems. The method may be used in applications where other sources of identification, such as instrumental variables or repeated measurements, are not available. Associated estimators take the form of two-stage least squares or generalized method of moments. Identification comes from a heteroscedastic covariance restriction that is shown to be a feature of many models of endogeneity or mismeasurement. Identification is also obtained for semiparametric partly linear models, and associated estimators are provided. Set identification bounds are derived for cases where point-identifying assumptions fail to hold. An empirical application estimating Engel curves is provided.

Vertical and horizontal resistive limit formulas for a rectangular-loop source on a conductive half-space

We derived analytic expressions for the time integrals of vertical and horizontal transient B-field responses on a conductive half-space excited by a rectangular-loop source. These formulas were applied in two ways in a fast transient electromagnetic (TEM) inversion scheme. Indefinite integrals were used to extrapolate measured TEM decays to early and late time to convert the observed data to resistive limits. Definite integrals over all time provided estimates for the resistive limit response of host-rock. An apparent conductivity of the host was calculated from the resistive limits via a simple least-squares formulation. These applications of the formulas were tested on synthetic and real TEM data.

Simulation of Multi-Component Mass Diffusion Pellet Model Using Least Squares Spectral Element Method for Steam Methane Reforming Process

Mass based pellet model has been solved using least-squares formulation to describe the evolution of species composition, pressure, velocity, total concentration, mass diffusion flux in porous pellets for the steam methane reforming (SMR) process. The diffusion-reaction problems are computationally intensive, requiring efficient numerical methods for dealing with them. This paper presents formulation and algorithm of least-squares spectral element method (LS-SEM) for solving multicomponent mass diffusion pellet models. The mass diffusion flux is described according to the rigorous Maxwell Stefan model. The effectiveness factors have been calculated for the SMR process and compared with the literature data. The model evaluations revealed that the least-squares method is well suited for solving the multicomponent mass diffusion pellet models for the SMR process, achieving exponential convergence.

A least-squares fem-bem coupling method for linear elasticity

This paper deals with a least-squares formulation of a second order transmission problem for linear elasticity. The problem in the unbounded exterior domain is rewritten with boundary integral equations on the boundary of the inner domain. In the interior domain we treat a linear elastic material which can also be nearly incompressible. The least-squares functional is given in terms of the H˜−1(Ω) and H1/2(Γ) norms. These norms are realized by solution operators of corresponding dual norm problems which are approximated using multilevel preconditioners.

On tracking the motion of rigid bodies through edge detection and least-squares fitting

A class of techniques is investigated for determining at least three components (two translational and one rotational) of the motion of a rigid body from silhouette images, with particular emphasis on motion within a fluid. The investigated techniques all employ edge detection followed by some form of least-squares fitting to the detected points in determining the movement of the body. Four techniques are discussed and, through both an artificial image analysis and calibrated sphere measurements, are shown to be capable of measuring displacements down to a few thousandths of a pixel under low image-noise conditions (\(\lesssim2\%\)). Measurements of two configurations in a high-enthalpy shock tunnel demonstrate the capabilities of the techniques under experimental conditions. In particular, a technique referred to as edge-tracking is introduced, which can be employed in situations where the model profile is unknown and/or only some fraction of it is visible. This latter quality is especially useful for measurements in high-enthalpy facilities, where test-gas luminosity can obscure a significant extent of the model outline. A further advantage of this technique is that, even for complex geometries, the fitting procedure can typically be reduced to solving a sequence of linear least-squares problems, rather than a nonlinear one, with a corresponding benefit in computational efficiency.

Solving a linear conservation law subject to initial and final conditions

An existence and uniqueness result for a linear conservation law subject to the initial and final conditions by using a spacetime least-squares formulation is proved. Some numerical simulations of a linear conservation law with a non-well-known velocity field are shown. An application to cardiac image reconstruction is presented.

Recovery of an interface from boundary measurement in an elliptic differential equation

We study the inverse problem of recovering an interior interface from a boundary measurement in an elliptic boundary value problem arising from a semiconductor transistor model. We set up a nonlinear least-squares formulation for solving the inverse problem, and establish the necessary derivatives with respect to the interface. We then propose both the Gauss---Newton iterative method and the conjugate gradient method for the least-squares problem, and present implementation of these methods using integral equations.

Determination of stress intensity factors for multi-material junctions

This study evaluates the junction-tip coordinates and the stress intensity factors (SIFs) of multi-material junctions using the image-correlation experiment and least-squares method. The major advantage of the proposed method is that the procedure is simple and systematic. First, complex displacement functions are deduced into a least-squares form, and then displacement fields from the image-correlation experiment are substituted into the least-squares equation to evaluate the SIFs. Compared with the SIFs from H-integrals using finite element results, the calculated least-squares SIFs are accurate if more than 10 eigenvalues are included. Furthermore, the least-squares SIFs are not sensitive to the maximum or minimum radius of the area from which data is included.

Comparative analysis of inverse coefficient problems for parabolic equations. Part III: Conjugate gradient method and coarse-fine grid algorithm

This article presents a computational analysis of the Conjugate Gradient Method (CGM), and a comparative analysis of the method (CGM) and coarse-fine grid algorithm (CFGA) for parabolic inverse coefficient problems (ICPs) based on boundary measured data. The adjoint problem approach is applied to obtain formal gradients of each ICPs as the L2-scalar product of the derivatives ux(x, t; k) and ϕx(x, t; k) of the corresponding direct and adjoint problems. Then the CGM is applied to the least-squares formulation of the inverse coefficient problems. Detailed numerical study of the method for each ICP is presented for various types of type input data concentrated at the boundary. Comparative computational analysis of the CGM and CFGA shows the limits of applications and effectiveness of each these methods.

An alternative methodology for the analysis of electrical resistivity data from a soil gas study

The aim of this paper is to present an alternative method for the analysis of resistivity data. The methodology was developed during a study to evaluate if electrical resistivity can be used as a tool for analysing subsurface gas dynamics and gas emissions from landfills. The main assumption of this study was that variations in time of resistivity data correspond to variations in the relative amount of gas and water in the soil pores. Field measurements of electrical resistivity, static chamber gas flux and weather data were collected at a landfill in Helsingborg, Sweden. The resistivity survey arrangement consisted of nine lines each with 21 electrodes in an investigation area of 16 x20 m. The ABEM Lund Imaging System provided vertical and horizontal resistivity profiles every second hour. The data were inverted in Res3Dinv using L-1-norm-based optimization method with a standard least-squares formulation. Each horizontal soil layer was then represented as a linear interpolated raster model. Different areas underneath the gas flux measurement points were defined in the resistivity model of the uppermost soil layer, and the vertical extension of the zones could be followed at greater depths in deeper layer models. The average resistivity values of the defined areas were calculated and plotted on a time axis, to provide graphs of the variation in resistivity with time in a specific section of the ground. Residual variation of resistivity was calculated by subtracting the resistivity variations caused by the diurnal temperature variations from the measured resistivity data. The resulting residual resistivity graphs were compared with field data of soil moisture, precipitation, soil temperature and methane flux. The results of the study were qualitative, but promising indications of relationships between electrical resistivity and variations in the relative amount of gas and water in the soil pores were found. Even though more research and better data quality is necessary for verification of the results presented here, we conclude that this alternative methodology of working with resistivity data seems to be a valuable and flexible tool for this application.

Extremum-seeking algorithm design for fed-batch cultures of microorganisms with overflow metabolism

Abstract Overflow metabolism characterizes cell strains that are likely to produce inhibiting metabolites as a result of an excess of substrate feeding and a saturated respiratory capacity. In this study, extremum seeking is used to assess the actual critical substrate level, which is a function of the cell respiratory capacity. The latter is modelled as a function representing the existence of two different metabolic modes, namely the respirative mode and the respiro-fermentative mode, as well as modulation factors taking account of inhibiting metabolites. A model-free approach is taken due to the high uncertainty in modelling bioprocesses, and two alternative gradient estimation procedures are considered and compared, e.g., a bank of filters and a recursive least-squares formulation. The RLS technique is simple to implement and provides quite satisfactory results in the presence of model uncertainties and measurement noise.

Determining 2D notch SIFs by the image-correlation method

This study evaluates the two-dimensional (2D) stress intensity factors (SIFs) of a sharp V-notch using the image-correlation experiment and least-square method for isotropic materials. First, the William's eigenfunction and complex displacement function approach are deduced into a least-square form, and then displacement fields from the image-correlation experiment are substituted into the least-square equation to evaluate the 2D SIFs. Compared with the SIFs from finite element and body force methods, the least-squares method can be used to calculate SIFs more accurately, if more than two displacement terms are included. The SIFs calculated from this least-squares method are not sensitive to the maximum or minimum radius of the area from which data is included. The major advantage of the proposed method is that the procedure is simple and systematic, so it can be applied to any finite element or experimental methods that obtain displacement fields.

First-order system least squares and the energetic variational approach for two-phase flow

This paper develops a first-order system least-squares (FOSLS) formulation for equations of two-phase flow. The main goal is to show that this discretization, along with numerical techniques such as nested iteration, algebraic multigrid, and adaptive local refinement, can be used to solve these types of complex fluid flow problems. In addition, from an energetic variational approach, it can be shown that an important quantity to preserve in a given simulation is the energy law. We discuss the energy law and inherent structure for two-phase flow using the Allen–Cahn interface model and indicate how it is related to other complex fluid models, such as magnetohydrodynamics. Finally, we show that, using the FOSLS framework, one can still satisfy the appropriate energy law globally while using well-known numerical techniques.

Reconstruction of the equilibrium of the plasma in a Tokamak and identification of the current density profile in real time

The reconstruction of the equilibrium of a plasma in a Tokamak is a free boundary problem described by the Grad–Shafranov equation in axisymmetric configuration. The right-hand side of this equation is a nonlinear source, which represents the toroidal component of the plasma current density. This paper deals with the identification of this nonlinearity source from experimental measurements in real time. The proposed method is based on a fixed point algorithm, a finite element resolution, a reduced basis method and a least-square optimization formulation. This is implemented in a software called Equinox with which several numerical experiments are conducted to explore the identification problem. It is shown that the identification of the profile of the averaged current density and of the safety factor as a function of the poloidal flux is very robust.

A layerwise mixed least-squares finite element model for static analysis of multilayered composite plates

A layerwise finite element model is developed in a mixed least-squares formulation for static analysis of multilayered composite plates. The model assumes a layerwise variable description of displacements, transverse stresses and in-plane strains, taken as independent variables. The mixed formulation allows to completely and a priori fulfil the known C z 0 requirements, which refer to the zig-zag form of displacements in the thickness direction and the interlaminar continuity of transverse stresses, due to compatibility and equilibrium reasons, respectively. This contrasts with layerwise displacement-based models that usually cannot a priori account for the interlaminar continuity of transverse stresses. In addition, the benefit of mixed least-squares formulation, as opposed to mixed weak form models, is that it leads to a variational unconstrained minimization problem, where the finite element approximating spaces can be chosen independently. Numerical examples are shown to assess the layerwise mixed least-squares model predictive capabilities compared to three-dimensional elasticity solutions and also other finite element results available in literature. Most notably, the present model is able to achieve accurate results in very good agreement with three-dimensional solutions and is shown to be insensitive to shear-locking.

A Least-Squares/Fictitious Domain Method for Linear Elliptic Problems with Robin Boundary Conditions

AbstractIn this article, we discuss a least-squares/fictitious domain method for the solution of linear elliptic boundary value problems with Robin boundary conditions. Let Ω and ω be two bounded domains of Rd such that ω̅⊂Ω. For a linear elliptic problem in Ω\ω̅ with Robin boundary condition on the boundary ϒ of ω, our goal here is to develop a fictitious domain method where one solves a variant of the original problem on the full Ω, followed by a well-chosen correction over ω. This method is of the virtual control type and relies on a least-squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. Numerical results obtained when applying our method to the solution of two-dimensional elliptic and parabolic problems are given; they suggest optimal order of convergence.

Illumination Recovery From Image With Cast Shadows Via Sparse Representation

In this paper, we propose using sparse representation for recovering the illumination of a scene from a single image with cast shadows, given the geometry of the scene. The images with cast shadows can be quite complex and, therefore, cannot be well approximated by low-dimensional linear subspaces. However, it can be shown that the set of images produced by a Lambertian scene with cast shadows can be efficiently represented by a sparse set of images generated by directional light sources. We first model an image with cast shadows composed of a diffusive part (without cast shadows) and a residual part that captures cast shadows. Then, we express the problem in an l(1)-regularized least-squares formulation, with nonnegativity constraints (as light has to be non-negative at any point in space). This sparse representation enjoys an effective and fast solution thanks to recent advances in compressive sensing. In experiments on synthetic and real data, our approach performs favorably in comparison with several previously proposed methods.

A non-biased form of least squares support vector classifier and its fast online learning

As an effective learning technique based on structural risk minimization, SVM has been confirmed an useful tool in many machine learning fields. With the increase in application requirement for some real-time cases, such as fast prediction and pattern recognition, the online learning based on SVM gradually becomes a focus. But the common SVM has disadvantages in classifier’s bias and the computational complexity of online modeling, resulting in the reduction in classifier’s generality and the low learning speed. Therefore, an non-biased least square support vector classifier(LSSVC) model is proposed in this paper by improving the form of structure risk. Also, a fast online learning algorithm using Cholesky factorization is designed based on this model according to the characteristic of the non-biased kernel extended matrix in the model’s dynamic change process. In this way, the calculation of Lagrange multipliers is simplified, and the time of online learning is greatly reduced. Simulation results testify that the non-biased LSSVC has good universal applicability and better generalization capability, at the same time, the algorithm has a great improvement on learning speed.

On the roles of minimization and linearization in least-squares finite element models of nonlinear boundary-value problems

In this paper we examine the roles of minimization and linearization in the least-squares finite element formulations of nonlinear boundary-values problems. The least-squares principle is based upon the minimization of the least-squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations (in the present case we consider L 2 norms). Since the least-squares method is independent of the discretization procedure and the solution scheme, the least-squares principle suggests that minimization should be performed prior to linearization, where linearization is employed in the context of either the Picard or Newton iterative solution procedures. However, in the least-squares finite element analysis of nonlinear boundary-value problems, it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations. The main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged. The assessment is performed mathematically, through an examination of the variational setting for the least-squares formulation of an abstract nonlinear boundary-value problem, and also computationally, through the numerical simulation of the least-squares finite element solutions of both a nonlinear form of the Poisson equation and also the incompressible Navier–Stokes equations. The assessment suggests that although the least-squares principle indicates that minimization should be performed prior to linearization, such an approach is often impractical and not necessary.