Least-squares Formulation Research Articles

A Hull Normal Based Approach for Cylindrical Size Assessment

Digital measurement devices, such as coordinate measuring machines, laser scanning devices, and digital imaging, can provide highly accurate and precise coordinate data representing the sampled surface. However, this discrete measurement process can only account for measured data points, not the entire continuous form, and is heavily influenced by the algorithm that interprets the measured data. The definition of cylindrical size for an external feature as specified by ASME Y14.5.1M-1994 [The American Society of Mechanical Engineers, 1995, Dimensioning and Tolerancing, ASME Standard Y14.5M-1994, ASME, New York, NY; The American Society of Mechanical Engineers, 1995, Mathematical Definition of Dimensioning and Tolerancing Principles, ASME Standard Y14.5.1M-1994, ASME, New York, NY] matches the analytical definition of a minimum circumscribing cylinder (MCC) when rule no. 1 [The American Society of Mechanical Engineers, 1995, Dimensioning and Tolerancing, ASME Standard Y14.5M-1994, ASME, New York, NY; The American Society of Mechanical Engineers, 1995, Mathematical Definition of Dimensioning and Tolerancing Principles, ASME Standard Y14.5.1M-1994, ASME, New York, NY] is applied to ensure a linear axis. Even though the MCC is a logical choice for size determination, it is highly sensitive to the sampling method and any uncertainties encountered in that process. Determining the least-sum-of-squares solution is an alternative method commonly utilized in size determination. However, the least-squares formulation seeks an optimal solution not based on the cylindrical size definition [The American Society of Mechanical Engineers, 1995, Dimensioning and Tolerancing, ASME Standard Y14.5M-1994, ASME, New York, NY; The American Society of Mechanical Engineers, 1995, Mathematical Definition of Dimensioning and Tolerancing Principles, ASME Standard Y14.5.1M-1994, ASME, New York, NY] and thus has been shown to be biased [Hopp, 1993, “Computational Metrology,” Manuf. Rev., 6(4), pp. 295–304; Nassef, and ElMaraghy, 1999, “Determination of Best Objective Function for Evaluating Geometric Deviations,” Int. J. Adv. Manuf. Technol., 15, pp. 90–95]. This work builds upon previous research in which the hull normal method was presented to determine the size of cylindrical bosses when rule no. 1 is applied [Turek, and Anand, 2007, “A Hull Normal Approach for Determining the Size of Cylindrical Features,” ASME, Atlanta, GA]. A thorough analysis of the hull normal method’s performance in various circumstances is presented here to validate it as a superior alternative to the least-squares and MCC solutions for size evaluation. The goal of the hull normal method is to recreate the sampled surface using computational geometry methods and to determine the cylinder’s axis and radius based upon it. Based on repetitive analyses of random samples of data from several measured parts and generated forms, it was concluded that the hull normal method outperformed all traditional solution methods. The hull normal method proved to be robust by having a lower bias and distributions that were skewed toward the true value of the radius, regardless of the amount of form error.

A least-square semismooth Newton method for the second-order cone complementarity problem

We present a nonlinear least-square formulation for the second-order cone complementarity problem based on the Fischer–Burmeister (FB) function and the plus function. This formulation has two-fold advantages. First, the operator involved in the over-determined system of equations inherits the favourable properties of the FB function for local convergence, for example, the (strong) semi-smoothness; second, the natural merit function of the over-determined system of equations share all the nice features of the class of merit functions f YF studied in [J.-S. Chen and P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Program. 104 (2005), pp. 293–327] for global convergence. We propose a semi-smooth Levenberg–Marquardt method to solve the arising over-determined system of equations, and establish the global and local convergence results. Among others, the superlinear (quadratic) rate of convergence is obtained under strict complementarity of the solution and a local error bound assumption, respectively. Numerical results verify the advantages of the least-square reformulation for difficult problems.

Cooperative Spectrum Sensing for Cognitive Radios Using Kriged Kalman Filtering

A cooperative cognitive radio (CR) sensing problem is considered, where a number of CRs collaboratively detect the presence of primary users (PUs) by exploiting the novel notion of channel gain (CG) maps. The CG maps capture the propagation medium per frequency from any point in space and time to each CR user. They are updated in real-time using Kriged Kalman filtering (KKF), a tool with well-appreciated merits in geostatistics. In addition, the CG maps enable tracking the transmit-power and location of an unknown number of PUs, via a sparse regression technique. The latter exploits the sparsity inherent to the PU activities in a geographical area, using an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm regularized, sparsity-promoting weighted least-squares formulation. The resulting sparsity-cognizant tracker is developed in both centralized and distributed formats, to reduce computational complexity and memory requirements of a batch alternative. Numerical tests demonstrate considerable performance gains achieved by the proposed algorithms .

Analysis and Computation of Compatible Least-Squares Methods for div-curl Equations

We develop and analyze least-squares finite element methods for two complementary div-curl elliptic boundary value problems. The first one prescribes the tangential component of the vector field on the boundary and is solved using curl-conforming elements. The second problem specifies the normal component of the vector field and is handled by div-conforming elements. We prove that both least-squares formulations are norm-equivalent with respect to suitable discrete norms, yield optimal asymptotic error estimates, and give rise to algebraic systems that can be solved by efficient algebraic multigrid methods. Numerical results that illustrate scalability of iterative solvers and optimal rates of convergence are also included.

Canonical Correlation Analysis for Multilabel Classification: A Least-Squares Formulation, Extensions, and Analysis

Canonical Correlation Analysis (CCA) is a well-known technique for finding the correlations between two sets of multidimensional variables. It projects both sets of variables onto a lower-dimensional space in which they are maximally correlated. CCA is commonly applied for supervised dimensionality reduction in which the two sets of variables are derived from the data and the class labels, respectively. It is well-known that CCA can be formulated as a least-squares problem in the binary class case. However, the extension to the more general setting remains unclear. In this paper, we show that under a mild condition which tends to hold for high-dimensional data, CCA in the multilabel case can be formulated as a least-squares problem. Based on this equivalence relationship, efficient algorithms for solving least-squares problems can be applied to scale CCA to very large data sets. In addition, we propose several CCA extensions, including the sparse CCA formulation based on the 1-norm regularization. We further extend the least-squares formulation to partial least squares. In addition, we show that the CCA projection for one set of variables is independent of the regularization on the other set of multidimensional variables, providing new insights on the effect of regularization on CCA. We have conducted experiments using benchmark data sets. Experiments on multilabel data sets confirm the established equivalence relationships. Results also demonstrate the effectiveness and efficiency of the proposed CCA extensions.

Application of the least squares approach to fixed beamformer design with frequency-invariant constraints

An application of the least squares (LS) approach to the design of frequency-invariant beamformers is proposed, based on three different formulations of the problem. The first one is the traditional linearly constrained LS formulation with its solution given by the Lagrange multipliers method. The second and the third ones, which are formulated as an unconstrained LS problem and a constrained total LS problem, respectively, can be considered as extensions of the traditional total LS method and their solutions are obtained by finding the minimum generalised eigenvector of two matrices. Compared with the conventional LS-based methods, a frequency-invariance controlling element is incorporated into the proposed cost functions. Design examples including both broadside and off-broadside main beams are provided with a satisfactory frequency invariant property and sidelobe attenuation. The proposed approach is general and can be applied to different array structures.

Efficiency Based Adaptive Local Refinement for First-Order System Least-Squares Formulations

In this paper, we propose new adaptive local refinement (ALR) strategies for first-order system least-squares finite elements in conjunction with algebraic multigrid methods in the context of nested iteration. The goal is to reach a certain error tolerance with the least amount of computational cost and nearly uniform distribution of the error over all elements. To accomplish this, the refinement decision at each refinement level is determined based on optimizing efficiency measures that take into account both error reduction and computational cost. Two efficiency measures are discussed: predicted error reduction and predicted computational cost. These methods are first applied to a two-dimensional (2D) Poisson problem with steep gradients, and the results are compared with the threshold-based methods described in [W. Dörfler, SIAM J. Numer. Anal., 33 (1996), pp. 1106–1124]. Next, these methods are applied to a 2D reduced model of the incompressible, resistive magnetohydrodynamic equations. These equations are used to simulate instabilities in a large aspect-ratio tokamak. We show that, by using the new ALR strategies on this system, we are able to resolve the physics using only 10 percent of the computational cost used to approximate the solutions on a uniformly refined mesh within the same error tolerance.

Reconstruction of Large, Irregularly Sampled Multidimensional Images. A Tensor-Based Approach

Many practical applications require the reconstruction of images from irregularly sampled data. The spline formalism offers an attractive framework for solving this problem; the currently available methods, however, are hard to deploy for large-scale interpolation problems in dimensions greater than two (3-D, 3-D+time) because of an exponential increase of their computational cost (curse of dimensionality). Here, we revisit the standard regularized least-squares formulation of the interpolation problem, and propose to perform the reconstruction in a uniform tensor-product B-spline basis as an alternative to the classical solution involving radial basis functions. Our analysis reveals that the underlying multilinear system of equations admits a tensor decomposition with an extreme sparsity of its one dimensional components. We exploit this property for implementing a parallel, memory-efficient system solver. We show that the computational complexity of the proposed algorithm is essentially linear in the number of measurements and that its dependency on the number of dimensions is significantly less than that of the original sparse matrix-based implementation. The net benefit is a substantial reduction in memory requirement and operation count when compared to standard matrix-based algorithms, so that even 4-D problems with millions of samples become computationally feasible on desktop PCs in reasonable time. After validating the proposed algorithm in 3-D and 4-D, we apply it to a concrete imaging problem: the reconstruction of medical ultrasound images (3-D+time) from a large set of irregularly sampled measurements, acquired by a fast rotating ultrasound transducer.

Full-Envelope Modular Adaptive Control of a FighterAircraft Using Orthogonal Least Squares

A new adaptive nonlinear flight controller is designed for a high fidelity, six degrees of freedom F-16 model for the entire flight envelope. The design is based on a modular approach which separates the design of the control law and the online identifier. The control law design is based on backstepping with nonlinear damping terms to robustify the design against parameter estimation errors and unknown bounded disturbances. The flight envelope is partitioned into hyperboxes, for each hyperbox a locally valid incremental model is estimated based on the linearized equations of motion. A continuous-time formulation of orthogonal least squares is used for identification of these locally valid models. The obtained local models are interpolated by means of B-splines to obtain a smooth model valid for the complete flight envelope. The performance of the resulting nonlinear adaptive control design is evaluated on the F-16 aircraft model for representative flight conditions, maneuvers, and failure cases.

INTERACTIVE REGISTRATION OF INTRACELLULAR VOLUMES WITH RADIAL BASIS FUNCTIONS

We propose a novel approach to 3D image registration of intracellular volumes. The approach extends a standard image registration framework to the curved cell geometry. An intracellular volume is mapped onto another intracellular domain by using two pairs of point set surfaces approximating their nuclear and plasma membranes. The mapping function consists of the affine transformation, tetrahedral barycentric interpolation, and least-squares formulation of radial basis functions for extracted cell geometry features. An interactive volume registration system is also developed based on our approach. We demonstrate that our approach is capable of creating cell models containing multiple organelles from observed data of living cells.

Galerkin least-squares approximations for upper-convected Maxwell fluid flows

In this work finite element approximations for upper-convected Maxwell fluid flows are presented. The mechanical model is approximated by a Galerkin least-squares formulation in extra-stress, pressure and velocity. This formulation has the advantage of remaining stable in locally elastic-dominated flow regions even employing a combination of equal-order finite element interpolations. The performance of the proposed formulation is evaluated by analyzing the flow around a cylinder kept by two parallel plates, for the Deborah number ranging from 0 to 0.9. The numerical results confirm the good features of the GLS formulation, since stable solutions are obtained for increasing elastic effects.

Simulation of chemical reactors using the least-squares spectral element method

In this work, we address the issue of solving detailed mathematical models of chemical reactors by using a method based on the least-squares formulation. The article presents the resolution of an axial dispersion model, and makes an analysis of relevant issues that are involved when solving this type of models using the least-squares spectral element method. A central issue is that C 1 elements are required if no additional variables are to be defined. We analyze two different alternatives for achieving global first-order differentiability without increasing the number of variables: using C 1 Hermite elements and imposing the constraint weakly using C 0 elements. We include numerical examples for the case of the transformation of natural gas into synthesis gas. The least-squares spectral element method showed no major difficulties in solving the model equations of the model, in spite of the steep variations obtained. We conclude that the optimal choice for achieving global first-order differentiability depends on the problem and the objectives of the calculation. For the model presented here, the Hermite approach gave the fastest results, although it required a high condition number of the problem matrix.

Inverse Problem of a Piezothermoelastic Cylinder Subject to Transient Heating

An inverse problem of a radially polarized piezoelectric hollow circular cylinder of crystal class 6 mm is investigated. It is assumed that a voltage induced by the action of a time-varying temperature applied to the inner surface of the cylinder is measured on the outer surface. The inverse problem entails a determination of the heating temperature from knowledge of the measured voltage. First, an exact solution to the problem is found by solving the equations of equilibrium and electrostatics for the cylinder subject to the prescribed boundary conditions. Then a least-squares residual method that incorporates Lagrange multipliers for satisfaction of the boundary equations is employed to derive an approximate analytic solution. Both formulations are utilized in order to calculate the unknown heating temperature, and the corresponding temperature, displacement and stress fields in a cylinder of cadmium selenide. Numerical results based upon the least-squares residual formulation are found to compare favorably with those obtained by the exact analysis.

Vast-scale Outdoor Navigation Using Adaptive Relative Bundle Adjustment

In this paper we describe a relative approach to simultaneous localization and mapping, based on the insight that a continuous relative representation can make the problem tractable at large scales. First, it is well known that bundle adjustment is the optimal non-linear least-squares formulation for this problem, in that its maximum-likelihood form matches the definition of the Cramer—Rao lower bound. Unfortunately, computing the maximum-likelihood solution is often prohibitively expensive: this is especially true during loop closures, which often necessitate adjusting all parameters in a loop. In this paper we note that it is precisely the choice of a single privileged coordinate frame that makes bundle adjustment costly, and that this expense can be avoided by adopting a completely relative approach. We derive a new relative bundle adjustment which, instead of optimizing in a single Euclidean space, works in a metric space defined by a manifold. Using an adaptive optimization strategy, we show experimentally that it is possible to solve for the full maximum-likelihood solution incrementally in constant time, even at loop closure. Our approach is, by definition, everywhere locally Euclidean, and we show that the local Euclidean estimate matches that of traditional bundle adjustment. Our system operates online in realtime using stereo data, with fast appearance-based loop closure detection. We show results on over 850,000 images that indicate the accuracy and scalability of the approach, and process over 330 GB of image data into a relative map covering 142 km of Southern England. To demonstrate a baseline sufficiency for navigation, we show that it is possible to find shortest paths in the relative maps we build, in terms of both time and distance. Query images from the web of popular landmarks around London, such as the London Eye or Trafalgar Square, are matched to the relative map to provide route planning goals.

Effective incorporating spatial information in a mutual information based 3D–2D registration of a CT volume to X-ray images

This paper addresses the problem of estimating the 3D rigid poses of a CT volume of an object from its 2D X-ray projection(s). We use maximization of mutual information, an accurate similarity measure for multi-modal and mono-modal image registration tasks. However, it is known that the standard mutual information measures only take intensity values into account without considering spatial information and their robustness is questionable. In this paper, instead of directly maximizing mutual information, we propose to use a variational approximation derived from the Kullback-Leibler bound. Spatial information is then incorporated into this variational approximation using a Markov random field model. The newly derived similarity measure has a least-squares form and can be effectively minimized by a multi-resolution Levenberg-Marquardt optimizer. Experiments were conducted on datasets from two applications: (a) intra-operative patient pose estimation from a limited number (e.g. 2) of calibrated fluoroscopic images, and (b) post-operative cup orientation estimation from a single standard X-ray radiograph with/without gonadal shielding. The experiment on intra-operative patient pose estimation showed a mean target registration accuracy of 0.8 mm and a capture range of 11.5 mm, while the experiment on estimating the post-operative cup orientation from a single X-ray radiograph showed a mean accuracy below 2° for both anteversion and inclination. More importantly, results from both experiments demonstrated that the newly derived similarity measures were robust to occlusions in the X-ray image(s).

Superfast-Trainable Multi-Class Probabilistic Classifier by Least-Squares Posterior Fitting

Kernel logistic regression (KLR) is a powerful and flexible classification algorithm, which possesses an ability to provide the confidence of class prediction. However, its training—typically carried out by (quasi-)Newton methods—is rather timeconsuming. In this paper, we propose an alternative probabilistic classification algorithm called Least-Squares Probabilistic Classifier (LSPC). KLR models the class-posterior probability by the log-linear combination of kernel functions and its parameters are learned by (regularized) maximum likelihood. In contrast, LSPC employs the linear combination of kernel functions and its parameters are learned by regularized least-squares fitting of the true class-posterior probability. Thanks to this linear regularized least-squares formulation, the solution of LSPC can be computed analytically just by solving a regularized system of linear equations in a class-wise manner. Thus LSPC is computationally very efficient and numerically stable. Through experiments, we show that the computation time of LSPC is faster than that of KLR by orders of magnitude, with comparable classification accuracy.

Determination of V-notch SIFs in multi-material anisotropic wedges by digital correlation experiments

In this paper, theoretical formulations based on the Stroh’s complex function approach were used to find the displacement field and H-integral of a sharp V-notch formed from several anisotropic materials. Displacements from the image-correlation experiments are then substituted into the least-squares formulation to find V-notch stress intensity factors (SIFs) in multi-material anisotropic wedges. Validations using the H-integral indicate that the experimental SIFs evaluated from the proposed method of acceptable accuracy. The major advantage is that the proposed method only requires displacements inside the specimen, and displacements near the notch tip, specimen boundaries, or notch surfaces are not necessary.

A least-squares collocation spectral elements formulation for solving PDEs in complex geometry domains.

A least squares collocation scheme is used to solve PDEs defined on a complex geometry domain. Triangular finite elements are used to build a macro-mesh of the whole domain and Fekete points are used as collocation points. We combine together the standard least squares method and spectral method to compute the global solution. The assembling process of local solutions is explained through an algorithm. The success of this method is studied throughout a test problem. Numerical results prove the accuracy of the method which takes into account the boundary conditions.

Identification of Contact Parameters from Stiff Multi-point Contact Robotic Operations

Computer simulations play an important role in the design and validation of constrained robotic operations. The fidelity of these simulations, however, depends on the specification of contact dynamics parameters, which often need to be determined experimentally. In this paper we investigate the identification of contact parameters from complex stiff multi-point contact scenarios encountered in typical robotic operations using a recently developed least-squares-based method. This method is extended to deal with geometric uncertainties by employing a non-linear separable least-squares formulation. The latter is solved using a variable projection method, and allows simultaneous identification of contact parameters and geometric uncertainties. The conditions for observability of geometric uncertainties are derived and a regularized formulation is proposed in case the identification of geometric uncertainties is ill-conditioned. The applicability of the original method and the benefits of the extended method with identification of geometric uncertainties for the identification of the contact parameters are illustrated by means of experimental data measured with the Special Purpose Dexterous Manipulator (SPDM) Task Verification Facility (STVF) manipulator at the Canadian Space Agency (CSA).

A modified least‐squares mixed finite element with improved momentum balance

AbstractThe main goal of this contribution is to provide an improved mixed finite element for quasi‐incompressible linear elasticity. Based on a classical least‐squares formulation, a modified weak form with displacements and stresses as process variables is derived. This weak form is the basis for a finite element with an advanced fulfillment of the momentum balance and therefore with a better performance. For the continuous approximation of stresses and displacements on the triangular and tetrahedral elements, lowest‐order Raviart–Thomas and linear standard Lagrange interpolations can be used. It is shown that coercivity and continuity of the resulting asymmetric bilinear form could be established with respect to appropriate norms. Further on, details about the implementation of the least‐squares mixed finite elements are given and some numerical examples are presented in order to demonstrate the performance of the proposed formulation. Copyright © 2009 John Wiley &amp; Sons, Ltd.