Least-squares Formulation Research Articles

Microwave measurements of dielectric properties using a gap-coupled multi-mode coaxial resonator

The paper describes the measuring cell based on a gap-coupled multi-modes coaxial resonator. The role of coupling effect in forming the spectra of longitudinal odd modes has been analyzed using the circuit model of the gap-coupled resonator. The relative dielectric constant εr′ and the loss tangent tanδ of a material under test are derived from the measured input reflectance employing the least-square formulation and non-linear optimization. Experimental results are presented for the samples of crude oil with varying water content in the range 0–25%. The use of the gap-coupled configuration offers a possibility of easily adjustable coupling coefficient resulting in an application of the same measuring cells for microwave characterizations both low loss and lossy liquids and solids in a wide frequency range.

Global approach for transient shear wave inversion based on the adjoint method: a comprehensive 2D simulation study

Shear wave imaging (SWI) maps soft tissue elasticity by measuring shear wave propagation with ultrafast ultrasound acquisitions (10 000 frames s−1). This spatiotemporal data can be used as an input for an inverse problem that determines a shear modulus map. Common inversion methods are local: the shear modulus at each point is calculated based on the values of its neighbour (e.g. time-of-flight, wave equation inversion). However, these approaches are sensitive to the information loss such as noise or the lack of the backscattered signal. In this paper, we evaluate the benefits of a global approach for elasticity inversion using a least-squares formulation, which is derived from full waveform inversion in geophysics known as the adjoint method. We simulate an acoustic waveform in a medium with a soft and a hard lesion. For this initial application, full elastic propagation and viscosity are ignored. We demonstrate that the reconstruction of the shear modulus map is robust with a non-uniform background or in the presence of noise with regularization. Compared to regular local inversions, the global approach leads to an increase of contrast (∼+3 dB) and a decrease of the quantification error (∼+2%). We demonstrate that the inversion is reliable in the case when there is no signal measured within the inclusions like hypoechoic lesions which could have an impact on medical diagnosis.

A robust high-resolution details preserving denoising algorithm for meshes

Finely captured meshes often contain details like sharp edges, corners and shallow features. In order to improve the quality of these meshes, we present a robust and efficient high-resolution details-preserving mesh denoising algorithm. Our method consists of three stages. For each triangular face and its face neighborhood, we improve a robust density-based clustering method and apply it to the face neighborhood to extract a subset of neighbors which belong to the same cluster as the central face. And then, we filter the central face normal iteratively within this subset to remove noise. Because the faces within the extracted subset are not distributed across high-resolution details, our normal filtering can preserve such details as much as possible. Finally, we update the vertex positions to be consistent with the filtered face normals using a least-squares formulation. Experiments on various types of meshes indicate that our method has advantages over previous surface denoising methods.

Flow of yield-stress liquids through an axisymmetric abrupt expansion-contraction

We analyze the effect of inertia on the flow of viscoplastic liquids through an axisymmetric expansion followed by a contraction. The fluid is assumed to behave according to the Generalized Newtonian Liquid (GNL) model whose viscosity function is given by the SMD equation [1]. The conservation equations of mass and momentum, together with the constitutive equation, are approximated by a mixed Galerkin least-squares (GLS) formulation in terms of pressure and velocity. A parametric study is performed to investigate the influence of inertia and viscoplasticity on this flow. It was found that the topology of the yield surface is strongly affected by the flow intensity, but only mildly influenced by the shear-thinning fluid behavior. Inertia dramatically changes the topology of the yield surfaces, such that the fore-aft symmetry observed for low-inertia flows in the middle tube is lost.

A least-squares formulation for the approximation of null controls for the Stokes system

This work deals with the approximation of distributed null controls for the Stokes equation. The existence of L2 controls has been obtained by Fursikov and Imanuvilov (1996) [5] via Carleman-type estimates. We introduce and analyze a least-squares formulation of the controllability problem, and we show that it allows the construction of convergent sequences of functions toward null controls for the Stokes system.

Efficient Estimation of Reflectance Parameters From Imaging Spectroscopy

In this paper, we address the problem of efficiently recovering reflectance parameters from a single multispectral or hyperspectral image. To do so, we propose a shapelet based estimator that employs shapelets to recover the shading in the image. The optimization setting presented is based upon a three-step process. The first of these concerns the recovery of the surface reflectance and the specular coefficients through a constrained optimization approach. Second, we update the illuminant power spectrum using a simple least-squares formulation. Third, the shading is computed directly once the updated illuminant power spectrum is obtained. This yields a computationally efficient method that achieves speed-ups of nearly an order of magnitude over its closest alternative without compromising performance. We provide results on illuminant power spectrum computation, shading recovery, skin recognition and replacement of the scene illuminant, and object reflectance in real-world images.

A least-squares method for the numerical solution of the Dirichlet problem for the elliptic monge − ampère equation in dimension two

We address in this article the computation of the convex solutions of the Dirichlet problem for the real elliptic Monge − Ampere equation for general convex domains in two dimensions. The method we discuss combines a least-squares formulation with a relaxation method. This approach leads to a sequence of Poisson − Dirichlet problems and another sequence of low dimensional algebraic eigenvalue problems of a new type. Mixed finite element approximations with a smoothing procedure are used for the computer implementation of our least-squares/relaxation methodology. Domains with curved boundaries are easily accommodated. Numerical experiments show the convergence of the computed solutions to their continuous counterparts when such solutions exist. On the other hand, when classical solutions do not exist, our methodology produces solutions in a least-squares sense.

EFFICIENT IMPLEMENTATION OF THE CAPON BEAMFORMING USING THE LEVENBERG-MARQUARDT SCHEME FOR TWO DIMENSIONAL AOA ESTIMATION

In this paper, we adopt the Levenberg-Marquardt (LM) algorithm to implement the nonlinear multivariable optimization for azimuth/elevation angle-of-arrival (AOA) estimation based on the Capon beamforming algorithm. The formulation is based on the fact that the cost function of the Capon algorithm can be expressed in a least-squares form. The performance in terms of the root mean square error (RMSE) and the computational complexity is illustrated via numerical results.

A weighted least squares finite element method for elliptic problems with degenerate and singular coefficients

We consider second order elliptic partial differential equations with coefficients that are singular or degenerate at an interior point of the domain. This paper presents formulation and analysis of a novel weighted-norm least squares finite element method for this class of problems. We propose a weighting scheme that eliminates the pollution effect and recovers optimal convergence rates. Theoretical results are carried out in appropriately weighted Sobolev spaces and include ellipticity bounds on the weighted homogeneous least squares functional, regularity bounds on the elliptic operator, and error estimates. Numerical experiments confirm the predicted error bounds.

Fitting the void: Data boundaries, point distributions and strain analysis

An objective method is developed for identifying the shape of the void generated by analysis of the spatial arrangement of point distributions using the Fry method and in particular Delaunay triangulation nearest neighbour data. The method works by numerically minimising a weighted non-linear least-squares formulation which tends to selectively apply higher weights to points below the boundary and less weight to points above the boundary. Sampling errors are estimated using a bootstrap procedure. A simulation study indicates that the method works best for closely packed object arrangements and tends to underestimate the strain axial ratio at high imposed strains. Overall the method is found to be objective and consistent.

An incremental least squares algorithm for large scale linear classification

In this work we consider the problem of training a linear classifier by assuming that the number of data is huge (in particular, data may be larger than the memory capacity). We propose to adopt a linear least-squares formulation of the problem and an incremental recursive algorithm which requires to store a square matrix (whose dimension is equal to the number of features of the data). The algorithm (very simple to implement) converges to the solution using each training data once, so that it effectively handles possible memory issues and is a viable method for linear large scale classification and for real time applications, provided that the number of features of the data is not too large (say of the order of thousands). The extensive computational experiments show that the proposed algorithm is at least competitive with the state-of-the-art algorithms for large scale linear classification.

Flow of elasto-viscoplastic thixotropic liquids past a confined cylinder

A numerical investigation of elasto-viscoplastic thixotropic liquids flowing past a cylinder confined in a parallel-plate channel is performed, using a recently proposed constitutive equation. The conservation equations are solved using a four-field Galerkin least-squares formulation in terms of the extra-stress, pressure, velocity, and structure parameter – a scalar quantity that represents the structuring level of the material microstructure. The focus is on determining the effect of both thixotropy and elasticity on the topology of yielded and unyielded regions of the channel. The effect on the cylinder drag coefficient is also studied. Relevant ranges of the dimensionless relaxation time and equilibrium time of the material structure are investigated. The numerical results reveal striking effects of thixotropy and elasticity. The trends observed are physically sound and in accordance with the related literature.

3D implicit mesh-less method for compressible flow calculations

An implicit mesh-less method is developed for calculation of compressible flows around three dimensional complex geometries. The algorithm is applied directly to the differential form of the governing equations using least-square formulation. A dual-time implicit time discretization scheme is developed, and the computational efficiency is enhanced by adopting accelerating techniques, such as local time stepping, residual smoothing and enthalpy damping. Two different artificial dissipation techniques are employed for stability preservation and it is shown that the scalar one is more efficient in terms of accuracy and computational time. The capabilities of the method are demonstrated by flow computations around different geometries under subsonic and transonic flow conditions. Results are presented which indicate good agreement with experimental and other reliable numerical data. The method is shown to reduce computational time by about 50% compared with the alternative explicit method.

Regression on parametric manifolds: Estimation of spatial fields, functional outputs, and parameters from noisy data

In this Note we extend the Empirical Interpolation Method (EIM) to a regression context which accommodates noisy (experimental) data on an underlying parametric manifold. The EIM basis functions are computed Offline from the noise-free manifold; the EIM coefficients for any function on the manifold are computed Online from experimental observations through a least-squares formulation. Noise-induced errors in the EIM coefficients and in linear-functional outputs are assessed through standard confidence intervals and without knowledge of the parameter value or the noise level. We also propose an associated procedure for parameter estimation from noisy data.

Assessment of a layerwise mixed least-squares model for analysis of multilayered piezoelectric composite plates

A layerwise mixed finite element model is developed based on the least-squares formulation for the coupled electromechanical static analysis of multilayered plates with piezoelectric and composite layers. The model assumes a layerwise variable description for displacements, transverse stresses and in-plane strains, along with the electrostatic potential, transverse electric displacement and in-plane electric field components, taken as independent variables. This original choice for the layerwise mixed formulation is intended to ensure the a priori and complete fulfilment of the interlaminar C0 continuity of both mechanical and electrical variables, which is due to compatibility and equilibrium conditions. Numerical applications are shown for assessment of the model predictive capabilities by comparison with available exact three-dimensional solutions, considering multilayered piezoelectric composite plates of various side-to-thickness ratios, under an applied load or surface potential.

Comparison of partial least squares and lasso regression techniques as applied to laser-induced breakdown spectroscopy of geological samples

A remote laser-induced breakdown spectrometer (LIBS) designed to simulate the ChemCam instrument on the Mars Science Laboratory Rover Curiosity was used to probe 100 geologic samples at a 9-m standoff distance. ChemCam consists of an integrated remote LIBS instrument that will probe samples up to 7m from the mast of the rover and a remote micro-imager (RMI) that will record context images. The elemental compositions of 100 igneous and highly-metamorphosed rocks are determined with LIBS using three variations of multivariate analysis, with a goal of improving the analytical accuracy. Two forms of partial least squares (PLS) regression are employed with finely-tuned parameters: PLS-1 regresses a single response variable (elemental concentration) against the observation variables (spectra, or intensity at each of 6144 spectrometer channels), while PLS-2 simultaneously regresses multiple response variables (concentrations of the ten major elements in rocks) against the observation predictor variables, taking advantage of natural correlations between elements. Those results are contrasted with those from the multivariate regression technique of the least absolute shrinkage and selection operator (lasso), which is a penalized shrunken regression method that selects the specific channels for each element that explain the most variance in the concentration of that element. To make this comparison, we use results of cross-validation and of held-out testing, and employ unscaled and uncentered spectral intensity data because all of the input variables are already in the same units. Results demonstrate that the lasso, PLS-1, and PLS-2 all yield comparable results in terms of accuracy for this dataset.However, the interpretability of these methods differs greatly in terms of fundamental understanding of LIBS emissions. PLS techniques generate principal components, linear combinations of intensities at any number of spectrometer channels, which explain as much variance in the response variables as possible while avoiding multicollinearity between principal components. When the selected number of principal components is projected back into the original feature space of the spectra, 6144 correlation coefficients are generated, a small fraction of which are mathematically significant to the regression. In contrast, the lasso models require only a small number (<24) of non-zero correlation coefficients (β values) to determine the concentration of each of the ten major elements. Causality between the positively-correlated emission lines chosen by the lasso and the elemental concentration was examined. In general, the higher the lasso coefficient (β), the greater the likelihood that the selected line results from an emission of that element. Emission lines with negative β values should arise from elements that are anti-correlated with the element being predicted. For elements except Fe, Al, Ti, and P, the lasso-selected wavelength with the highest β value corresponds to the element being predicted, e.g. 559.8nm for neutral Ca. However, the specific lines chosen by the lasso with positive β values are not always those from the element being predicted. Other wavelengths and the elements that most strongly correlate with them to predict concentration are obviously related to known geochemical correlations or close overlap of emission lines, while others must result from matrix effects. Use of the lasso technique thus directly informs our understanding of the underlying physical processes that give rise to LIBS emissions by determining which lines can best represent concentration, and which lines from other elements are causing matrix effects.

Methodology and theory for partial least squares applied to functional data

The partial least squares procedure was originally developed to estimate the slope parameter in multivariate parametric models. More recently it has gained popularity in the functional data literature. There, the partial least squares estimator of slope is either used to construct linear predictive models, or as a tool to project the data onto a one-dimensional quantity that is employed for further statistical analysis. Although the partial least squares approach is often viewed as an attractive alternative to projections onto the principal component basis, its properties are less well known than those of the latter, mainly because of its iterative nature. We develop an explicit formulation of partial least squares for functional data, which leads to insightful results and motivates new theory, demonstrating consistency and establishing convergence rates.

Least-squares spectral element solution of incompressible Navier–Stokes equations with adaptive refinement

Least-squares spectral element solution of steady, two-dimensional, incompressible flows are obtained by approximating velocity, pressure and vorticity variable set on Gauss–Lobatto–Legendre nodes. Constrained Approximation Method is used for h- and p-type nonconforming interfaces of quadrilateral elements. Adaptive solutions are obtained using a posteriori error estimates based on least squares functional and spectral coefficient. Effective use of p-refinement to overcome poor mass conservation drawback of least-squares formulation and successful use of h- and p-refinement together to solve problems with geometric singularities are demonstrated. Capabilities and limitations of the developed code are presented using Kovasznay flow, flow past a circular cylinder in a channel and backward facing step flow.

Nonconforming Least-Squares Method for Elliptic Partial Differential Equations with Smooth Interfaces

In this paper a least-squares based method is proposed for elliptic interface problems in two dimensions, where the interface is smooth. The underlying method is spectral element method. The least-squares formulation is based on the minimization of a functional as defined in (4.1). The jump in the solution and its normal derivative across the interface are enforced (in an appropriate Sobolev norm) in the functional. The solution is obtained by solving the normal equations using preconditioned conjugate gradient method. Essentially the method is nonconforming, so a block diagonal matrix is constructed as a preconditioner based on the stability estimate where each diagonal block is decoupled. A conforming solution is obtained by making a set of corrections to the nonconforming solution as in Schwab (p and h---p Finite Element Methods, Clarendon Press, Oxford, 1998) and an error estimate in H 1-norm is given which shows the exponential convergence of the proposed method.

Robust affine iterative closest point algorithm with bidirectional distance

This study proposes a robust affine iterative closest point (ICP) algorithm based on bidirectional distance for the registration of m -dimensional ( m -D) point sets. Since the affine registration problem can be formulated as a least square (LS) problem by incorporating an affine transformation, this study first analyses the ill-posed problem of the affine registration and turn it into well-posed one by introducing the bidirectional distance into the LS formulation. Then, the corresponding affine ICP algorithm is proposed to solve the well-posed problem. By using the bidirectional distance, the proposed algorithm can directly estimate the affine transformation and converge monotonically to a local minimum from any given initial parameters. To obtain the desired global minimum, good initial parameters can be estimated by independent component analysis (ICA) technique. The proposed approach makes no geometric assumptions on point sets, so it is a general framework for affine registration of m -D point sets. Experimental results demonstrate its robustness and accuracy compared with the current state-of-the-art approaches.