Least-squares Formulation Research Articles

Globally Conservative, Hybrid Self-Adjoint Angular Flux and Least-Squares Method Compatible with Voids

In this paper, we derive a method for the second-order form of the transport equation that is both globally conservative and compatible with voids using the continuous finite element method. The main idea is to use the least-squares (LS) form of the transport equation in the void regions and the self-adjoint angular flux (SAAF) form elsewhere. While the SAAF formulation is globally conservative, the LS formulation needs correction in voids. The price to pay for this fix is the loss of symmetry of the bilinear form. We first derive this conservative LS (CLS) formulation in a void. Second, we combine the SAAF and CLS forms and end up with an hybrid SAAF-CLS method having the desired properties. We show that extending the theory to near-void regions is a minor complication and can be done without affecting the global conservation of the scheme. Being angular discretization-agnostic, this method can be applied to both discrete ordinates (SN) and spherical harmonics (PN) methods. However, since a globally conservative and void-compatible second-order form already exists for SN [Wang et al., Nucl. Sci. Eng., Vol. 176, p. 201 (2014)] but not for PN, we focus most of our attention on the latter angular discretization. We implement and test our method in Rattlesnake within the Multiphysics Object Oriented Simulation Environment (MOOSE) framework. The results are also compared to those of other methods.

Online Estimation of Multiple Harmonic Signals

In this paper, we propose a time-recursive multipitch estimation algorithm using a sparse reconstruction framework, assuming that only a few pitches from a large set of candidates are active at each time instant. The proposed algorithm does not require any training data, and instead utilizes a sparse recursive least-squares formulation augmented by an adaptive penalty term specifically designed to enforce a pitch structure on the solution. The amplitudes of the active pitches are also recursively updated, allowing for a smooth and more accurate representation. When evaluated on a set of ten music pieces, the proposed method is shown to outperform other general purpose multipitch estimators in either accuracy or computational speed, although not being able to yield performance as good as the state-of-the art methods, which are being optimally tuned and specifically trained on the present instruments. However, the method is able to outperform such a technique when used without optimal tuning, or when applied to instruments not included in the training data.

Least-Squares Proper Generalized Decompositions for Weakly Coercive Elliptic Problems

Proper generalized decompositions (PGDs) are a family of methods for efficiently solving high-dimensional PDEs, which seek to find a low-rank approximation to the solution of the PDE a priori. Convergence of PGD algorithms can only be proven for problems which are continuous, symmetric, and strongly coercive. In the particular case of problems which are only weakly coercive we have the additional issue that weak coercivity estimates are not guaranteed to be inherited by the low-rank PGD approximation. This can cause stability issues when employing a Galerkin PGD approximation of weakly coercive problems. In this paper we propose the use of PGD algorithms based on least-squares formulations which always lead to symmetric and strongly coercive problems and hence provide stable and provably convergent algorithms. Taking the Stokes problem as a prototypical example of a weakly coercive problem, we develop and compare rigorous least-squares PGD algorithms based on continuous least-squares estimates for two dif...

A Fluidity-Based First-Order System Least-Squares Method for Ice Sheets

This paper describes two first-order system least-squares (FOSLS) formulations of a nonlinear Stokes flow model for glaciers and ice sheets. In Glen's law, the most commonly used constitutive equation for ice rheology, the ice viscosity becomes infinite as the velocity gradients (strain rates) approach zero, which typically occurs near the ice surface where deformation rates are low, or when the basal slip velocities are high. The computational difficulties associated with the infinite viscosity are often overcome by an arbitrary modification of Glen's law that bounds the maximum viscosity. In this paper, two FOSLS formulations (the viscosity formulation and fluidity formulation) are presented. The viscosity formulation is a FOSLS representation of the standard nonlinear Stokes problem. The new fluidity formulation exploits the fact that only the product of the viscosity and strain rate appears in the nonlinear Stokes problem, a quantity that, in fact, approaches zero as the strain rate goes to zero. The fluidity formulation is expressed in terms of a new set of variables and overcomes the problem of infinite viscosity. The new formulation is well posed, and linearizations are essentially $H^1$-elliptic around approximations for which the viscosity is bounded. A nested iteration (NI) Newton-FOSLS-AMG (algebraic multigrid) approach is used to solve both nonlinear Stokes problems, in which most of the iterations are performed on the coarsest grid. Both formulations demonstrate optimal finite element convergence. However, the fluidity formulation is more accurate. The fluidity formulation results in linear systems that are more efficiently solved by AMG.

Increasing the robustness and applicability of full-waveform inversion: An optimal transport distance strategy

Full-waveform inversion starts being used as a standard stage of the seismic-imaging workflow, at the exploration scale, for the reconstruction of high-resolution wave velocity models. However, its successful application still relies on the estimation of an accurate enough initial velocity model, as well as on the design of a suitable hierarchical workflow, allowing it to feed the inversion process progressively with data. These two requirements are mandatory to avoid the cycle-skipping or phase-ambiguity problem when comparing observed and synthetic data. This difficulty is due to the definition of the full-waveform inversion problem as the least-squares minimization of the data misfit. The resulting misfit function has local minima, which correspond to the interpretation of the seismic data up to one or several phase shifts. In this article, we review an alternative formulation of full-waveform inversion based on the optimal transport distance we have proposed in recent studies. We propose to use a particular instance of the optimal transport problem, which is adapted to the interpretation of real seismic data and for which we design an efficient low-complexity numerical strategy. Numerical results in 2D and 3D configurations (BP 2004, Chevron 2014 benchmark model, SEG/EAGE overthrust model) show that this reformulation should yield a more convex misfit function, less prone to cycle skipping. In this study, we present a simple illustration on the Marmousi model, which illustrates how this new distance strongly relaxes the requirement on the initial model design. Starting from a rather simplistic approximation of the initial model, the method is able to reconstruct a meaningful estimation of the Marmousi model, while the standard least-squares formulation is trapped into a local, meaningless minimum.

Economic consequences of aviation system disruptions: A reduced-form computable general equilibrium analysis

The state of the art approach to economic consequence analysis (ECA) is computable general equilibrium (CGE) modeling. However, such models contain thousands of equations and cannot readily be incorporated into computerized systems to yield rapid estimates of economic impacts of various types of transportation system failures due to natural hazards, terrorist attacks or technological accidents. This paper presents a reduced-form approach to simplify the analytical content of CGE models and make them more transparent and enhance their utilization potential. The reduced-form CGE analysis is conducted by first running simulations one hundred times, varying key parameters, such as the magnitude of the initial shock, duration, location, behavioral responses, and resilience, according to a Latin Hypercube sampling procedure. Statistical analysis is then applied to the “synthetic data” results in the form of both ordinary least-squares and quantile regression. The analysis yields linear equations that are incorporated into a computerized system and utilized along with Monte Carlo simulation methods for propagating uncertainties in economic consequences. Although our demonstration and discussion focuses on aviation system disruptions caused by terrorist attacks, the approach can be applied to a broad range of threat scenarios.

Optimal-order isogeometric collocation at Galerkin superconvergent points

In this paper we investigate numerically the order of convergence of an isogeometric collocation method that builds upon the least-squares collocation method presented in Anitescu et al. (2015) and the variational collocation method presented in Gomez and De Lorenzis (2016). The focus is on smoothest B-splines/NURBS approximations, i.e, having global Cp−1 continuity for polynomial degree p. Within the framework of Gomez and De Lorenzis (2016), we select as collocation points a subset of those considered in Anitescu et al. (2015), which are related to the Galerkin superconvergence theory. With our choice, that features local symmetry of the collocation stencil, we improve the convergence behavior with respect to Gomez and De Lorenzis (2016), achieving optimal L2-convergence for odd degree B-splines/NURBS approximations. The same optimal order of convergence is seen in Anitescu et al. (2015), where, however a least-squares formulation is adopted. Further careful study is needed, since the robustness of the method and its mathematical foundation are still unclear.

Blending design of composite panels with lamination parameters

ABSTRACTIn this paper, a new optimisation method incorporating lamination parameters and a guide-based blending model is proposed. Lamination parameters for a guide laminate and ply number of each panel are employed as design variables for optimisation with a parallel real-coded genetic algorithm incorporating structure behaviour and manufacturing constraints. During the optimisation process, with a form of least squares fitting adopted, another genetic algorithm is used to obtain the guide stacking sequence of the guide laminate from the guide lamination parameters, and then the laminate configurations of each panel are obtained from the guide stacking sequence and number of plies for each panel. The proposed framework is demonstrated via design of an 18-panel horseshoe configuration, where each panel is optimised individually with a buckling constraint. Numerical results indicate that the present algorithm is capable of obtaining fully blended designs.

Exploring canonical correlation analysis with subspace and structured sparsity for web image annotation

Canonical correlation analysis (CCA) has been extensively exploited for modelling Internet multimedia. However, two major challenges are raised for the classical CCA. First, CCA frequently fails to remove noisy and irrelevant features. Second, CCA cannot effectively capture the correlation between semantic labels, which is especially beneficial for annotating web images. In this paper, we propose a new framework that integrates structural sparsity and low-rank shared subspace into the least-squares formulation of CCA. Under this framework, multiple label interactions can be uncovered by the shared common structure of the input space. Meanwhile, a few highly discriminative features can be decided via the structural sparse norm. Owing to the presence of non-smooth structured sparsity, a new efficient iterative algorithm is derived with guaranteed convergence. The empirical studies over several popular web image data collections consistently deliver the effectiveness of our new formulation in comparison with competing algorithms.

A Robust and Multi-Weighted Approach to Estimating Topographically Correlated Tropospheric Delays in Radar Interferograms

Spatial and temporal variations in the vertical stratification of the troposphere introduce significant propagation delays in interferometric synthetic aperture radar (InSAR) observations. Observations of small amplitude surface deformations and regional subsidence rates are plagued by tropospheric delays, and strongly correlated with topographic height variations. Phase-based tropospheric correction techniques assuming a linear relationship between interferometric phase and topography have been exploited and developed, with mixed success. Producing robust estimates of tropospheric phase delay however plays a critical role in increasing the accuracy of InSAR measurements. Meanwhile, few phase-based correction methods account for the spatially variable tropospheric delay over lager study regions. Here, we present a robust and multi-weighted approach to estimate the correlation between phase and topography that is relatively insensitive to confounding processes such as regional subsidence over larger regions as well as under varying tropospheric conditions. An expanded form of robust least squares is introduced to estimate the spatially variable correlation between phase and topography by splitting the interferograms into multiple blocks. Within each block, correlation is robustly estimated from the band-filtered phase and topography. Phase-elevation ratios are multiply- weighted and extrapolated to each persistent scatter (PS) pixel. We applied the proposed method to Envisat ASAR images over the Southern California area, USA, and found that our method mitigated the atmospheric noise better than the conventional phase-based method. The corrected ground surface deformation agreed better with those measured from GPS.

Distributed Sequential Location Estimation of a Gas Source via Convex Combination in WSNs

Localization of the hazardous gas source plays an important role in the protection of public security, since it can save a lot of time for subsequent rescue works. For gas source localization (GSL), a large number of gas sensor nodes can be rapidly deployed to construct a wireless sensor network (WSN) and cover the whole concerned area. Although least-squares (LS) methods can solve the problem of GSL in WSNs regardless of the distribution of measurement noises, centralized LS methods are not power efficient and robust since they require the gathering and processing of large-scale measurements on a central node. In this paper, we propose a novel distributed method for GSL in WSNs, which is performed on a sequence of sensor nodes successively. Each sensor node in the sequence conducts an individual estimation and a convex combination. The individual estimation is inspired by the LS formulation of the problem of GSL in WSNs. The proposed method is fully distributed and computationally efficient, and it does not rely on the absolute location of the sensor nodes. Extensive simulation results and a set of experimental results demonstrate that the success rate and localization accuracy of the proposed method are generally higher than those of the trust-region-reflective method.

An adaptive least-squares FEM for the Stokes equations with optimal convergence rates

This paper introduces the first adaptive least-squares finite element method (LS-FEM) for the Stokes equations with optimal convergence rates based on the newest vertex bisection with lowest-order Raviart-Thomas and conforming $$P_1$$P1 discrete spaces for the divergence least-squares formulation in 2D. Although the least-squares functional is a reliable and efficient error estimator, the novel refinement indicator stems from an alternative explicit residual-based a posteriori error control with exact solve. Particular interest is on the treatment of the data approximation error which requires a separate marking strategy. The paper proves linear convergence in terms of the levels and optimal convergence rates in terms of the number of unknowns relative to the notion of a non-linear approximation class. It extends and generalizes the approach of Carstensen and Park (SIAM J. Numer. Anal. 53:43---62 2015) from the Poisson model problem to the Stokes equations.

A multi-label feature extraction algorithm via maximizing feature variance and feature-label dependence simultaneously

Dimensionality reduction is an important pre-processing procedure for multi-label classification to mitigate the possible effect of dimensionality curse, which is divided into feature extraction and selection. Principal component analysis (PCA) and multi-label dimensionality reduction via dependence maximization (MDDM) represent two mainstream feature extraction techniques for unsupervised and supervised paradigms. They produce many small and a few large positive eigenvalues respectively, which could deteriorate the classification performance due to an improper number of projection directions. It has been proved that PCA proposed primarily via maximizing feature variance is associated with a least-squares formulation. In this paper, we prove that MDDM with orthonormal projection directions also falls into the least-squares framework, which originally maximizes Hilbert–Schmidt independence criterion (HSIC). Then we propose a novel multi-label feature extraction method to integrate two least-squares formulae through a linear combination, which maximizes both feature variance and feature-label dependence simultaneously and thus results in a proper number of positive eigenvalues. Experimental results on eight data sets show that our proposed method can achieve a better performance, compared with other seven state-of-the-art multi-label feature extraction algorithms.

Parametric Regression on the Grassmannian.

We address the problem of fitting parametric curves on the Grassmann manifold for the purpose of intrinsic parametric regression. We start from the energy minimization formulation of linear least-squares in Euclidean space and generalize this concept to general nonflat Riemannian manifolds, following an optimal-control point of view. We then specialize this idea to the Grassmann manifold and demonstrate that it yields a simple, extensible and easy-to-implement solution to the parametric regression problem. In fact, it allows us to extend the basic geodesic model to (1) a "time-warped" variant and (2) cubic splines. We demonstrate the utility of the proposed solution on different vision problems, such as shape regression as a function of age, traffic-speed estimation and crowd-counting from surveillance video clips. Most notably, these problems can be conveniently solved within the same framework without any specifically-tailored steps along the processing pipeline.

Modeling connected regions in arbitrary planar point clouds by robust B-spline approximation

This paper presents an algorithm for robustly approximating the boundary of a domain, latent in a planar set of scattered points, by a B-spline curve. The algorithm is characterized by three key features: First, we propose a distance measure, called the Asymmetric Distance (AD), which allows for handling outliers inside the curve and finding the outer boundary or concave hull by specifying very natural parameters like smoothness and accuracy. Second, we provide a solution to the problem of unknown required degrees of freedom by Error-Adaptive Knot Insertion (EAKI). During the iterations of our re-weighted least-squares formulation, we check for regions of high error on the curve and locally increase the degrees of freedom if necessary. Third, we present a method to handle deep and narrow concavities, called Concavity Filling (CF). The curve is examined for areas of large distances to the closest data points. In these regions, we explicitly strap the curve to internal points to force it to bend inwards and fill the concavity. Compared with the state of the art, our method shows fundamental improvement in terms of robustness and applicability to real-world data. For 3D reconstruction of organized and unorganized point clouds, prevalent in robotic RGBD perception, we achieve higher robustness compared to state-of-the-art methods and compression rates up to a factor of 300. We have integrated our code into the Point Cloud Library (PCL) and created a tutorial that guides through the steps of the algorithm (see footnote 1).

Economic Consequences of Aviation System Disruptions: A Reduced-Form Computable General Equilibrium Analysis

The state of the art approach to economic consequence analysis (ECA) is computable general equilibrium (CGE) modeling. However, such models contain thousands of equations and cannot readily be incorporated into computerized systems used by policy analysts to yield estimates of economic impacts of various types of transportation system failures due to natural hazards, terrorist attacks or technological accidents. This paper presents a reduced-form approach to simplify the analytical content of CGE models to make them more transparent and enhance their utilization potential. The reduced-form CGE analysis is conducted by first running simulations one hundred times, varying key parameters, such as magnitude of the initial shock, duration, location, remediation, and resilience, according to a Latin Hypercube sampling procedure. Statistical analysis is then applied to the “synthetic data” results in the form of both ordinary least squares and quantile regression. The analysis yields linear equations that are incorporated into a computerized system and utilized along with Monte Carlo simulation methods for propagating uncertainties in economic consequences. Although our demonstration and discussion focuses on aviation system disruptions caused by terrorist attacks, the approach can be applied to a broad range of threat scenarios.

Regression, least squares, and the general version of inclusive fitness.

A general version of inclusive fitness based on regression is rederived with minimal mathematics and directly from the verbal interpretation of its terms that motivated the original formulation of the inclusive fitness concept. This verbal interpretation is here extended to provide the two relationships required to determine the two coefficients -c and b. These coefficients retain their definition as expected effects on the fitness of an individual, respectively of a change in allelic state of this individual, and of correlated change in allelic state of social partners. The known least-squares formulation of the relationships determining b and c can be immediately deduced and shown to be equivalent to this new formulation. These results make clear that criticisms of the mathematical tools (in particular least-squares regression) previously used to derive this version of inclusive fitness are misdirected.

On the solution of the advection equation and advective dominated reactor models by weighted residual methods

The least-squares, tau and orthogonal collocation methods are employed to give approximate solutions to, e.g., the advective equation and an advective dominated fixed packed bed reactor model. It has been argued in the literature that the least-squares formulation is a unified method for the approximate solution of differential equations governing diverse areas in engineering and science. In particular, it has been argued that the least-squares method is suited for first-order differential operators. To investigate this, five numerical test problems have been constructed.The least-squares method showed rather poor numerical accuracy for the solution of a chemical process operated in a fixed packed bed reactor. The classical orthogonal collocation and tau formulations showed superior numerical accuracy to that of the least-squares method for this problem. For other test cases, i.e. first-order ordinary differential equation and finite-element solution of a one-dimensional advection equation with a sharp gradient in the solution, the least-squares method obtained better numerical results than the orthogonal collocation and tau methods.The least-squares method is not necessarily the better choice for advection dominated problems as a number of different problems favor the orthogonal collocation and/or tau methods. However, for the advective terms of some problems the least-squares method can be used to stabilize the Galerkin discretization. The choice of the better approximation method for an advection dominated problem is case dependent.

Two Time-Scaled Battery Model Identification With Application to Battery State Estimation

Electrified propulsion systems have now become an increasingly popular option for automotive companies to meet the more stringent emissions standards. A well-designed battery state estimation (BSE) system, which includes state-of-charge and state-of-health estimation, is one of the most important aspects of a successful electrified propulsion system design. Among different methods, model-based state estimation has proven to be very successful in their accuracy and implementability. A relatively newer approach to model-based BSE is to identify the battery model parameters (typically a low-order control-oriented model) in real time. This allows the battery model parameter to adjust to changing characteristics of the battery, and thus further improving the robustness of the design. However, standard identification algorithms used have very limited capability in performing this identification successfully due to the frequency response characteristics of the battery. In this brief, we describe a two time-scaled battery model parameter identification method, where the slower and faster battery dynamics are identified separately. Compared with standard approach to real-time battery model identification, where no such separation is made, this method can generate a model whose frequency response is much closer to that of the actual battery. Furthermore, this method uses the standard least squares regression method, which can be easily implemented in real time in the form of recursive least squares. Using this identification method, we show how battery SoC can be estimated. Laboratory battery cell data is used to illustrate the difference between this method and the more standard approach. Then, battery pack collected from a test vehicle is used to demonstrate the SoC estimation capability.

Plane flow of thixotropic elasto-viscoplastic materials through a 1:4 sudden expansion

A numerical investigation of an elasto-viscoplastic thixotropic fluid flowing through a 1:4 plane expansion 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 thixotropy, elasticity and viscoplasticity on the topology of yielded and unyielded regions of the expansion, on the field of structuring level, and on the field of elastic strain. Relevant ranges of the relaxation time, yield stress, and thixotropy characteristic time are investigated. The numerical results reveal significant effects of these parameters. The trends observed are physically sound and in accordance with the related literature.