Least-squares Formulation Research Articles

A mixed-integer least-squares formulation of the GNSS snapshot positioning problem

AbstractThis paper presents a formulation of snapshot positioning as a mixed-integer least-squares problem. In snapshot positioning, one estimates a position from code-phase (and possibly Doppler-shift) observations of global navigation satellite system (GNSS) signals without knowing the time of departure (timestamp) of the codes. Solving the problem allows a receiver to determine a fix from short radio-frequency snapshots missing the timestamp information embedded in the GNSS data stream. This is used to reduce the time to first fix in some receivers, and it is used in certain wildlife trackers. This paper presents two new formulations of the problem and an algorithm that solves the resulting mixed-integer least-squares problems. We also show that the new formulations can produce fixes even with huge initial errors, much larger than permitted in Van Diggelen's widely-cited coarse-time navigation method.

Three-dimensional inversion for sparse potential data using first-order system least squares with application to gravity anomalies in Western Queensland

SUMMARY We present an inversion algorithm tailored for point gravity data. As the data are from multiple surveys, it is inconsistent with regards to spacing and accuracy. An algorithm design objective is the exact placement of gravity observations to ensure no interpolation of the data is needed prior to any inversion. This is accommodated by discretization using an unstructured tetrahedral finite-element mesh for both gravity and density with mesh nodes located at all observation points and a first-order system least-squares (FOSLS) formulation for the gravity modelling equations. Regularization follows the Bayesian framework where we use a differential operator approximation of an exponential covariance kernel, avoiding the usual requirement of inverting large dense covariance matrices. Rather than using higher order basis functions with continuous derivatives across element faces, regularization is also implemented with a FOSLS formulation using vector-valued property function (density and its gradient). Minimization of the cost function, comprised of data misfit and regularization, is achieved via a Lagrange multiplier method with the minimum of the gravity FOSLS functional as a constraint. The Lagrange variations are combined into a single equation for the property function and solved using an integral form of the pre-conditioned conjugate gradient method (I-PCG). The diagonal entries of the regularization operator are used as the pre-conditioner to minimize computational costs and memory requirements. Discretization of the differential operators with the finite-element method (FEM) results in matrix systems that are solved with smoothed aggregation algebraic multigrid pre-conditioned conjugate gradient (AMG-PCG). After their initial setup, the AMG-PCG operators and coarse grid solvers are reused in each iteration step, further reducing computation time. The algorithm is tested on data from 23 surveys with a total of 6519 observation points in the Mt Isa–Cloncurry region in north–west Queensland, Australia. The mesh had about 2.5 million vertices and 16.5 million cells. A synthetic case was also tested using the same mesh and error measures for localized concentrations of high and low densities. The inversion results for different parameters are compared to each other as well as to lower order smoothing. Final inversion results are shown with and without depth weighting and compared to previous geological studies for the Mt Isa–Cloncurry region.

Least-squares ReLU neural network (LSNN) method for linear advection-reaction equation

This paper studies least-squares ReLU neural network method for solving the linear advection-reaction problem with discontinuous solution. The method is a discretization of an equivalent least-squares formulation in the set of neural network functions with the ReLU activation function. The method is capable of approximating the discontinuous interface of the underlying problem automatically through the free hyper-planes of the ReLU neural network and, hence, outperforms mesh-based numerical methods in terms of the number of degrees of freedom. Numerical results of some benchmark test problems show that the method can not only approximate the solution with the least number of parameters, but also avoid the common Gibbs phenomena along the discontinuous interface. Moreover, a three-layer ReLU neural network is necessary and sufficient in order to well approximate a discontinuous solution with an interface in R2 that is not a straight line.

Kernel Recursive Least Square Approach for Power System Harmonic Estimation

Harmonic frequencies existing in power systems produce harmful effects. Signals containing harmonic effect produce power quality problems such as heating, premature wear, motor pulsations, and so on. Commonly, to cancel the harmonic effects in power systems is based on traditional filtering methods. However, such methodologies tend to fail in the presence of high convexity, which produces undesirable results. Under such circumstances, the appropriate harmonic identification is an important task for removing and reducing the harmonic effects, avoiding the potential damage in electrical power systems. Several published research in the literature has addressed this issue using traditional techniques as Recursive Least Square (RSL), Evolutionary Computation Techniques (ECT), or even the hybridization of both achieving interesting results. However, such techniques present some shortcomings, such as low accuracy and principally high computational time. In this paper, a Kernel Recursive Least Square (KRLS) approach for harmonic estimation is proposed as an alternative methodology. The KRLS uses the least-square formulation in feature space by using kernel mapping for determining the harmonic signals, different experimental simulations demonstrate the high accuracy and robustness, such as the considerably low computational time in different noisy conditions of the proposed method regarding similar approaches.

Distributed power flow and distributed optimization—Formulation, solution, and open source implementation

Solving the power flow problem in a distributed fashion empowers different grid operators to compute the overall grid state without having to share grid models—this is a practical problem to which industry does not have off-the-shelf answers. We propose two physically consistent problem formulations (a feasibility and a least-squares formulation) amenable to two solution methods from distributed optimization: the Alternating direction method of multipliers ( admm ), and the Augmented Lagrangian based Alternating Direction Inexact Newton method ( aladin ); the latter comes with convergence guarantees. In addition, we provide open source matlab code for rapid prototyping for distributed power flow (rapid pf ): a fully matpower -compatible software that facilitates the laborious task of formulating power flow problems as distributed optimization problems. Simulation results for systems ranging from 53 buses (with 3 regions) up to 4662 buses (with 5 regions) show that the least-squares formulation solved with aladin requires just about half a dozen coordinating steps before the power flow problem is solved.

Contact Stability and Contact Safety of a Magnetic Resonance Imaging-Guided Robotic Catheter Under Heart Surface Motion.

Contact force quality is one of the most critical factors for safe and effective lesion formation during catheter based atrial fibrillation ablation procedures. In this paper, the contact stability and contact safety of a novel magnetic resonance imaging (MRI)-actuated robotic cardiac ablation catheter subject to surface motion disturbances are studied. First, a quasi-static contact force optimization algorithm, which calculates the actuation needed to achieve a desired contact force at an instantaneous tissue surface configuration is introduced. This algorithm is then generalized using a least-squares formulation to optimize the contact stability and safety over a prediction horizon for a given estimated heart motion trajectory. Four contact force control schemes are proposed based on these algorithms. The first proposed force control scheme employs instantaneous heart position feedback. The second control scheme applies a constant actuation level using a quasi-periodic heart motion prediction. The third and the last contact force control schemes employ a generalized adaptive filter-based heart motion prediction, where the former uses the predicted instantaneous position feedback, and the latter is a receding horizon controller. The performance of the proposed control schemes is compared and evaluated in a simulation environment.

Least-squares formulations for eigenvalue problems associated with linear elasticity

We study the approximation of the spectrum of least-squares operators arising from linear elasticity. We consider a two-field (stress/displacement) and a three-field (stress/displacement/vorticity) formulation; other formulations might be analyzed with similar techniques.We prove a priori estimates and we confirm the theoretical results with simple two-dimensional numerical experiments.

Further results on a space-time FOSLS formulation of parabolic PDEs

In [2019, Space-time least-squares finite elements for parabolic equations, arXiv:1911.01942] by Führer and Karkulik, well-posedness of a space-time First-Order System Least-Squares formulation of the heat equation was proven. In the present work, this result is generalized to general second order parabolic PDEs with possibly inhomogenoeus boundary conditions, and plain convergence of a standard adaptive finite element method driven by the least-squares estimator is demonstrated. The proof of the latter easily extends to a large class of least-squares formulations.

A least‐squares finite element formulation to solve incompressible non‐Newtonian fluid flow

AbstractThe Modeling of non‐Newtonian fluids is an essential issue for many industrial applications, for example in the field of chemistry, bioengineering and medical science. In this contribution we present a least‐squares (LS) finite element approach to model steady flow of incompressible non‐Newtonian fluids based on the Navier‐Stokes equations. The application of the least‐squares finite element method (LSFEM) especially in the case of fluid mechanics is motivated by some theoretical advantages compared to the well‐known (mixed) Galerkin method. The LSFEM is not restricted to the LBB‐condition and results in a minimization problem with symmetric positive definite equation systems also for differential equations with non self‐adjoint operators. Furthermore, the LSFEM provides an inherent a posteriori error estimator without additional costs, which enables an efficiency enhancement through adaptive mesh refinements.In this contribution the first‐order LS formulation in terms of stresses and velocities, as introduced in [4], is extended to consider the nonlinear dependence of the viscosity on the shear rate of the fluid. Therefore, the well‐known Carreau model for generalized Newtonian fluids, see for instance [1], is investigated. For the approximation of the independent variables, velocities and stresses, we apply conforming discretizations in H1 and H(div) using standard Lagrange interpolation polynomials and vector‐valued Raviart‐Thomas interpolations functions. A flow through a square domain with exact solution is considered to validate the proposed schemes with respect to accuracy and efficiency.

The Condition Number of Riemannian Approximation Problems

We consider the local sensitivity of least-squares formulations of inverse problems. The sets of inputs and outputs of these problems are assumed to have the structures of Riemannian manifolds. The problems we consider include the approximation problem of finding the nearest point on a Riemannian embedded submanifold from a given point in the ambient space. We characterize the first-order sensitivity, i.e., condition number, of local minimizers and critical points to arbitrary perturbations of the input of the least-squares problem. This condition number involves the Weingarten map of the input manifold, which measures the amount by which the input manifold curves in its ambient space. We validate our main results through experiments with the $n$-camera triangulation problem in computer vision.

Output-Relevant Common Trend Analysis for KPI-Related Nonstationary Process Monitoring With Applications to Thermal Power Plants

Operation safety and efficiency are two main concerns in power plants. It is important to detect the anomalies in power plants, and further judge whether they affect key performance indicators (KPIs), such as the thermal efficiency. These two goals can be achieved by KPI-related nonstationary process monitoring. Although the thermal efficiency cannot be accurately measured online, it can be strongly characterized by some online measurable variables, including the exhaust gas temperature and oxygen content of flue gas. These critical variables closely related to the thermal efficiency are termed as output variables. Inspired from nonstationary common trends between input and output variables in thermal power plants, the output-relevant common trend analysis (OCTA) method is proposed, in this article, to model the input–output relationship. In OCTA, input and output variables are decomposed into nonstationary common trends and stationary residuals, and the model parameters are estimated by solving an optimization problem. It is pointed out that OCTA is a generalized form of partial least squares (PLS). The superior monitoring performance of OCTA is illustrated by case studies on a real power plant in Zhejiang Provincial Energy Group of China. Compared with the other PLS-based recursive algorithms, OCTA can effectively detect the anomalies in power plants and accurately determine whether they have an impact on the thermal efficiency or not.

A least-squares formulation of the Moving Discontinuous Galerkin Finite Element Method with Interface Condition Enforcement

A least-squares formulation of the Moving Discontinuous Galerkin Finite Element Method with Interface Condition Enforcement (LS-MDG-ICE) is presented. This method combines MDG-ICE, which uses a weak formulation that separately enforces a conservation law and the corresponding interface condition and treats the discrete geometry as a variable, with the Discontinuous Petrov–Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan to systematically generate optimal test functions from the trial spaces of both the discrete flow field and discrete geometry. For inviscid flows, LS-MDG-ICE detects and fits a priori unknown interfaces, including shocks. For convection-dominated diffusion, LS-MDG-ICE resolves internal layers, e.g., viscous shocks, and boundary layers using anisotropic curvilinear r-adaptivity in which high-order shape representations are anisotropically adapted to accurately resolve the flow field. As such, LS-MDG-ICE solutions are oscillation-free, regardless of the grid resolution and polynomial degree. Finally, for both linear and nonlinear problems in one dimension, LS-MDG-ICE is shown to achieve optimal-order convergence of the L2 solution error with respect to the exact solution when the discrete geometry is fixed and super-optimal convergence when the discrete geometry is treated as a variable.

Multi-Beam Multiplexing Design for Arbitrary Directions Based on the Interleaved Subarray Architecture

Through a common set of analogue coefficients and a simple digital coding scheme in the form of ones and minus ones, a previous technique can only multiplex two beams whose directions satisfy a fixed relationship. In this work, two novel designs are proposed, which together with the corresponding inter-subarray coding schemes, can achieve multi-beam multiplexing for arbitrary directions to serve corresponding users. In the first design, based on the relationship of directions between the two required beams, the adjacent antenna spacing is regarded as a variable to be determined, while in the second design, the adjacent antenna spacing is fixed and an alternate optimisation procedure is proposed to solve the problem based on a least-square formulation. Designed examples based on uniform linear arrays and uniform planar arrays are provided to demonstrate the effectiveness of the proposed methods.

Compensated Load Flow Solutions for Distribution System State Estimation

State estimation of distribution systems typically relies on measurement sets with very low redundancy levels. In this paper, this fact is exploited by first solving a conventional load flow, using exclusively a critical set of measurements, and then compensating the solution to account for the few redundant measurements available. This leads to a suboptimal but sufficiently accurate estimate. It is shown how the sparse triangular factorization of the load flow Jacobian matrix can be fully exploited throughout the compensation-based procedure, preventing in this way the ill-conditioning associated with the gain matrix arising in the conventional least-squares formulation. Simulation results are provided for measurement configurations customarily found in distribution systems, showing the potential advantages of the proposed methodology.

Towards credible CFD simulations for floating offshore wind turbines

The credibility in the results needs to be demonstrated in order to be able to use computational fluid dynamics (CFD) as an engineering tool. This may be obtained by sufficient verification and validation studies involving error and uncertainty quantification. This study investigates how to perform credible CFD simulations of floating offshore wind turbines (FOWTs). Three methods to estimate discretisation errors were compared for three different problems related to FOWTs: wave propagation in 2D, wave loads on a circular cylinder, and surge decay of a semi-submersible FOWT. The three discretisation error estimation methods are a least-squares formulation using the observed order of convergence in combination with a data quality measure for different spatial and temporal refinement, another least-squares fit method using the theoretical order of convergence for constant Courant number grid refinement studies, and the factor of safety method applying a ratio between observed and theoretical order of convergence. We compared the final results to an analytical solution of the 2D wave signal, and to experimental data for the wave loads and surge decay motions, for validation purposes. The results of this work show the advantages and disadvantages of the three error estimation methods. The uncertainty bars for the discretisation uncertainty of the numerical simulations were mostly larger than the comparison error with the model test data and analytical solution.

Design optimization of functionally graded plates under thermo-mechanical loadings to minimize stress, deformation and mass

This work addresses the multiobjective design optimization of metal-ceramic functionally graded (FG) plates, which are composed of a main functionally graded material (FGM) layer and may include metal and/or ceramic faces, under thermo-mechanical loadings. The design variables are the thickness of the FGM layer, the index of its power-law distribution of metal-ceramic volume fractions, and if included, the thickness of the metal and/or ceramic faces. The three objectives focus on mass, maximum transverse displacement and maximum value of the Tsai-Hill failure criteria to measure the stress field, aiming to minimize all together. Both thermal and mechanical problems are solved simultaneously using a layerwise mixed model based on least-squares formulation with multi-field independent variables, namely, displacements, temperature, transverse stresses, transverse heat flux, in-plane strains and in-plane components of the thermal gradient. The FGM layer z-continuous effective properties are fully described via high-order z-expansions, similarly to finite element approximations. The multiobjective optimization problem is solved by Direct MultiSearch (DMS) derivative-free method, which uses the notion of Pareto dominance to retain a list of feasible non-dominated solutions. Numerical results provide optimal designs of FG plates under thermo-mechanical loadings, exploring distinct metal-ceramic constituent materials and different side-to-thickness ratios, including three-dimensional approximate solutions for validation.

Local least–squares element differential method for solving heat conduction problems in composite structures

In this article, a completely new numerical method called the Local Least-Squares Element Differential Method (LSEDM), is proposed for solving general engineering problems governed by second order partial differential equations. The method is a type of strong-form finite element method. In this method, a set of differential formulations of the isoparametric elements with respect to global coordinates are employed to collocate the governing differential equations and Neumann boundary conditions of the considered problem to generate the system of equations for internal nodes and boundary nodes of the collocation element. For each outer boundary or element interface, one equation is generated using the Neumann boundary condition and thus a number of equations can be generated for each node associated with a number of element interfaces. The least-squares technique is used to cast these interface equations into one equation by optimizing the local physical variable at the least-squares formulation. Thus, the solution system has as many equations as the total number of nodes of the present heat conduction problem. The proposed LSEDM can ultimately guarantee the conservativeness of the heat flux across element surfaces and can effectively improve the solution stability of the element differential method in solving problems with hugely different material properties, which is a challenging issue in meshfree methods. Numerical examples on two- and three-dimensional heat conduction problems are given to demonstrate the stability and efficiency of the proposed method.

Graduated Non-Convexity for Robust Spatial Perception: From Non-Minimal Solvers to Global Outlier Rejection

Semidefinite Programming (SDP) and Sums-of-Squares (SOS) relaxations have led to certifiably optimal non-minimal solvers for several robotics and computer vision problems. However, most non-minimal solvers rely on least-squares formulations, and, as a result, are brittle against outliers. While a standard approach to regain robustness against outliers is to use robust cost functions, the latter typically introduce other non-convexities, preventing the use of existing non-minimal solvers. In this paper, we enable the simultaneous use of non-minimal solvers and robust estimation by providing a general-purpose approach for robust global estimation, which can be applied to any problem where a non-minimal solver is available for the outlier-free case. To this end, we leverage the Black-Rangarajan duality between robust estimation and outlier processes (which has been traditionally applied to early vision problems), and show that graduated non-convexity (GNC) can be used in conjunction with non-minimal solvers to compute robust solutions, without requiring an initial guess. Although GNC's global optimality cannot be guaranteed, we demonstrate the empirical robustness of the resulting robust non-minimal solvers in applications, including point cloud and mesh registration, pose graph optimization, and image-based object pose estimation (also called shape alignment). Our solvers are robust to 70-80% of outliers, outperform RANSAC, are more accurate than specialized local solvers, and faster than specialized global solvers. We also propose the first certifiably optimal non-minimal solver for shape alignment using SOS relaxation.

Highly accurate space-time coupled least-squares finite element framework in studying wave propagation

Simulation of stress wave propagation through solid medium is commonly carried using Galerkin weak-form cast over decoupled space and time domains. In this paper, accuracy of this commonly utilized framework is compared to that of the variationally-consistent least-squares form of the wave equation cast over space-time domain. The two formulations are tested for numerical dispersion and numerical diffusion, through two test cases. The first case studies the dispersion in harmonic shear wave propagation through a soil column over a wide range of forcing frequencies. The second test case investigates numerical diffusion in an axial wave propagation generated by constant force; which is removed after a certain time to allow free vibration to take place. Low numerical dispersion and numerical diffusion as well as high rates of convergence are the main advantages of the coupled least-squares (CLS) computational framework; when compared to the decoupled Galerkin (DG) framework. Based on studies presented here, CLS has low dispersion; yielding errors with one to two orders of magnitude less than that of DG. Also, the numerical diffusion present in DG framework causes a %40 error in DG’s prediction of the stress-wave intensity. Furthermore, accumulative error during evolution is virtually nonexistent for CLS, whereas, the error steadily increases as the solution evolves in DG framework. It is also demonstrated that CLS feature of temporal meshing allows for faster computations.

A sharp-interface Cartesian grid method for time-domain acoustic scattering from complex geometries

This paper is concerned with the development of a Cartesian grid method for predicting the acoustic field scattered by complex geometries immersed in an infinite fluid. The sound wave in the fluid is formulated by the linearized Euler equations (LEEs), which are computed on a fixed Cartesian grid by using a fourth-order dispersion-relation-preserving (DRP) scheme. The solid object naturally cuts the underlying Cartesian grid with its geometrical boundary, which is modeled by the Lagrangian method. A sharp-interface Cartesian grid model based on the ghost-cell immersed boundary method is established to impose the non-penetrating conditions on the complex boundary of the solid. The values of the acoustic variables at the ghost points are constructed by a linear extrapolation in conjunction with a constrained moving least-squares (CMLS) interpolation method. The interpolation method is capable of eliminating the numerical instability encountered in the conventional moving least-squares (MLS) formulation. The perfectly matched layers are adopted to absorb the out-going sound waves without any reflections from the computational boundary to the interior domain. The proposed method is verified by several benchmark problems in computational aeroacoustics, including the initial value problem of a pressure perturbation scattered by a rigid cylinder, and Gaussian distributed acoustic sources scattered by a single cylinder or two cylinders. Application of the proposed method to the problem of acoustic waves scattered by solids with complex geometries is also presented to demonstrate the effectiveness and robustness of the method.