Normal Equations Research Articles

A regularized interior point method for sparse optimal transport on graphs

In this work, the authors address the Optimal Transport (OT) problem on graphs using a proximal stabilized Interior Point Method (IPM). In particular, strongly leveraging on the induced primal–dual regularization, the authors propose to solve large scale OT problems on sparse graphs using a bespoke IPM algorithm able to suitably exploit primal–dual regularization in order to enforce scalability. Indeed, the authors prove that the introduction of the regularization allows to use sparsified versions of the normal Newton equations to inexpensively generate IPM search directions. A detailed theoretical analysis is carried out showing the polynomial convergence of the inner algorithm in the proposed computational framework. Moreover, the presented numerical results showcase the efficiency and robustness of the proposed approach when compared to network simplex solvers.

A predictive TDM-PON resource allocation using the ANN method based on equilibrium points of discrete dynamical systems

Resource allocation is one of the challenges in the time division multiple access (TDM) passive optical network (PON) area. Much research has been done to find an efficient resource allocation method. Using machine learning (ML) and finding proper ML algorithms in predictive dynamic bandwidth allocation remains an open research question. This paper aims to predict resource allocation accurately and efficiently based on historical requests. Artificial neural networks (ANN) with historically allocated bandwidth from different time series are used to build the prediction model. This model is constructed by the normal equation method. The proposed model needs a small dataset for its training phase, while its accuracy is higher than other models. The discrete dynamical system builds to find an equilibrium point, which is a stable point in the system. The proposed model can be implemented as a pre-training, or it can be dynamically refined when the model is running by using some of its output as feedback. The system evaluation verifies that the proposed DBA (EQ-ANN) elevates the quality of service (QoS) metrics, such as system throughput, packet drop probability, and packet means delay. The results show that the system throughput improves by up to 5%, and packet loss improves by up to 15%. Moreover, the prediction accuracy of the system reached 99.48%.

Approximating sparse Hessian matrices using large-scale linear least squares

Large-scale optimization algorithms frequently require sparse Hessian matrices that are not readily available. Existing methods for approximating large sparse Hessian matrices have limitations. To try and overcome these, we propose a novel approach that reformulates the problem as the solution of a large linear least squares problem. The least squares problem is sparse but can include a number of rows that contain significantly more entries than other rows and are regarded as dense. We exploit recent work on solving such problems using either the normal equations or an augmented system to derive a robust approach for computing approximate sparse Hessian matrices. Example sparse Hessians from the CUTEst test problem collection for optimization illustrate the effectiveness and robustness of the new method.

Satellite Gravity Field Recovery Using Variance‐Covariance Information From Ocean Tide Models

AbstractMonthly gravity field recovery using data from the GRACE and GRACE Follow‐On missions includes errors limiting the spatial and temporal resolution of the estimated gravity fields. The major error contributions, besides the noise of the accelerometer instruments, arise from temporal aliasing errors due to imperfections in the non‐tidal atmospheric and oceanic de‐aliasing models and ocean tide models. We derive uncertainty information for the eight major tidal constituents from five different ocean tide models and introduce it into the gravity field recovery process in terms of a constrained normal equation system while expanding the parameter space by additional tidal parameters to be adjusted. We prove the effectiveness of the ocean tide variance‐covariance information through realistic simulations and we assess its potential based on microwave and laser interferometer observations from the GRACE Follow‐On mission. We show that errors are reduced by more than 20% ocean wRMS for a Gaussian filter radius of 300 km if uncertainty information for ocean tides is considered and stochastic modeling of instrument errors is applied, compared to the latest GFZ release 6.1. Our results also show the limited visibility of the effectiveness of the ocean tide variance‐covariance information due to the dominating error contribution of non‐tidal atmospheric and oceanic mass variations. Additionally, we investigate the option of estimating ocean tide parameters over a 1‐year period while including ocean tide uncertainty information in order to improve ocean tide background modeling.

Linear superposition for a novel nonlocal PT -symmetric mKdV-sG system

A novel nonlocal -symmetric modified Korteweg–de Vries-sine Gordon (mKdV-sG) system is established via the consistent correlated bang (CCB) method from the normal mKdV-sG equation. The obtained nonlocal mKdV-sG system is Lax integrable and can be reduced to the normal mKdV-sG equation with suitable function selections. Using symmetry-antisymmetry separation method, a linear superposition solution theorem for the nonlocal mKdV-sG system is concluded. Abundant nonlinear wave structures of the obtained nonlocal system, such as the periodic waves, the kinks, the breathers and periodic-periodic interaction waves, periodic-kink interaction waves, periodic-breathers interaction waves solutions are obtained from the linear superposition solution theorem.

The type-oriented algorithm for baseline correction of acceleration time histories

This paper presents the development of an exhaustive approach to baseline correction of acceleration signals that amalgamates several existing methods to provide a more powerful and flexible toolkit for scientific software implementation. The proposed type-oriented baseline correction algorithm employs the traditional methodology of subtracting a least squares polynomial from the raw time series, but it is seeded by a specified combination of correction types that can vary in both polynomial order and kinematic variable considered (i.e., acceleration, velocity, or displacement). The result is a dramatic expansion in the number of corrections possible with current commercial software, which tend to consider only one kinematic variable while also restricting the polynomial order used. The proposed algorithm incorporates seven unique correction types based on permutations of kinematic variables adjusted in sequence. Different numerical methods are evaluated to solve the system of normal equations of the least squares polynomial approximation of the one-dimensional real function. The result is an algorithm that achieves practically unlimited polynomial order. The external behavior of the algorithm is evaluated by presenting a set of parametric tests that use different correction types. It is shown that the optimal correction of some vibration signals, particularly those with low frequency content occurring over many cycles, cannot be obtained without adjusting the time histories by high order best-fits of more than one kinematic variable. The importance of baseline correction in the context of finite element analysis is discussed, and several implementation examples are provided in different programming languages.

AutoQS v1: automatic parametrization of QuickSampling based on training images analysis

Abstract. Multiple-point geostatistics are widely used to simulate complex spatial structures based on a training image. The practical applicability of these methods relies on the possibility of finding optimal training images and parametrization of the simulation algorithms. While methods for automatically selecting training images are available, parametrization can be cumbersome. Here, we propose to find an optimal set of parameters using only the training image as input. The difference between this and previous work that used parametrization optimization is that it does not require the definition of an objective function. Our approach is based on the analysis of the errors that occur when filling artificially constructed patterns that have been borrowed from the training image. Its main advantage is to eliminate the risk of overfitting an objective function, which may result in variance underestimation or in verbatim copy of the training image. Since it is not based on optimization, our approach finds a set of acceptable parameters in a predictable manner by using the knowledge and understanding of how the simulation algorithms work. The technique is explored in the context of the recently developed QuickSampling algorithm, but it can be easily adapted to other pixel-based multiple-point statistics algorithms using pattern matching, such as direct sampling or single normal equation simulation (SNESIM).

Hyper-reduction for Petrov–Galerkin reduced order models

Projection-based Reduced Order Models are based on the idea of minimizing the discrete residual of a “full order model” (FOM) while at the same time constraining the unknowns to live in a space of reduced dimension. For problems with symmetric positive definite (SPD) Jacobians, this minimization can be achieved optimally by projecting the full order residual onto the approximation basis (Galerkin Projection). This approach is sub-optimal for problems with non-SPD Jacobians since it only guarantees that the projection of the residual onto the chosen basis is minimized and not the residual itself. One possible alternative in such cases is to directly minimize the 2-norm of the residual. This minimization can be achieved either by using QR factorization to solve the resulting least-squares problem or by employing the method of the normal equations (LSPG) to the same end. The first approach involves constructing and factorizing a rectangular tall and skinny matrix of size proportional to the number of unknowns of the FOM. The LSPG method avoids the use of the large matrix by directly constructing the product by its transpose. Unfortunately, constructing such a product element by element is not feasible and requires the use of a complementary mesh, which adds an extra layer of complexity to the hyper-reduction process when performing mesh sampling. The main idea of this work is to propose an alternative technique based on the idea of Petrov–Galerkin minimization. Essentially, we choose a left basis so that a least-squares minimization procedure can be carried out on a reduced problem while guaranteeing that the discrete full order residual is minimized. The resulting procedure is applicable to problems with both SPD and non-SPD Jacobians. Additionally, the resulting minimization problem can be assembled element by element, avoiding the use of the complementary mesh and simplifying implementation in the context of finite elements. The resulting technique is amenable to hyper-reduction by the use of the Empirical Cubature Method and can be readily applied in the context of nonlinear reduction procedures.

Stress resultants governing equations of any beam theory by direct manipulation of 3D equilibrium

In this article, the author shows that the equilibrium equations of any one-dimensional theory can be obtained by direct manipulation of the indefinite 3D elasticity equations: It is found that there is no need to refer to Newton-type or Lagrange-type equations to write the equilibrium in terms of stress resultants. Such ‘direct manipulation’ is illustrated by considering various problems, from the simplest to the most complex. First, the bar case is analyzed and the well-known equilibrium normal stress resultant equation is obtained: second, the Timoshenko beam theory is tackled; finally, the case of arbitrary higher order beam theories based on the Carrera Unified Formulation is treated. The equations of a new, unprecedented higher-order beam theory are derived directly from the indefinite equilibrium equations as an example of the capabilities of the proposed method. The manipulation introduced consists of the following points: (i) the 3D elasticity is first written for each degree of freedom (dof) of the beam theories; (ii) the respective stress resultants are used to replace stress components in the latter equations; (iii) the derivatives on the beam axis z do not affect the nature of the indefinite equilibrium equations and these derivatives are applied directly to the respective stress resultants; (iv) the derivatives on the section coordinate x – y should 1 – change the sign of the respective stress resultants, 2 – be applied to the functions of x, y used to define the stress resultants. These points have been proved by referring to the principle of virtual work and splitting the differential operators of the strain-displacement relations into two parts: over the section derivatives along the beam axis derivatives. Some possible further generalizations are discussed in the appendix; in particular, the extension to is briefly discussed.

Biomedical simulations of hybrid nano fluid flow through a balloon catheterized stenotic artery with the effects of an inclined magnetic field and variable thermal conductivity

One of the current challenges of the coronary angioplasty technique for treating atherosclerosis in the artery is to target the desired area using nanoparticles. This study aims to investigate the hybrid Au-Fe3O4/blood nanofluid flow that passes through a narrowed section of a stenotic artery. The behavior of the Au-Fe3O4/blood flow in the stenosis region is mathematically examined, taking into account an inclined magnetic field and temperature-dependent thermal conductivity. Additionally, this work analyses the effect of nanolayers on nanofluid flow in the presence of a catheter with a balloon (angioplasty) on its wall. Tube and balloon catheter models are considered individually to compare the results of flow characteristics and heat transmission. Channel boundaries comprise wall properties with thermal slip and second-order velocity slip conditions. A similarity transformation procedure is used to convert the governing equations into highly nonlinear normal differential equations, which are analytically computed using the homotopy perturbation technique. The obtained results reveal an increased velocity profile due to the magnetic force, while an inverse trend occurs due to the thermal conductivity of the nanofluid. Furthermore, interfacial nanolayers of nanoparticles significantly alter the blood temperature to prevent and manage cardiovascular diseases. This rheological model may be useful for cancer diagnosis, tumor-selective photothermal therapy, medical simulation devices, and other applications.

Effects of non-tidal loading applied in VLBI-only terrestrial reference frames

We investigate the impact of the reduction of non-tidal loading (NTL) in the computation of secular terrestrial reference frames (TRFs) from Very Long Baseline Interferometry (VLBI) observations. There are no conventional models for NTL in the geodetic community yet, but the Global Geophysical Fluid Center prepared a set of corresponding site displacements for the 2020 realizations of the International Terrestrial Reference System. We make use of these data, which comprise the total NTL consisting of non-tidal atmospheric, oceanic, and hydrological loading. The displacement series contain linear trends (i.e., offsets plus drifts), and since these affect the estimated linear station positions and the realized geodetic datum in a secular TRF, we remove the trends before reducing the NTL in our computations. The displacements are applied at two different levels of the parameter estimation process: the observation and the normal equation level. This way, we can analyze whether the latter offers a suitable approximation if the original observations have not been reduced by NTL. We find that the TRF statistics are hardly affected by the NTL. The largest impact is given for the secular motion of antennas with short observation time spans. The application level is basically irrelevant for the linear antenna positions, but it leads to differences in the rates of the jointly estimated Earth orientation parameters (EOPs). Secular TRF solutions and session solutions deviate with respect to the parameterization of the antenna coordinates, and thus also with respect to the correlations between the estimated antenna parameters and the EOPs. Due to this, the consistently estimated EOP series also show differences. However, for both solution types the reduction of the NTL leads to a change of the annual signal in the EOP series.

Direct solution of the normal equations in interior point methods for convex transportation problems

Recent research has shown that very large convex transportation problems can be solved efficiently with interior point methods by finding a good pre-conditioner for an iterative linear-equation solver. In this article, it is demonstrated that the specific structure of the constraints allows use of a direct method for solving those linear equations with the same order of worst-case time complexity.

Analysis of the Results Determining the Positions and Velocities of Satellite Laser Ranging Stations during Earthquakes in 2010–2011

This paper presents an analysis of the results determining the positions and the velocities of the 21 selected satellite laser ranging stations which performed observations from January 2008 to December 2012. This particularly interesting period of five years was selected, during which two strong earthquakes occurred near the stations. The focus was directed on the stations where the effects of the earthquakes were observed, i.e., Koganei (7308), Simosato (7838), and Changchun (7237) as a result of the tsunami in Japan on 11 February 2011, as well as Concepcion (7405IR) and San Juan (7406) as a result of the earthquake in Chile on 27 February 2010. The station positions were computed using the GSFC NASA GEODYN-II orbital software from the observations results of the satellites LAGEOS-1 and LAGEOS-2. The geocentric coordinates of the stations were determined from the normal equations of both satellites. The station velocities were computed from the positions determined for the observation epoch using the linear regression method. For each station, the following parameters were determined: mean total coordinates stability, standard deviation of determined positions and mean deviation from ITRF for topocentric components. For the best stations, the stability ranged from 4.0 mm to 6.5 mm, with the standard deviation of determined positions ranging from 0.9 mm to 1.7 mm. For two stations, a quadratic change in the station position was detected instead of a normal linear one after strong earthquakes.

Performance Analysis of Centroid-based Routing Protocol in Wireless Sensor Networks

In this paper, a clustering-based energy-efficient routing protocol is designed to overcome the hotspot problem in Wireless Sensor networks (WSNs). We have divided the sensor nodes into several clusters, each cluster having a Cluster Head (CH) or a leader node and also implemented a centroid-based mobile sink. The sink moves to a location called “Energy Dependent Centroid Point” where the CHs need minimal energy to transmit data to the sink. We have defined this “Centroid Point” by an equation where the residual energies of CHs are used as a “weight factor” on a normal centroid equation. We have simulated the proposed algorithm and analyzed the results by varying the values of the energy weight factor in the centroid equation up to certain limits (α = 0, 1, 2, and 3). The simulation results are compared with WSN’s existing routing protocols having a mobile sink and a static sink. The comparison results show that our proposed centroid model with α = 0, where the mobile sink moves independent of the residual energies of CHs gives better performances in terms of network lifetime, energy loss, and total data packet transmission.

Investigating horizontal-axis wind turbine aerodynamics using cascade flows

The simplest aerodynamic model of horizontal-axis wind turbines is the blade element momentum theory, which assumes that the blades behave as airfoils, but a correct two-dimensional representation is an infinite cascade of lifting bodies. This study analyzes the conventional and impulse forms of the forces on cascades of airfoils at spacings and pitch angles typical of wind turbine applications. OpenFOAM software was used to simulate steady, incompressible flow at a Reynolds number of 6×106 through cascades of NACA 0012 airfoils. The force equations agree well (less than 1% error) with the forces determined directly from OpenFOAM for four spacing ratios. We concentrate on the “wake vorticity” term, which is ignored in blade element momentum analysis. At a pitch angle of 90°, this term balances the viscous drag when the angle of attack is zero. At zero pitch, which models the outer region of a wind turbine blade at a high tip speed ratio, the term can account for 27% of the axial thrust when the angle of attack is about 4°. The normal force equation, like the angular momentum equation for wind turbines, has no viscous term, which forces the body drag to contribute to the circulation in the wake. It is shown that the airfoil assumption is conservative in that cascade elements have higher lift-to-drag ratios than airfoils at the same angle of attack. An associated result is that separation occurs at higher angles of attack on a cascade element compared to an airfoil.

CP decomposition for tensors via alternating least squares with QR decomposition

AbstractThe CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent low‐rank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical ill‐conditioning, which can compromise the results of the decomposition. In this paper, we develop versions of the CP‐ALS algorithm using the QR decomposition and the singular value decomposition, which are more numerically stable than the normal equations, to solve the linear least squares problems. Our algorithms utilize the tensor structure of the CP‐ALS subproblems efficiently, have the same complexity as the standard CP‐ALS algorithm when the input is dense and the rank is small, and are shown via examples to produce more stable results when ill‐conditioning is present. Our MATLAB implementation achieves the same running time as the standard algorithm for small ranks, and we show that the new methods can obtain lower approximation error.

Cascade-enhanced transport efficiency of biochemical systems.

Recent developments in nonequilibrium thermodynamics, known as thermodynamic uncertainty relations, limit the system's accuracy by the amount of free-energy consumption. A transport efficiency, which can be used to characterize the capacity to control the fluctuation by means of energy cost, is a direct result of the thermodynamic uncertainty relation. According to our previous research, biochemical systems consume much lower energy cost by noise-induced oscillations to keep almost equal efficiency to maintain precise processes than that by normal oscillations. Here, we demonstrate that the performance of noise-induced oscillations propagating can be further improved through a cascade reaction mechanism. It has been discovered that it is possible to considerably enhance the transport efficiency of the biochemical reactions attained at the terminal cell, allowing the cell to use the cascade reaction mechanism to operate more precisely and efficiently. Moreover, an optimal reaction coupling strength has been predicted to maximize the transport efficiency of the terminal cell, uncovering a concrete design strategy for biochemical systems. By using the local mean field approximation, we have presented an analytical framework by extending the stochastic normal form equation to the system perturbed by external signals, providing an explanation of the optimal coupling strength.

Development of simplified air drag models including crosswinds for commercial heavy vehicle combinations

Accurate range prediction requires good knowledge of the prevailing wind conditions and how they affect the energy consumption of the ego vehicle. A few different simplified vehicle air drag models that explicitly include the effect from crosswinds are presented and compared through some objective criteria. The models are developed from the normal air drag equation where the effect from wind is implicit and therefore often forgotten or neglected. The purpose is to find a low-complexity model complementing CFD models and wind tunnel tests, that can be used for range estimation and predictive energy management algorithms. To simplify online estimation, a requirement is that the air drag models only contain a few tuning parameters. The models are validated against CFD calculations for a few vehicle combinations and the best models show good accuracy for air attack angles up to at least 60 degrees. It is shown that the parameters of the simplified models can loosely be connected to some basic geometrical attributes of a vehicle combination so it should be possible to give at least a rough estimate of the parameters of a simplified model based on these geometrical attributes. This is useful for making a first estimate of the aerodynamic properties of a vehicle combination after major changes in the exterior, e.g. when adding a trailer. It also highlights that the size and the shape of the vehicle side may be mainly responsible for the high longitudinal air drag sensitivity to crosswinds for large vehicle combinations.

An Interior Point–Inspired Algorithm for Linear Programs Arising in Discrete Optimal Transport

Discrete optimal transport problems give rise to very large linear programs (LPs) with a particular structure of the constraint matrix. In this paper, we present a hybrid algorithm that mixes an interior point method (IPM) and column generation, specialized for the LP originating from the Kantorovich optimal transport problem. Knowing that optimal solutions of such problems display a high degree of sparsity, we propose a column generation–like technique to force all intermediate iterates to be as sparse as possible. The algorithm is implemented nearly matrix-free. Indeed, most of the computations avoid forming the huge matrices involved and solve the Newton system using only a much smaller Schur complement of the normal equations. We prove theoretical results about the sparsity pattern of the optimal solution, exploiting the graph structure of the underlying problem. We use these results to mix iterative and direct linear solvers efficiently in a way that avoids producing preconditioners or factorizations with excessive fill-in and at the same time guaranteeing a low number of conjugate gradient iterations. We compare the proposed method with two state-of-the-art solvers and show that it can compete with the best network optimization tools in terms of computational time and memory use. We perform experiments with problems reaching more than four billion variables and demonstrate the robustness of the proposed method. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Funding: F. Zanetti received funding from the University of Edinburgh, in the form of a PhD scholarship. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0184 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0184 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Nasal nitric oxide in healthy Chinese children aged 6-18 years.

To obtain the normal values of fractional concentration of nasal nitric oxide in Chinese children aged 6-18 years, so as to provide reference for clinical diagnosis. 2,580 out of 3,200 children (1,359 males and 1,221 females), whom were included from 12 centers around China were taken tests, their height and weight were also recorded. Data were used to analyze the normal range and influencing factors of fractional concentration of nasal nitric oxide values. Data was measured using the Nano Coulomb Breath Analyzer (Sunvou-CA2122, Wuxi, China), according to the American Thoracic Society/European Respiratory Society (ATS/ERS) recommendations. We calculated the normal range and prediction equation of fractional concentration of nasal nitric oxide values in Chinese children aged 6-18 years. The mean FnNO values of Chinese aged 6-18 yrs was 454.5 ± 176.2 ppb, and 95% of them were in the range of 134.5-844.0 ppb. The prediction rule of FnNO values for Chinese children aged 6-11 yrs was: FnNO = 298.881 + 17.974 × age. And for children aged 12-18 yrs was: FnNO = 579.222-30.332 × (male = 0, female = 1)-5.503 × age. Sex and age were two significant predictors of FnNO values for Chinese children(aged 12-18 yrs). Hopefully this study can provide some reference value for clinical diagnosis in children.