Newton Method Research Articles

TDOA-based UWB indoor 1D localization via weighted sliding window filtering

Accurate localization of a person or object is a significant aspect in the field of industrial internet of things (IoT). Ultra-wideband (UWB) localization system, a promising technology, has been widely studied due to its high accuracy in 2D and 3D scenarios. Moreover, in narrow corridors and deep-well environments, the 1D localization is also an important application with the advantage of resource saving, however which attracts less attention. In this paper, we focus on the problem of 1D localization and propose a practical localization model with time difference of arrivals (TDOA). To solve the nonlinear 1D localization problem, we design an algorithm that integrates the dichotomous method and Newton's iterative method, which realizes an efficient solution. Meanwhile, in order to cope with the random errors in the real scenes, we introduce a sliding window algorithm to filter out the undesirable solutions. By observing the localization error map, we design a weighted sliding window algorithm to achieve further improvement of the localization accuracy. In addition, we derive the Cramér-Rao lower bound (CRLB) for 1D localization of TDOA. Simulations and practical experiments show that our proposed algorithm greatly benefits the localization accuracy of 1D TDOA and provides strong support for its practical application.

A hybrid inexact regularized Newton and negative curvature method

A hybrid inexact regularized Newton and negative curvature method

Comparison of a quasi Newton method using Broyden’s update formula and an adjoint method for determining local magnetic material properties of electrical steel sheets

PurposePerforming accurate numerical simulations of electrical drives, the precise knowledge of the local magnetic material properties is of utmost importance. Due to the various manufacturing steps, e.g. heat treatment or cutting techniques, the magnetic material properties can strongly vary locally, and the assumption of homogenized global material parameters is no longer feasible. This paper aims to present the general methodology and two different solution strategies for determining the local magnetic material properties using reference and simulation data.Design/methodology/approachThe general methodology combines methods based on measurement, numerical simulation and solving an inverse problem. Therefore, a sensor-actuator system is used to characterize electrical steel sheets locally. Based on the measurement data and results from the finite element simulation, the inverse problem is solved with two different solution strategies. The first one is a quasi Newton method (QNM) using Broyden's update formula to approximate the Jacobian and the second is an adjoint method. For comparison of both methods regarding convergence and efficiency, an artificial example with a linear material model is considered.FindingsThe QNM and the adjoint method show similar convergence behavior for two different cutting-edge effects. Furthermore, considering a priori information improved the convergence rate. However, no impact on the stability and the remaining error is observed.Originality/valueThe presented methodology enables a fast and simple determination of the local magnetic material properties of electrical steel sheets without the need for a large number of samples or special preparation procedures.

The architectural application of shells whose boundaries subtend a constant solid angle

Surface geometry plays a central role in the design of bridges, vaults and shells, using various techniques for generating a geometry which aims to balance structural, spatial, aesthetic and construction requirements.In this paper we propose the use of surfaces defined such that given closed curves subtend a constant solid angle at all points on the surface and form its boundary. Constant solid angle surfaces enable one to control the boundary slope and hence achieve an approximately constant span-to-height ratio as the span varies, making them structurally viable for shell structures. In addition, when the entire surface boundary is in the same plane, the slope of the surface around the boundary is constant and thus follows a principal curvature direction. Such surfaces are suitable for surface grids where planar quadrilaterals meet the surface boundaries. They can also be used as the Airy stress function in the form finding of shells having forces concentrated at the corners.Our technique employs the Gauss-Bonnet theorem to calculate the solid angle of a point in space and Newton's method to move the point onto the constant solid angle surface. We use the Biot-Savart law to find the gradient of the solid angle. The technique can be applied in parallel to each surface point without an initial mesh, opening up for future studies and other applications when boundary curves are known but the initial topology is unknown.We show the geometrical properties, possibilities and limitations of surfaces of constant solid angle using examples in three dimensions.

Modified Newton–CAPRESB method for solving a class of systems of nonlinear equations with complex symmetric Jacobian matrices

In this paper, we contemplate addressing nonlinear problems involving complex symmetric Jacobian matrices. Firstly, we establish a parameter-free method called modified Newton–CAPRESB (MN–CAPRESB) method by harnessing the modified Newton method as the outer iteration and the CAPRESB (Chebyshev accelerated preconditioned square block) method as the inner iteration. Secondly, the local and semilocal convergence theorems of MN–CAPRESB method are proved under some conditions. Eventually, the numerical experiments of two kinds of complex nonlinear equations are presented to validate the feasibility of MN–CAPRESB method compared to other existing iteration methods.

Feasibility Verification and Upper Bound Computation in Global Minimization Using Approximate Active Index Sets

We propose a new upper bounding procedure for global minimization problems with continuous variables and possibly nonconvex inequality and equality constraints. Upper bounds are crucial for standard termination criteria of spatial branch-and-bound (SBB) algorithms to ensure that they can enclose globally minimal values sufficiently well. However, whereas for most lower bounding procedures from the literature, convergence on smaller boxes is established, this does not hold for several methods to compute upper bounds even though they often perform well in practice. In contrast, our emphasis is on the convergence. We present a new approach to verify the existence of feasible points on boxes, on which upper bounds can then be determined. To this end, we resort to existing convergent feasibility verification approaches for purely equality and box constrained problems. By considering carefully designed modifications of subproblems based on the approximation of active index sets, we enhance such methods to problems with additional inequality constraints. We prove that our new upper bounding procedure finds sufficiently good upper bounds so that termination of SBB algorithms is guaranteed after a finite number of iterations. Our theoretical findings are illustrated by computational results on a large number of standard test problems. These results show that compared with interval Newton methods from the literature, our proposed method is more successful in feasibility verification for both, a full SBB implementation (42 instead of 26 test problems) and exhaustive sequences of boxes around known feasible points (120 instead of 29 test problems). History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. 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.2023.0162 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0162 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

A new efficient two-step iterative method for solving absolute value equations

PurposeThis study presents a two-step numerical iteration method specifically designed to solve absolute value equations. The proposed method is valuable and efficient for solving absolute value equations. Several numerical examples were taken to demonstrate the accuracy and efficiency of the proposed method.Design/methodology/approachWe present a two-step numerical iteration method for solving absolute value equations. Our two-step method consists of a predictor-corrector technique. The new method uses the generalized Newton method as the predictor step. The four-point open Newton-Cotes formula is considered the corrector step. The convergence of the proposed method is discussed in detail. This new method is highly effective for solving large systems due to its simplicity and effectiveness. We consider the beam equation, using the finite difference method to transform it into a system of absolute value equations, and then solve it using the proposed method.FindingsThe paper provides empirical insights into how to solve a system of absolute value equations.Originality/valueThis paper fulfills an identified need to study absolute value equations.

A generalization of B-differentiable equations method for elastoplastic contact problems with dual mortar discretization

In this study, a novel method based on the B-differentiable equations (BDEs) is proposed to solve the three-dimensional (3D) static and dynamic elastoplastic contact problems with friction. The contact conditions are formulated as the BDEs system, which can precisely satisfy Coulomb’s law of friction. Based on the mortar segment-to-segment contact model, the contact constraints are enforced by dual Lagrange multipliers for nonconforming meshes of the contact boundary. The non-smooth and nonlinear system equations composed of equilibrium and contact equations are simultaneously solved by the B-differentiable damping Newton method (BDNM), which has global convergence property. In each iteration, firstly, some contact displacements and contact forces, called substituted contact variables, are expressed by others according to the different contact states and contact conditions. The equilibrium equation is then solved, where the stiffness matrix is updated by substituting the contact variables. At last, the substituted contact variables will be restored. In the proposed method, the contact flexibility matrix is no longer needed, which saves the time for calculating the contact flexibility matrix due to changes in the stiffness matrix resulting from the update of stresses in the iteration step. By comparing with the results obtained by finite element commercial software ANSYS, the accuracy of the proposed algorithm is verified, and the property of convergence and efficiency are demonstrated for the elastoplastic contact problems by several numerical examples.

Investigation of Feasible and Marginal Operating Regimes of Electric Power Systems

The paper is devoted to the analysis of the feasibility domain of electric power systems. The problems of calculating feasible and marginal regimes of power systems, analyzing the geometry of the feasibility domain, and generating samples in this region are considered. Parallels are drawn with the works of B.T. Polyak on the analysis of the image of a quadratic map, modification of the Newton method and the development of methods for generating asymptotically uniform samples in areas with complex geometry. Particular attention is paid to Newton’s method with the transversality condition (TENR), its application for constructing a boundary oracle procedure and utilization for generating samples in the power system feasibility domain.

Ultra-fast calculation method of incident angle based on underwater acoustic round-trip positioning

In underwater acoustic round-trip positioning, the incident angle calculation in constant gradient acoustic ray tracing significantly hinders the efficiency of positioning calculation. To address this issue, we proposed an ultra-fast calculation method for determining incident angle based on underwater acoustic round-trip positioning. Specifically, the method employs the k-mean++ clustering algorithm to streamline the sound velocity profile. Additionally, a function formula for the incident angle is constructed, and its partial derivative with respect to Snell's constant is derived. Finally, Newton's method is applied to iteratively calculate the incident angles of the signal emission and reception, respectively. The simulation test results demonstrate that the proposed method reduces the positioning calculation time by 80.123%, 80.105%, 80.136%, and 80.104% on four transponders compared to the traditional method. In the field experiment, the proposed method is superior to the traditional method by 81.693% in positioning calculation time. These findings indicate the significant superiority of the proposed method in positioning calculation efficiency compared to the traditional method.

Newtonian Program Analysis of Probabilistic Programs

Due to their quantitative nature, probabilistic programs pose non-trivial challenges for designing compositional and efficient program analyses. Many analyses for probabilistic programs rely on iterative approximation. This article presents an interprocedural dataflow-analysis framework, called NPA-PMA, for designing and implementing (partially) non-iterative program analyses of probabilistic programs with unstructured control-flow, nondeterminism, and general recursion. NPA-PMA is based on Newtonian Program Analysis (NPA), a generalization of Newton's method to solve equation systems over semirings. The key challenge for developing NPA-PMA is to handle multiple kinds of confluences in both the algebraic structures that specify analyses and the equation systems that encode control flow: semirings support a single confluence operation, whereas NPA-PMA involves three confluence operations (conditional, probabilistic, and nondeterministic). Our work introduces ω-continuous pre-Markov algebras (ωPMAs) to factor out common parts of different analyses; adopts regular infinite-tree expressions to encode probabilistic programs with unstructured control-flow; and presents a linearization method that makes Newton's method applicable to the setting of regular-infinite-tree equations over ωPMAs. NPA-PMA allows analyses to supply a non-iterative strategy to solve linearized equations. Our experimental evaluation demonstrates that (i) NPA-PMA holds considerable promise for outperforming Kleene iteration, and (ii) provides great generality for designing program analyses.

Means-based class of modified Weerakoon-Fernando method for determining the zero of nonlinear equation

This paper introduces a generalized class of modified Weerakoon and Fernando iterative method (WFM) for determining the zero of equations that are nonlinear. The idea applied in the modification, involves the composition of a perturbed Newton method with a corrector iterative function that is based on parametric bi-variate rational function of degree one and a power means generating function. The convergence analysis on the developed class of methods, shows that it is of convergence order four with efficiency index of 1.5874. Some existing modified WFM were established to be concrete members of the developed class of methods. Numerical results obtained when the new methods were implemented are provided to compare their effectiveness and efficiency.

The Hessian by blocks for neural network by backward propagation

The back-propagation algorithm used with a stochastic gradient and the increase in computer performance are at the origin of the recent Deep learning trend. For some problems, however, the convergence of gradient methods is still very slow. Newton's method offers potential advantages in terms of faster convergence. This method uses the Hessian matrix to guide the optimization process but increases the computational cost at each iteration. Indeed, although the expression of the Hessian matrix is explicitly known, previous work did not propose an efficient algorithm for its fast computation. In this work, we first propose a backward algorithm to compute the exact Hessian matrix. In addition, the introduction of original operators, for the calculation of second derivatives, facilitates the reading and allows the parallelization of the backward-looking algorithm. To study the practical performance of Newton's method, we apply the proposed algorithm to train two classical neural networks for regression and classification problems and display the associated numerical results.

High-order shape interpolation

We propose a simple yet effective method to interpolate high-order meshes. Given two manifold high-order triangular (or tetrahedral) meshes with identical connectivity, our goal is to generate a continuum of curved shapes with as little distortion as possible in the mapping from the source mesh to the interpolated mesh. Our algorithm contains two steps: (1) linearly blend the pullback metric of the identity mapping and the input mapping between two Bézier elements on a set of sampling points; (2) project the interpolated metric into the metric space between Bézier elements using the Newton method for nonlinear optimization. We demonstrate the feasibility and practicability of the method for high-order meshes through extensive experiments in both 2D and 3D.

The Mathematical Modeling, Diffusivity, Energy, and Enviro-Economic Analysis (MD3E) of an Automatic Solar Dryer for Drying Date Fruits

Date fruit drying is a process that consumes a significant amount of energy due to the long duration required for drying. To better understand how moisture flows through the fruit during drying and to speed up this process, drying studies must be conducted in conjunction with mathematical modeling, energy analysis, and environmental economic analysis. In this study, twelve thin-layer mathematical models were designed utilizing experimental data for three different date fruit varieties (Sakkoti, Malkabii, and Gondaila) and two solar drying systems (automated solar dryer and open-air dryer). These models were then validated using statistical analysis. The drying period for the date fruit varieties varied between 9 and 10 days for the automated solar dryer and 14 to 15 days for open-air drying. The moisture diffusivity coefficient values, determined using Fick’s second law of diffusion model, ranged from 7.14 × 10−12 m2/s to 2.17 × 10−11 m2/s. Among the twelve thin-layer mathematical models, we chose the best thin drying model based on a higher R2 and lower χ2 and RMSE. The Two-term and Modified Page III models delivered the best moisture ratio projections for date fruit dried in an open-air dryer. For date fruit dried in an automated solar dryer, the Two-term Exponential, Newton (Lewis), Approximation diffusion or Diffusion Method, and Two-term Exponential modeling provided the best moisture ratio projections. The energy and environmental study found that the particular amount of energy used varied from 17.936 to 22.746 kWh/kg, the energy payback time was 7.54 to 7.71 years, and the net CO2 mitigation throughout the lifespan ranged from 8.55 to 8.80 tons. Furthermore, economic research showed that the automated solar dryer’s payback period would be 2.476 years.

Mastering the art: Proficient finite element implementation and robust evaluation of a strain-hardening porous ductile material crack growth prediction model at finite strain

Accurate prediction of crack extension is crucial for ensuring the structural integrity of metal components exposed to diverse loads. While the Gurson model and its extensions are widely accepted for delineating ductile fracture stages, especially in porous materials with a rigid-perfectly plastic matrix, their efficacy diminishes in the presence of strain-hardening matrices. To address this limitation, our study introduces a robust numerical implementation and assessment of Perrin's model. In contrast to Gurson's approach, Perrin's model incorporates two distinct strain-hardening parameters derived from an intricate analysis of a strain-hardening hollow sphere subjected to axisymmetric loading.Moving beyond theoretical discussions, this paper actively engages in the rigorous evaluation of Perrin's model under various scenarios, encompassing isotropic, kinematic, and mixed isotropic-kinematic hardening conditions. The model's effectiveness is convincingly demonstrated through meticulous comparisons between numerical simulations and real-world experimental observations conducted on pre-cracked specimens.Facing challenges related to achieving global elastic-plastic convergence in large-scale simulations, our research introduces a sophisticated approach. Leveraging stiffness tangent moduli ensure the maintenance of quadratic convergence in global Newton method iterations. This not only enhances the reliability of our simulations but also underscores the practical applicability of our findings.The study also focuses on the necking and crack propagation of a cylindrical specimen through meticulous computational modeling. Our mesh captures intricate details and addresses stress gradients, revealing coplanar crack propagation contrary to prior beliefs. The confirmation of the cup-cone fracture effect challenges previous interpretations, emphasizing the critical role of void nucleation. This reinforces the credibility and applicability of our approach in advancing the understanding of material fracture behavior.In summary, this study significantly contributes to existing knowledge in fracture prediction models, offering a robust framework for predicting ductile fracture under substantial deformations. The presented findings pave the way for advancements in structural engineering, providing invaluable insights for optimizing the performance and resilience of metal structures in real-world applications.

A double‐layer Jacobi method for partial differential equation‐constrained nonlinear model predictive control

AbstractThis paper presents a real‐time optimization method for nonlinear model predictive control (NMPC) of systems governed by partial differential equations (PDEs). The NMPC problem to be solved is formulated by discretizing the PDE system in space and time by using the finite difference method. The proposed method is called the double‐layer Jacobi method, which exploits both the spatial and temporal sparsities of the PDE‐constrained NMPC problem. In the upper layer, the NMPC problem is solved by ignoring the temporal couplings of either the state or costate (Lagrange multiplier corresponding to the state equation) equations so that the spatial sparsity is preserved. The lower‐layer Jacobi method is a linear solver dedicated to PDE‐constrained NMPC problems by exploiting the spatial sparsity. Convergence analysis indicates that the convergence of the proposed method is related to the prediction horizon. Results of a numerical experiment of controlling a heat transfer process show that the proposed method can be two orders of magnitude faster than the conventional Newton's method exploiting the banded structure of NMPC problems.

Multifrequency Electrical Impedance Tomography Images for Swine Pneumonia Detection Using Conductivity Differences

This article introduces a new MfEIT UDESC Mark I system, which consists of a 32-electrode setup featuring a modified Howland current source, low cost, portability, and non-radiation. The system is capable of reconstructing electrical conductivity tomographic images at a rate of 30.624 frames per second, taking about 5 min for imaging. The current source employs a 0.5 mA adjacent current application pattern with frequencies ranging from 10 kHz to 1 MHz. This article outlines the hardware, firmware, and software design specifications, which include the design of the current source, calibration procedures, and image reconstruction process. Tomographic images of conductivity were reconstructed in ex vivo healthy pig lungs and those with pneumonia, as a proof of concept for future applications in live pigs. The high spectral power density, combined with real-time system calibration provides clinical advantages in veterinary medicine. The goal is to identify lung areas affected by Mycoplasma hyopneumoniae in pigs through the analysis of electrical conductivity difference, offering a valuable tool to assist veterinarians to obtain images of respiratory diseases. The modified reconstruction method GREIT (EIDORS) was evaluated with experimental data and was compared with the Gauss–Newton and Total Variation methods, where GREIT 2D proved to be superior.

Global convergence analysis of Caputo fractional Whittaker method with real world applications

The present article deals with the effect of convexity in the study of the well-known Whittaker iterative method, because an iterative method converges to a unique solution \(t^*\) of the nonlinear equation \(\psi(t)=0\) faster when the function's convexity is smaller. Indeed, fractional iterative methods are a simple way to learn more about the dynamic properties of iterative methods, i.e., for an initial guess, the sequence generated by the iterative method converges to a fixed point or diverges. Often, for a complex root search of nonlinear equations, the selective real initial guess fails to converge, which can be overcome by the fractional iterative methods. So, we have studied a Caputo fractional double convex acceleration Whittaker's method (CFDCAWM) of order at least (\(1+2\zeta\)) and its global convergence in broad ways. Also, the faster convergent CFDCAWM method provides better results than the existing Caputo fractional Newton method (CFNM), which has (\(1+\zeta\)) order of convergence. Moreover, we have applied both fractional methods to solve the nonlinear equations that arise from different real-life problems.

Simplified Algorithm for the Worldvolume HMC and the Generalized Thimble HMC

Abstract The Worldvolume Hybrid Monte Carlo method (WV-HMC method) is a reliable and versatile algorithm towards solving the sign problem. Like the tempered Lefschetz thimble method, this method removes the ergodicity problem inherent in algorithms based on Lefschetz thimbles. In addition to this advantage, the WV-HMC method significantly reduces the computational cost because one need not compute the Jacobian of deformation in generating configurations. A crucial step in this method is the RATTLE algorithm, where the Newton method is used at each molecular dynamics step to project a transported configuration onto a submanifold (worldvolume) in the complex space. In this paper, we simplify the RATTLE algorithm by employing a simplified Newton method (the fixed-point method) along with iterative solvers for orthogonal decompositions of vectors, and show that this algorithm further reduces the computational cost. We also apply this algorithm to the HMC algorithm for the generalized thimble method (GT-HMC method). We perform a numerical test for the convergence of the simplified RATTLE algorithm, and show that the convergence depends on the system size only weakly. The application of this simplified algorithm to various models will be reported in subsequent papers.