Newton Iteration Research Articles

An accelerated inexact Newton-type regularizing algorithm for ill-posed operator equations

We propose and analyze a new iterative regularization approach, called IN-SETPG, for efficiently solving nonlinear ill-posed operator equations in the Hilbert-space setting. IN-SETPG consists of an outer iteration and an inner iteration. The outer iteration is terminated by the discrepancy principle and consists of an inexact Newton regularization method, while the inner iteration is performed by a sequential subspace optimization method based on the two-point gradient iteration. The key idea behind IN-SETPG is that, unlike the standard Landweber method, it uses multiple search directions per iteration in combination with an adaptive step size in order to reduce the total number of iterations. The regularization property of IN-SETPG has been established, i.e., the iterate converges to a solution of the nonlinear problem with exact data when the noise level tends to zero. Various numerical experiments are presented to demonstrate that, compared with the original inexact Newton iteration, IN-SETPG can achieve better reconstruction results and remarkable acceleration.

Physical invariants-preserving compact difference schemes for the coupled nonlinear Schrödinger-KdV equations

In this paper, we develop efficient compact difference schemes for the coupled nonlinear Schrödinger-KdV (CNLS-KdV) equations to conserve all the physical invariants, namely, the energy of oscillations, the number of plasmon, the number of particle and the momentum. By combining the exponential scalar auxiliary variable (E-SAV) approach, we reconstruct the original CNLS-KdV equations and adopt the compact difference method and Crank-Nicolson method to develop energy stable scheme. The E-SAV compact difference scheme preserves the total energy and the number of particle. We further introduce two Lagrange multipliers for the E-SAV reformulation system to develop compact difference scheme, which preserves exactly the number of plasmon and the momentum. At each time step for the second scheme, we only need to solve linear systems with constant coefficients and nonlinear quadratic algebraic equations which can be efficiently solved by Newton's iteration. Numerical experiments are given to show the effectiveness, accuracy and performance of the proposed schemes.

A comparative study of sensitivity computations in ESDIRK-based optimal control problems

This paper compares the impact of iterated and direct approaches to sensitivity computation in fixed step-size explicit singly diagonally implicit Runge–Kutta (ESDIRK) methods when applied to optimal control problems (OCPs). We strictly use the principle of internal numerical differentiation (IND) for the iterated approach, i.e., reusing iteration matrix factorizations, the number of Newton-type iterations, and Newton iterates, to compute the sensitivities. The direct method computes the sensitivities without using the Newton schemes. We compare the impact of these sensitivity computations in OCPs for the quadruple tank system (QTS). We discretize the OCPs using multiple shooting and solve these with a sequential quadratic programming (SQP) solver. We benchmark the iterated and direct approaches against a base case. This base case applies the ESDIRK methods with exact Newton schemes and a direct approach for sensitivity computations. In these OCPs, we vary the number of integration steps between control intervals and evaluate the performance based on the number of SQP and QPs iterations, KKT violations, function evaluations, Jacobian updates, and iteration matrix factorizations. We also provide examples using the continuous-stirred tank reactor (CSTR), and the IPOPT algorithm instead of the SQP. For OCPs solved using SQP, the QTS results show the direct method converges only once, while the iterated approach and base case converges in all situations. Similar results are seen with the CSTR. Using IPOPT, both the iterated approach and base case converge in all cases. In contrast, the direct method only converges in all cases regarding the CSTR.

MATRICS: The implicit matrix-free Eulerian hydrodynamics solver

Context. There exists a zoo of different astrophysical fluid dynamics solvers, most of which are based on an explicit formulation and hence stability-limited to small time steps dictated by the Courant number expressing the local speed of sound. With this limitation, the modeling of low-Mach-number flows requires small time steps that introduce significant numerical diffusion, and a large amount of computational resources are needed. On the other hand, implicit methods are often developed to exclusively model the fully incompressible or 1D case since they require the construction and solution of one or more large (non)linear systems per time step, for which direct matrix inversion procedures become unacceptably slow in two or more dimensions. Aims. In this work, we present a globally implicit 3D axisymmetric Eulerian solver for the compressible Navier–Stokes equations including the energy equation using conservative formulation and a fully simultaneous approach. We use the second-order-in-time backward differentiation formula for temporal discretization as well as the κ scheme for spatial discretization. We implement different limiter functions to prohibit the occurrence of spurious oscillations in the vicinity of discontinuities. Our method resembles the well-known monotone upwind scheme for conservation laws (MUSCL). We briefly present efficient solution methods for the arising sparse and nonlinear system of equations. Methods. To deal with the nonlinearity of the Navier–Stokes equations we used a Newton iteration procedure in which the required Jacobian matrix-vector product was reconstructed with a first-order finite difference approximation to machine precision in a matrix-free way. The resulting linear system was solved either completely matrix-free with a combination of a sufficient Krylov solver and an approximate Jacobian preconditioner or semi-matrix-free with the incomplete lower upper factorization technique as a preconditioner. The latter was dependent on a standalone approximation of the Jacobian matrix, which was optionally calculated and needed solely for the purpose of preconditioning. Results. We show our method to be capable of damping sound waves and resolving shocks even at Courant numbers larger than one. Furthermore, we prove the method’s ability to solve boundary value problems like the cylindrical Taylor-Couette flow (TC), including viscosity, and to model transition flows. To show the latter, we recover predicted growth rates for the vertical shear instability, while choosing a time step orders of magnitude larger than the explicit one. Finally, we verify that our method is second order in space by simulating a simplistic, stationary solar wind.

A simplified reliability-based design method for multiple design variables considering different limit states

A generalized reliability model comprising the objective, constraint, and judgment functions is established for the reliability index approach (RIA), taking parameters’ properties of engineering practice and negative reliability index into consideration. Based on this, the reliability-based design (RBD) problem with multiple design variables is translated into the solution to the nonlinear equations, and a simplified method consisting of a simple variant of the Newton iteration method and the finite difference method (FDM) is proposed. Numerical examples are presented to verify the efficiency of the proposed reliability approach and to determine the incremental step size for FDM. RBD of a simply supported beam is illustrated and the variabilities of design variables are investigated considering the uncertainties in the manufacturing process and practical operations. Results reveal that the variations of the design variables should not be ignored. Moreover, analysis results show that the design value might not intuitively increase with the increase of its coefficient of variation (CoV), and it might not increase with the increasing reliability requirement for problems involving multiple variables. The reasons for this phenomenon are very complicated, and it is a systematic problem. One should be aware of this phenomenon, and specific analysis is required for specific problems.

Analysis of nonlinear problems using the Dual Mesh Control Domain Method with arbitrary meshes

The Dual Mesh Control Domain Method (DMCDM), proposed and developed by the second author, is a recent advancement in the field of numerical methods. This method represents a blend of the Finite Element Method (FEM) and the Finite Volume Method (FVM), each having its primary domains of popularity. The DMCDM incorporates the concepts of interpolation, duality, and numerical integration from the former and the idea of satisfying the integral (global) form of the differential (or balance) equations from the latter. In this paper, we present a fresh and systematic approach to solving two-dimensional nonlinear problems, including advection–diffusion problems, with arbitrary primal meshes. We develop an approach where element-level equations are derived and assembled in a manner similar to that of the FEM, as opposed to using nodal equations, as has been the convention in all of the previous works involving the DMCDM. Additionally, we extend the previous works to include higher-order interpolation, specifically quadratic interpolation. In short, we provide formulations for four-node bilinear and nine-node biquadratic elements of quadrilateral shape, enabling the application of the DMCDM to arbitrary primal and dual meshes. Finally, we explore the Newton iteration method for linearizing the nonlinear system of equations that arises from the DMCDM discretization. We present numerical results and comparisons against available exact and numerical solutions to verify the accuracy and robustness of the DMCDM.

Improvement of augmented free-water flash algorithm based on HV mixing rule and simulated annealing optimization algorithm

The calculation of three-phase equilibrium in CO2-hydrocarbon-water mixtures holds significant importance within numerical simulations, particularly in applications such as CO2-enhanced oil recovery and carbon dioxide sequestration. However, due to the non-ideality of CO2 and the polarity of water, three-phase equilibrium calculations often encounter convergence challenges. The augmented free-water flash algorithm [6], specifically focusing on CO2 dissolution in the aqueous phase, addresses the convergence challenges encountered by traditional three-phase flash algorithms. Despite its accuracy diminishes under conditions of high pressure and high CO2 concentrations, limiting its capability to accurately describe CO2 dissolution in oil-gas-water multiphase systems. The GE (excess Gibbs free energy) type mixing rule can be effectively integrated with equations of state, including PR and SRK, by utilizing activity models like NRTL and UNIFAC, which can be applied to the calculation of thermodynamic parameters for both nonpolar and polar systems and high-pressure complex systems. In this study, based on the augmented free-water assumption, we have developed a new augmented free-water three-phase flash algorithm for CO2/hydrocarbon/water mixtures. This algorithm integrates the PR equation of state with the NRTL activity model, employing the HV mixing rule for CO2-water interactions and the Van der Waals rule for other non-polar components. Furthermore, to enhance computational efficiency, the algorithm incorporates the simulated annealing global optimization algorithm, replacing Newton iteration to more effectively identify the global minima of the complex objective function. The example calculations show that the improved augmented free-water flash algorithm has higher accuracy and efficiency than the traditional augmented free-water flash algorithm. The augmented free-water three-phase flash algorithm proposed in this paper offers a precise model for phase equilibrium calculations essential for numerical simulations in CO2-enhanced oil recovery and carbon dioxide sequestration.

Estimation of Soil Characteristic Parameters for Electric Mountain Tractor Based on Gauss–Newton Iteration Method

Future field work tasks will require mountain tractors to pass through rough terrain with limited human supervision. The wheel–soil interaction plays a critical role in rugged terrain mobility. In this paper, an algorithm for the estimation of soil characteristic parameters based on the Simpson numerical integration method and Gauss–Newton iteration method is presented. These parameters can be used for passability prediction or in a traction control algorithm to improve tractor mobility and to plan safe operation paths for autonomous navigation systems. To verify the effectiveness of the solving algorithm, different initial values and soils were selected for simulation calculations of soil characteristic parameters such as internal friction angle, settlement index, and the joint parameter of soil cohesion modulus and friction modulus. The results show that the error was kept within 2%, and the calculation time did not exceed 0.84 s, demonstrating high robustness and real-time performance. To test the applicability of the algorithm model, further research was conducted using different wheel parameters of electric mountain tractors under wet clay conditions. The results show that these parameters also have high accuracy and stability with only a few iterations. Thus, the estimation algorithm can meet the requirements of quickly and accurately identifying soil characteristic parameters during tractor operation. A criterion for the passability of wheeled tractors through unknown terrain is proposed, utilizing identified soil parameters.

Discrete-time Euler-smoothing methods for time-varying convex constrained optimization

In this paper, we propose two kinds of prediction–correction methods, discrete-time Euler-smoothing methods, for solving the general time-varying convex constrained optimization problem. By using the complementarity function and smoothing technique, the Karush–Kuhn–Tucker (KKT) trajectory of the problem can be reformulated as a system of smooth equations and considering the dynamics of the system of smooth equations in a discrete constant time-sampling periods scheme. The proposed methods mainly consist of the prediction step derived by Euler approximation and the correction step generated by one or multiple Newton iterations. Under some reasonable conditions, the approximate solution trajectories produced by the methods achieve the asymptotic error bound behaving as O(h4) with respect to optimal trajectory both in the primal and dual space as the smoothing parameter approaches to zero, where h is the sampling period. Finally, the experimental results show that our methods are efficient, and the steady-state tracking errors are much smaller than the other prediction–correction methods under the same conditions.

Enhanced Moving Source Localization with Time and Frequency Difference of Arrival: Motion-Assisted Method for Sub-Dimensional Sensor Networks

Localizing a moving source by Time Difference of Arrival (TDOA) and Frequency Difference of Arrival (FDOA) commonly requires at least N+1 sensors in N-dimensional space to obtain more than N pairs of TDOAs and FDOAs, thereby establishing more than 2N equations to solve for 2N unknowns. However, if there are insufficient sensors, the localization problem will become underdetermined, leading to non-unique solutions or inaccuracies in the minimum norm solution. This paper proposes a localization method using TDOAs and FDOAs while incorporating the motion model. The motion between the source and sensors increases the equivalent length of the baseline, thereby improving observability even when using the minimum number of sensors. The problem is formulated as a Maximum Likelihood Estimation (MLE) and solved through Gauss–Newton (GN) iteration. Since GN requires an initialization close to the true value, the MLE is transformed into a semidefinite programming problem using Semidefinite Relaxation (SDR) technology, while SDR results in a suboptimal estimate, it is sufficient as an initialization to guarantee the convergence of GN iteration. The proposed method is analytically shown to reach the Cramér–Rao Lower Bound (CRLB) accuracy under mild noise conditions. Simulation results confirm that it achieves CRLB-level performance when the number of sensors is lower than N+1, thereby corroborating the theoretical analysis.

Newton iteration for lexicographic Gröbner bases in two variables

We present an m-adic Newton iteration with quadratic convergence for lexicographic Gröbner basis of zero dimensional ideals in two variables. We rely on a structural result about the syzygies in such a basis due to Conca and Valla, that allowed them to explicitly describe these Gröbner bases by affine parameters; our Newton iteration works directly with these parameters.

On the Quadratic Convergence of Newton’s Method for Mean Field Games with Non-separable Hamiltonian

AbstractWe analyze asymptotic convergence properties of Newton’s method for a class of evolutive Mean Field Games systems with non-separable Hamiltonian arising in mean field type models with congestion. We prove the well posedness of the Mean Field Game system with non-separable Hamiltonian and of the linear system giving the Newton iterations. Then, by forward induction and assuming that the initial guess is sufficiently close to the solution of problem, we show a quadratic rate of convergence for the approximation of the Mean Field Game system by Newton’s method. We also consider the case of a nonlocal coupling, but with separable Hamiltonian, and we show a similar rate of convergence.

A new family of fourth-order energy-preserving integrators

For Hamiltonian systems with non-canonical structure matrices, a new family of fourth-order energy-preserving integrators is presented. The integrators take a form of a combination of Runge–Kutta methods and continuous-stage Runge–Kutta methods and feature a set of free parameters that offer greater flexibility and efficiency. Specifically, we demonstrate that by carefully choosing these free parameters, a simplified Newton iteration applied to the integrators of order four can be parallelizable. This results in faster and more efficient integrators compared with existing fourth-order energy-preserving integrators.

Investigation on the operating characteristics of a three-phase crystalline energy storage and heating system based on lithium bromide

In the current global energy landscape, energy storage has the potential to become a key technical support for global carbon neutrality. Drawing insights from a comprehensive overview of existing energy storage systems, this paper proposes a three-phase crystalline energy storage and heating system characterized by intermittent operation. The unique thermodynamic property of lithium bromide gifts the system with remarkable energy storage density and heating capacity. To further investigate its operating characteristics, this study employs the Newton iteration method, with energy storage density as a pivotal technical indicator for evaluation. The results reveal that the existence of crystals mainly benefits nighttime heating supply. By adjusting input factors, it becomes evident that increasing the temperature of high-temperature heat source can enhance the heating supply by 16.92 %, which also leads to an increase in energy storage density to 2986.85 MJ/m3. Furthermore, the raising temperature of low-temperature heat source results in a substantial 61.78 % escalation in heating supply. Additionally, the dynamic investment payback period of the system is estimated to be 2.4 years, signifying its economic viability. The findings of this paper can provide valuable theoretical support for the application of crystalline energy storage systems.

Simulation of Drying‐Rewetting Processes in Numerical Groundwater Models Using a New Picard Iteration‐Based Method

AbstractWhen simulating groundwater flow in unconfined and convertible aquifers using a groundwater model with the block‐centered finite‐difference approach, such as MODFLOW, it frequently encounters drying and rewetting of cells. Although many drying and rewetting simulation methods have been proposed in the past, balancing simulation accuracy and convergence capability all at once is difficult. MODFLOW‐2005, which has second‐order accuracy, employs a trial‐and‐error method, but it suffers from computational instability when large quantities of grid cells are dried. MODFLOW‐NWT adopts the upstream‐weighting approach and Newton iteration method to ensure the stability of the drying and rewetting simulations. However, the upstream‐weighting approach has only first‐order accuracy, and the Newton iteration method is complex to implement because it necessitates the establishment of an additional Jacobian matrix. The methods employed by MODFLOW‐NWT are also available in MODFLOW 6, therefore it inherits both the strengths and weaknesses of MODFLOW‐NWT. In this study, a new method, Picard iteration‐based always active cell (PAAC), is proposed. Similar to MODFLOW‐NWT, the PAAC method also uses dry cells as active cells. The PAAC method, however, does not use the upstream‐weighting approach and has second‐order accuracy. Moreover, it ensures good convergence stability even under the Picard iteration method. In addition to discussing the algorithm, five cases were used to comprehensively compare the simulation effects of the PAAC method with MODFLOW‐2005 and MODFLOW‐NWT, including an analytical solution, repeated drying‐rewetting of multi‐layer grids, pumping well problem, perched aquifer problem and a nearly dry single‐layer grid, which verified the practicability of the PACC method.

A hybrid path-following approach for constrained nonlinear-buckling optimization of variable stiffness composite shells with shape imperfections

Nonlinear buckling optimization of variable stiffness (VS) composite shells is computationally expensive due to frequent nonlinear analyses. Although the generalized path-following method can serve to reduce the redundant computation, it is weakened by two kinds of discontinuity. First, the design variables may not always be searched continuously in the constrained optimization. Second, the critical buckling states may not be continuous at singular points on the search path. To address this issue, a hybrid path-following approach is proposed to trace the critical buckling states in the nonlinear-buckling optimization with constraints on the realizability of the lamination parameters and the amplitude range of geometric imperfections. Firstly, the optimal search direction is determined with the sensitivity analysis of the critical buckling equations in the isogeometric continuum shell model. Then, a continuous following strategy is adopted to provide a proper initial estimate for the Newton iterations at a disconnected point. Moreover, an investigation into the singularity of the critical buckling equations is conducted to identify the discontinuous points of the critical states in a line-search. Upon detection of the singularity, a segment of path in state space is inserted to recover the continuity. Numerical experiments show that the continuity of the critical states in the optimization is well maintained, which leads to a more efficient optimization for the VS shell with shape imperfections.

Fractional Block Method for the Solution of Fractional Order Differential Equations

The construction of the fourth-order 2-point Fractional Block Backward Differentiation Formula (2FBBDF(4)) to solve the fractional order differential equations (FDEs) is presented in this paper. The method is developed using the fractional linear multistep method (FLMM) linked with the linear difference operator. This paper aims to approximate the fractional order problems via 2FBBDF(4), which is normally used to solve ordinary differential equations. The criteria for the stability of the method are analyzed in order to solve FDE problems. Consequently, the method is determined to be \textit{A}-stable for different values of α within the interval (0,1) . Then, Newton's iteration is implemented in this method to solve the problems. Multiple numerical examples of linear, nonlinear, and system FDEs are provided for the scenario where the order α lies within the range of 0 and 1 . Ultimately, the numerical results confirm that the proposed method performs at a similar level to the existing methods.

A model of erosion rate prediction for component with complex geometry based on numerical simulation

The structural components prone to erosion damage often exhibit irregular and complex shapes. Predicting erosion rates for such irregular geometries is a field that needs more attention, since it can be used to evaluate the performance of the component. This study aims to develop an evaluation method for predicting erosion rates of components with complex geometry. Standard erosion tests were performed on small plates to investigate the effects of different impact angles and particle velocities on the erosion rate of the target material. Based on these test results, the Newton iteration method was employed to determine the parameters of the Tabakoff and Grant erosion model. Moreover, these parameters were used in computational fluid dynamics simulation of components with complex geometries. Besides, a model was developed to convert the erosion rate density into erosion rate in numerical simulation, allowing quantitative comparison of simulation results with experimental data. The calibration of the erosion model parameters was conducted by using 1060 aluminum alloy specimens. The predicted erosion rate changes with impact angle showed consistency good agreement with the testing data. Subsequently, the erosion rate of a simplified similar structure for aircraft engine blade made of 1060 aluminum alloy was predicted and verified.

A Lie group variational integrator in a closed-loop vector space without a multiplier

Abstract. As a non-tree multi-body system, the dynamics model of four-bar mechanism is a differential algebraic equation. The constraints breach problem leads to many problems for computation accuracy and efficiency. With the traditional method, constructing an ODE-type dynamics equation for it is difficult or impossible. In this exploration, the dynamics model is built with geometry mechanic theory. The kinematic constraint variation relation of a closed-loop system is built in matrix and vector space with Lie group and Lie algebra theory respectively. The results indicate that the attitude variation between the driven body and the follower body has a linear recursion relation, which is the basis for dynamics modelling. With the Lie group variational integrator method, the closed-loop system Lagrangian dynamics model is built in vector space, with Legendre transformation. The dynamics model is reduced to be the Hamilton type. The kinematic model and dynamics model are solved using Newton iteration and the Runge–Kutta method respectively. As a special case of a crank and rocker mechanism, the dynamics character of a parallelogram mechanism is presented to verify the good structure conservation character of the closed-loop geometry dynamics model.

Remarks on Solving Methods of Nonlinear Equations

Abstract: In the field of mechanical engineering, many practical problems can be converted into nonlinear problems, such as the meshing problem of mechanical transmission. So the solution of nonlinear equations has important theoretical research and practical application significance. Whether the traditional Newton iteration method or the intelligent optimization algorithm after the popularization of computers, both them have been greatly enriched and developed through the continuous in-depth research of scholars at home and abroad, and a series of improved algorithms have emerged. This paper mainly reviews the research status of solving nonlinear equations from two aspects of traditional iterative method and intelligent optimization algorithm, systematically reviews the research achievements of domestic and foreign scholars, and puts forward prospects for future research directions.