Newton Method Research Articles

A novel Newton method for inverse elastic scattering problems

This work is concerned with an inverse elastic scattering problem of identifying the unknown rigid obstacle embedded in an open space filled with a homogeneous and isotropic elastic medium. A Newton-type iteration method relying on the boundary condition is designed to identify the boundary curve of the obstacle. Based on the Helmholtz decomposition and the Fourier–Bessel expansion, we explicitly derive the approximate scattered field and its derivative on each iterative curve. Rigorous mathematical justifications for the proposed method are provided. Numerical examples are presented to verify the effectiveness of the proposed method.

Enhanced spectral collocation Gegenbauer approach for the time‐fractional Fisher equation

In this study, we utilize a specific combination of orthogonal Gegenbauer polynomials as basis functions that satisfy homogeneous boundary conditions in the spatial variable and homogeneous initial conditions in the temporal variable. We derive an expression for the fractional derivative of the temporal basis and another expression for the second‐order derivative of the spatial basis. Subsequently, we discretize the nonlinear Fisher problem using the spectral collocation method to transform the problem into the solution of a nonlinear system of algebraic equations. We then solve this resulting system using Newton's method with an initial guess approaching zero. Furthermore, we conduct an error analysis of the proposed method and assess its applicability through three test problems, providing comparisons for validation.

Reconstruction of inhomogeneous media by an iteration algorithm with a learned projector

This paper is concerned with the inverse problem of reconstructing an inhomogeneous medium from the acoustic far-field data at a fixed frequency in two dimensions. This inverse problem is severely ill-posed (and also strongly nonlinear), and certain regularization strategy is thus needed. However, it is difficult to select an appropriate regularization strategy which should enforce some a priori information of the unknown scatterer. To address this issue, we plan to use a deep learning approach to learn some a priori information of the unknown scatterer from certain ground truth data, which is then combined with a traditional iteration method to solve the inverse problem. Specifically, we propose a deep learning-based iterative reconstruction algorithm for the inverse problem, based on a repeated application of a deep neural network and the iteratively regularized Gauss–Newton method (IRGNM). Our deep neural network (called the learned projector in this paper) mainly focuses on learning the a priori information of the shape of the unknown contrast with a normalization technique in the training processes and is trained to act like a projector which is helpful for projecting the solution into some feasible region. Extensive numerical experiments show that our reconstruction algorithm provides good reconstruction results even for the high contrast case and has a satisfactory generalization ability.

A new truncated-perturbed Gauss–Newton method for underdetermined nonlinear least squares problems

In this paper, we study the solvability of a truncated-perturbed Gauss–Newton method for solving underdetermined nonlinear least squares problems. Our aim is to address a new analysis of a semilocal convergence to the aforementioned method. In particular, the main theorem is established under a kind of Hölder-relaxed condition, and two special cases of this are obtained. Furthermore, the computational behavior of the considered method is illustrated with some numerical tests.

A new MM algorithm for root‐finding problems

The minorization–maximization (MM) algorithm is an optimization technique for iteratively calculating the maximizer of a concave target function rather than a root–finding tool. In this paper, we in the first time develop the MM algorithm as a new method for seeking the root of a univariate nonlinear equation . The key idea is to transfer the root–finding issue to iteratively calculate the maximizer of a concave target function by designing a new MM algorithm. According to the ascent property of the MM algorithm, we know that the proposed algorithm converges to the root and does not depend on any initial values, in contrast to Newton's method. Several statistical examples are provided to demonstrate the proposed algorithm.

A comparison of the convergence rates of Hestenes’ conjugate Gram-Schmidt method without derivatives with other numerical optimization methods

Abstract This article describes an approach known as the conjugate Gram-Schmidt method for estimating gradients and Hessian using function evaluations and difference quotients, and uses the Gram-Schmidt conjugate direction algorithm to minimize functions and compares it to other techniques for solving ∇f = 0. Comparable minimization algorithms are also used to demonstrate convergence rates using quotient and root convergence factors, as described by Ortega and Rheinbolt to determine the optimal minimization technique to obtain similar results to the Newton method, between the Gram-Schmidt approach and other minimizing approaches. A survey of the existing literature in order to compare Hestenes’ Gram-Schmidt conjugate direction approach without derivative to other minimization methods is conducted and the further analytical and computational details are provided.

Combined methods for solving degenerate unconstrained optimization problems

UDC 519.853.6 : 519.613.2 We present constructive second- and fourth-order methods for solving degenerate unconstrained optimization problems. The fourth-order method applied in the present work is a combination of the Newton method and the method that uses fourth-order derivatives. Our approach is based on the decomposition of ℝ n into the direct sum of the kernel of a Hessian matrix and its orthogonal complement. The fourth-order method is applied to the kernel of the Hessian matrix, whereas the Newton method is applied to its orthogonal complement. This method proves to be efficient in the case of a one-dimensional kernel of the Hessian matrix. In order to get the second-order method, Newton's method is combined with the steepest-descent method. We study the efficiency of these methods and analyze their convergence rates. We also propose a new adaptive combined quasi-Newton-type method (ACQNM) based on the use of the second- and fourth-order methods in the degenerate case. The efficiency of ACQNM is demonstrated by analyzing an example of some most common test functions.

A theoretical method for oblique and curved detonation waves

In this paper, a theoretical solution method for the post-wave parameters of detonation is proposed and developed with a series of analyses and applications. Based on Newton's method, the objective function for shock-coupled chemical reactions is constructed along with its derivative. Two verification examples demonstrate that the method can calculate accurate post-wave parameters quickly and is suitable for single-step and detailed mechanistic chemical reactions. In addition, the method provides sensitivities between various aerodynamic parameters to offer a fresh perspective for detonation, polar analysis with sensitivity is built as a result. Moreover, the method can predict the transition pattern of the detonation, and the validity is supported by the comparison of different examples. Rather than being limited to oblique detonation, the post-wave parameters of the curved detonation can also be calculated correctly, which indicates the excellent applicability of the method. This method can also be applied to the thermodynamic efficiency of detonation combustion and its sensitivity, which demonstrates the unique advantages of this method. Furthermore, the method can be rewritten as a solution for wedge angle under the given wave angle by changing the independent variable. This solution is validated by the simulation results, which implies that the method can be used as a simple inverse design method in oblique detonation engines. In general, the proposed method is an effective theoretical solution, analytical tool, and inverse design method for detonation.

Controlling parameters proportional integral derivative of DC motor using a gradient-based optimizer

In this paper, a gradient-based optimizer (GBO) algorithm is presented to optimize the parameters of a proportional integral derivative (PID) controller in DC motor control. The GBO algorithm which mathematically models and mimics is inspired by the gradient-based Newton method. It was developed to address various optimization issues. To determine the performance of the proposed method, a comparison method with the ant colony optimization (ACO) method. It was compared using the integral of time multiplied absolute error (ITAE). They are most popularly used in the literature. From the test results, the proposed method is promising and has better effectiveness. The proposed method, namely GBO-PID, shows the best performance.

Nonlinear vibrations and critical angular velocity information of the high-speed rotating eccentric disk made of Gori-metamaterials: Introducing data-driven solution for solving the nonlinear problems

Eccentric discs in rotational motion are commonly utilized in various technical fields, including gas turbine engines, flywheels, gears, and brakes. So, improving its critical angular velocity and frequency characteristics is a challenging issue for engineers. So, in this work for the first time, nonlinear vibrations and buckling analysis of the high-speed rotating eccentric disks using mathematical simulation and data-driven solutions are presented. One of the suggestions for improving its mechanical properties is considering metamaterials in the construction of the eccentric disk. Metamaterial is a novel synthetic material that has distinctive physical and mechanical capabilities that are unattainable in natural materials due to its well-designed structure. The properties of the eccentric disks are controlled by the amount of graphene and the degree of folding of graphene origami (GOri) across the thickness of the eccentric disks. These properties, such as Poisson's ratio, vary depending on the position and can be estimated using micromechanical models assisted by genetic programming (GP). Using von-Karman nonlinearity, transformed differential quadrature method (TDQM), and Newton's method the nonlinear governing equations are obtained and solved, respectively. The results show that when the radius ratio of the rotating eccentric disk increases by 8%, the critical point is decreased from 420 HZ to 370 HZ, a reduction of around 12 %. Another suggestion for improving its mechanical properties is controlling the geometrics and physics of the presented structure according to the designer's purposes.

Optimization in complex spaces with the mixed Newton method

Optimization in complex spaces with the mixed Newton method

Nonlinearly preconditioned semismooth Newton algorithms for nonlinear nonsmooth systems

We aim to develop efficient and robust algorithms for nonsmooth nonlinear systems arising from complementarity problems. The semismooth Newton algorithm is popular due to its reliability and efficiency. However, it struggles with issues with imbalanced nonlinearities of the problems, leading to degraded convergence rates or failure despite help from the globalization techniques like linesearch or trust region. We introduce a right nonlinearly preconditioned semismooth Newton algorithm to address this difficulty. The critical success ingredient is that before each global Newton update, a nonlinear preconditioning step implicitly removes the so-called ‘bad components’ causing trouble via nonlinear subspace correction, inspired by Gaussian elimination but adapted nonlinearly to balance system nonlinearities. Additionally, our method integrates with a domain decomposition framework, enhancing parallelism. Numerical results on two classes of problems demonstrate significantly improved convergence over standard semismooth Newton methods.

Correction to: A Gauss–Newton method for mixed least squares-total least squares problems

Correction to: A Gauss–Newton method for mixed least squares-total least squares problems

Collision-Avoiding Positioning of Clothing Parts for Preventing Onset Tanglements.

Positioning clothing parts ( S s) such as sleeves and collars has been in the realm of manual task that had to be done meticulously in order to prevent unnecessary tanglements during the simulation. This paper proposes an optimization-based method to computerize the above S-positioning task. For that, we embed each S to an abstracting cylinder C such that S-positioning can be done by adjusting only 3 ∼4 DOFs (e.g., translating/rotating C or adjusting its radius) instead of per-vertex-full-DOFs. Then, we formulate an objective function E by scoring undesirableness of S' s position (e.g., S penetrating the body, S making cloth-to-cloth intersection). In organizing E into the loop of the Newton's method, the main challenge was to calculate the symbolic gradient and hessian, for which this paper makes several novel contributions. The resultant S-positioning method works quite successfully; S* s (the output of the S-positioning method ) are tanglement-free thus running the simulator to that configuration produces acceptable draping quickly; Experiments show that, in obtaining acceptable draping, the proposed method produces about ×9.7 speed up compared to when not using it.

Comments on finite termination of the generalized Newton method for absolute value equations

Comments on finite termination of the generalized Newton method for absolute value equations

A two-grid characteristic block-centered finite difference method for Darcy-Forchheimer slightly compressible miscible displacement problem

In this paper, a two-grid characteristic block-centered finite difference method is developed for solving a nonlinear Darcy-Forchheimer slightly compressible miscible displacement problem in porous media. The method is proposed for solving the nonlinear system in two steps. In the first step, the nonlinear equations are approximated on a coarse grid using the characteristic block-centered finite difference method. In the second step, the nonlinear system is linearized using Newton's method with a small positive parameter to ensure the differentiability of the nonlinear term |u|u in the Darcy-Forchheimer equation. The proposed two-grid method is rigorously analyzed, in which a priori error estimates of the velocity, pressure, concentration and its flux are provided, and the rates of convergence are O(ε+Δt+H3+h2). Finally, numerical experiments are conducted to demonstrate the effectiveness of the proposed methods by comparing the efficiency, especially in terms of CPU time.

Highly accurate wavelet solution for the two-dimensional Bratu's problem

A modified wavelet interpolation Galerkin method (WIGM) with improved flexibility of nodal distribution is formulated and utilized to solve the Bratu's problem. Such a method treats the nonlinear term as an independent function and directly approximates it through the proposed wavelet interpolation, resulting in a low-cost discretization of the Bratu's equation with exponential nonlinearity. An efficient implementation of Newton's method to find the lower and upper branches of solution as well as the turning point is then presented. Numerical results demonstrate that the present WIGM can accurately estimate both two solution branches and turning point for the Bratu's problem. Compared with many other existing methods, the proposed WIGM also offers marked superiority in terms of accuracy when using the same node distribution. Moreover, a convergent solution to the resulting nonlinear discrete system obtained by the WIGM can be observed in a few iterations even when the initial guess significantly deviates from the exact solution. Finally, a highly accurate approximation of both two-branched solutions and the turning point for the two-dimensional Bratu's problem is obtained by the proposed WIGM.

Governing equations of plate theories by direct manipulation of three-dimensional elasticity equilibrium

– This article shows that the governing equations of any plate theory can be directly obtained by manipulating the three-dimensional (3D) equilibrium indefinite equations. Stress components are transformed into plate stress resultants by implementing a few steps proposed in this article. Such a manipulation is shown for simple and complex deformation states. Various known plate theories are considered, such as membrane and bending ones. The most general equations are obtained by referring to higher-order theories based on the Carrera Unified Formulation. To show the method’s effectiveness, the differential equations of known (Reissner–Mindlin, Hildelbrand–Reissner–Thomas) and unprecedented plate theories are derived by simply working on the 3D-equilibrium equations. The introduced manipulation consists of the following steps: (1) The 3D equilibrium elasticity is formally written for each of the degrees of freedom of the considered plate theories; (2) the stress resultants are used to replace stress components in the latter equations; (3) the derivatives of the stress components along the plate reference surface coordinate x, y do not affect the nature of the indefinite equilibrium equations, and these derivatives are directly moved to the related stress resultants; (4) the derivatives over the thickness coordinate z should: (i) change the sign of the related stress resultants and (ii) be applied to the base functions of z used to define the stress resultants (as well as the assumption for the displacement along the thickness). In conclusion, this article presents a straightforward method that can effectively replace the use of Newton and Lagrange methods to derive the equilibrium equations of any plate theory. The four steps introduced in this article demonstrate the simplicity and efficiency of this approach.

Elucidating nonlinear pharmacokinetics of telmisartan: Integration of target-mediated drug disposition and OATP1B3-mediated hepatic uptake in a physiologically based model.

Telmisartan, a selective inhibitor of angiotensin II receptor type 1 (AT1), demonstrates nonlinear pharmacokinetics (PK) when orally administered in ascending doses to healthy volunteers, but the underlying mechanisms remain unclear. This study presents a physiologically based pharmacokinetic model integrated with target-mediated drug disposition (TMDD-PBPK model) to explore the mechanism of its nonlinear PK. We employed the Cluster-Gauss Newton method for top-down analysis, estimating the invivo Km,OATP1B3 (Michaelis-Menten constant for telmisartan hepatic uptake via Organic Anion Transporting Polypeptide 1B3) to be 2.0-5.7 nM. This range is significantly lower than the reported invitro value of 810 nM, obtained in 0.3% human serum albumin (HSA) conditions. Further validation was achieved through invitro assessment in plated human hepatocytes with 4.5% HSA, showing a Km of 4.5 nM. These results underscore the importance of albumin-mediated uptake effect for the hepatic uptake of telmisartan. Our TMDD-PBPK model, developed through a "middle-out" approach, underwent sensitivity analysis to identify key factors in the nonlinear PK of telmisartan. We found that the nonlinearity in the area under the concentration-time curve (AUC) and/or maximum concentration (Cmax) of telmisartan is sensitive to Km,OATP1B3 across all dosages. Additionally, the dissociation constant (Kd) for telmisartan binding to the AT1 receptor, along with its receptor abundance, notably influences PK at lower doses (below 20 mg). In conclusion, the nonlinear PK of telmisartan appears primarily driven by hepatic uptake saturation across all dose ranges and by AT1-receptor binding saturation, notably at lower doses.

Optimal control on ABC fractal fractional order model of micro-plastic pollution in soil and its effect on the nutrient cycle

Micro-plastic pollution is the major ultimatum to contaminate agricultural soils and empirical situation to lessening the soil quality and fertility. In this work the soil pollution due to micro-plastic in the Atangana-Baleanu-Caputo fractal dimension and fractional order inquiry with four compartments namely, Microplastic (P(t)), Soil pollution (S(t)), Remediate or recycling (R(t)) and Soil Nutrients cycle (N(t)). The qualitative analysis of the steady states is explained as locally stable, globally stable and Hyer Ulam stable. The expected equilibrium point appears in a pollution-free equilibrium point and pollution-extinct equilibrium point. Two significant best control strategies (u1(t) and u2(t)) are found to reduce soil pollution. Sensitivity analyses bring off the nature of various deportment of Basic reproduction numbers. Numerical approximation extracted with the help of Newton's Polynomial method. The purpose of precise expected results is demonstrated as a numerical simulation for various orders (0≤α≤0.6) and dimensions.