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The computation of contact force between elastic bodies is the key to analyzing non-stationary vibrations in mechanical system and warrants further research. To establish a contact force calculation approach suitable for elastic bodies under external forces, the contact state is categorized into three types by analyzing energy conversion during the elastic contact process: collision contact without external forces, collision contact with zero initial relative velocity, and collision contact considering external forces. For the first type, a hysteresis damping factor is derived based on energy restitution coefficients and numerical analysis. which maintains high accuracy even with low restitution coefficients. Next, an iterative calculation approach combining the bisection and Newton's methods is developed for the second type. On this basis, a numerical computation process is subsequently proposed for the third type. To demonstrate the utility of the proposed approach, dynamic characteristics analysis of simultaneous collisions and multiple continuous collisions are presented as case studies. The results confirm the effectiveness of this calculation approach. [ABSTRACT FROM AUTHOR]

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 x∗$$ {x}^{\ast } $$ of a univariate nonlinear equation g(x)=0$$ g(x)=0 $$. 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 x∗$$ {x}^{\ast } $$ and does not depend on any initial values, in contrast to Newton's method. Several statistical examples are provided to demonstrate the proposed algorithm. [ABSTRACT FROM AUTHOR]

The numerical solution of optimal control problems through second-order methods is examined in this paper. Controlled processes are described by a system of nonlinear ordinary differential equations. There are two specific characteristics of the class of control actions used. The first one is that controls are searched for in a given class of functions, which depend on unknown parameters to be found by minimizing an objective functional. The parameter values, in general, may be different at different time intervals. The second feature of the considered problem is that the boundaries of time intervals are also optimized with fixed values of the parameters of the control actions in each of the intervals. The special cases of the problem under study are relay control problems with optimized switching moments. In this work, formulas for the gradient and the Hessian matrix of the objective functional with respect to the optimized parameters are obtained. For this, the technique of fast differentiation is used. A comparison of numerical experiment results obtained with the use of first- and second-order optimization methods is presented. [ABSTRACT FROM AUTHOR]

In this article, we apply three numerical methods to study the L∞-convergence of the Newton-multigrid method for parabolic quasi-variational inequalities with a nonlinear right-hand side. To discretize the problem, we utilize a finite element method for the ofaator and Euler scheme for the time. To obtain the system discretization of the problem, we reformulate the parabolic quasi-variational inequality as a Hamilton–Jacobi–Bellman equation. For linearizing the problem on the coarse grid, we employ Newton's method as an external interior iteration of the Jacobian system. On the smooth grid, we apply the multigrid method as an interior iteration on the Jacobian system. Finally, we provide a proof for the L∞-convergence of the Newton-multigrid method for parabolic quasi-variational inequalities with a nonlinear right-hand, by giving a numerical example for this problem. [ABSTRACT FROM AUTHOR]

We derived an explicit form of the Jacobian for discrete approximations of a nonlinear initial boundary value problems (IBVPs) in matrix-vector form. The Jacobian is used in Newton's method to solve the corresponding nonlinear system of equations. The technique was exemplified on the incompressible Navier-Stokes equations discretized using summation-by-parts (SBP) difference operators and weakly imposed boundary conditions using the simultaneous approximation term (SAT) technique. The convergence rate of the iterations is verified by using the method of manufactured solutions. The methodology in this paper can be used on any numerical discretization of IBVPs in matrix-vector form, and it is particularly straightforward for approximations in SBP-SAT form.

In this paper, we developed a novel numerical method for solving general nonlinear fractional ordinary differential equations (FODEs). First, we transformed the nonlinear FODEs into the equivalent Volterra integral equations. We then developed a time-stepping algorithm for the numerical solution of the Volterra integral equations based on the third-order Taylor expansion for approximating the integrands in the Volterra integral equations on a chosen mesh with the mesh parameter $ h $. This approximation led to implicit nonlinear algebraic equations in the unknowns at each given mesh point, and an iterative algorithm based on Newton's method was developed to solve the resulting implicit equations. A convergence analysis of this numerical scheme showed that the error between the exact solution and numerical solution at each mesh point is $ \mathcal{O}(h^{3}) $, independent of the fractional order. Finally, four numerical examples were solved to verify the theoretical results and demonstrate the effectiveness of the proposed method.

We propose a novel distributed method for non-convex optimization problems with coupling equality and inequality constraints. This method transforms the optimization problem into a specific form to allow distributed implementation of modified gradient descent and Newton's methods so that they operate as if they were distributed. We demonstrate that for the proposed distributed method: (i) communications are significantly less time-consuming than oracle calls, (ii) its convergence rate is equivalent to the convergence of Newton's method concerning oracle calls, and (iii) for the cases when oracle calls are more expensive than communication between agents, the transition from a centralized to a distributed paradigm does not significantly affect computational time. The proposed method is applicable when the objective function is twice differentiable and constraints are differentiable, which holds for a wide range of machine learning methods and optimization setups. [ABSTRACT FROM AUTHOR]

This paper describes a method for solving smooth nonconvex minimization problems subject to bound constraints with good worst-case complexity guarantees and practical performance. The method contains elements of two existing methods: the classical gradient projection approach for bound-constrained optimization and a recently proposed Newton-conjugate gradient algorithm for unconstrained nonconvex optimization. Using a new definition of approximate second-order optimality parametrized by some tolerance ϵ (which is compared with related definitions from previous works), we derive complexity bounds in terms of ϵ for both the number of iterations required and the total amount of computation. The latter is measured by the number of gradient evaluations or Hessian-vector products. We also describe illustrative computational results on several test problems from low-rank matrix optimization. [ABSTRACT FROM AUTHOR]

Chaotic maps, despite their deterministic nature, can introduce controlled randomness into optimization algorithms. This chaotic map behaviour helps overcome the lack of mathematical validation in traditional stochastic methods. The chaotic optimization algorithm (COA) uses chaotic maps that help it achieve faster convergence and escape local optima. The effective use of these maps to find the global optimum would be possible only with a complete understanding of them, especially their fixed points. In chaotic maps, fixed points repeat indefinitely, disrupting the map's characteristic unpredictability. While using chaotic maps for global optimization, it is crucial to avoid starting the search at fixed points and implement corrective measures if they arise in between the sequence. This paper outlines strategies for addressing fixed points and provides a numerical evaluation (using Newton's method) of the fixed points for 20 widely used chaotic maps. By appropriately handling fixed points, researchers and practitioners across diverse fields can avoid costly failures, improve accuracy, and enhance the reliability of their systems. [ABSTRACT FROM AUTHOR]

We derived an explicit form of the Jacobian for discrete approximations of a nonlinear initial boundary value problems (IBVPs) in matrix-vector form. The Jacobian is used in Newton’s method to solve the corresponding nonlinear system of equations. The technique was exemplified on the incompressible Navier-Stokes equations discretized using summation-by-parts (SBP) difference operators and weakly imposed boundary conditions using the simultaneous approximation term (SAT) technique. The convergence rate of the iterations is verified by using the method of manufactured solutions. The methodology in this paper can be used on any numerical discretization of IBVPs in matrix-vector form, and it is particularly straightforward for approximations in SBP-SAT form. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to develop a practical strategy to accelerate Newton's method in the vicinity of singular points. We present an adaptive safeguarding scheme with a tunable parameter, which we call adaptive γ-safeguarding, that one can use in tandem with Anderson acceleration to improve the performance of Newton's method when solving problems at or near singular points. The key features of adaptive γ-safeguarding are that it converges locally for singular problems, and it can detect nonsingular problems automatically, in which case the Newton-Anderson iterates are scaled towards a standard Newton step. The result is a flexible algorithm that performs well for singular and nonsingular problems, and can recover convergence from both standard Newton and Newton-Anderson with the right parameter choice. This leads to faster local convergence compared to both Newton's method, and Newton-Anderson without safeguarding, with effectively no additional computational cost. We demonstrate three strategies one can use when implementing Newton-Anderson and γ-safeguarded Newton-Anderson to solve parameterdependent problems near singular points. For our benchmark problems, we take two parameter-dependent incompressible flow systems: flow in a channel and Rayleigh-Bénard convection. [ABSTRACT FROM AUTHOR]

In this article, a numerical method is proposed and investigated for an initial boundary value problem governed by a fractional differential generalization of the nonlinear transient filtration law which describes fluid motion in a porous medium. This type of equation is widely used to describe complex filtration processes such as fluid movement in horizontal wells in fractured geological formations. To construct the numerical method, a high-order approximation formula for the fractional derivative in the sense of Caputo is applied, and a combination of the finite difference method with the finite element method is used. The article proves the uniqueness and continuous dependence of the solution on the input data in differential form, as well as the stability and convergence of the proposed numerical scheme. The linearization of nonlinear terms is carried out by the Newton method, which allows for achieving high accuracy in solving complex problems. The research results are confirmed by a series of numerical tests that demonstrate the applicability of the developed method in real engineering problems. The practical significance of the presented approach lies in its ability to accurately and effectively model filtration processes in shale formations, which allows engineers and geologists to make more informed decisions when designing and operating oil fields. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we survey the recent development of inversion‐free Newton's method, which directly avoids computing the inversion of Hessian, and demonstrate its applications in estimating parameters of models such as linear and logistic regression. A detailed review of existing methodology is provided, along with comparisons of various competing algorithms. We provide numerical examples that highlight some deficiencies of existing approaches, and demonstrate how the inversion‐free methods can improve performance. Motivated by recent works in literature, we provide a unified subsampling framework that can be combined with the inversion‐free Newton's method to estimate model parameters including those of linear and logistic regression. Numerical examples are provided for illustration. [ABSTRACT FROM AUTHOR]

We develop a new globally convergent optimization method for solving a constrained minimization problem underlying the minimum density power divergence estimator for univariate Gaussian data in the presence of outliers. Our hybrid procedure combines classical Newton's method with a gradient descent iteration equipped with a step control mechanism based on Armijo's rule to ensure global convergence. Extensive simulations comparing the resulting estimation procedure with the more prominent robust competitor, Minimum Covariance Determinant (MCD) estimator, across a wide range of breakdown point values suggest improved efficiency of our method. Application to estimation and inference for a real-world dataset is also given. [ABSTRACT FROM AUTHOR]

The examination of nonlinear equations is essential in diverse domains such as science, business, and engineering because of the widespread occurrence of nonlinear phenomena. The primary obstacle in computational science is to create numerical algorithms that are both computationally efficient and possess a high convergence rate. This work tackles these problems by presenting a three-step nonlinear, time-efficient numerical method for solving nonlinear models. The selected method exhibits a convergence of sixth order and an efficiency index of 1.4310, with the added benefit of only requiring five function evaluations per iteration. This study diverges from earlier research by emphasizing the use of first-order derivatives instead of higher-order derivatives. As a result, the method becomes more versatile and applicable. It is possible to estimate solutions that are locally different in Banach spaces by looking at convergence on both local and semi-local levels. The stability and performance of the scheme are also checked by using polynomiography to do a visual analysis and compare it to other schemes in terms of convergence, speed, and CPU time. [ABSTRACT FROM AUTHOR]

In this article, we propose an adaptive mesh-refining based on the multi-level algorithm and derive a unified a posteriori error estimate for a class of nonlinear problems. We have shown that the multi-level algorithm on adaptive meshes retains quadratic convergence of Newton's method across different mesh levels, which is numerically validated. Our framework facilitates to use the general theory established for a linear problem associated with given nonlinear equations. In particular, existing a posteriori error estimates for the linear problem can be utilized to find reliable error estimators for the given nonlinear problem. As applications of our theory, we consider the pseudostress-velocity formulation of Navier–Stokes equations and the standard Galerkin formulation of semilinear elliptic equations. Reliable and efficient a posteriori error estimators for both approximations are derived. Finally, several numerical examples are presented to test the performance of the algorithm and validity of the theory developed. [ABSTRACT FROM AUTHOR]

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.

In this work, we give a thorough examination of Newton's technique. We show that certain places outperform others in terms of where a good initial approximation may be made to assure convergence. Furthermore, to assure quicker and better convergence, certain criteria must be imposed on the function, such as dealing with additional terms from the Taylor series to achieve a technique comparable to Newton's method, but with a degree of convergence greater than two. We compare the use of Newton's technique for solving equations with a single variable to the solution of equations with many variables. While we widen our discussion to include the solution to complex-valued functions, our primary focus is on locating the roots of unity. Some new theories have been proven which is an addition to this topic, and their results are shown in the examples at the end of the paper. We investigate the incorrect choice of the starting approximation for the nth root of unity in the complex plane. When utilizing Newton's technique on a complex plane, we employ various stunning fractal graphs to explain the features and behavior of the roots of interest. [ABSTRACT FROM AUTHOR]

Root-finding methods solve equations and identify unknowns in physics, engineering, and computer science. Memory-based root-seeking algorithms may look back to expedite convergence and enhance computational efficiency. Real-time systems, complicated simulations, and high-performance computing demand frequent, large-scale calculations. This article proposes two unique root-finding methods that increase the convergence order of the classical Newton-Raphson (NR) approach without increasing evaluation time. Taylor's expansion uses the classical Halley method to create two memory-based methods with an order of 2.4142 and an efficiency index of 1.5538. We designed a two-step memory-based method with the help of Secant and NR algorithms using a backward difference quotient. We demonstrate memory-based approaches' robustness and stability using visual analysis via polynomiography. Local and semilocal convergence are thoroughly examined. Finally, proposed memory-based approaches outperform several existing memory-based methods when applied to models including a thermistor, path traversed by an electron, sheet-pile wall, adiabatic flame temperature, and blood rheology nonlinear equation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We discuss a nonlinear domain-decomposition preconditioning method for fully implicit simulations of multicomponent porous media flow based on the additive Schwarz preconditioned exact Newton method (ASPEN). The method efficiently accelerates nonlinear convergence by resolving unbalanced nonlinearities in a local stage and long-range interactions in a global stage. ASPEN can improve robustness and significantly reduce the number of global iterations compared with standard Newton, but extra work introduced in the local steps makes each global iteration more expensive. We discuss implementation aspects for the local and global stages. We show how the global-stage Jacobian can be transformed to the same form as the fully implicit system, so that one can use standard linear preconditioners and solvers. We compare the computational performance of ASPEN to standard Newton on a series of test cases, ranging from conceptual cases with simplified geometry or flow physics to cases representative of real assets. Our overall conclusion is that ASPEN is outperformed by Newton when this method works well and converges in a few iterations. On the other hand, ASPEN avoids time-step cuts and has significantly lower runtimes in time steps where Newton struggles. A good approach to computational speedup is therefore to adaptively switch between Newton and ASPEN throughout a simulation. A few examples of switching strategies are outlined. [ABSTRACT FROM AUTHOR]

In Baraldi (Math Program 20:1–40, 2022), we introduced an inexact trust-region algorithm for minimizing the sum of a smooth nonconvex function and a nonsmooth convex function in Hilbert space—a class of problems that is ubiquitous in data science, learning, optimal control, and inverse problems. This algorithm has demonstrated excellent performance and scalability with problem size. In this paper, we enrich the convergence analysis for this algorithm, proving strong convergence of the iterates with guaranteed rates. In particular, we demonstrate that the trust-region algorithm recovers superlinear, even quadratic, convergence rates when using a second-order Taylor approximation of the smooth objective function term. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of finding the optimal step length for the Newton method on the class of self-concordant functions, with the decrease in function value as criterion. We formulate this problem as an optimal control problem and use optimal control theory to solve it. [ABSTRACT FROM AUTHOR]

We seek the solution of the coupled continuous-time algebraic Riccati equations, arising in the optimal control of Markovian jump linear systems. Newton's method is applied to construct the solution, under a mild and natural stabilizability assumption, leading to some coupled Lyapunov equations. Iterative methods of O (n 3) computational complexity for the coupled Lyapunov equations and the corresponding Newton's methods for the coupled continuous-time algebraic Riccati equations are analyzed. Illustrative examples are presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse problem of vibro-acoustography, a technique for enhancing ultrasound imaging by making use of nonlinear effects. It amounts to determining two spatially variable coefficients in a system of PDEs describing propagation of two directed sound beams and the wave resulting from their nonlinear interaction. To justify the use of Newton's method for solving this inverse problem, on one hand, we verify well-definedness and differentiability of the forward operator corresponding to two versions of the PDE model; on the other hand, we consider an all-at-once formulation of the inverse problem and prove convergence of Newton's method for its solution. [ABSTRACT FROM AUTHOR]

The purpose of this study is to propose an efficient numerical method for solving the inverse nonlinear problem of the movement of the compressor rotor collar in a fluid film thrust bearing. Methods. A periodic thermoelastohydrodynamic (PTEHD) mathematical model of hydrodynamic and thermal processes in a bearing is constructed under the condition of the rotor collar motion. Within the framework of the model, an inverse nonlinear problem of determining the position of the collar under a given external load is formulated. An iterative solution method is proposed, which utilizes the solution of the direct problem. To reduce computational costs, a modified Dekker–Brent method is employed in conjunction with a modified Newton’s method. Results. Numerical experiments have been conducted, demonstrating the effectiveness of the proposed approaches. The suggested methods significantly reduce the required computational resources by minimizing the number of calls to the target function in the optimization problem. A software suite has been developed that allows for the calculation of the nonlinear system of rotor motion under various physical and geometric parameters. Conclusion. An efficient set of numerical methods for solving the inverse nonlinear problem of the motion of the rotor collar in the compressor fluid film thrust bearing is proposed. The method’s effectiveness lies in substantial savings of computational resources. The method’s efficiency has been demonstrated in numerical experiments.

В статье представлено исследование вычислительной эффективности параллельной версии численного алгоритма для решения уравнения Риккати с производной дробного перменного порядка типа Герасимова-Капуто. Численный алгоритм представляет собой нелокальную неявную конечно-разностную схему, которая сводится к системе нелинейных алгебраических уравнений и решается с помощью модифицированного метода Ньютона. Нелокальность численной схемы создает высокую вычислительную нагрузку на вычислительные ресурсы, из-за чего возникает необходимость в реализации эффективных параллельных алгоритмов их решения. Исследуемый на эффективность численный алгоритм реализован на языке C из-за его универсальности при работе с памятью. Распаралеливание проводилось с помощью технологии OpenMP. Проводится серия вычислительных экспериментов на вычислительном сервере NVIDIA DGX STATION (Институт математики имени В.И. Романовского, г. Ташкент, Узбекистан) и ноутбуке HP Pavilion Gaming Laptop Z270X, где решалась задача Коши для дробного уравнения Риккати с непостоянными коэффициентами. На основе среднего времени вычисления вычисляются: ускорение, эффективность и стоимость алгоритма. Из анализа данных видно, что OpenMP параллельная программная реализация нелокальной неявной конечно-разностной схемы показывает ускорение работы от 9-12 раз в зависимости от количества задействованных ядер CPU.

We develop the local convergence of the six order Contraharmonic-mean Newton’s method (CHN) to solve Banach space valued equations. Our analysis approach is two fold: The first way uses Taylor’s series and derivatives of higher orders. The second one uses only the first derivatives. We examine the theoretical results by solving a boundary value problem also using the examples relating the proposed method with other’s methods such as Newton’s, Kou’s and Jarratt’s to show that the proposed method performs better. The conjugate maps for second-degree polynomial are verified. We also calculate the fixed points (extraneous). The article is completed with the study of basins of attraction, which support and further validate the theoretical and numerical results.

An important service in the wireless systems for the human daily life is the information of a mobile user location. Wireless sensor network is a structure that can be used to determine the mobile user position. The time-difference-of-arrival (TDoA) technique is often considered for wireless localization due to the low cost of the sensor network. In this work, the error covariance matrices are derived for predicting the error performance of the conventional closed-form constrained total least squares (CTLS) estimator. More importantly, three new Newton’s methods are proposed for computing the CTLS solution in the TDoA localization. In addition, theoretical performance of both new Newton-based approaches is provided in closed forms. Numerical simulation is conducted to compare the derived theoretical prediction with the corresponding actual estimation error. It is illustrated that the two new Newton-based techniques provide better performance, in terms of lower bias and root mean square error, less computational time, and more reliability, than the former Newton-based algorithms. Furthermore, the derived expressions for the theoretical error covariance well coincide with the actual random estimation results.

The numerical solution of optimal control problems through second-order methods is examined in this paper. Controlled processes are described by a system of nonlinear ordinary differential equations. There are two specific characteristics of the class of control actions used. The first one is that controls are searched for in a given class of functions, which depend on unknown parameters to be found by minimizing an objective functional. The parameter values, in general, may be different at different time intervals. The second feature of the considered problem is that the boundaries of time intervals are also optimized with fixed values of the parameters of the control actions in each of the intervals. The special cases of the problem under study are relay control problems with optimized switching moments. In this work, formulas for the gradient and the Hessian matrix of the objective functional with respect to the optimized parameters are obtained. For this, the technique of fast differentiation is used. A comparison of numerical experiment results obtained with the use of first- and second-order optimization methods is presented.

We develop the local convergence of the six order Contraharmonic-mean Newton's method (CHN) to solve Banach space valued equations. Our analysis approach is two fold: The first way uses Taylor's series and derivatives of higher orders. The second one uses only the first derivatives. We examine the theoretical results by solving a boundary value problem also using the examples relating the proposed method with other's methods such as Newton's, Kou's and Jarratt's to show that the proposed method performs better. The conjugate maps for second-degree polynomial are verified. We also calculate the fixed points (extraneous). The article is completed with the study of basins of attraction, which support and further validate the theoretical and numerical results. [ABSTRACT FROM AUTHOR]

We seek the hull of a ship with a given volume which minimizes the water resistance with uncertainties on the cruising speed. The water resistance is based on Michell's wave resistance functional, and the speed is a random variable whose probability distribution is known. We first handle the case where the support of the hull is given and then we also optimize this support for a given area. In each case, an optimal hull is shown to exist. The numerical simulations are costly so we adapt to our problem Newton's method for shape optimization. The Dirichlet energy is used as a test case. The numerical results for the robust optimal hull are compared with the case where the cruising speed is known. [ABSTRACT FROM AUTHOR]

In this study, we present three methods for approximating functions using interpolation polynomials. In engineering, there are many situations where it is necessary to approximate the value of a function at a certain point, knowing only a finite set of experimental data or when the function itself is presented in a form that is not easily usable. We use specific methods in numerical analysis such as Lagrange interpolation, the method of dividing differences and Neville's method. Given a function f defined on an interval [a,b], the purpose of this work is to construct a function that approximates the function f at a predetermined value. The expression of the approximation functions will be calculated and we will evaluate the obtained error for each approximation. [ABSTRACT FROM AUTHOR]

This paper develops a new class of nonlinear acceleration algorithms based on extending conjugate residual-type procedures from linear to nonlinear equations. The main algorithm has strong similarities with Anderson acceleration as well as with inexact Newton methods--depending on which variant is implemented. We prove theoretically and verify experimentally, on a variety of problems from simulation experiments to deep learning applications, that our method is a powerful accelerated iterative algorithm. The code is available at https://github.com/Data-driven-numericalmethods/Nonlinear-Truncated-Conjugate-Residual. [ABSTRACT FROM AUTHOR]

Many numerical methods have been developed since 1961, but unresolved issues remain. This study developed a numerical method to address these issues and determine the coefficients and properties of rotational waves with a shear current in a finite water depth. The number of unknown constants was reduced significantly by introducing a wavelength-independent coordinate system. The reference depth was calculated independently using the shooting method. Therefore, there was no need for partial derivatives with respect to the wavelength and the reference depth, which simplified the numerical formulation. This method had less than half of the unknown constants of the other method because Newton's method only determines the coefficients. The breaking limit was calculated for verification, and the result agreed with the Miche formula. The water particle velocities were calculated, and the results were consistent with the experimental data. Dispersion relations were calculated, and the results are consistent with other numerical findings. The convergence of this method was examined. Although the required series order was reduced significantly, the total error was smaller, with a faster convergence speed.

We propose a novel distributed method for non-convex optimization problems with coupling equality and inequality constraints. This method transforms the optimization problem into a specific form to allow distributed implementation of modified gradient descent and Newton’s methods so that they operate as if they were distributed. We demonstrate that for the proposed distributed method: (i) communications are significantly less time-consuming than oracle calls, (ii) its convergence rate is equivalent to the convergence of Newton’s method concerning oracle calls, and (iii) for the cases when oracle calls are more expensive than communication between agents, the transition from a centralized to a distributed paradigm does not significantly affect computational time. The proposed method is applicable when the objective function is twice differentiable and constraints are differentiable, which holds for a wide range of machine learning methods and optimization setups.

In this article, a numerical method is proposed and investigated for an initial boundary value problem governed by a fractional differential generalization of the nonlinear transient filtration law which describes fluid motion in a porous medium. This type of equation is widely used to describe complex filtration processes such as fluid movement in horizontal wells in fractured geological formations. To construct the numerical method, a high-order approximation formula for the fractional derivative in the sense of Caputo is applied, and a combination of the finite difference method with the finite element method is used. The article proves the uniqueness and continuous dependence of the solution on the input data in differential form, as well as the stability and convergence of the proposed numerical scheme. The linearization of nonlinear terms is carried out by the Newton method, which allows for achieving high accuracy in solving complex problems. The research results are confirmed by a series of numerical tests that demonstrate the applicability of the developed method in real engineering problems. The practical significance of the presented approach lies in its ability to accurately and effectively model filtration processes in shale formations, which allows engineers and geologists to make more informed decisions when designing and operating oil fields.