Residue Regulating Homotopy Method for Strongly Nonlinear Oscillator

It is highly desired yet challenging to obtain analytical approximate solutions to strongly nonlinear oscillators accurately and efficiently. Here we propose a new approach, which combines the homtopy concept with a “residue-regulating” technique to construct a continuous homotopy from an initial guess solution to a high-accuracy analytical approximation of the nonlinear problems, namely the residue regulating homotopy method (RRHM). In our method, the analytical expression of each order homotopy-series solution is associated with a set of base functions which are pre-selected or generated during the previous order of approximations, while the corresponding coefficients are solved from deformation equations specified by the nonlinear equation itself and auxiliary residue functions. The convergence region, rate and final accuracy of the homotopy are controlled by a residue-regulating vector and an expansion threshold. General procedures of implementing RRHM are demonstrated using the Duffing and Van der Pol-Duffing oscillators, where approximate solutions containing abundant frequency components are successfully obtained, yielding significantly better convergence rate and performance stability compared to the other conventional homotopy-based methods.

Assessing the robustness and scalability of the accelerated pseudo-transient method towards exascale computing

The development of highly efficient, robust and scalable numerical algorithms lags behind the rapid increase in massive parallelism of modern hardware. We address this challenge with the accelerated pseudo-transient iterative method and present here a physically motivated derivation. We analytically determine optimal iteration parameters for a variety of basic physical processes and confirm the validity of theoretical predictions with numerical experiments. We provide an efficient numerical implementation of pseudo-transient solvers on graphical processing units (GPUs) using the Julia language. We achieve a parallel efficiency over 96 % on 2197 GPUs in distributed memory parallelisation weak scaling benchmarks. 2197 GPUs allow for unprecedented terascale solutions of 3D variable viscosity Stokes flow on 49953 grid cells involving over 1.2 trillion degrees of freedom. We verify the robustness of the method by handling contrasts up to 9 orders of magnitude in material parameters such as viscosity, and arbitrary distribution of viscous inclusions for different flow configurations. Moreover, we show that this method is well suited to tackle strongly nonlinear problems such as shear-banding in a visco-elasto-plastic medium. A GPU-based implementation can outperform CPU-based direct-iterative solvers in terms of wall-time even at relatively low resolution. We additionally motivate the accessibility of the method by its conciseness, flexibility, physically motivated derivation and ease of implementation. This solution strategy has thus a great potential for future high-performance computing applications, and for paving the road to exascale in the geosciences and beyond.

A study of uncertain multi-objective nonlinear programming problems for rough intervals

In real conditions, the parameters of multi-objective nonlinear programming (MONLP) problem models can’t be determined exactly. Hence in this paper, we concerned with studying the uncertainty of MONLP problems. We propose algorithms to solve rough and fully-rough-interval multi-objective nonlinear programming (RIMONLP and FRIMONLP) problems, to determine optimal rough solutions value and rough decision variables, where all coefficients and decision variables in the objective functions and constraints are rough intervals (RIs). For the RIMONLP and FRIMONLP problems solving methodology are presented using the weighting method and slice-sum method with Kuhn-Tucker conditions, We will structure two nonlinear programming (NLP) problems. In the first one of this NLP problem, all of its variables and coefficients are the lower approximation (LAI) it’s RIs. The second NLP problems are upper approximation intervals (UAI) of RIs. Subsequently, both NLP problems are sliced into two crisp nonlinear problems. NLP is utilized because numerous real systems are inherently nonlinear. Also, rough intervals are so important for dealing with uncertainty and inaccurate data in decision-making (DM) problems. The suggested algorithms enable us to the optimal solutions in the largest range of possible solution. Finally, Illustrative examples of the results are given.

New Algorithm to Solve Mixed Integer Quadratically Constrained Quadratic Programming Problems Using Piecewise Linear Approximation

Techniques and methods of linear optimization underwent a significant improvement in the 20th century which led to the development of reliable mixed integer linear programming (MILP) solvers. It would be useful if these solvers could handle mixed integer nonlinear programming (MINLP) problems. Piecewise linear approximation (PLA) is one of most popular methods used to transform nonlinear problems into linear ones. This paper will introduce PLA with brief a background and literature review, followed by describing our contribution before presenting the results of computational experiments and our findings. The goals of this paper are (a) improving PLA models by using nonuniform domain partitioning, and (b) proposing an idea of applying PLA partially on MINLP problems, making them easier to handle. The computational experiments were done using quadratically constrained quadratic programming (QCQP) and MIQCQP and they showed that problems under PLA with nonuniform partition resulted in more accurate solutions and required less time compared to PLA with uniform partition.

Implicit implementation of the nonlocal operator method: an open source code

In this paper, we present an open-source code for the first-order and higher-order nonlocal operator method (NOM) including a detailed description of the implementation. The NOM is based on so-called support, dual-support, nonlocal operators, and an operate energy functional ensuring stability. The nonlocal operator is a generalization of the conventional differential operators. Combined with the method of weighed residuals and variational principles, NOM establishes the residual and tangent stiffness matrix of operate energy functional through some simple matrix without the need of shape functions as in other classical computational methods such as FEM. NOM only requires the definition of the energy drastically simplifying its implementation. The implementation in this paper is focused on linear elastic solids for sake of conciseness through the NOM can handle more complex nonlinear problems. The NOM can be very flexible and efficient to solve partial differential equations (PDEs), it’s also quite easy for readers to use the NOM and extend it to solve other complicated physical phenomena described by one or a set of PDEs. Finally, we present some classical benchmark problems including the classical cantilever beam and plate-with-a-hole problem, and we also make an extension of this method to solve complicated problems including phase-field fracture modeling and gradient elasticity material.

Application of Multilayer Neural Network in Sports Psychology

The research on multilayer neural network theory has developed rapidly in recent years. It has parallel processing capabilities and fault tolerance and has aroused the interest of many researchers. The neural network has made great progress in the field of control, especially in model identification and control. It has been quickly applied in the fields of device design, optimized operation, and fault analysis and diagnosis. Neural network control, as an automated control technology in the 21st century, has been fully proved by theories and practices at home and abroad, and it is very useful in complex process control. Sports psychology is a discipline that studies the psychological characteristics and laws of people engaged in sports, and it is also a new development in sports science. The main task of sports psychology is to study people’s psychological processes when participating in sports, such as feeling, perception, appearance, thinking, memory, emotion, and characteristics of will and its role and significance in sports. An important feature of multilayer neural networks is to achieve results that match the expected output through network learning. It has strong self-learning, self-adaptability, and fault tolerance. The multilayer neural network system evaluation method is unique with its extraordinary ability to deal with complex nonlinear problems and does not involve human intervention. This article presents a multilayer neural network algorithm, which classifies the samples of athletes, and studies the physical education training process, the psychological characteristics of related personnel in sports competitions, such as the psychological characteristics of the formation of sports skills, and the psychological training of athletes before the game.

Existence results for nonlinear problems with $\varphi$- Laplacian operators and nonlocal boundary conditions

Using Leray-Schauder degree theory we study the existence of at least one solution for the boundary value problem of the type\[\left\{\begin{array}{lll}(\varphi(u' ))' = f(t,u,u') & & \\u'(0)=u(0), \ u'(T)= bu'(0), & & \quad \quad \end{array}\right.\] where $\varphi: \mathbb{R}\rightarrow \mathbb{R}$ is a homeomorphism such that $\varphi(0)=0$, $f:\left[0, T\right]\times \mathbb{R} \times \mathbb{R}\rightarrow \mathbb{R} $ is a continuous function, $T$ a positive real number, and $b$ some non zero real number.

Accelerating Symmetric Rank-1 Quasi-Newton Method with Nesterov’s Gradient for Training Neural Networks

Gradient-based methods are popularly used in training neural networks and can be broadly categorized into first and second order methods. Second order methods have shown to have better convergence compared to first order methods, especially in solving highly nonlinear problems. The BFGS quasi-Newton method is the most commonly studied second order method for neural network training. Recent methods have been shown to speed up the convergence of the BFGS method using the Nesterov’s acclerated gradient and momentum terms. The SR1 quasi-Newton method, though less commonly used in training neural networks, is known to have interesting properties and provide good Hessian approximations when used with a trust-region approach. Thus, this paper aims to investigate accelerating the Symmetric Rank-1 (SR1) quasi-Newton method with the Nesterov’s gradient for training neural networks, and to briefly discuss its convergence. The performance of the proposed method is evaluated on a function approximation and image classification problem.

A Reliable Treatment for Nonlinear Differential Equations

In this paper, we use the concept of homotopy, Laplace transform, and He’s polynomials, to propose the auxiliary Laplace homotopy parameter method (ALHPM). We construct a homotopy equation consisting on two auxiliary parameters for solving nonlinear differential equations, which switch nonlinear terms with He’s polynomials. The existence of two auxiliary parameters in the homotopy equation allows us to guarantee the convergence of the obtained series. Compared with numerical techniques, the method solves nonlinear problems without any discretization and is capable to reduce computational work. We use the method for different types of singular Emden–Fowler equations. The solutions, constructed in the form of a convergent series, are in excellent agreement with the existing solutions.

Levenberg–Marquardt Backpropagation for Numerical Treatment of Micropolar Flow in a Porous Channel with Mass Injection

In this research work, an effective Levenberg–Marquardt algorithm-based artificial neural network (LMA-BANN) model is presented to find an accurate series solution for micropolar flow in a porous channel with mass injection (MPFPCMI). The LMA is one of the fastest backpropagation methods used for solving least-squares of nonlinear problems. We create a dataset to train, test, and validate the LMA-BANN model regarding the solution obtained by optimal homotopy asymptotic (OHA) method. The proposed model is evaluated by conducting experiments on a dataset acquired from the OHA method. The experimental results are obtained by using mean square error (MSE) and absolute error (AE) metric functions. The learning process of the adjustable parameters is conducted with efficacy of the LMA-BANN model. The performance of the developed LMA-BANN for the modelled problem is confirmed by achieving the best promise numerical results of performance in the range of E-05 to E-08 and also assessed by error histogram plot (EHP) and regression plot (RP) measures.