Symmetric Nonlinear Equations Research Articles

A variational framework for Cahn–Hilliard-type diffusion coupled with Allen–Cahn-type multi-phase transformations in elastic and dissipative solids

This article presents a variational framework for coupled chemo-mechanical solids undergoing irreversible micro-structural changes at infinitesimal strains. The coupled problem is characterised by phenomena such as phase transitions, micro-structure coarsening and swelling. It is an extension of our previous work on variational inelasticity for a conserved chemo-mechanical setting to a unified conserved and non-conserved setting which include multi-phase transformations. The variational framework, again governed by continuous-time, discrete-time and discrete-space–time incremental variational principles, is outlined for coupled diffusion-phase transformation phenomena in elastic and dissipative solids. For the sake of simplicity, focus is restricted to isothermal conditions. It is shown that the governing macro- and micro-balance equations of the coupled problem appear as Euler equations of these minimisation and saddle point principles. In contrast to our previous work, extended variational principles (with the gradient of the chemical potential and phase fractions) are constructed that account for diffusion-phase transformation coupling. This is achieved by Legendre transformations. Note that the local–global solution strategy is still preserved and the resulting system of symmetric non-linear algebraic equations are solved by Newton–Raphson-type iterative methods. The applicability of the proposed framework is demonstrated by numerical simulations that qualitatively characterise lower bainitic micro-structure.

An efficient modified residual-based algorithm for large scale symmetric nonlinear equations by approximating successive iterated gradients

In this paper, a new descent approximate modified residual algorithm is developed to solve a large scale system of nonlinear symmetric equations, where the basic strategy to improve its numerical performance is to approximately compute the gradients and the difference of gradients. The error bounds of this approximation are presented, and in virtue of this approximation, a conjugate gradient algorithm for solving large-scale optimization problems in the literature is extended to solve the system of nonlinear symmetric equations without needs of computing and storing the Jacobian matrices or their approximate matrices. It is proved that the obtained search directions in our developed algorithm are sufficiently decent with respect to the so-called approximate modified residues. Under mild assumptions, global and local convergence results of the developed algorithm are proved. Numerical tests indicate that the developed algorithm outperforms the other similar ones available in the literature.

Solvability of the symmetric nonlinear functional differential equations

Conditions for the existence of a symmetric solution of nonlinear functional differential equations with symmetries are established.

Variational formulation of Cahn–Hilliard-type diffusion coupled with crystal plasticity

This article presents a relatively general framework for the description of the Cahn–Hilliard-type diffusion in solids undergoing infinitesimal elastic and plastic deformations. The coupled chemo-mechanical problem, characterised by phenomena such as phase segregation, microstructure coarsening and swelling, is treated using the variational framework which is governed by continuous-time, discrete-time and discrete-space–time incremental variational principles. It is shown that the governing equations of the coupled problem can be derived as Euler equations of minimisation and saddle point principles. A point of departure from the existing works is the coupling of crystal plasticity to the problem of diffusion and optimising the potential with respect to the plastic variables such that they are solved locally at the integration points. This is done using a return map algorithm which results in a reduced global problem. The variational framework results in a system of symmetric non-linear algebraic equations that are solved by Newton–Raphson-type iterative methods. This is a novel and attractive feature with respect to numerical implementation, as models resulting from the proposed variational framework are computationally less expensive in comparison with non-symmetric formulations. The numerical simulations presented at the end predict the applicability of models resulting from the proposed variational framework for multiple scenarios.

On Nonlinear Ψ-Caputo Fractional Integro Differential Equations Involving Non-Instantaneous Conditions

We propose a solution to the symmetric nonlinear Ψ-Caputo fractional integro differential equations involving non-instantaneous impulsive boundary conditions. We investigate the existence and uniqueness of the solution for the proposed problem. Banach contraction theorem is employed to prove the uniqueness results, while Krasnoselkii’s fixed point technique is used to prove the existence results. Additionally, an example is used to explain the results. In this manner, our results represent generalized versions of some recent interesting contributions.

Double direction three-term spectral conjugate gradient method for solving symmetric nonlinear equations

We propose a new three-term spectral conjugate gradient method via double direction approach by considering the first direction to be the one proposed by Halilu and Waziri (2018) and the other obtained by extending the direction proposed by Birgin and Mertinez (2001) to three-term spectral conjugate gradient (CG) direction. The propose method generates a descent direction via an inexact line search. The global convergence of the propose algorithm was established under appropriate conditions and numerical experiments on some benchmark test problems demonstrate its efficiency over some existing ones.

An Inexact Optimal Hybrid Conjugate Gradient Method for Solving Symmetric Nonlinear Equations

This article presents an inexact optimal hybrid conjugate gradient (CG) method for solving symmetric nonlinear systems. The method is a convex combination of the optimal Dai–Liao (DL) and the extended three-term Polak–Ribiére–Polyak (PRP) CG methods. However, two different formulas for selecting the convex parameter are derived by using the conjugacy condition and also by combining the proposed direction with the default Newton direction. The proposed method is again derivative-free, therefore the Jacobian information is not required throughout the iteration process. Furthermore, the global convergence of the proposed method is shown using some appropriate assumptions. Finally, the numerical performance of the method is demonstrated by solving some examples of symmetric nonlinear problems and comparing them with some existing symmetric nonlinear equations CG solvers.

Application of an Improved Ultrasound Full-Waveform Inversion in Bone Quantitative Measurement

Inspired by the large number of applications for symmetric nonlinear equations, an improved full waveform inversion algorithm is proposed in this paper in order to quantitatively measure the bone density and realize the early diagnosis of osteoporosis. The isotropic elastic wave equation is used to simulate ultrasonic propagation between bone and soft tissue, and the Gauss–Newton algorithm based on symmetric nonlinear equations is applied to solve the optimal solution in the inversion. In addition, the authors use several strategies including the frequency-grid multiscale method, the envelope inversion and the new joint velocity–density inversion to improve the result of conventional full-waveform inversion method. The effects of various inversion settings are also tested to find a balanced way of keeping good accuracy and high computational efficiency. Numerical inversion experiments showed that the improved full waveform inversion (FWI) method proposed in this paper shows superior inversion results as it can detect small velocity–density changes in bones, and the relative error of the numerical model is within 10%. This method can also avoid interference from small amounts of noise and satisfy the high precision requirements for quantitative ultrasound measurements of bone.

A Modified PRP-CG Type Derivative-Free Algorithm with Optimal Choices for Solving Large-Scale Nonlinear Symmetric Equations

Inspired by the large number of applications for symmetric nonlinear equations, this article will suggest two optimal choices for the modified Polak–Ribiére–Polyak (PRP) conjugate gradient (CG) method by minimizing the measure function of the search direction matrix and combining the proposed direction with the default Newton direction. In addition, the corresponding PRP parameters are incorporated with the Li and Fukushima approximate gradient to propose two robust CG-type algorithms for finding solutions for large-scale systems of symmetric nonlinear equations. We have also demonstrated the global convergence of the suggested algorithms using some classical assumptions. Finally, we demonstrated the numerical advantages of the proposed algorithms compared to some of the existing methods for nonlinear symmetric equations.

A globally convergent BFGS method for symmetric nonlinear equations

<p style='text-indent:20px;'>A BFGS type method is presented to solve symmetric nonlinear equations, which is shown to be globally convergent under suitable conditions. Compared with some existing Gauss-Newton-based BFGS methods whose iterative matrix approximates the Gauss-Newton matrix, an important feature of the proposed method lies in that the iterative matrix is an approximation of the Jacobian, which greatly reduces condition number of the iterative matrix. Numerical results are reported to support the theory.</p>

D-Stability of the Initial Value Problem for Symmetric Nonlinear Functional Differential Equations

This paper presents a method of establishing the D-stability terms of the symmetric solution of scalar symmetric linear and nonlinear functional differential equations. We determine the general conditions of the unique solvability of the initial value problem for symmetric functional differential equations. Here, we show the conditions of the symmetric property of the unique solution of symmetric functional differential equations. Furthermore, in this paper, an illustration of a particular symmetric equation is presented. In this example, all theoretical investigations referred to earlier are demonstrated. In addition, we graphically demonstrate two possible linear functions with the required symmetry properties.

Conditional Lie-Bäcklund Symmetries and Differential Constraints of Radially Symmetric Nonlinear Convection-Diffusion Equations with Source

A conditional Lie-Bäcklund symmetry method and differential constraint method are developed to study the radially symmetric nonlinear convection-diffusion equations with source. The equations and the admitted conditional Lie-Bäcklund symmetries (differential constraints) are identified. As a consequence, symmetry reductions to two-dimensional dynamical systems of the resulting equations are derived due to the compatibility of the original equation and the additional differential constraint corresponding to the invariant surface equation of the admitted conditional Lie-Bäcklund symmetry.

A modified BFGS type quasi-Newton method with line search for symmetric nonlinear equations problems

In this paper, using approximate gradient of the norm square metric function, we present an inexact MBFGS method with line search for solving symmetric nonlinear equations, which is a generalization of the MBFGS method proposed by Li and Fukushima (2001) for solving smooth unconstrained optimization. The used approximate gradient of the proposed method only requires computing two residuals at every iteration and is easy to obtain by utilizing symmetric structure of the system. We establish global convergence of the proposed method and report some numerical results.

Improved quasi-newton method via SR1 update for solving symmetric systems of nonlinear equations

The systems of nonlinear equations emerges from many areas of computing, scientific and engineering research applications. A variety of an iterative methods for solving such systems have been developed, this include the famous Newton method. Unfortunately, the Newton method suffers setback, which includes storing matrix at each iteration and computing Jacobian matrix, which may be difficult or even impossible to compute. To overcome the drawbacks that bedeviling Newton method, a modification to SR1 update was proposed in this study. With the aid of inexact line search procedure by Li and Fukushima, the modification was achieved by simply approximating the inverse Hessian matrix with an identity matrix without computing the Jacobian. Unlike the classical SR1 method, the modification neither require storing matrix at each iteration nor needed to compute the Jacobian matrix. In finding the solution to non-linear problems of the form 40 benchmark test problems were solved. A comparison was made with other two methods based on CPU time and number of iterations. In this study, the proposed method solved 37 problems effectively in terms of number of iterations. In terms of CPU time, the proposed method also outperformed the existing methods. The contribution from the methodology yielded a method that is suitable for solving symmetric systems of nonlinear equations. The derivative-free feature of the proposed method gave its advantage to solve relatively large-scale problems (10,000 variables) compared to the existing methods. From the preliminary numerical results, the proposed method turned out to be significantly faster, effective and suitable for solving large scale symmetric nonlinear equations.

Symmetric nonlinear functional differential equations at resonance

Symmetric nonlinear functional differential equations at resonance

A class of conjugate gradient methods for convex constrained monotone equations

The recent designed non-linear conjugate gradient method of Dai and Kou [SIAM J Optim. 2013;23:296–320] is very efficient currently in solving large-scale unconstrained minimization problems due to its simpler iterative form, lower storage requirement and its closeness to the scaled memoryless BFGS method. Just because of these attractive properties, this method was extended successfully to solve higher dimensional symmetric non-linear equations in recent years. Nevertheless, its numerical performance in solving convex constrained monotone equations has never been explored. In this paper, combining with the projection method of Solodov and Svaiter, we develop a family of non-linear conjugate gradient methods for convex constrained monotone equations. The proposed methods do not require the Jacobian information of equations, and even they do not store any matrix in each iteration. They are potential to solve non-smooth problems with higher dimensions. We prove the global convergence of the class of the proposed methods and establish its R-linear convergence rate under some reasonable conditions. Finally, we also do some numerical experiments to show that the proposed methods are efficient and promising.

A NEW QUASI-NEWTON METHOD BASED ON ADJOINT BROYDEN UPDATES FOR SYMMETRIC NONLINEAR EQUATIONS

In this paper, we propose a new rank two quasi-Newton method based on adjoint Broyden updates for solving symmetric nonlinear equations, which can be seen as a class of adjoint BFGS method. The new rank two quasi-Newton update not only can guarantee that <TEX>$B_{k+1}$</TEX> approximates Jacobian <TEX>$F^{\prime}(x_{k+1})$</TEX> along direction <TEX>$s_k$</TEX> exactly, but also shares some nice properties such as positive deniteness and least change property with BFGS method. Under suitable conditions, the proposed method converges globally and superlinearly. Some preliminary numerical results are reported to show that the proposed method is effective and competitive.

The inexact-Newton via GMRES subspace method without line search technique for solving symmetric nonlinear equations

In this paper, we propose an inexact-Newton via GMRES (generalized minimal residual) subspace method without line search technique for solving symmetric nonlinear equations. The iterative direction is obtained by solving the Newton equation of the system of nonlinear equations with the GMRES algorithm. The global convergence and local superlinear convergence rate of the proposed algorithm are established under some reasonable conditions. Finally, the numerical results are reported to show the effectiveness of the proposed algorithm.

Enhanced Derivative-Free Conjugate Gradient Method for Solving Symmetric Nonlinear Equations

In this article, an enhanced conjugate gradient approach for solving symmetric nonlinear equations is propose without computing the Jacobian matrix. This approach is completely derivative and matrix free. Using classical assumptions the proposed method has global convergence with nonmonotone line search. Some reported numerical results shows the approach is promising.&lt;p style="margin: 0px; text-indent: 0px; -qt-block-indent: 0; -qt-paragraph-type: empty;"&gt; &lt;/p&gt;

Limited memory technique using trust regions for nonlinear equations

A new trust region subproblem that accelerates convergence for solving symmetric nonlinear equations is defined. To avoid repeatedly computing the trust region subproblem, a line search technique without derivative information is used in the trust region algorithm. Moreover, a limited memory BFGS update is employed to generate an approximated matrix rather than a normal Jacobian matrix or quasi-Newton matrix. Under mild conditions, the global convergence and the superlinear convergence of the given algorithm are established. The numerical results indicate that this method can be beneficial for solving the presented problems.