Solutions For Systems Of Nonlinear Equations Research Articles

Decomposition-based multiobjective optimization for nonlinear equation systems with many and infinitely many roots

Although the development of Pareto-dominance-based multiobjective optimization algorithms has enabled the solution of nonlinear equation systems, few studies have been conducted on the use of multiobjective optimization techniques for the solution of nonlinear equation systems. Accordingly, in this study, we use decomposition-based multiobjective optimization to solve nonlinear equation system, an attempt is made at deploying the decomposition-based multiobjective optimization to solve nonlinear equation systems, including those with infinite roots. In our novel approach, a given system is transformed into a bi-objective optimization problem using reference points. An improved decomposition-based multiobjective optimization is then applied to solve a transformed bi-objective optimization problem. To ensure this optimization suits the characteristics of the problem, we develop an adaptive multiobjective differential evolution and local search approach. The roots of the original nonlinear equation system can then be identified, together with the Pareto-optimal solutions of the transformed problem. We conducted extensive experiments to compare the performance of our novel approach with that of 16 state-of-the-art algorithms in solving 30 nonlinear equation systems derived from industrial applications. The results clearly show that our novel approach achieves better performance with a shorter execution time than that of the selected single-objective-based evolutionary approaches or Pareto-dominance-based multiobjective optimization algorithms.

Rotation parameter estimation error of deflectometer basic unit

Subject of Research. The paper proposes a method for determining the rotation parameters of the basic unit of an optical- electronic deflectometer, having an effect on the deflection measurement error of large-scale objects such as a floating dock. Method. We proposed an algorithm and mathematical model structure for the effect of the image coordinate estimation error of the collimating mark on the rotation parameter estimation error of the deflectometer basic unit with the use of the elements of vector algebra and matrix analysis. Main Results. We have proved that the rotation parameters of the basic unit can be determined as a result of solution of nonlinear equation systems based on the Levenberg–Marquardt optimization algorithm. Studies on a mathematical model have given the possibility to estimate the effect of the image coordinate estimation error of the collimating mark on the rotation parameter estimation error of the basic unit. Practical Relevance. The results of this work will enable us to develop an algorithm for compensation of the basic unit rotation parameters due to the effect of external factors and, as a result, to reduce the error in determining the spatial coordinates of the controlled object (deflection of the floating dock).

An improved adaptive decomposition method for forest parameters estimation using polarimetric SAR interferometry image

ABSTRACT Forest parameters estimation using polarimetric synthetic aperture radar interferometry (PolInSAR) images is one of the greatest interests in remote sensing applications. Applying the model-based decomposition concept to PolInSAR data opened a new way for forest parameters estimation. However, the method tends to underestimate the forest height due to reflection symmetry assumption and required the numerical solution of nonlinear equation system. In order to overcome these limitations, an improved adaptive decomposition technique with PolInSAR data is proposed. In this approach, an accurate topographical phase and asymmetry volume scattering model are applied to the model-based decomposition technique for polarimetric SAR interferometry image. The accurate topographical phase is first estimated and then used as the initial input parameter to our numerical method. This approach is not only avoiding large error generated by the constant topographical phase in fluctuating forest areas but also improve the accuracy of forest height estimation and the magnitude associated with each mechanism. The performance of proposed method is demonstrated with simulated data from PolSARproSim software and SIR-C/X-SAR spaceborne PolInSAR images over the Tien-Shan areas, China. Experimental results indicate that forest parameters could be effectively extracted by proposed method.

A New Solution Approach for Non-Linear Equation Systems with Grey Wolf Optimizer

The aim of this study is to bring a new perspective for the solutions of non-linear equation systems. So this study handles the non-linear equation systems as a constrained optimization problem, while generally is handled unconstrained optimization problem or multi objective optimization problem. The object is to minimize the sum of the squares of nonlinear equations under the nonlinear equality constraints. A recently developed heuristic optimization algorithm called Grey Wolf Optimizer (GWO) is proposed for the solution of nonlinear equation systems. Two results were obtained. Firstly, it has been seen that GWO can be an alternative solution technique for the solution of nonlinear equation systems. Secondly, modelling the systems of nonlinear equations as constrained optimization gives better results.

A Weighted Biobjective Transformation Technique for Locating Multiple Optimal Solutions of Nonlinear Equation Systems

Due to the fact that a nonlinear equation system (NES) may contain multiple optimal solutions, solving NESs is one of the most important challenges in numerical computation. When applying evolutionary algorithms to solve NESs, two issues should be considered: 1) how to transform an NES into a kind of optimization problem and 2) how to develop an optimization algorithm to solve the transformed optimization problem. In this paper, we tackle the first issue by transforming an NES into a weighted biobjective optimization problem. By the above transformation, not only do all the optimal solutions of an original NES become the Pareto optimal solutions of the transformed biobjective optimization problem, but also their images are different points on a linear Pareto front in the objective space. In addition, we suggest an adaptive multiobjective differential evolution, the goal of which is to effectively locate the Pareto optimal solutions of the transformed biobjective optimization problem. Once these solutions are found, the optimal solutions of the original NES can also be obtained correspondingly. By combining the weighted biobjective transformation technique with the adaptive multiobjective differential evolution, we propose a generic framework for the simultaneous locating of multiple optimal solutions of NESs. Comprehensive experiments on 38 NESs with various features have demonstrated that our framework provides very competitive overall performance compared with several state-of-the-art methods.

An analytical solution to lateral buckling control of subsea pipelines by distributed buoyancy sections

Lateral buckling is the primary buckling form of the subsea pipelines. A promising recent practice is to the distributed buoyancy sections to control the lateral buckling of subsea pipeline. In this study, an analytical solution is deduced for the lateral buckling of the pipeline with buoyancy sections in the entire design region. However, numerical difficulty was encountered during the solution of the design equations. To overcome this difficulty, we adopt a response surface method (RSM) for the solution of nonlinear equation system. A strategy of multi-response-surface is proposed by dividing the entire design region into several partial domains and establishing different kinds of response surfaces for each region. A framework for lateral buckling control of subsea pipelines using distributed buoyancy sections is presented. Several illustrative examples demonstrate that the proposed method has a high efficiency and accuracy in solving the problem of the lateral buckling control.

Locating Multiple Optimal Solutions of Nonlinear Equation Systems Based on Multiobjective Optimization

Nonlinear equation systems may have multiple optimal solutions. The main task of solving nonlinear equation systems is to simultaneously locate these optimal solutions in a single run. When solving nonlinear equation systems by evolutionary algorithms, usually a nonlinear equation system should be transformed into a kind of optimization problem. At present, various transformation techniques have been proposed. This paper presents a simple and generic transformation technique based on multiobjective optimization for nonlinear equation systems. Unlike the previous work, our transformation technique transforms a nonlinear equation system into a biobjective optimization problem that can be decomposed into two parts. The advantages of our transformation technique are twofold: 1) all the optimal solutions of a nonlinear equation system are the Pareto optimal solutions of the transformed problem, which are mapped into diverse points in the objective space, and 2) multiobjective evolutionary algorithms can be directly applied to handle the transformed problem. In order to verify the effectiveness of our transformation technique, it has been integrated with nondominated sorting genetic algorithm II to solve nonlinear equation systems. The experimental results have demonstrated that, overall, our transformation technique outperforms another state-of-the-art multiobjective optimization based transformation technique and four single-objective optimization based approaches on a set of test instances. The influence of the types of Pareto front on the performance of our transformation technique has been investigated empirically. Moreover, the limitation of our transformation technique has also been identified and discussed in this paper.

Computation of maximal turning points to nonlinear equations by nonsmooth optimization

We develop a new approach to finding bifurcations of solutions of nonlinear equation systems, which is based on the detection of extreme values of a new type of variational functional associated with the considered problems. The variational functional is obtained constructively by the extended functional method which can be applied to a wide class of parametric problems including nonlinear partial differential equations. Sufficient and necessary conditions for the existence of a maximal turning point by the approach are proved. Based on these, an algorithm of the quasi-direction of steepest ascent is introduced. Simulation experiments are used to illustrate the behaviour of the method and to discuss its advantages and disadvantages in comparison with the alternatives.

Factored solution of nonlinear equation systems

This article generalizes a recently introduced procedure to solve nonlinear systems of equations, radically departing from the conventional Newton–Raphson scheme. The original nonlinear system is first unfolded into three simpler components: (i) an underdetermined linear system, (ii) a one-to-one nonlinear mapping with explicit inverse, and (iii) an overdetermined linear system. Then, instead of solving such an augmented system at once, a two-step procedure is proposed in which two equation systems are solved at each iteration, one of them involving the same symmetric matrix throughout the solution process. The resulting factored algorithm converges faster than Newton's method and, if carefully implemented, can be computationally competitive for large-scale systems. It can also be advantageous for dealing with infeasible cases.

A time--memory trade-off approach for the solution of nonlinear equation systems

We propose a memory-based method for the solution of a specific type of nonlinear equation systems. We observe that when the equations in a system can be separated into 2 parts, where each subset contains fewer parameters than the whole set of equations, the system can be solved faster with a preprocessing phase. We show that reduced rounds of AES produce such a system under a chosen plaintext scenario. This observation enables us to solve that system within a practically applicable complexity of 2^{37} operations where a brute force approach requires 2^{72} trials. The method can be used for the solution of other equation systems of the same structure. In the optimal case where we can divide the equations into 2, a problem that contains n binary variables can be solved at time O(n/2. 2^{n/2}) operations and using O(2^{n/2}) units of memory rather than O(2^n) trials of the equation system.

Application of the DTM to Nonlinear Cases Arising in Fluid Flows with Variable Viscosity

This paper employs the di erential transformation method to investigate two nonlinear ordinary di erential systems for plane coquette ow having variable viscosity and thermal conductivity. The concept of di erential transformation is brie y introduced, and then di erential transformation method is employed to derive solutions of nonlinear equation systems. The results of di erential transformation method are compared with those ones obtained by Adomian decomposition method to verify the accuracy of proposed method. The results reveal that the di erential transformation method can achieve suitable results in predicting the solution of such problems.

On the local convergence of adjoint Broyden methods

The numerical solution of nonlinear equation systems is often achieved by so-called quasi-Newton methods. They preserve the rapid local convergence of Newton’s method at a significantly reduced cost per step by successively approximating the system Jacobian though low-rank updates. We analyze two variants of the recently proposed adjoint Broyden update, which for the first time combines the classical least change property with heredity on affine systems. However, the new update does require, the evaluation of so-called adjoint vectors, namely products of the transposed Jacobian with certain dual direction vectors. The resulting quasi-Newton method is linear contravariant in the sense of Deuflhard (Newton methods for nonlinear equations. Springer, Heidelberg, 2006) and it is shown here to be locally and q-superlinearly convergent. Our numerical results on a range of test problems demonstrate that the new method usually outperforms Newton’s and Broyden’s method in terms of runtime and iterations count, respectively.

APPLICATION OF DIFFERENTIAL TRANSFORMATIONS FOR THE MODELING MOTION OF AEROSTATIC VEHICLES AND FOR THE SYNTHESIS OF CONTROL ALGORITHMS

The approach to simulation of flight dynamics and numerically-analytical method of airship control algorithms are offered. It’s based on differential transformations of initial mathematical model of airship motion. The given approach allows for elimination of viewing time function for their differential spectra in the image field. It gives the possibility to reduce a problem of closed algorithm synthesis of vehicle control to the solution of non-linear equation system concerning control variable.

Solution of nonlinear equations and computation of multiple solutions of a simple reaction-diffusion equation

AbstractThe first part of this paper starts with a brief discussion of some methods for solution of nonlinear equations which have interested the first author over the last twenty years or so. In the second part we discuss a recent research involvement, the success of which relies heavily on the numerical solution of nonlinear equation systems. We briefly describe path-following methods and then present an application to a simple steady-state reaction-diffusion equation arising in combustion theory. Results for some regular geometric shapes are shown and compared with those from an approximate method.

A Comparative Study on Methods for Convergence Acceleration of Iterative Vector Sequences

We discuss several methods for accelerating the convergence of the iterative solution of nonlinear equation systems commonly in use and point to interrelations between them. In particular we investigate two of the most sophisticated schemes, namely the Anderson mixing and the Broyden update, both generalized to the consideration of arbitrarily many previous iterations. For the Broyden method we give a new derivation which is much simpler than that recently proposed by Vanderbilt and Louie. We show that if the additional parameters invented by these authors in order to increase flexibility are used to optimize the convergence of the iteration process they in fact cancel out. In addition we prove that in this (optimal) case the Anderson mixing and the Broyden update as applied to the inverse Jacobian are fully identical. Thus we come to the conclusion that neither of these schemes is superior. Moreover, we show that Broyden update of the inverse Jacobian is superior to updating the Jacobian itself. Finally we propose an extension of the Anderson mixing which avoids the numerical difficulties all these methods are faced with.

Solving nonlinear equation systems via global partition and search: Some experimental results

A general framework is proposed for the solution of nonlinear equation systems, applying multiextremal (global) optimization methodology. Following a brief presentation of the background needed, numerical results are reported for two classes of easily reproducible pseudo-random test problems.

Occurrence of secular terms in the Carleman embedding

Secular terms occur in many perturbative solutions of nonlinear equation systems. In this work, an investigation is made of which cases they may occur in as the result of the application of the linear Carleman embedding to a system of nonlinear equations. The solution for the embedded system is written in a form that makes it convenient to see how these terms originate. Their occurrence for the general case is discussed and the results are exemplified by working out the Hénon–Heiles system.

Future trends in nonlinear structural analysis

In the following paragraphs possible future trends in the nonlinear analysis of structures are contemplated. If any predictions at all are to be made, they must be based on the assumption that research and development in the area will be responsive to the needs of the engineering community, perhaps primarily within the aerospace and nuclear industries. This need is definitely for more efficient general purpose programs derived in such a way that they are reasonably easy to use. Some new technology is presented in the recent literature, in the form of new ideas in disdretization procedures and in automation of the methods for solution of nonlinear equation systems. The prediction is simply that these are going to be further developed, thoroughly evaluated and eventually included in commonly available software.

Use of barycentric coordinates to solve certain computational problems

IT is shown in this paper that, using barycentric coordinates, simple formulae and useful algorithms can be derived to solve several conputational problems for functions of numerous variables, viz. interpolation of functions, solution of nonlinear eQuation systems and finding extrema of functions.