Narrow Results

This research work introduces a novel hybrid conjugate gradient algorithm that combines three descent directions, namely $d_k^{\mathrm{\text{DY}}}$, $d_k^{\mathrm{\text{HS}}}$, and $d_k^{\mathrm{\text{DL}}}$, in a quasi-convex manner. The algorithm aims to construct a direction that closely approximates the Newton direction using the secant equation. To ensure convergence, appropriate hybridization parameters are determined, which not only approximate the Newton direction but also remain bounded. The descent property of the algorithm has been proven, and its global convergence has been demonstrated. Experimental results confirm the efficiency of our new algorithm, surpassing the (DY), (HS), (DL) algorithms in many cases.

Newton method is a widely used iteration method in solving nonlinear algebraic equations. In this method, a linear algebraic equations need to be solved in every step. The coefficient matrix of the algebraic equations is so-called Jacobian matrix, which needs to be determined at every step. For a complex non-linear system, usually no explicit form of Jacobian matrix can be found. Several methods are introduced to obtain an approximated matrix, which are classified as Jacobian-free method. The finite difference method is used to approximate the derivatives in Jacobian matrix, and a small parameter is needed in this process. Some problems may arise because of the interaction of this parameter and round-off errors. In the present work, we show that this kind of Newton method may encounter difficulties in solving non-linear partial differential equation (PDE) on fine mesh. To avoid this problem, the continuous Newton method is presented, which is a modification of classical Newton method for non-linear PDE.

This paper analyzes the convergence properties and convergence region of a class of trajectory-based power flow methods. The convergence region of the trajectory-based method is a connected set and possesses the near-by property. The convergence region of Newton method and trajectory-based method in solving power flow problems are numerically investigated. Since the convergence region of trajectory-based method corresponds to the stability region of the nonlinear dynamic system, the stability regions of two dynamic systems are computed. The numerical results indicate that the stability region of the dynamic system possesses better geometry features than the convergence region of Newton method. These properties make the trajectory-based power flow method robust, especially on heavy loading conditions.

In this paper, weak formulations and finite element discretizations of the governing partial differential equations of three-dimensional nonlinear acoustics in absorbing fluids are presented. The fluid equations are considered in an Eulerian framework, rather than a displacement framework, since in the latter case the corresponding finite element formulations suffer from spurious modes and numerical instabilities. When taken with the governing partial differential equations of a solid body and the continuity conditions, a coupled formulation is derived. The change in solid/fluid interface conditions when going from a linear acoustic fluid to a nonlinear acoustic fluid is demonstrated. Finite element discretizations of the coupled problem are then derived, and verification examples are presented that demonstrate the correctness of the implementations. We demonstrate that the time step size necessary to resolve the wave decreases as steepening occurs. Finally, simulation results are presented on a resonating acoustic cavity, and a coupled elastic/acoustic system consisting of a fluid-filled spherical tank.

We consider a mathematical model which describes the frictional contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is modeled with normal compliance condition of such a type that the penetration is restricted with unilateral constraint. The friction is modeled with a nonmonotone law in which the friction bound depends both on the tangential displacement and on the value of the penetration. In order to approximate the contact conditions, we consider a regularized problem wherein the contact is modeled by a standard normal compliance condition without finite penetrations. For each problem, we derive a variational formulation and an existence result of the weak solutions of regularized problems is obtained. Next, we prove the convergence of the weak solutions of regularized problems to the weak solution of the initial nonregularized problem. Finally, we provide a numerical validation of this convergence result. To this end we introduce a discrete scheme for the numerical approximation of the frictional contact problems. The solution of the resulting nonsmooth and nonconvex frictional contact problems is found, basing on approximation by a sequence of nonsmooth convex programming problems. Some numerical simulation results are presented in the study of an academic two-dimensional example.

We extend Kim's complex exponential function and come up with a theory about Julia sets of Newton method for general exponential equation. We analyze the behavior of the roots of some complex exponential equation, and prove the Julia Set's symmetry, boundedness and embedding topology distribution structure of attraction regions in theory.

In recent years, Image Registration has attracted lots of attention due to its capabilities and numerous applications. Various methods have been exploited to map two images with the same concept but different conditions. Considering the finding of the mentioned map as an optimization problem, mathematical-based optimization methods have been extensively employed due to their real-time performances. In this paper, we employed the Newton method to optimize two defined cost functions. These cost functions are Sum of Square Difference and Cross-Correlation. These presented algorithms have fast convergence and accurate features. Also, we propose an innovative treatment in order to attend to one of the free parameter-rotations or scale as a sole variable and the other one as the constant value. The assignment is replaced through the iterations for both parameters. The intuition is to turn a two-variable optimization problem into a single variable one in every step. Our simulation on benchmark images by the means of Root Mean Square Error and Mutual Information as the goodness criteria, that have been extensively used in similar studies, has shown the robustness and affectivity of the proposed method.

In this work, a modified Newton–Özban composition of convergence order six for solving nonlinear systems is presented. The first two steps of proposed scheme are based on third-order method given by Özban [Özban, A. Y. [2004] “Some new variants of Newton’s method,” Appl. Math. Lett. 17, 677–682.] for solving scalar equations. Computational efficiency of the presented method is discussed and compared with well-known existing methods. Numerical examples are studied to demonstrate the accuracy of the proposed method. The basins of attraction of some of the existing methods along with the proposed method are given to exhibit their performance.

In this paper, a class of efficient iterative methods with increasing order of convergence for solving systems of nonlinear equations is developed and analyzed. The methodology uses well-known third-order Potra–Pták iteration in the first step and Newton-like iterations in the subsequent steps. Novelty of the methods is the increase in convergence order by an amount three per step at the cost of only one additional function evaluation. In addition, the algorithm uses a single inverse operator in each iteration, which makes it computationally more efficient and attractive. Local convergence is studied in the more general setting of a Banach space under suitable assumptions. Theoretical results of convergence and computational efficiency are verified through numerical experimentation. Comparison of numerical results indicates that the developed algorithms outperform the other similar algorithms available in the literature, particularly when applied to solve the large systems of equations. The basins of attraction of some of the existing methods along with the proposed method are given to exhibit their performance.