A shift splitting modified Newton-type (SSMN) iteration method is introduced for solving large sparse generalized absolute value equations (GAVEs). The SSMN method is established by replacing the regularized splitting of the coefficient matrix of the linear part, which is employed in the modified Newton-type (MN) iteration method, with the shift splitting of the matrix. The conditions for the convergence of the proposed method are discussed in depth for the cases when the coefficient matrix is a general matrix, a symmetric positive definite matrix, and an $H_{+}$-matrix. Through two numerical examples, we find that the SSMN and MN methods complement each other. The optimal performances of the two methods depend on the definiteness of the coefficient matrix of the linear part. The MN method is more efficient when the coefficient matrix is positive definite, whereas the SSMN method has a better performance when the coefficient matrix is indefinite.
We propose a new modification of Newton iteration for finding some nonnegative Z-eigenpairs of a nonnegative tensor. The method has local quadratic convergence to a nonnegative eigenpair of a nonnegative tensor, under the usual assumption guaranteeing the local quadratic convergence of the original Newton iteration.
Globalization concepts for Newton-type iteration schemes are widely used when solving nonlinear problems numerically. Most of these schemes are based on a predictor/corrector step size methodology with the aim of steering an initial guess to a zero of $f$ without switching between different attractors. In doing so, one is typically able to reduce the chaotic behavior of the classical Newton-type iteration scheme. In this note we propose a globalization methodology for general Newton-type iteration concepts which changes into a simplified Newton iteration as soon as the transformed residual of …
A Multilevel Newton Iteration Method for Eigenvalue Problems
Memorial University of Newfoundland（纽芬兰纪念大学）；Tianjin University of Finance and Economics（天津财经大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
The nonlinear eigen-problem $ Ax+F(x)=λx$ is studied where $A$ is an $n\times n$ irreducible Stieltjes matrix. Under certain conditions, this problem has a unique positive solution. We show that, starting from a multiple of the positive eigenvector of $A$, the Newton-like iteration for this problem converges monotonically. Numerical results illustrate the effectiveness of this Newton-like method.
The cost of a partitioned fluid-structure interaction scheme is typically assessed by the number of coupling iterations required per time step, while ignoring the Newton loops within the nonlinear sub-solvers. In this work, we discuss why these single-field iterations deserve more attention when evaluating the coupling's efficiency and how to find the optimal number of Newton steps per coupling iteration.
Fast convergent, accurate, computationally efficient, parallelizable, and robust matrix inversion and parameter estimation algorithms are required in many time-critical and accuracy-critical applications such as system identification, signal and image processing, network and big data analysis, machine learning and in many others. This paper introduces new composite power series expansion with optionally chosen rates (which can be calculated simultaneously on parallel units with different computational capacities) for further convergence rate improvement of high order Newton-Schulz iteration. N…
Recently, two families of HSS-based iteration methods are constructed for solving the system of absolute value equations (AVEs), which is a class of non-differentiable NP-hard problems. In this study, we establish the Picard-CSCS iteration method and the nonlinear CSCS-like iteration method for AVEs involving the Toeplitz matrix. Then, we analyze the convergence of the Picard-CSCS iteration method for solving AVEs. By using the theory about nonsmooth analysis, we particularly prove the convergence of the nonlinear CSCS-like iterationsolver for AVEs. The advantage of these methods is that they …
Inexact Newton Methods are widely used to solve systems of nonlinear equations. The convergence of these methods is controlled by the relative linear tolerance, $η_ν$, that is also called the forcing term. A very small $η_ν$ may lead to oversolving the Newton equation. Practical reservoir simulation uses inexact Newton methods with fixed forcing term, usually in the order of $10^{-3}$ or $10^{-4}$. Alternatively, variable forcing terms for a given inexact Newton step have proved to be quite successful in reducing the degree of oversolving in various practical applications. The cumulative numbe…
There are thousands of papers on rootfinding for nonlinear scalar equations. Here is one more, to talk about an apparently new method, which I call ``Inverse Cubic Iteration'' (ICI) in analogy to the Inverse Quadratic Iteration in Richard Brent's zeroin method. The possibly new method is based on a cubic blend of tangent-line approximations for the inverse function. We rewrite this iteration for numerical stability as an average of two Newton steps and a secant step: only one new function evaluation and derivative evaluation is needed for each step. The total cost of the method is therefore on…
We present novel coupling schemes for partitioned multi-physics simulation that combine four important aspects for strongly coupled problems: implicit coupling per time step, fast and robust acceleration of the corresponding iterative coupling, support for multi-rate time stepping, and higher-order convergence in time. To achieve this, we combine waveform relaxation -- a known method to achieve higher order in applications with split time stepping based on continuous representations of coupling variables in time -- with interface quasi-Newton coupling, which has been developed throughout the l…
The nonlinear HSS-like iteration method for absolute value equations
School of Mathematics and Statistics, Lanzhou University（兰州大学数学与统计学院）；School of Mathematics and Statistics, Hexi University（河西学院数学与统计学院）；School of Chemistry and Chemical Engineering, Hexi University（河西学院化学化工学院）
Systematic overview of Newton-Schulz and Durand iterations with convergence analysis and factorizations is presented in the chronological sequence in unified framework. Practical recommendations for the choice of the order and factorizations of the algorithms and integration into Richardson iteration are given. The simplest combination of Newton-Schulz and Richardson iteration is applied to the parameter estimation problem associated with the failure detection via evaluation of the frequency content of the signals in electrical network. The detection is performed on real data for which the sof…
In this paper, two numerical approaches based on the Newton iteration method with spectral algorithms are introduced to solve the Thomas-Fermi equation. That Thomas-Fermi equation is a nonlinear singular ordinary differential equation (ODE) with boundary condition in infinite. In these schemes, the Newton method is combined with a spectral method where in one of those, by Newton method we convert nonlinear ODE to a sequence of linear ODE then, solve them using the spectral method. In another one, by the spectral method the nonlinear ODE be converted to system of nonlinear algebraic equations, …
Picard-Based Acceleration of Newton Continuation for Mean Field Game PDE Systems
New York University Shanghai（纽约大学上海）；NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai（纽约大学上海华东师范大学数学科学联合研究院）；Department of Mathematics, New York University Shanghai（纽约大学上海数学系）；Department of Data Science, New York University Shanghai（纽约大学上海数据科学系）；Department of Mathematics, New York University（纽约大学数学系）
We present an iterative method for the search of extreme entries in low-rank tensors which is based on a power iteration combined with a binary search. In this work we use the HT-format for low-rank tensors but other low-rank formats can be used verbatim. We present two different approaches to accelerate the basic power iteration: an orthogonal projection of Rayleigh-Ritz type, as well as an acceleration of the power iteration itself which can be achieved due to the diagonal structure of the underlying eigenvalue problem. Finally the maximizing index is determined by a binary search based on t…
This paper studies the performance Newton's iteration applied with Anderson acceleration for solving the incompressible steady Navier-Stokes equations. We manifest that this method converges superlinearly with a good initial guess, and moreover, a large Anderson depth decelerates the convergence speed comparing to a small Anderson depth. We observe that the numerical tests confirm these analytical convergence results, and in addition, Anderson acceleration sometimes enlarges the domain of convergence for Newton's method.
