Discrete Newton Method Research Articles

Efficient Newton‐multigrid FEM solver for multifield nonlinear coupled problems applied to thixoviscoplastic flows

AbstractThis note is concerned with efficient Newton‐multigrid finite element method (FEM) solver for multifield nonlinear flow problems. In our approach, for efficient FEM solver, we advantageously use the delicate symbiosis aspects of the problem settings for FEM approximations, and the algorithmic tools to obtain the numerical solutions. We concretize our ideas on thixoviscoplastic flow problems. It is a two‐field coupled nonlinear problem. And beside the integrated nonlinearity within momentum and microstructure equations, thixoviscoplastic problems induce a nonlinear two‐way coupling. As far as FEM numerical solutions are concerned, we set the problem in a suitable variational form to use the corresponding wellposedness analysis to develop FEM techniques for the solver. Indeed, the wellposedness study is not an intellectual exercise, rather it is the foundation for the approximate thixoviscoplastic problem as well as for the development of an efficient solver. We base our investigations for the solver on our wellposedness and error analysis results of thixoviscoplastic flow problems published in Proc. Appl. Math. Mech. We continue our series, and proceed to develop a monolithic Newton‐multigrid thixoviscoplastic solver. The solver is based on Newton's method and geometric multigrid techniques to treat the coupling of the problem. So, we use Local Pressure Schur Complement (LPSC) concept to solve the coupled problem on mesh's elements, and proceed with outer blocks Gauss‐Seidel iteration to update the global solutions. Furthermore, we handle the nonlinearity of the problem with the combined adaptive discrete Newton's and multigrid methods. The adaptivity within discrete Newton's method is based on the adaptive step‐length control for the discrete differencing in the Jacobian calculations, while the convergence of linear multigrid solver is made to match the convergence requirement of nonlinear solver, accordingly. And the solver's update parameters are solely dependent on the actual convergence rate of the nonlinear problem. We provide the numerical results of solver performance for thixoviscoplastic lid‐driven cavity flow.

A Proactive Reliable Mechanism-Based Vehicular Fog Computing Network

As vehicles are becoming more and more intelligent, mobile data traffic in vehicular ad hoc network (VANET) has been increasing dramatically. This makes the communication capacity of VANET systems and the computing resources of vehicles insufficient. In the meantime, location-aware large-scale distributed services with very low latency and high reliability are demanded by most of the novel functions, such as accident alarming, and congestion warning, in the intelligent transportation system. To meet these claimed characteristics of VANET, we first present a novel architecture that integrates vehicular fog computing and vehicle-to-vehicle (V2V) communication technologies. Lower latency and higher quality services can be supplied to vehicles by nearby fog servers, which are virtualized from vehicles that locate close enough and communicate using the V2V link. However, like all collaborative systems, computing reliability is vital to collaborative VANET. In this article, we design a novel energy-efficient proactive replication mechanism. Follower vehicles calculate with a lazy rate act as backups of host vehicles to ensure the reliability of the system. Considering the time sensitivity of computing requirements in VANET, the upper bound on the total number of failures is proposed through theoretical analysis. Then, the lower bound on the lazy calculating rate of followers is derived by balancing the tradeoffs between delay and energy. A fast algorithm for searching this lower bound based on the discrete Newton method is also proposed. Results of numerical experiments show that our new mechanism is effective in energy saving and reliability enhancing.

Discrete Newton methods for the evacuation problem

A dynamic network is a directed graph in which arcs have capacities and transit times. In this paper, we consider the evacuation problem in dynamic networks. In this problem, we are given a dynamic network with a single sink vertex in which each vertex except the sink vertex has a supply. Then the goal of this problem is to find the minimum time limit T such that we can send all the supplies to the sink vertex by T. In this paper, we propose a discrete Newton method for the evacuation problem. First, we prove that the number of iterations of this method is at most the number of vertices of the input dynamic network. Then we propose theoretical and practical implementation of this method. The theoretical implementation is based on submodular function minimization, and the practical implementation is based on maximum flow computations in time-expanded networks. Finally, we compare the proposed practical implementation with an algorithm using a binary search in time-expanded networks in computational experiments.

Submodular fractional programming for balanced clustering

We address the balanced clustering problem where cluster sizes are regularized with submodular functions. The objective function for balanced clustering is a submodular fractional function, i.e., the ratio of two submodular functions, and thus includes the well-known ratio cuts as special cases. In this paper, we present a novel algorithm for minimizing this objective function (submodular fractional programming) using recent submodular optimization techniques. The main idea is to utilize an algorithm to minimize the difference of two submodular functions, combined with the discrete Newton method. Thus, it can be applied to the objective function involving any submodular functions in both the numerator and the denominator, which enables us to design flexible clustering setups. We also give theoretical analysis on the algorithm, and evaluate the performance through comparative experiments with conventional algorithms by artificial and real datasets.

Algorithm 896

We present 14 basic Fortran subroutines for large-scale unconstrained and box constrained optimization and large-scale systems of nonlinear equations. Subroutines PLIS and PLIP, intended for dense general optimization problems, are based on limited-memory variable metric methods. Subroutine PNET, also intended for dense general optimization problems, is based on an inexact truncated Newton method. Subroutines PNED and PNEC, intended for sparse general optimization problems, are based on modifications of the discrete Newton method. Subroutines PSED and PSEC, intended for partially separable optimization problems, are based on partitioned variable metric updates. Subroutine PSEN, intended for nonsmooth partially separable optimization problems, is based on partitioned variable metric updates and on an aggregation of subgradients. Subroutines PGAD and PGAC, intended for sparse nonlinear least-squares problems, are based on modifications and corrections of the Gauss-Newton method. Subroutine PMAX, intended for minimization of a maximum value (minimax), is based on the primal line-search interior-point method. Subroutine PSUM, intended for minimization of a sum of absolute values, is based on the primal trust-region interior-point method. Subroutines PEQN and PEQL, intended for sparse systems of nonlinear equations, are based on the discrete Newton method and the inverse column-update quasi-Newton method, respectively. Besides the description of methods and codes, we propose computational experiments which demonstrate the efficiency of the proposed algorithms.

A Discrete Newton's Method for Gain Based Predistorter

Gain based predistorter (PD) is a highly effective and simple digital baseband predistorter which compensates for the nonlinear distortion of PAs. Lookup table (LUT) is the core of the gain based PD. This paper presents a discrete Newton’s method based adaptive technique to modify LUT. We simplify and convert the hardship of adaptive updating LUT to the roots finding problem for a system of two element real equations on athematics. And we deduce discrete Newton’s method based adaptive iterative formula used for updating LUT. The iterative formula of the proposed method is in real number field, but secant method previously published is in complex number field. So the proposed method reduces the number of real multiplications and is implemented with ease by hardware. Furthermore, computer simulation results verify gain based PD using discrete Newton’s method could rectify nonlinear distortion and improve system performance. Also, the simulation results reveal the proposed method reaches to the stable statement in fewer iteration times and less runtime than secant method.

On Finite Element Methods for Fully Nonlinear Elliptic Equations of Second Order

For the first time, we present for the general case of fully nonlinear elliptic differential equations of second order a nonstandard $C^1$ finite element method (FEM). We consider, throughout the paper, two cases in parallel: For convex, bounded, polyhedral domains in ${\bf R}^n$, or for $C^2$ bounded domains in ${\bf R}^2$, we prove stability and convergence for the corresponding conforming or nonconforming $C^1$ FEM, respectively. The results for equations and systems of orders 2 and $2m$ and quadrature approximations appear elsewhere. The classical theory of discretization methods is applied to the differential operator or the combined differential and the boundary operator. The consistency error for satisfied or violated boundary conditions on polyhedral or curved domains has to be estimated. The stability has to be proved in an unusual way. This is the hard core of the paper. Essential tools are linearization, a compactness argument, the interplay between the weak and strong form of the linearized operator, and a new regularity result for solutions of finite element equations. An essential basis for our proofs are Davydov's results for $C^1$ FEs on polyhedral domains in ${\bf R}^n$ or of local degree 5 for $C^2$ domains in ${\bf R}^2$. Better convergence and extensions to ${\bf R}^n$ for $C^2$ domains are to be expected from his forthcoming results on curved domains. Our proof for the second case in ${\bf R}^n$, includes the first essentially as a special case. The method applies to quasi-linear elliptic problems not in divergence form as well. A discrete Newton method is shown to converge locally quadratically, essentially independently of the actual grid size by the mesh independence principle.

Implicit FEM‐FCT algorithms and discrete Newton methods for transient convection problems

AbstractA new generalization of the flux‐corrected transport (FCT) methodology to implicit finite element discretizations is proposed. The underlying high‐order scheme is supposed to be unconditionally stable and produce time‐accurate solutions to evolutionary convection problems. Its nonoscillatory low‐order counterpart is constructed by means of mass lumping followed by elimination of negative off‐diagonal entries from the discrete transport operator. The raw antidiffusive fluxes, which represent the difference between the high‐ and low‐order schemes, are updated and limited within an outer fixed‐point iteration. The upper bound for the magnitude of each antidiffusive flux is evaluated using a single sweep of the multidimensional FCT limiter at the first outer iteration. This semi‐implicit limiting strategy makes it possible to enforce the positivity constraint in a very robust and efficient manner. Moreover, the computation of an intermediate low‐order solution can be avoided. The nonlinear algebraic systems are solved either by a standard defect correction scheme or by means of a discrete Newton approach, whereby the approximate Jacobian matrix is assembled edge by edge. Numerical examples are presented for two‐dimensional benchmark problems discretized by the standard Galerkin finite element method combined with the Crank–Nicolson time stepping. Copyright © 2007 John Wiley & Sons, Ltd.

Discrete Newton's method with local variations for solving large-scale nonlinear systems

A globally convergent discrete Newton method is proposed for solving large-scale nonlinear systems of equations. Advantage is taken from discretization steps so that the residual norm can be reduced while the Jacobian is approximated, besides the reduction at Newtonian iterations. The Curtis–Powell–Reid (CPR) scheme for discretization is used for dealing with sparse Jacobians. Global convergence is proved and numerical experiments are presented.

The Effectiveness of Newton's Method for ImprovingAb InitioForce Fields with Applications to CO2and H2CO

Ab initioforce field parameters are refined using modifications to Newton's method in which a figure of merit function is approximated as a Taylor Series truncated at second order. We investigate two versions of Newton's method: the Discrete Newton method where the Hessian is approximated by finite differences and the Gauss–Newton method where the Hessian is approximated as the product of first derivatives. The applicability of both rests on the capability of currentab initiomethods to calculate quartic force fields that accurately reproduce experimental observables such as band centers and inertial constants. As examples, we calculate refined potential energy surfaces for CO2and H2CO. We show that an algorithm depending solely on Newton's methods requires only a small number of iterations. Additionally, Newton-based methods provide a great deal of information about the sensitivity of the force field parameters to the observables being fit. We also demonstrate that the less computationally demanding Gauss–Newton method gives results similar to the less approximate Discrete Newton method.

The Discrete Newton Method of the Three Parameters Inversion in Electrical Logging

Abstract Log analysts are always interested in the problem of determining the invaded zone resistivity, Rxo, the virgin formation resistivity and the diameter of the invaded zone, D, simultaneously, using the logging tool responses of the deep and shallow laterolog 3. In this paper, the problem of multiple parameter identification in electrical logging is reduced to a partial differential equation. Using the Discrete Newton method and the regularization method, the mathematical formulas of inverting three parameters are derived. Thus, an iterative method of simultaneously solving Rxo, RT and D is obtained. To test this method, some numerical computations have been made. The computational results show that when no noise is added, the mean relative errors of inverting Rxo, RT and D are 0.67%, 0.47%, and 2.28%, respectively. When 10% noise is added to the resistivities of the adjacent formation, the mean relative errors are 2.65%, 0.92%, and 5.39%, respectively. This method has been used to deal with the real measured data. This method can be used widely to direct log interpretation and eliminate the interpretation chart.

Observer design for nonlinear systems with discrete-time measurements

This paper focuses on the development of asymptotic observers for nonlinear discrete-time systems. It is argued that instead of trying to imitate the linear observer theory, the problem of constructing a nonlinear observer can be more fruitfully studied in the context of solving simultaneous nonlinear equations. In particular, it is shown that the discrete Newton method, properly interpreted, yields an asymptotic observer for a large class of discrete-time systems, while the continuous Newton method may be employed to obtain a global observer. Furthermore, it is analyzed how the use of Broyden's method in the observer structure affects the observer's performance and its computational complexity. An example illustrates some aspects of the proposed methods; moreover, it serves to show that these methods apply equally well to discrete-time systems and to continuous-time systems with sampled outputs. >

Sparse newton-like methods in equation oriented flowsheeting

Numerical solution methods for the simultaneous solution of large sparse sets of nonlinear algebraic equations arising from flowsheeting problems are compared. The methods are tested on a set of mathematical problems and on six flowsheet simulations. Newton and quasi-Newton methods have been significantly improved by paying attention to particular details of implementation. The best implementations of sparse quasi-Newton methods, in particular, a new sparse scale invariant method, are in general more efficient and more robust than discrete Newton methods particularly when used on flowsheeting problems.

Global aspects of the continuous and discrete Newton method: A case study

Global aspects of the continuous and discrete Newton method: A case study

Numerical Solution of Large Sets of Algebraic Nonlinear Equations

This paper describes the application of the partitioned updating quasi-Newton methods for the solution of high-dimensional systems of algebraic nonlinear equations. This concept was introduced and successfully tested in nonlinear optimization of partially separable functions (see [6]). Here its application to the case of nonlinear equations is explored. Nonlinear systems of this nature arise in many large-scale applications, including finite elements and econometry. It is shown that the method presents some advantages in efficiency over competing algorithms, and that use of the partially separable structure of the system can lead to significant improvements also in the more classical discrete Newton method.

Numerical solution of large sets of algebraic nonlinear equations

This paper describes the application of the partitioned updating quasi-Newton methods for the solution of high-dimensional systems of algebraic nonlinear equations. This concept was introduced and successfully tested in nonlinear optimization of partially separable functions (see [6]). Here its application to the case of nonlinear equations is explored. Nonlinear systems of this nature arise in many large-scale applications, including finite elements and econometry. It is shown that the method presents some advantages in efficiency over competing algorithms, and that use of the partially separable structure of the system can lead to significant improvements also in the more classical discrete Newton method.

Discrete Defect Corrections: Basic Ideas

AbstractIn this paper we combine Newton methods, mesh refinement strategies, and discrete Newton methods to develop an efficien process for approximating the solutions of operator equations.

Discrete Newton methods and iterated defect corrections

In this paper we present a general theory for "discrete Newton methods", "iterated defect corrections via neighbouring problems" and "deferred corrections" based on asymptotic expansions of the dis...

Numerical realization of the discrete Newton method

Numerical realization of the discrete Newton method

Some Efficient Algorithms for Solving Systems of Nonlinear Equations

We compare the Ostrowski efficiency of some methods for solving systems of nonlinear equations without explicitly using derivatives. The methods considered include the discrete Newton method, Shamanskii’s method, the two-point secant method, and Brown’s methods. We introduce a class of secant methods and a class of methods related to Brown’s methods, but using orthogonal rather than stabilized elementary transformations. The idea of these methods is to avoid finding a new approximation to the Jacobian matrix of the system at each step, and thus increase the efficiency. Local convergence theorems are proved, and the efficiencies of the methods are calculated. Numerical results are given, and some possible extensions are mentioned.