Number Of Newton Iterations Research Articles

A New Fully Implicit Two-Phase Pore-Network Model by Utilizing Regularization Strategies

In this paper, we address the expensive computational cost resulting from limited time-step sizes during numerical simulations of two-phase flow in porous media using dynamic pore-network models. To overcome this issue, we propose a numerical method for dynamic pore-network models using a fully implicit approach. The proposed method introduces a regularization strategy considering the historical fluid configuration at the pore throat, which smooths the discontinuities in local conductivity caused by invasion and snap-off events. The results demonstrate the superiority of the proposed method in terms of accuracy, efficiency and consistency in comparison with other numerical schemes. With similar computational cost, determined by time-step sizes and number of Newton iterations, the developed method in this work yields more accurate results compared to similar schemes presented in the literature. Additionally, our results highlight the enhanced robustness of the our scheme, as it exhibits reduced sensitivity to variations in time-step sizes.

Newton’s method in practice, II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees

We present an algorithm, based on Newton’s method, for finding all roots of univariate complex polynomials so that the observed complexity is linear in the degree, up to logarithmic factors. Unlike the usual Newton method, which finds at most one root at a time, it is global in the sense that it attempts to find all roots of polynomials simultaneously.We demonstrate the feasibility of this algorithm by employing it to find all roots of several families of polynomials of degrees up to more than one billion (109). In all cases, the observed (empirical) complexity for finding all roots of a polynomial of degree d – measured either as the number of Newton iterations or computing time – was between O(dlnd) and O(dln3d), with small constants.

Near optimal subdivision algorithms for real root isolation

Isolating real roots of a square-free polynomial in a given interval is a fundamental problem in computational algebra. Subdivision based algorithms are a standard approach to solve this problem. For instance, Sturm's method, or various algorithms based on the Descartes's rule of signs. For the benchmark problem of isolating all the real roots of a polynomial of degree n and root separation σ, the size of the subdivision tree of most of these algorithms is bounded by O(log⁡1/σ) (assume σ<1). Moreover, it is known that this is optimal for subdivision algorithms that perform uniform subdivision, i.e., the width of the interval decreases by some constant. Recently Sagraloff (2012) and Sagraloff–Mehlhorn (2016) have developed algorithms for real root isolation that combine subdivision with Newton iteration to reduce the size of the subdivision tree to O(n(log⁡(nlog⁡1/σ))).We describe a subroutine that reduces the size of the subdivision tree of any subdivision algorithm for real root isolation. The subdivision tree size of our algorithm using predicates based on either the Descartes's rule of signs or Sturm sequences is bounded by O(nlog⁡n), which is close to the optimal value of O(n). The corresponding bound for the algorithm EVAL, which uses certain interval-arithmetic based predicates, is O(n2log⁡n). Our analysis differs in two key aspects from earlier approaches. First, we use the general technique of continuous amortization from Burr–Krahmer–Yap (2009), and extend it to handle non-uniform subdivisions; second, we use the geometry of clusters of roots instead of root bounds. The latter aspect enables us to derive a bound on the subdivision tree that is independent of the root separation σ. The number of Newton iterations is bounded by O(nlog⁡log⁡(1/σ)).

Numerical Impact of Using Different $E$ –$J$ Relationships for 3-D Simulations of AC Losses in MgB2Superconducting Wires

AC losses in superconductors are generated every time a time-varying current/field is present. Engineers must be able to predict these losses as accurately as possible during the design phase of power applications. The electrodynamics of superconductors can be formulated as a nonlinear eddy current problem in which the resistivity of the superconducting region is a highly nonlinear function of the current density. In 3-D finite-element simulations, it leads to time-consuming simulations and convergence issues. In this paper, we compare two different $E$ – $J$ constitutive equations, namely: 1) power law model and 2) the percolation model (PM), programmed within both the $H$ – $\phi $ and $T$ – $\phi $ formulations. Based on the 3-D case of a three-filament twisted superconducting wire, the numerical performance of all these formulations/material models is compared in terms of accuracy, computation times, number of time steps, and number of Newton iterations for different relaxation methods. It is shown that the combination of the $T$ – $\phi $ formulation and the $E$ – $J$ PM works fine and should be further developed, as it seems to constitute the best modeling option from both a numerical and physical point of view.

A modified iterated projection method adapted to a nonlinear integral equation

The classical way to tackle a nonlinear Fredholm integral equation of the second kind is to adapt the discretization scheme from the linear case. The Iterated projection method is a popular method since it shows, in most cases, superconvergence and it is easy to implement. The problem is that the accuracy of the approximation is limited by the mesh size discretization. Better approximations can only be achieved for fine discretizations and the size of the linear system to be solved then becomes very large: its dimension grows up with an order proportional to the square of the mesh size. In order to overcome this difficulty, we propose a novel approach to first linearize the nonlinear equation by a Newton-type method and only then to apply the Iterated projection method to each of the linear equations issued from the Newton method. We prove that, for any value (large enough) of the discretization parameter, the approximation tends to the exact solution when the number of Newton iterations tends to infinity, so that we can attain any desired accuracy. Numerical experiments confirm this theoretical result.

A scalable algorithm for MAP estimators in Bayesian inverse problems with Besov priors

We present a scalable solver for approximating the maximum a posteriori (MAP) point of Bayesian inverse problems with Besov priors based on wavelet expansions with random coefficients. It is a subspace trust region interior reflective Newton conjugate gradient method for bound constrained optimization problems. The method combines the rapid locally-quadratic convergence rate properties of Newton's method, the effectiveness of trust region globalization for treating ill-conditioned problems, and the Eisenstat--Walker idea of preventing oversolving. We demonstrate the scalability of the proposed method on two inverse problems: a deconvolution problem and a coefficient inverse problem governed by elliptic partial differential equations. The numerical results show that the number of Newton iterations is independent of the number of wavelet coefficients $n$ and the computation time scales linearly in $n$. It will be numerically shown, under our implementations, that the proposed solver is two times faster than the split Bregman approach, and it is an order of magnitude less expensive than the interior path following primal-dual method. Our results also confirm the fact that the Besov $\mathbb{B}_{11}^1$ prior is sparsity promoting, discretization-invariant, and edge-preserving for both imaging and inverse problems governed by partial differential equations.

Parallel fully implicit two‐grid methods for distributed control of unsteady incompressible flows

SUMMARYIn this paper, we investigate some fully coupled parallel two‐grid Lagrange–Newton–Krylov–Schwarz algorithms for the suboptimal distributed control of unsteady incompressible flows governed by the Navier–Stokes equations. The algorithms include two major parts: a two‐grid Newton method for the nonlinear part of the problem and a two‐level Schwarz preconditioner for the linear part of the problem. Most of the existing approaches for distributed control problems are based on the so‐called reduced space method, which is easier to implement, but may have convergence issues in some situations. In the full space approach, we couple the state variables, the control variables, and the adjoint variables in a single large system of nonlinear equations. The coupled system is considerably more ill‐conditioned than its sub‐systems; however, with the powerful two‐grid approach, we are able to solve these difficult systems efficiently on large scale parallel computers. We show numerically that such an approach is scalable in the sense that the number of Newton iterations and the number of linear iterations are both nearly independent of the grid size, the number of processors, and the Reynolds numbers. We present numerical experiments for some suboptimal control problems obtained on supercomputers with more than two thousand processors. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Parallel Two-Grid Semismooth Newton-Krylov-Schwarz Method for Nonlinear Complementarity Problems

We develop scalable parallel domain decomposition algorithms for nonlinear complementarity problems including, for example, obstacle problems and free boundary value problems. Semismooth Newton is a popular approach for such problems, however, the method is not suitable for large scale calculations because the number of Newton iterations is not scalable with respect to the grid size; i.e., when the grid is refined, the number of Newton iterations often increases drastically. In this paper, we introduce a family of Newton-Krylov-Schwarz methods based on a smoothed grid sequencing method, a semismooth inexact Newton method, and a two-grid restricted overlapping Schwarz preconditioner. We show numerically that such an approach is totally scalable in the sense that the number of Newton iterations and the number of linear iterations are both nearly independent of the grid size and the number of processors. In addition, the method is not sensitive to the sharp discontinuity often associated with obstacle problems. We present numerical results for several large scale calculations obtained on machines with hundreds of processors.

IMPROVING STABILITY IN THE TIME-STEPPING ANALYSIS OF STRUCTURAL NONLINEAR DYNAMICS

A change in the representation of discrete motion equations for nonlinear structural dynamics of two-dimensional bodies is developed. The objective is to write the motion equation in a less nonlinear form. This leads to a significant increase in the range of stability of the time integration process and a reduction in the number of Newton iterations required in the time integration step.

Changing the representation and improving stability in time-stepping analysis of structural non-linear dynamics

In the present paper, we propose a change in the representation of the discrete motion equations in structural non-linear dynamics to obtain an improvement in the stability of time numerical integrations. In particular, natural local state variables are indicated for a finite-element approach to beam problems. The results, relative to Newmark approximations for the variations in the displacement and velocity vectors, show a significant increase in the range of stability of the time-integration process and a reduction in the number of Newton iterations required in the time-integration steps. It is also shown that the proposed approach inherently obeys the law of conservation of energy.

Parallel iterative linear solvers for multistep Runge-Kutta methods

This paper deals with solving stiff systems of differential equations by implicit Multistep Runge-Kutta (MRK) methods. For this type of methods, nonlinear systems of dimension sd arise, where s is the number of Runge-Kutta stages and d the dimension of the problem. Applying a Newton process leads to linear systems of the same dimension, which can be very expensive to solve in practice. With a parallel iterative linear system solver, especially designed for MRK methods, we approximate these linear systems by s systems of dimension d, which can be solved in parallel on a computer with s processors. In terms of Jacobian evaluations and LU-decompositions, the k-step s-stage MRK applied with this technique is on s processors equally expensive as the widely used k-step Backward Differentiation Formula on 1 processor, whereas the stability properties are better than that of BDF. A simple implementation of both methods shows that, for the same number of Newton iterations, the accuracy delivered by the new method is higher than that of BDF.

Improvement in norm-reducing Newton methods for circuit simulation

The general convergence problems encountered when applying Newton's method to the circuit simulation domain are discussed. The authors identify and explore one source of difficulties for these methods and discuss a solution. The basic transconductance element, widely used to construct FET and bipolar transistor models, results in a pathological failure case for L/sub 2/-based norm-reducing methods due to the unidirectional nature between its input and output mode. Their particular solution retains the generic nature of norm-reducing methods but replaces the L/sub 2/-norm with a nonconsistent point of view. This norm determines which equations should converge first, prioritizes them, and guides the damping of the Newton updates accordingly. From a mathematical point of view, the steepest-descent direction in the Nu-norm is parallel to each Newton update at the iterate point and, therefore, allows more effective damping of the updates. The result has been an order-of-magnitude reduction in the number of Newton iterations. The performance of this norm on a series of high-electron-mobility transistor (HEMT) circuits is presented. The nonconsistency of the Nu-norm and its impact on global convergence properties for norm-reducing methods are discussed. >

An efficient algorithm reduces computing time in solving a system of stiff ordinary differential equations

The problem of stiff differential equations arises in many computer-aided design techniques, particularly in the transient analysis of network simulation. Special multistep methods are used to solve the first-order stiff nonlinear differential equations. Instability and a large number of steps are encountered during simulation. Different techniques such as step and order selection schemes, procedures for changing step and order, may reduce the number of steps while preserving stability. An improved algorithm is presented using BDF formulas given by Brayton et at. and leads to reducing computer time by controlling the number of integration steps, but also the number of Newton iterations, the number of Jacobian matrix evaluations and other parameters, without producing additional errors or instability phenomena. Experimental results are shown.

A general mesh independence principle for Newton's method applied to second order boundary value problems

Recent work has established that for certain classes of nonlinear boundary value problems, the number of Newton iterations applied to the related standard discrete problem for a given tolerance is independent of the mesh size when the mesh is sufficiently fine. This paper develops an extension of the mesh independence principle by relaxing the assumption on the differential equation, its boundary conditions, and the related difference approximation.

Optimal Starting Approximations for Newton's Method

Various writers have dealt with the subject of optimal starting approxi- mations for square-root calculation by Newton's method. Three optimality criteria that have been used can be shown to lead to closely related approximations. This fact makes it surprisingly easy to choose a starting approximation of some pre- scribed form so that the maximum relative error after any number of Newton iterations is as small as possible. U 1. Introduction. The choice of polynomial and rational starting approximations for square-root calculation by Newton's method has been the subject of various investigations (e.g., (1)-(5)). These approach the problem from several different points of view. In this paper we will show how approximations obtained from these different viewpoints are related and how some of them can be derived from others. The problem of evaluating V/x for any x > 0 is easily reduced to the problem of evaluating V/x for x in some closed interval (a, b) such that 0 < a < b. Here (a, b) depends on the radix of the floating-point number system of the computer to be used; typical possibilities are (1/16, 1) and (-, 2). The following procedure is used to compute an approximate value for VIx. Using a polynomial or rational approxima- tion f(x) to V/x, valid in (a, b), compute a starting value yo = f(x) and then obtain Y1, Y22 .2 Yn by means of the relation Yk+1 = 12 (Yk + Xl8k), k = O. 1, * ,n - 1 . Then Yn V/x. It is customary not to test for convergence, since the number of iterations required in practice is quite small. Instead, the number of iterations n is

Sterbenz, P. H

Various writers have dealt with the subject of optimal starting approximations for square-root calculation by Newton’s method. Three optimality criteria that have been used can be shown to lead to closely related approximations. This fact makes it surprisingly easy to choose a starting approximation of some prescribed form so that the maximum relative error after any number of Newton iterations is as small as possible.