Damped Newton Method Research Articles

A Unified Algorithm for Channel Imbalance and Antenna Phase Center Position Calibration of a Single-Pass Multi-Baseline TomoSAR System

The multi-baseline synthetic aperture radar (SAR) tomography (TomoSAR) system is employed in such applications as disaster remote sensing, urban 3-D reconstruction, and forest carbon storage estimation. This is because of its 3-D imaging capability in a single-pass platform. However, a high 3-D resolution of TomoSAR is based on the premise that the channel imbalance and antenna phase center (APC) position are precisely known. If this is not the case, the 3-D resolution performance will be seriously degraded. In this paper, a unified algorithm for channel imbalance and APC position calibration of a single-pass multi-baseline TomoSAR system is proposed. Based on the maximum likelihood method, as well as the least squares and the damped Newton method, we can calibrate the channel imbalance and APC position. The algorithm is suitable for near-field conditions, and no phase unwrapping operation is required. The effectiveness of the proposed algorithm has been verified by simulation and experimental results.

Integration of voltage dependent power injections of distributed generators into the power flow by using a damped Newton method

Voltage dependent active and reactive power injections are used to improve the grid integration of distributed generators, such as photovoltaic systems. In state-of-the-art literature these power injections are considered during power flow calculations via an external loop, which requires to perform multiple (up to ten or more) power flow calculations in order to obtain one static operation point. In this paper it is shown how to integrate voltage dependent power injections into the nonlinear grid equations in order to solve for a static working point by means of performing one Newton power flow calculation. That makes the external loop unnecessary and save computational time. Furthermore, it is shown for a two-bus system that the integration of voltage dependent power injections into the nonlinear grid equations can exhibit ill-conditioned power mismatch functions. Therefore, a damped Newton method is used in order to avoid numerical oscillations. Simulations of the IEEE 118-bus test system and a real German 234-bus test system show that the damped method converges faster and is computationally more efficient than the external approach currently used in literature and state-of-the-art simulation tools in power systems. In addition, the implemented power flow algorithm is validated in the laboratory for a two-bus system and a numerical example of a low-voltage feeder is given.

A damped Newton variational inversion method for SAR wind retrieval

AbstractThe variational inversion for synthetic aperture radar (SAR) wind retrieval can take all sources' errors into account, but its iterative computation is very time consuming. For the wind vectors, (u, v) components are commonly used for variational inversion, but they are not intuitive for practical applications. In this paper, we modify the decomposition of wind vectors in the cost function into wind speed and wind direction and adopt the damped Newton method (DNVAR) to solve the cost function. Experimental results on simulated data show that DNVAR can effectively reduce background wind vector errors. Additionally, the average number of iterations is reduced drastically compared to prior arts. Furthermore, a detailed comparison between direct SAR wind retrieval (DIRECT) and DNVAR is performed. Simulations reveal that the DNVAR errors are smaller than the background wind vector errors in all considered cases. Thus, DNVAR could be employed to retrieve SAR sea surface wind. For practical applications, when the background wind speed is within moderate and high wind speed range, DNVAR has higher accuracy and is thus preferred. Otherwise, both DNVAR and DIRECT are feasible, considering the unknown actual errors of both background wind vectors and geophysical model function. Experimental results on Envisat/advanced synthetic aperture radar data show that the wind speed accuracy of DIRECT is largely affected by the background wind direction errors, but DNVAR can reduce the wind direction errors with minor effect on the wind speed errors in comparison with the background wind errors.

Randomized Block Proximal Damped Newton Method for Composite Self-Concordant Minimization

In this paper we consider the composite self-concordant (CSC) minimization problem, which minimizes the sum of a self-concordant function $f$ and a (possibly nonsmooth) proper closed convex function $g$. The CSC minimization is the cornerstone of the path-following interior point methods for solving a broad class of convex optimization problems. It has also found numerous applications in machine learning. The proximal damped Newton (PDN) methods have been well studied in the literature for solving this problem that enjoy a nice iteration complexity. Given that at each iteration these methods typically require evaluating or accessing the Hessian of $f$ and also need to solve a proximal Newton subproblem, the cost per iteration can be prohibitively high when applied to large-scale problems. Inspired by the recent success of block coordinate descent methods, we propose a randomized block proximal damped Newton (RBPDN) method for solving the CSC minimization. Compared to the PDN methods, the computational cos...

Numerical simulation of the effects of coating on thermal elastohydrodynamic lubrication in cam/tappet contact

Investigating the effects of coating on cam/tappet thermal elastohydrodynamic lubrication through numerical simulation has great significance in the design of coated cam/tappet conjunctions. This paper presents a numerical model for the prediction of the thermal elastohydrodynamic lubrication of a coated cam/tappet and the results of a study of the effects of coating parameters on cam/tappet lubrication performance. In the model, the Reynolds equation is solved by the damped Newton method to obtain the pressure distribution, and energy equations are used to obtain the temperature distribution. The total elastic deformation is calculated by the finite element method. The effects of the coating’s mechanical properties on pressure and temperature were found to be significant, as were the effects of the coating’s thermal properties on temperature. These effects were found to increase with increasing coating thickness. A soft coating with low thermal inertia has the greatest ability to reduce friction loss, and the higher the inlet temperature is, the lower the friction loss is. The influence of coating of both the cam and tappet on friction loss is greater than the effect of coating of the tappet only, which is greater than the effect of coating of the cam only.

An affine covariant composite step method for optimization with PDEs as equality constraints

We propose a composite step method, designed for equality constrained optimization with partial differential equations. Focus is laid on the construction of a globalization scheme, which is based on cubic regularization of the objective and an affine covariant damped Newton method for feasibility. We show finite termination of the inner loop and fast local convergence of the algorithm. Numerical results are shown for optimal control problems subject to a nonlinear heat equation.

A semismooth Newton method for tensor eigenvalue complementarity problem

In this paper, we consider the tensor eigenvalue complementarity problem which is closely related to the optimality conditions for polynomial optimization, as well as a class of differential inclusions with nonconvex processes. By introducing an NCP-function, we reformulate the tensor eigenvalue complementarity problem as a system of nonlinear equations. We show that this function is strongly semismooth but not differentiable, in which case the classical smooth methods cannot apply. Furthermore, we propose a damped semismooth Newton method for tensor eigenvalue complementarity problem. A new procedure to evaluate an element of the generalized Jacobian is given, which turns out to be an element of the B-subdifferential under mild assumptions. As a result, the convergence of the damped semismooth Newton method is guaranteed by existing results. The numerical experiments also show that our method is efficient and promising.

Implicit finite difference solution for the magneto-hydro-dynamic unsteady free convective flow and heat transfer of a third-grade fluid past a porous vertical plate

Magneto-hydro-dynamic (MHD) unsteady flow and heat transfer of a third grade fluid passing an infinite vertical porous plate with uniform suction applied at the plate has been studied. The governing nonlinear partial differential equations are reduced to a system of nonlinear algebraic equations using implicit finite difference schemes and then the system is solved using damped-Newton method. The effect of the various physical parameters on the velocity and temperature are studied. One of the important observations is that when the third order elastic parameter increases the velocity decreases, whereas an opposite effect is observed when second order elastic parameter increases. Present method is valid for all values of elastic parameters where as the earlier approximation methods like perturbation and power series methods are valid for small values of elastic parameter. Further the advantage of our method is that it does not require repeated derivation for every change in the boundary conditions.

Generalized Newton’s method for solving nonlinear and nondifferentiable algebraic systems

In this paper a model based on non-smooth equations is proposed for solving a non-linear and non-differential equation obtained by discretization of a quasi-variational inequality that models the frictional contact problem. The main aim of this paper is to show that the Newton method based on the plenary hull of the Clarke generalized Jacobians (the non-smooth damped Newton method) can be implemented for solving Lipschitz non-smooth equation.

Continuation Newton methods

Severely nonlinear problems can only be solved by some homotopy continuation method. An example of a homotopy method is the continuous Newton method which, however, must be discretized which leads to the damped step version of Newton’s method.The standard Newton iteration method for solving systems of nonlinear equations F(u)=0 must be modified in order to get global convergence, i.e. convergence from any initial point. The control of steplengths in the damped step Newton method can lead to many small steps and slow convergence. Furthermore, the applicability of the method is restricted in as much as it assumes a nonsingular and everywhere differentiable mapping F(⋅).Classical continuation methods are surveyed. Then a new method in the form of a coupled Newton and load increment method is presented and shown to have a global convergence already from the start and second order of accuracy with respect to the load increment step and with less restrictive regularity assumptions than for the standard Newton method. The method is applied for an elastoplastic problem with hardening.

A mathematical programming approach for frictional contact problems with the extended finite element method

The extended finite element method (XFEM) is a numerical method for modeling discontinuities within the classical finite element framework. This method can treat arbitrary cracks independent of the mesh and crack growth with minimal remeshing. In this paper, a nonlinear complementarity model for frictional elastic contact problems with the XFEM is presented. The conditions that describe frictional contact are formulated as a system of non-smooth equations based on variational inequality theory, and the non-smooth damped Newton method is given based on the definitions of generalized derivative to directly solve the system of equations. Finally, numerical examples are presented to demonstrate the correctness and effectiveness of this method.

An implicit stabilized finite element method for the compressible Navier–Stokes equations using finite calculus

A new implicit stabilized formulation for the numerical solution of the compressible Navier–Stokes equations is presented. The method is based on the finite calculus (FIC) scheme using the Galerkin finite element method (FEM) on triangular grids. Via the FIC formulation, two stabilization terms, called streamline term and transverse term, are added to the original conservation equations in the space-time domain. The non-linear system of equations resulting from the spatial discretization is solved implicitly using a damped Newton method benefiting from the exact Jacobian matrix. The matrix system is solved at each iteration with a preconditioned GMRES method. The efficiency of the proposed stabilization technique is checked out in the solution of 2D inviscid and laminar viscous flow problems where appropriate solutions are obtained especially near the boundary layer and shock waves. The work presented here can be considered as a follow up of a previous work of the authors Kouhi, Oñate (Int J Numer Methods Fluids 74:872–897, 2014). In that paper, the stabilized Galerkin FEM based on the FIC formulation was derived for the Euler equations together with an explicit scheme. In the present paper, the extension of this work to the Navier–Stokes equations using an implicit scheme is presented.

2-D stationary gas dynamics in a barred galaxy

Abstract A code for solving the 2-D isothermal Euler equations of gas dynamics in a rotating disc is presented. The gravitational potential represents a weak bar and controls the flow. A damped Newton method solves the second-order upwind discretisation of the equations for a steady-state solution, using a consistent linearisation and a direct solver. Successive grid refinement, starting from a finite-volume grid with 8 by 8 cells, is applied to find solutions on subsequently finer meshes. On coarser meshes, a first-order spatial discretisation is used. The method obtains quadratic convergence once the solution approaches the steady state. The initial search is quick with the first-order scheme and slower with the second-order discretisation, up to 256 by 256 cells. Beyond, with 512 by 512 cells, the number of iterations becomes too large to be of practical use. Potential causes are discussed. The code can be applied as a tool for generating flow models if used on not too fine meshes.

Global convergence of damped semismooth Newton methods for ℓ1 Tikhonov regularization

We are concerned with Tikhonov regularization of linear ill-posed problems with ℓ1 coefficient penalties. Griesse and Lorenz (2008 Inverse Problems 24 035007) proposed a semismooth Newton method for the efficient minimization of the corresponding Tikhonov functionals. In the class of high-precision solvers for such problems, semismooth Newton methods are particularly competitive due to their superlinear convergence properties and their ability to solve piecewise affine equations exactly within finitely many iterations. However, the convergence of semismooth Newton schemes is only local in general. In this work, we discuss the efficient globalization of B(ouligand)-semismooth Newton methods for ℓ1 Tikhonov regularization by means of damping strategies and suitable descent with respect to an associated merit functional. Numerical examples are provided which show that our method compares well with existing iterative, globally convergent approaches.

A damped semismooth Newton method for the Brugnano–Casulli piecewise linear system

The piecewise linear system is a nonsmooth but semismooth equation. In this paper, a damped semismooth Newton method is presented for solving a class of piecewise linear systems. Under appropriate conditions, both monotone convergence and finite termination properties are investigated for the proposed method.

Homotopy Method for Finding the Steady States of Oscillators

Shooting, finite difference, or harmonic balance techniques in conjunction with a damped Newton method are widely employed for the numerical calculation of limit cycles of (free-running, autonomous) oscillators. In some cases, however, nonconvergence occurs when the initial estimate of the solution is not close enough to the exact one. Generally, the higher the quality factor of the oscillator the tighter are the constraints for the initial estimate. A 2-D homotopy method is presented in this paper that overcomes this problem. The resulting linear set of equations is underdetermined, leading to a nullspace of rank two. This underdetermined system is solved in a least squares sense for which a rigorous mathematical basis can be derived. An efficient algorithm for solving the least squares problem is derived where sparse matrix techniques can be used. As continuation methods are only employed for obtaining a sufficient initial guess of the limit cycle, a coarse grid discretization is sufficient to make the method runtime efficient.

Viscosity Model Data Fitting for Polymer Melt Based on Golden Section Method

The viscosity models of polymer melts are adopted popularly in the numerical simulation of plastic injection molding.The data fitting to these models is an important task. Based on golden section method, a kind of fitting method is proposed in order to reduce the restriction on initial values and to eliminate ill-conditioned matrixes. A direct search strategy is applied, with which the derivatives of an objective function need not be calculated. The search process is executed by round, and each round of search is carried out following a group of orthogonal feasible directions in turn. Based on golden section method, a one-dimensional nonlinear search is performed at a parametric interval in each feafible direction, so that a group of stage optimal solutions are attained. The convergence of a search process is guaranteed by continuously shrinking of the intervals and nonincreasing of the objective function values. Better extreme points can be found more easily by this kind of search method that make a current optimal point as the center of a searching interval, and fitting precision is enhanced accordingly. Moreover, a set of simple functions for calculating initial values is supplied for Cross-Arrhenius model. These functions can be employed in all kinds of fitting algorithms to this model. The fitting example shows that this method allows a wide range of feasible initial values, and its fitting precision is higher than Damped Newton's Method and Genetic Algorithm. It can engage in second optimization to the results made by other fitting methods. This fitting process can be accomplished in several seconds with a microcomputer for this example.

A PRECONDITIONED ITERATIVE SOLUTION SCHEME FOR NONLINEAR PARABOLIC SYSTEMS ARISING IN AIR POLLUTION MODELING

A preconditioned iterative solution method is presented for nonlinear parabolic transport systems. The ingredients are implicit Euler discretization in time and finite element discretization in space, then an outer-inner (outer damped inexact Newton method with inner preconditioned conjugate gradient) iteration, further, as a main part, preconditioning via an l-tuple of independent elliptic operators. Numerical results show that the suggested method works properly for a test problem in air pollution modeling.

Solution of Complementarity Problems via Piecewise Linearization

AbstractMy master thesis concerns the solution linear complementarity problems (LCP). The Lemke algorithm, the most commonly used algorithm for solving a LCP until this day, was compared with the piecewise Newton method (PLN algorithm). The piecewise Newton method is an algorithm to solve a piecewise linear system on the basis of damped Newton methods. The linear complementarity problem is formulated as a piecewise linear system for the applicability of the PLN algorithm. Then, different application examples will be presented, solved with the PLN algorithm. As a result of the findings (of my master thesis) it can be assumed that – under the condition of coherent orientation – the PLN‐algorithm requires fewer iterations to solve a linear complementarity problem than the Lemke algorithm. The coherent orientation for piecewise linear problems corresponds for linear complementarity problems to the P‐matrix‐property. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An improved inversion-free method for solving the matrix equation [formula omitted

In this paper, based on the damped Newton method, we present an inversion-free iterative method with a stepsize parameter for solving the nonlinear matrix equation X+A∗X−αA=Q, where Q is a Hermitian positive definite matrix and α≥1. The proposed method is an extension of Algorithm 2.2 in [Z. Peng, S.M. El-Sayed, X. Zhang, Iterative methods for the extremal positive definite solution of the matrix equation X+A∗X−αA=Q, J. Comput. Appl. Math. 200 (2007) 520–527]. Preliminary numerical results show that the proposed method can improve numerical effect.