Inexact Newton Approach Research Articles

Self-consistent solution of cosmological radiation-hydrodynamics and chemical ionization

We consider a PDE system comprising compressible hydrodynamics, flux-limited diffusion radiation transport and chemical ionization kinetics in a cosmologically-expanding universe. Under an operator-split framework, the cosmological hydrodynamics equations are solved through the piecewise parabolic method, as implemented in the Enzo community hydrodynamics code. The remainder of the model, including radiation transport, chemical ionization kinetics, and gas energy feedback, form a stiff coupled PDE system, which we solve using a fully-implicit inexact Newton approach, and which forms the crux of this paper. The inner linear Newton systems are solved using a Schur complement formulation, and employ a multigrid-preconditioned conjugate gradient solver for the inner Schur systems. We describe this approach and provide results on a suite of test problems, demonstrating its accuracy, robustness, and scalability to very large problems.

Newton-Krylov Algorithm for Aerodynamic Design Using the Navier-Stokes Equations

A Newton‐Krylov algorithm is presented for two-dimensional Navier‐Stokes aerodynamic shape optimization problems. The algorithm is applied to both the discrete-adjoint and the discrete e ow-sensitivity methods for calculating the gradient of the objective function. The adjoint and e ow-sensitivity equations are solved using a novel preconditioned generalized minimum residual (GMRES)strategy. Together with a complete linearization of the discretized Navier‐Stokes and turbulence model equations, this results in an accurate and efecient evaluation of the gradient. Furthermore, fast e ow solutions are obtained using the same preconditioned GMRES strategy in conjunction with an inexact Newton approach. The performance of the new algorithm is demonstrated for several design examples,includinginversedesign,lift-constraineddragminimization, liftenhancement, and maximization of lift-to-dragratio. In all examples, the normof the gradientisreduced by several ordersof magnitude, indicating that alocalminimumhasbeen obtained. Bytheuseoftheadjoint method,thegradient isobtained infromone-e fth to one-half of the time required to converge a eow solution.

A New SQP Algorithm for Large-Scale Nonlinear Programming

An efficient new SQP algorithm capable of solving large-scale problems is described. It generates descent directions for an $\ell_1$ plus log-barrier merit function and uses a line-search to obtain a sufficient decrease of this function. The unmodified exact Hessian matrix of the Lagrangian function is normally used in the QP subproblem, but this is set to zero if it fails to yield a descent direction for the merit function. The QP problem is solved by an interior-point method using an inexact Newton approach, iterating to an accuracy just sufficient to produce a descent direction in the early stages and tightening the accuracy as we approach a solution. We prove finite termination of the algorithm, at an $\epsilon$-optimal Fritz-John point if feasibility is attained. We also show that if any iterate is close enough to an isolated connected subset of local minimizers, then the iterates converge to this subset. The rate of convergence is Q-quadratic if the subset is an isolated minimizer which satisfies a second-order sufficiency condition, but Q-quadratic convergence to an $\epsilon$-optimal point can still be achieved without any conditions beyond Lipschitz continuity of second-order derivatives. The implementation SQPIPM is designed for problems with many degrees of freedom and is shown to perform well compared with other codes on a range of standard problems.

Analysis of large nonlinearly loaded antenna arrays under multitone excitation

In this paper, the responses of large nonlinearly loaded antenna arrays under multitone excitation are analyzed using the inexact Newton approach and infinite periodic structure Green's function. Numerical operations are shown to be very efficient. The analysis is suitable for large, nonlinearly loaded antenna arrays under multitone excitation. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 25: 319–323, 2000.

Two efficient algorithms for the analyses of a nonlinearly loaded antenna and antenna array in the frequency domain

In this paper, two efficient algorithms for the steady-state scattering responses of a nonlinearly loaded antenna and antenna array in the frequency domain are studied. One is the reflection algorithm (RA), and the other is the inexact Newton approach (INA). The analysis is equivalent to a nonlinear circuit problem with the circuit parameters of antenna element obtained by the moment method. This equivalent nonlinear circuit is then solved using the RA or the INA. In general, the RA is suitable to strongly nonlinearly circuit element. While the INA is suitable for multifrequency excitation. In dealing with the nonlinearly loaded antenna array, the infinite periodic structure Green's function, including the array mutual coupling effects, is applied to simplify the analysis. Both of the two algorithms are easy to be combined with the periodic Green's function to deal with the large nonlinearly loaded antenna array problem.

Fast and robust inexact Newton approach to the harmonic-balance analysis of nonlinear microwave circuits

The authors discuss a novel approach to nonlinear microwave circuit simulation by the harmonic-balance (HE) technique. The nonlinear system is solved by an inexact Newton method, and the GMRES iteration is used at each step to find a suitable inexact Newton update. The peculiar structure of the Jacobian matrix allows the basis vectors of the Krylov subspace to be computed mostly by the FFT. The resulting simulation tool is fast and robust, and outperforms conventional HE techniques when applied to large-size nonlinear analysis problems.

A theory of secant preconditioners

In this paper we analyze the use of structured quasi-Newton formulae as preconditioners of iterative linear methods when the inexact-Newton approach is employed for solving nonlinear systems of equations. We prove that superlinear convergence and bounded work per iteration is obtained if the preconditioners satisfy a Dennis-Moré condition. We develop a theory of Least-Change Secant Update preconditioners and we present an application concerning a structured BFGS preconditioner.

On the Successive Projections Approach to Least-Squares Problems

In this paper, we suggest a generalized Gauss–Seidel approach to sparse linear and nonlinear least-squares problems. The algorithm, closely related to one given by Elfving (1980), uses the work of Curtis, Powell, and Reid (1974) as extended by Coleman and Moré (1983) to divide the variables into nondisjoint groups of structurally orthogonal columns and then projects the updated residual into each column subspace of the Jacobian in turn. In the linear case, this procedure can be viewed as an alternate ordering of the variables in the Gauss–Seidel method. Preliminary tests indicate that this leads quickly to cheap solutions of limited accuracy for linear problems, and that this approach is promising for an inexact Gauss–Newton analog of the inexact Newton approach of Dembo, Eisenstat, and Steihaug (1982).