An accelerated Newton algorithm based on Krylov subspaces is applied to solving nonlinear equations of structural equilibrium. The algorithm uses a low-rank least-squares analysis to advance the search for equilibrium at the degrees of freedom DOFs where the largest changes in structural state occur; then it corrects for smaller changes at the remaining DOFs using a modified Newton computation....

An optimization technique for designing cosine-modulated filter banks using the frequency-response masking approach is proposed. In the given method, we perform minimization in the least-squares and minimax senses, subject to direct and aliasing transfer-function constraints. For optimization, a quasi-Newton algorithm with line search is used, based on sequential quadratic programming. Simplifi...

We describe regularization tools for training large-scale artiicial feed-forward neural networks. We propose algorithms that explicitly use a sequence of Tikhonov regularized nonlinear least squares problems. For large-scale problems, methods using new special purpose automatic diierentiation are used in a conjugate gradient method for computing a truncated Gauss-Newton search direction. The al...

A new fully-variational approach is studied for elliptic grid generation (EGG). It is based on a general algorithm developed in a companion paper [10] that involves using Newton’s method to linearize an appropriate equivalent first-order system, first-order system least squares (FOSLS) to formulate and discretize the Newton step, and algebraic multigrid (AMG) to solve the resulting matrix equat...

Stability analysis algorithms coupled with a robust steady state solver are used to understand the behavior of the 2D model problem of thermal convection in a 8 : 1 differentially heated cavity. The system is discretized using a Galerkin=Least Squares Finite Element formulation, and solved to steady state on parallel computers using a fully coupled Newton method and iterative linear solvers. An...

Many iterative methods that are used to solve Ax = b can be derived as quasi-Newton methods for minimizing the quadratic function 1 2 xAAx−xAb. In this paper, several such methods are considered, including conjugate gradient least squares (CGLS), Barzilai-Borwein (BB), residual norm steepest descent (RNSD) and Landweber (LW). Regularization properties of these methods are studied by analyzing t...

We report on progress in developing a magnetotelluric inversion method for two-dimensional anisotropic conductivity distribution. A standard two-dimensional model is discretized into a number of rectangular cells, each with a constant conductivity tensor, and the solution of the inverse problem is obtained by minimizing a global objective functional consisting of data misfit, a structural const...

We consider methods for regularising the least-squares solution of the linear system Ax = b. In particular, we propose iterative methods for solving large problems in which a trust-region bound ‖x‖ ≤ ∆ is imposed on the size of the solution, and in which the least value of linear combinations of ‖Ax−b‖q2 and a regularisation term ‖x‖ p 2 for various p and q = 1, 2 is sought. In each case, one o...

A nonlinear additive Schwarz preconditioned inexact Newton method (ASPIN) was introduced recently for solving large sparse highly nonlinear system of equations obtained from the discretization of nonlinear partial differential equations. In this paper, we discuss some extensions of ASPIN for solving the steady-state incompressible Navier-Stokes equations with high Reynolds numbers in the veloci...

We discuss a nite element method based on the mixed least-squares formulation. The cost functional turns out to be a polynomial so that its gradient and Hessian can be computed eeciently. A multi-level Newton iteration is introduced for minimizing the cost functional that can converge from a rough initial guess. Error estimates are derived which not only are optimal in certain connguration, but...