The Gauss-Newton algorithm is an iterative method regularly used for solving nonlinear least squares problems. It is particularly well-suited to the treatment of very large scale variational data assimilation problems that arise in atmosphere and ocean forecasting. The procedure consists of a sequence of linear least squares approximations to the nonlinear problem, each of which is solved by an...

The Gauss–Newton algorithm is an iterative method regularly used for solving nonlinear least squares problems. It is particularly well suited to the treatment of very large scale variational data assimilation problems that arise in atmosphere and ocean forecasting. The procedure consists of a sequence of linear least squares approximations to the nonlinear problem, each of which is solved by an...

We describe stochastic Newton and stochastic quasi-Newton approaches to efficiently solve large linear least-squares problems where the very large data sets present a significant computational burden (e.g., the size may exceed computer memory or data are collected in real-time). In our proposed framework, stochasticity is introduced in two different frameworks as a means to overcome these compu...

Least squares methods are effective for solving systems of partial differential equations. In the case of nonlinear systems the equations are usually linearized by a Newton iteration or successive substitution method, and then treated as a linear least squares problem. We show that it is often advantageous to form a sum of squared residuals first, and then compute a zero of the gradient with a ...

An algorithm for finding sums of hermitian squares decompositions for polynomials in noncommuting variables is presented. The algorithm is based on the “Newton chip method”, a noncommutative analog of the classical Newton polytope method, and semidefinite programming.

We present a structured algorithm for solving constrained nonlinear least squares problems, and establish its local two-step Q-superlinear convergence. The approach is based on an adaptive structured scheme due to Mahdavi-Amiri and Bartels of the exact penalty method of Coleman and Conn for nonlinearly constrained optimization problems. The structured adaptation also makes use of the ideas of N...

This paper introduces tensor methods for solving large, sparse nonlinear least squares problems where the Jacobian either is analytically available or is computed by nite diier-ence approximations. Tensor methods have been shown to have very good computational performance for small to medium-sized, dense nonlinear least squares problems. In this paper we consider the application of tensor metho...

The smoothness-constrained least-squares method is widely used for two-dimensional (2D) and three-dimensional (3D) inversion of apparent resistivity data sets. The Gauss–Newton method that recalculates the Jacobian matrix of partial derivatives for all iterations is commonly used to solve the least-squares equation. The quasi-Newton method has also been used to reduce the computer time. In this...