An iterative algorithm for solving ill-conditioned linear least squares problems

Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.

A Novel Low-Dimensional Method for Analytically Solving Partial Differential Equations

This paper is concerned with a low-dimensional dynamical system model for analytically solving partial differential equations(PDEs).The model proposed is based on a posterior optimal truncated weighted residue(POT-WR)method,by which an infinite dimensional PDE is optimally truncated and analytically solved in required condition of accuracy.To end that,a POT-WR condition for PDE under consideration is used as a dynamically optimal control criterion with the solving process.A set of bases needs to be constructed without any reference database in order to establish a space to describe low-dimensional dynamical system that is required.The Lagrangian multiplier is introduced to release the constraints due to the Galerkin projection,and a penalty function is also employed to remove the orthogonal constraints.According to the extreme principle,a set of ordinary differential equations is thus obtained by taking the variational operation of the generalized optimal function.A conjugate gradient algorithm by FORTRAN code is developed to solve the ordinary differential equations.The two examples of one-dimensional heat transfer equation and nonlinear Burgers’equation show that the analytical results on the method proposed are good agreement with the numerical simulations and analytical solutions in references,and the dominant characteristics of the dynamics are well captured in case of few bases used only.