An Arrow-Hurwicz-Uzawa type flow as least squares solver for network linear equations

We study the approach to obtaining least squares solutions to systems of linear algebraic equations over networks by using distributed algorithms. Each node has access to one of the linear equations and holds a dynamic state. The aim for the node states is to reach a consensus as a least squares solution of the linear equations by exchanging their states with neighbors over an underlying interaction graph. A continuous-time distributed least squares solver over networks is developed in the form of the famous Arrow–Hurwicz–Uzawa flow. A necessary and sufficient condition is established on the graph Laplacian for the continuous-time distributed algorithm to give the least squares solution in the limit, with an exponentially fast convergence rate. The feasibility of different fundamental graphs is discussed including path graph and random graph. Moreover, a discrete-time distributed algorithm is developed by Euler’s method, converging exponentially to the least squares solution at the node states with suitable step size and graph conditions.This work was supported by the DAAD with funds of the German Federal Ministry of Education and Research (BMBF), by the Australian Research Council (ARC) under grants DP-130103610 and DP-160104500

A nonlinear weighted least-squares finite element method for Stokes equations

AbstractThe paper concerns a nonlinear weighted least-squares finite element method for the solutions of the incompressible Stokes equations based on the application of the least-squares minimization principle to an equivalent first order velocity–pressure–stress system. Model problem considered is the flow in a planar channel. The least-squares functional involves the L2-norms of the residuals of each equation multiplied by a nonlinear weighting function and mesh dependent weights. Using linear approximations for all variables, by properly adjusting the importance of the mass conservation equation and a carefully chosen nonlinear weighting function, the least-squares solutions exhibit optimal L2-norm error convergence in all unknowns. Numerical solutions of the flow pass through a 4 to 1 contraction channel will also be considered

MINRES-QLP: a Krylov subspace method for indefinite or singular symmetric systems

CG, SYMMLQ, and MINRES are Krylov subspace methods for solving symmetric systems of linear equations. When these methods are applied to an incompatible system (that is, a singular symmetric least-squares problem), CG could break down and SYMMLQ's solution could explode, while MINRES would give a least-squares solution but not necessarily the minimum-length (pseudoinverse) solution. This understanding motivates us to design a MINRES-like algorithm to compute minimum-length solutions to singular symmetric systems. MINRES uses QR factors of the tridiagonal matrix from the Lanczos process (where R is upper-tridiagonal). MINRES-QLP uses a QLP decomposition (where rotations on the right reduce R to lower-tridiagonal form). On ill-conditioned systems (singular or not), MINRES-QLP can give more accurate solutions than MINRES. We derive preconditioned MINRES-QLP, new stopping rules, and better estimates of the solution and residual norms, the matrix norm, and the condition number.Comment: 26 pages, 6 figure

Algorithm for linear least squares problems with equality and nonnegativity constraints

A new algorithm for solving a linear least-squares problem with linear constraints is presented. The constraints can be equality constraint equations and nonnegativity constraints on selected variables. This problem, while appearing to be quite special, is the core problem arising in the solution of the general linearly constrained linear least-squares problem. The reduction process of the general problem to the core problem can be done in many ways. Three such techniques are discussed. The method employed for solving the core problem is based on combining the equality constraints with differentially weighted least-squares equations to form an augmented least-squares system. This weighted least-squares system is solved with nonnegativity constraints on selected variables. Seven small examples, including a constrained least-squares curve fitting example, are presented. A reference to user instructions for subprograms to compute solutions of constrained least-squares problems is included. 3 figures, 9 tables

L-structure least squares solutions of reduced biquaternion matrix equations with applications

This paper presents a framework for computing the structure-constrained least squares solutions to the generalized reduced biquaternion matrix equations (RBMEs). The investigation focuses on three different matrix equations: a linear matrix equation with multiple unknown L-structures, a linear matrix equation with one unknown L-structure, and the general coupled linear matrix equations with one unknown L-structure. Our approach leverages the complex representation of reduced biquaternion matrices. To showcase the versatility of the developed framework, we utilize it to find structure-constrained solutions for complex and real matrix equations, broadening its applicability to various inverse problems. Specifically, we explore its utility in addressing partially described inverse eigenvalue problems (PDIEPs) and generalized PDIEPs. Our study concludes with numerical examples.Comment: 30 page

A rational conjugate gradient method for linear ill-conditioned problems

We consider linear ill-conditioned operator equations in a Hilbert space setting. Motivated by the aggregation method, we consider approximate solutions constructed from linear combinations of Tikhonov regularization, which amounts to finding solutions in a rational Krylov space. By mixing these with usual Krylov spaces, we consider least-squares problem in these mixed rational spaces. Applying the Arnoldi method leads to a sparse, pentadiagonal representation of the forward operator, and we introduce the Lanczos method for solving the least-squares problem by factorizing this matrix. Finally, we present an equivalent conjugate-gradient-type method that does not rely on explicit orthogonalization but uses short-term recursions and Tikhonov regularization in each second step. We illustrate the convergence and regularization properties by some numerical examples

Singulärwert-Zerlegung und Methode der kleinsten Quadrate

The method of least squares is an important instrument to determine the optimal linear estimators in regression models. By means of the singular value decomposition we can find the least squares estimators without differentiation, without solving the normal equations and without assumptions on the rank of the data matrix. Even in case of multicollinearity we can find the simple and natural solutions. The results in the paper are not new, they have been developed mainly in numerical publications, but they are hardly to be found in statistical textbooks