Least elements revisited

Least element theory is extended, for linear problems, to general multiple-objective problems, and the pre-Leontief matrix and left-hand matrix inverse notions are generalized.This work was completed while the author was visiting the Center for Advanced Studies and the Department of Systems Engineering at the University of Virginia. The author is indebted to the referees for their helpful comments.     

Least squares with a quadratic constraint

Summary We present the theory of the linear least squares problem with a quadratic constraint. New theorems characterizing properties of the solutions are given. A numerical application is discussed.     

Numerical solution of the Fokker Planck equation using moving finite elements

Numerical solution of the Fokker Planck equation for the probability density function of a stochastic process by traditional finite difference or finite element methods produces erroneous oscillations and negative values whenever the drift is large compared to the diffusion. Upwinding schemes to eliminate the oscillations introduce false numerical diffusion because it is impossible to make the one step drift large enough to match the original equation without making the one step diffusion too large. A variation of the moving finite element method is presented that overcomes these difficulties by using basis functions that satisfy the drift part of the equation by moving along the trajectories of the deterministic dynamical system associated with the stochastic process. A Galerkin type method can then be used to find the coefficients in the remaining pure diffusion equation. Solutions of two test equations are presented to illustrate the effectiveness of the method.     

Least squares data fitting with implicit functions

This paper discusses the computational problem of fitting data by an implicitly defined function depending on several parameters. The emphasis is on the technique of algebraic fitting off(x, y; p) = 0 which can be treated as a linear problem when the parameters appear linearly. Various constraints completing the problem are examined for their effectiveness and in particular for two applications: fitting ellipses and functions defined by the Lotka-Volterra model equations. Finally, we discuss geometric fitting as an alternative, and give examples comparing results.     

Least squares and approximate equidistribution in multidimensions

In this article it is shown that, under a natural condition, least squares minimization of the residual of the divergence of a vector field is equivalent to that of a least squares measure of equidistribution of the residual. More specifically, consider the conservation law div f = 0, when the vector field f is approximated by a conforming piecewise differentiable function F on a partition of a polygonal region Ω into triangles. Then, we show that, if F has a prescribed flux across the outer boundary ∂Ω of Ω, minimization of the l₂ norm of the average residual of div F over all internal parameters of the partition (including nodal positions as well as solution amplitudes) is equivalent to minimization of the l₂ norm of the differences in the average residuals of F , taken over all pairs of triangles of the partition. The result is of importance in the approximate solution of conservation laws, where alignment of the mesh is often of considerable benefit in deriving extra accuracy. The property is readily extended to systems of conservation laws. Moreover it holds for the average vorticity residual of F over a triangle as well as for l₂‐type norms combining both the divergence and the vorticity (as in the case of the Cauchy‐Riemann equations). © 1999 Wiley & Sons. Inc. Numer Methods Partial Differential Eq 15:605–615, 1999     

Normal and tangential continuous elements for least‐squares mixed finite element methods

We consider a finite element discretization of the primal first‐order least‐squares mixed formulation of the second‐order elliptic problem. The unknown variables are displacement and flux, which are approximated by equal‐order elements of the usual continuous element and the normal continuous element, respectively. We show that the error bounds for all variables are optimal. In addition, a field‐based least‐squares finite element method is proposed for the 3D‐magnetostatic problem, where both magnetic field and magnetic flux are taken as two independent variables which are approximated by the tangential continuous and the normal continuous elements, respectively. Coerciveness and optimal error bounds are obtained. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

Least squares based iterative parameter estimation algorithm for multivariable controlled ARMA system modelling with finite measurement data

Difficulties of identification for multivariable controlled autoregressive moving average (ARMA) systems lie in that there exist unknown noise terms in the information vector, and the iterative identification can be used for the system with unknown terms in the information vector. By means of the hierarchical identification principle, those noise terms in the information vector are replaced with the estimated residuals and a least squares based iterative algorithm is proposed for multivariable controlled ARMA systems. The simulation results indicate that the proposed algorithm is effective.     

Aggregation of microglia in 2D with string gradient weighted moving finite elements

Alzheimer's disease (AD) is a severe neurodegenerative disorder characterised by cognitive impairment and dementia. In the AD‐affected brain, microglia cells are up‐regulated and accumulate at senile plaques, the most prominent pathological feature of AD. In order to further study and predict the movement of activated microglia, we utilised their chemotactic properties. Specifically, we formulated the string gradient weighted moving finite element method for a system of partial differential equations in two dimensions, which includes nonlinear diffusion of a different variable found in chemotaxis models. The method was applied successfully to solve highly nonlinear chemorepulsion–chemorepellent models in two dimensions, and the results were compared with one‐dimensional results found previously in the literature. We conclude that the string gradient weighted moving finite element method is easily applied to chemotaxis models, in particular movement and aggregation of microglia, resulting in the ability to study the models extended in two dimensions efficiently. Our study highlights the feasibility and power of mathematical modelling to advance our understanding of pathophysiological processes in neurodegenerative diseases, including AD. Copyright © 2012 John Wiley & Sons, Ltd.     

Least squares methods for Volterra equations and generalizations

In this article least squares approximations to Volterra integral equations are considered, both with exact integration and with quadrature. Optimal error estimates are derived, and it is shown that the same order of convergence is obtained in both cases with only modest requirements on the quadrature rule used in the latter. The most important practical setting for least squares is the case of convolution kernels, and these are also studied in this article. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 679–693, 1998     

Least squares completions for nonlinear differential algebraic equations

Summary A method has been proposed for numerically solving lower dimensional, nonlinear, higher index differential algebraic equations for which more classical methods such as backward differentiation or implicit Runge-Kutta may not be appropriate. This method is based on solving nonlinear DAE derivative arrays using nonlinear singular least squares methods. The theoretical foundations, generality, and limitations of this approach remain to be determined. This paper carefully examines several key aspects of this approach. The emphasis is on general results rather than specific results based on the structure of various applications.Research supported in part by the U.S. Army Research Office under DAALO3-89-D-0003 and the National Science Foundation under ECS-9012909 and DMS-9122745     

Least squares method for statistical analysis of polyrhythmics

The least squares estimate methods for estimation of characteristics of almost periodically correlated random processes as mathematical models of stochastic oscillations with polyrhythmical structure are considered. The bias and variance of mean and correlation function estimates are analyzed. Asymptotic formulas for them are deduced.     

Least squares approximation by splines with free knots

Suppose we are given noisy data which are considered to be perturbed values of a smooth, univariate function. In order to approximate these data in the least squares sense, a linear combination of B-splines is used where the tradeoff between smoothness and closeness of the fit is controlled by a smoothing term which regularizes the least squares problem and guarantees unique solvability independent of the position of knots. Moreover, a subset of the knot sequence which defines the B-splines, the so-calledfree knots, is included in the optimization process.The resulting constrained least squares problem which is linear in the spline coefficients but nonlinear in the free knots is reduced to a problem that has only the free knots as variables. The reduced problem is solved by a generalized Gauss-Newton method. The method developed can be combined with a knot removal strategy in order to obtain an approximating spline with as few parameters as possible.Dedicated to Professor Dr.-Ing. habil. Dr. h.c. Helmut Heinrich on the occasion of his 90th birthdayResearch of the second author was partly supported by Deutsche Forschungsgemeinschaft under grant Schm 968/2-1.     

Least trimmed squares regression,least median squares regression,and mathematical programming

In this paper, we study LTS and LMS regression, two high breakdown regression estimators, from an optimization point of view. We show that LTS regression is a nonlinear optimization problem that can be treated as a concave minimization problem over a polytope. We derive several important properties of the corresponding objective function that can be used to obtain algorithms for the exact solution of LTS regression problems, i.e., to find a global optimum to the problem. Because of today's limited problem-solving capabilities in exact concave minimization, we give an easy-to-implement pivoting algorithm to determine regression parameters corresponding to local optima of the LTS regression problem. For the LMS regression problem, we briefly survey the existing solution methods which are all based on enumeration. We formulate the LMS regression problem as a mixed zero-one linear programming problem which we analyze in depth to obtain theoretical insights required for future algorithmic and computational work.     

Solution of Burgers' equation for large Reynolds number using finite elements with moving nodes

Burgers' equation often arises in the mathematical modelling used to solve problems in fluid dynamics involving turbulence. Numerical difficulties arise in the solution for the case of large Reynolds number. To obtain high accuracy, finite element methods are important. The aim of this paper is to summarize relevant past work and to use a moving node finite element method to obtain a solution of Burgers' equation under certain prescribed conditions. The results for high Reynolds number are compared with accurate results obtained by other authors.     

Least squares problems with inequality constraints as quadratic constraints

Linear least squares problems with box constraints are commonly solved to find model parameters within bounds based on physical considerations. Common algorithms include Bounded Variable Least Squares (BVLS) and the Matlab function lsqlin. Here, the goal is to find solutions to ill-posed inverse problems that lie within box constraints. To do this, we formulate the box constraints as quadratic constraints, and solve the corresponding unconstrained regularized least squares problem. Using box constraints as quadratic constraints is an efficient approach because the optimization problem has a closed form solution. The effectiveness of the proposed algorithm is investigated through solving three benchmark problems and one from a hydrological application. Results are compared with solutions found by lsqlin, and the quadratically constrained formulation is solved using the L-curve, maximum a posteriori estimation (MAP), and the χ² regularization method. The χ² regularization method with quadratic constraints is the most effective method for solving least squares problems with box constraints.     

Least squares and bounded variation regularization with nondifferentiable functionals

Let A be an operator from a real Banach space into a real Hilbert space. In this paper we study least squares regularization methods for the ill-posed operator equation A(u) = f using nonlinear nondifferentiable penalty functionals. We introduce a notion of distributional approximation, and use constructs of distributional approximations to establish convergence and stability of approximations of bounded variation solutions of the operator equation. We also show that the results provide a framework for a rigorous analysis of numerical methods based on Euler-Lagrange equations to solve the minimization problem. This justifies many of the numerical implementation schemes of bounded variation minimization that have been recently proposed.     

Least squares data fitting using shape preserving piecewise approximations

In recent years there has been a great deal of interest in the preservation of data properties in an interpolating function, and many good algorithms are available for this problem.In this paper a basis is constructed for a tensioned spline that gives a numerically stable algorithm for theL ₂ fitting of data that can preserve monotonicity and/or convexity. The motivation for this work is the fitting of data from a sewerage farm.