Related papers: A least-squares collocation method for nonlinear h…
In this note we discuss several questions concerning the implementation of overdetermined least-squares collocation methods for higher-index differential algebraic equations (DAEs). Since higher-index DAEs lead to ill-posed problems in…
The present paper continues our investigation of an implementation of a least-squares collocation method for higher-index differential-algebraic equations. In earlier papers, we were able to substantiate the choice of basis functions and…
In this paper, we study the problem of finding the least square solutions of over-determined linear algebraic equations over networks in a distributed manner. Each node has access to one of the linear equations and holds a dynamic state. We…
Nonlinear least-squares problems are a special class of unconstrained optimization problems in which their gradient and Hessian have special structures. In this paper, we exploit these structures and proposed a matrix-free algorithm with a…
Non-linear least squares solvers are used across a broad range of offline and real-time model fitting problems. Most improvements of the basic Gauss-Newton algorithm tackle convergence guarantees or leverage the sparsity of the underlying…
We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face…
This study shows how to obtain least-squares solutions to initial and boundary value problems to nonhomogeneous linear differential equations with nonconstant coefficients of any order. However, without loss of generality, the approach has…
We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equations in two steps. We first obtain a numerical approximation…
We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for the numerical solution of partial differential equations. We start with a detailed explanation of the method for the…
For three decades, carrier-phase observations have been used to obtain the most accurate location estimates using global navigation satellite systems (GNSS). These estimates are computed by minimizing a nonlinear mixed-integer least-squares…
Least squares form one of the most prominent classes of optimization problems, with numerous applications in scientific computing and data fitting. When such formulations aim at modeling complex systems, the optimization process must…
In this paper, we present the isogeometric least-squares collocation (IGA-L) method, which determines the numerical solution by making the approximate differential operator fit the real differential operator in a least-squares sense. The…
In this paper, we develop regularized discrete least squares collocation and finite volume methods for solving two-dimensional nonlinear time-dependent partial differential equations on irregular domains. The solution is approximated using…
A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It…
A method for moving least squares interpolation and differentiation is presented in the framework of orthogonal polynomials on discrete points. This yields a robust and efficient method which can avoid singularities and breakdowns in the…
In this paper, we introduce an iterative numerical method to solve systems of nonlinear equations. The third-order convergence of this method is analyzed. Several examples are given to illustrate the efficiency of the proposed method.
The problem of finding roots or solutions of a nonlinear partial differential equation may be formulated as the problem of minimizing a sum of squared residuals. One then defines an evolution equation so that in the asymptotic limit a…
We present a generalized formulation for reweighted least squares approximations. The goal of this article is twofold: firstly, to prove that the solution of such problem can be expressed as a convex combination of certain interpolants when…
Modelling real world systems frequently requires the solution of systems of nonlinear equations. A number of approaches have been suggested and developed for this computational problem. However, it is also possible to attempt solutions…
We propose a novel stochastic gradient descent method for solving linear least squares problems with partially observed data. Our method uses submatrices indexed by a randomly selected pair of row and column index sets to update the iterate…