We present a reduced basis method for parametrized linear elliptic partial differential equations (PDEs) in least-squares finite element framework. A rigorous and reliable error estimate is developed, shown to bound the with respect exact solution of PDE, contrast estimates that measure finite-dimensional (high-fidelity) approximation. It first-order formulation key ingredient. The demonstrated...

A mesh-free method based on moving least squares approximation (MLS)  and weak form of governing equations including two dimensional equations of motion and Maxwell’s equation is used to analyze the free vibration of functionally graded piezoelectric material (FGPM) beams. Material properties in beam are determined using a power law distribution. Essential boundary conditions are imposed by the...

The standard approaches to solving overdetermined linear systems Ax ≈ b construct minimal corrections to the vector b and/or the matrix A such that the corrected system is compatible. In ordinary least squares (LS) the correction is restricted to b, while in data least squares (DLS) it is restricted to A. In scaled total least squares (Scaled TLS) [15], corrections to both b and A are allowed, ...

The Levenberg-Marquardt algorithm is one of the most popular algorithms for the solution of nonlinear least squares problems. In this paper, we propose and analyze the global and local convergence results of a novel Levenberg-Marquadt method for solving general nonlinear least squares problems. The proposed algorithm enjoys strong convergence properties (global convergence as well as quadratic ...

We construct and analyze a least-squares procedure for approximately solving the initial-value problem for the linearized equations of coupled sound and heat flow, in a bounded domain Í2 in Ä , with homogeneous Dirichlet boundary conditions. The method is based on Crank-Nicolson time differencing. To approximately solve the resulting system of boundary value problems at each time step, a least-...

Since meshless methods have been introduced to alleviate the difficulties arising in conventional finite element method, many papers on applications of meshless methods to boundary element method have been published. However, most of these papers use moving least squares approximation functions that have difficulties in prescribing essential boundary conditions. Recently, in order to strengthen...