Ritz-least squares method for finding a control parameter in a one-dimensional parabolic inverse problem

2016 ◽  
Vol 22 (2) ◽  
Author(s):  
Meisam Noei Khorshidi ◽  
Sohrab Ali Yousefi
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Control Parameter ◽  
Least Squares Method ◽  
Ritz Method ◽  
Source Control ◽  
Algebraic Equations ◽  
Least Squares Approximation ◽  
One Dimensional ◽  
Satisfier Function

AbstractAn inverse problem concerning a diffusion equation with source control parameter is considered. The approximation of the problem is based on the Ritz method with satisfier function. The Ritz method together with the least squares approximation (Ritz-least squares method) are utilized to reduce the inverse problem to the solution of algebraic equations. We extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique.

scholarly journals Moving Least Squares Method for a One-Dimensional Parabolic Inverse Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Baiyu Wang
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
High Efficiency ◽  
Least Squares Method ◽  
Parabolic Problems ◽  
Moving Least Squares ◽  
One Dimensional ◽  
Moving Least Squares Method ◽  
The Stability

This paper investigates the numerical solution of a class of one-dimensional inverse parabolic problems using the moving least squares approximation; the inverse problem is the determination of an unknown source term depending on time. The collocation method is used for solving the equation; some numerical experiments are presented and discussed to illustrate the stability and high efficiency of the method.

scholarly journals One-dimensional constrained inversion study of TEM and application in coal goafs’ detection

2020 ◽  
Vol 12 (1) ◽  
pp. 1533-1540
Author(s):  
Si Yuanlei ◽  
Li Maofei ◽  
Liu Yaoning ◽  
Guo Weihong
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Data Interpretation ◽  
Volume Effect ◽  
Underground Space ◽  
Attenuation Curve ◽  
One Dimensional ◽  
Transient Electromagnetic ◽  
Geological Conditions ◽  
Geological Information

AbstractTransient electromagnetic method (TEM) is often used in urban underground space exploration and field geological resource detection. Inversion is the most important step in data interpretation. Because of the volume effect of the TEM, the inversion results are usually multi-solvable. To reduce the multi-solvability of inversion, the constrained inversion of TEM has been studied using the least squares method. The inversion trials were performed using two three-layer theoretical geological models and one four-layer theoretical geological model. The results show that one-dimensional least squares constrained inversion is faster and more effective than unconstrained inversion. The induced electromotive force attenuation curves of the inversion model indicate that the same attenuation curve may be used for different geological conditions. Therefore, constrained inversion using known geological information can more accurately reflect the underground geological information.

scholarly journals Determination of parameters for Cauchy’s problem for systems of ODEs with application to biological modelling

BIOMATH ◽  
2016 ◽  
Vol 5 (1) ◽  
pp. 1604231
Author(s):  
A.N. Pete ◽  
Peter Mathye ◽  
Igor Fedotov ◽  
Michael Shatalov
Keyword(s):  
Experimental Data ◽  
Mathematical Models ◽  
Least Squares ◽  
Numerical Method ◽  
Least Squares Method ◽  
Algebraic Equations ◽  
Systems Of Odes ◽  
Determination Of Parameters ◽  
Biological Modelling

An inverse numerical method that estimate parameters of dynamic mathematical models given some information about unknown trajectories at some time is applied to examples taken from Biology and Ecology. The method consisting of determining an over-determined system of algebraic equations using experimental data. The solution of the over-determined system is then obtained using, for example the least-squares method. To illustrate the effectiveness of the method an analysis of examples and corresponding numerical example are presented.

scholarly journals Electrical Dipole Source Localization using Hybrid Least Squares Method in combination with ICA

2019 ◽  
Vol 5 (1) ◽  
pp. 361-363
Author(s):  
Fars Samann ◽  
Andreas Rausch ◽  
Thomas Schanze
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Source Localization ◽  
Biomedical Engineering ◽  
Least Squares Method ◽  
Dipole Source ◽  
Dipole Source Localization ◽  
Initial Values ◽  
Non Linear ◽  
Eeg Inverse Problem

AbstractIn biomedical engineering, dipole source localization is commonly used to identify brain activities from scalp recorded potentials, which is known as inverse problem of electroencephalography (EEG) source localization. However, this problem is fundamental in biomedical engineering, medicine and neuroscience. The EEG inverse problem is non-linear, in addition, it is ill-posed and the solver can be unstable, i.e. the solution is non-unique and it is highly sensitive to small changes of the measured signal (noise). For solving the EEG inverse problem iterative methods, like Levenberg-Marquardt algorithm, are usually considered. However, these techniques require good initial values and many electrodes N, since a large redundancy supports the finding of the right solution. Therefore, in this paper, a hybrid method of linear and non-linear modelling and least squares approach are proposed to overcome of these problems: the solutions calculated by means of a linear approximation of EEG inverse problems serve as initial values for solving the original non-linear model. In addition, independent component analysis (ICA) is combined with the proposed hybrid least squares method to separate different dipole sources from multiple EEG signals. The performance of the hybrid least squares method with and without ICA is measured in term of root mean square error. The simulation results show that the proposed method can estimate the location of dipole source with acceptable accuracy under high noise condition and small N comparing with linear least squares method considering larger N. Finally, it should be mentioned that the proposed method promises advantages in finding solutions of the EEG inverse problem effectively.

Solitary waves of the RLW equation via least squares method

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ozlem Ersoy Hepson ◽  
Idris Dag ◽  
Bülent Saka ◽  
Buket Ay
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Least Square Method ◽  
Least Square ◽  
Spline Functions ◽  
Algebraic Equations ◽  
B Spline ◽  
B Splines ◽  
Rlw Equation ◽  
Set Up

Abstract Integration using least squares method in space and Crank–Nicolson approach in time is managed to set up an algorithm to solve the RLW equation numerically. Trial functions in the least square method consist of a combination of the quartic B-spline functions. Integration of the RLW equation gives a system of algebraic equations. The solutions consisting of a combination of the quartic B-splines are given for some initial and boundary value problems of RLW equation.

A mixed least-squares method for an inverse problem of a nonlinear beam equation

1999 ◽  
Vol 15 (1) ◽  
pp. 19-32 ◽  
Author(s):  
Richard E Ewing ◽  
Tao Lin ◽  
Yanping Lin
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Least Squares Method ◽  
Beam Equation ◽  
Nonlinear Beam

Solution of problems in dynamic elasticity via a mixed least squares method

2015 ◽  
Vol 32 (3) ◽  
pp. 687-704 ◽  
Author(s):  
Maria Tchonkova
Keyword(s):  
Least Squares ◽  
Analytical Solutions ◽  
Least Squares Method ◽  
Three Dimensional ◽  
Time Step ◽  
Theoretical Formulation ◽  
Content Type ◽  
One Dimensional ◽  
Dynamic Elasticity ◽  
Backward Euler

Purpose – The purpose of this paper is to present an original mixed least squares method for solving problems in dynamic elasticity. Design/methodology/approach – The proposed approach involves two different types of unknowns: velocities and stresses. The approximate solution to the dynamic elasticity equations is obtained via a minimization of a least squares functional, consisting of two terms: a term, which includes the squared residual of a weak form of the time rate of the constitutive relationships, expressed in terms of velocities and stresses, and a term, which depends on the squared residual of the equations of motion. At each time step the functional is minimized with respect to the velocities and stresses, which for the purpose of this study, are approximated by equal order piece-wise linear polynomial functions. The time discretization is based upon the backward Euler scheme. The displacements are computed from the obtained velocities in terms of a finite difference interpolation. The proposed theoretical formulation is given the general three-dimensional case and is tested numerically on the solution of one-dimensional wave equations. Findings – To test the performance of the method, it has been implemented in an original computer code, using object-oriented logic and written from scratch. Two one-dimensional problems from the mathematical physics, with well-known exact analytical solutions, have been solved. The numerical examples include a forced vibrating spring, fixed at its both ends and a rod, vibrating under its own weight, when one of its ends is fixed and the other is traction-free. The performed convergence study suggests that the method is convergent for both: velocities and stresses. The obtained results show excellent agreement between the exact and analytical solutions for displacement modes, velocities and stresses. It is observed that this method appears to be stable for the different mesh sizes and time step values. Originality/value – The mixed least squares formulation, described in this paper, serves as a basis for interesting future developments and applications to two and three-dimensional problems in dynamic elasticity.

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