scholarly journals Stable Identification of Sources Located on Interface of Nonhomogeneous Media

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1932
Author(s):  
José Julio Conde Mones ◽  
Emmanuel Roberto Estrada Aguayo ◽  
José Jacobo Oliveros Oliveros ◽  
Carlos Arturo Hernández Gracidas ◽  
María Monserrat Morín Castillo
Keyword(s):  
Identification Problem ◽  
Stable Form ◽  
Numerical Instability ◽  
Stable Method ◽  
Smoothing Methods ◽  
Nonhomogeneous Media ◽  
Irregular Region ◽  
Ill Posed ◽  
The Media ◽  
Homogeneous Media

This paper presents a stable method for the identification of sources located on the separation interface of two homogeneous media (where one of them is contained by the other one), from measurement yielded by those sources on the exterior boundary of the media. This is an ill-posed problem because numerical instability is presented, i.e., minimal errors in the measurement can result in significant changes in the solution. To obtain the proposed stable method the identification problem is categorized into three subproblems, two of which present numerical instability and regularization methods must be applied to obtain their solution in a stable form. To manage the numerical instability due to the ill-posedness of these subproblems, the Tikhonov regularization and sequential smoothing methods are used. We illustrate this methodology in a circular and irregular region to demonstrate the feasibility of the proposed method, which yields convergent and stable solutions for input data with and without noise.

A MODIFIED FINITE DIFFERENCES METHOD FOR ANALYSIS OF ULTRASONIC PROPAGATION IN NONHOMOGENEOUS MEDIA

2010 ◽  
Vol 18 (01) ◽  
pp. 31-45 ◽  
Author(s):  
LUCIANO ALONSO RENTERIA ◽  
JUAN M. PEREZ ORIA
Keyword(s):  
Helmholtz Equation ◽  
Finite Differences ◽  
Computational Cost ◽  
Industrial Applications ◽  
Ultrasonic Waves ◽  
Nonhomogeneous Media ◽  
Finite Differences Method ◽  
High Gradients ◽  
Ultrasonic Propagation ◽  
Homogeneous Media

The propagation of ultrasonic waves is generally studied in homogeneous media, although in certain industrial applications the conditions of propagation differ from the ideal conditions and the predicted results are not valid. This work is focused on the resolution of the Helmholtz equation for the study of the ultrasonic propagation in nonhomogeneous media. In this way, the solution of the Helmholtz equation has been obtained by means of Finite Differences, using a nonconventional scheme that substantially improves the results obtained with other techniques such as standard Finite Differences or Finite Elements. Moreover, it decreases the computational cost in the calculation of the coefficients about 85%. The effects on the ultrasonic echoes in propagation environments with high gradients of propagation's speed have been analyzed by simulation using the method presented, and the results obtained have been experimentally validated through a set of measurements.

scholarly journals Range-relaxed criteria for choosing the Lagrange multipliers in nonstationary iterated Tikhonov method

2018 ◽  
Vol 40 (1) ◽  
pp. 606-627 ◽  
Author(s):  
R Boiger ◽  
A Leitão ◽  
B F Svaiter
Keyword(s):  
Lagrange Multipliers ◽  
Approximate Solutions ◽  
Identification Problem ◽  
Linear Operators ◽  
Type Method ◽  
Geometrical Properties ◽  
Regularization Parameters ◽  
New Strategy ◽  
Parameter Identification Problem ◽  
Ill Posed

Abstract In this article we propose a novel nonstationary iterated Tikhonov (NIT)-type method for obtaining stable approximate solutions to ill-posed operator equations modeled by linear operators acting between Hilbert spaces. Geometrical properties of the problem are used to derive a new strategy for choosing the sequence of regularization parameters (Lagrange multipliers) for the NIT iteration. Convergence analysis for this new method is provided. Numerical experiments are presented for two distinct applications: (I) a two-dimensional elliptic parameter identification problem (inverse potential problem); and (II) an image-deblurring problem. The results obtained validate the efficiency of our method compared with standard implementations of the NIT method (where a geometrical choice is typically used for the sequence of Lagrange multipliers).

scholarly journals Wavelet-Galerkin Method for Identifying an Unknown Source Term in a Heat Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Fangfang Dou
Keyword(s):  
Heat Equation ◽  
Galerkin Method ◽  
Identification Problem ◽  
Stability And Convergence ◽  
Wavelet Galerkin Method ◽  
Unknown Source ◽  
Frequency Components ◽  
Ill Posed ◽  
Meyer Wavelets ◽  
The Stability

We consider the problem of identification of the unknown source in a heat equation. The problem is ill posed in the sense that the solution (if it exists) does not depend continuously on the data. Meyer wavelets have the property that their Fourier transform has compact support. Therefore, by expanding the data and the solution in the basis of the Meyer wavelets, high-frequency components can be filtered away. Under the additional assumptions concerning the smoothness of the solution, we discuss the stability and convergence of a wavelet-Galerkin method for the source identification problem. Numerical examples are presented to verify the efficiency and accuracy of the method.

scholarly journals Retention of Nanoparticles: From Laboratory Cores to Outcrop Scale

Geofluids ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-16 ◽  
Author(s):  
Harpreet Singh ◽  
Farzam Javadpour
Keyword(s):  
Porous Media ◽  
Laboratory Experiments ◽  
Multiple Scales ◽  
Homogeneous Medium ◽  
Laboratory Scale ◽  
Small Scale ◽  
Length Scale ◽  
The Media ◽  
Small Change ◽  
Homogeneous Media

Laboratory experiments on small scale core plugs have shown controlled nanoparticles (NPs) retention. The length scale of subsurface media where NPs must be transported is an important factor that should be accounted for in a comprehensive manner when translating laboratory results to field scale. This study investigates the fraction of NPs retained inside porous media as a function of length scale of the media. A two-dimensional numerical model was used to simulate the retention of NPs at multiple scales of porous media, starting from laboratory scale cores to heterogeneous outcrop scales. Retention of NPs is modeled based on the concept of reversible and irreversible retention, by using the laboratory scale determined parameters. Our results show that the fraction of retained NPs increases nonlinearly with the length scale of the homogeneous media. The results also show that if the heterogeneity of the medium is consistent across scales, the fraction of retained NPs would behave just like homogeneous medium. In this study, small change in heterogeneity at two outcrop scales affects the retention of NPs, suggesting that heterogeneity may significantly impact the retention behavior of NPs that may not necessarily follow the behavior predicted from homogeneous cores (or periodically heterogeneous medium).

Error estimates for the simplified iteratively regularized Gauss–Newton method in Banach spaces under a Morozov-type stopping rule

2018 ◽  
Vol 26 (3) ◽  
pp. 311-333 ◽  
Author(s):  
Pallavi Mahale ◽  
Sharad Kumar Dixit
Keyword(s):  
Banach Spaces ◽  
Error Estimates ◽  
Newton Method ◽  
A Priori ◽  
Identification Problem ◽  
Stopping Rule ◽  
Optimal Error Estimates ◽  
Optimal Error ◽  
Parameter Identification Problem ◽  
Ill Posed

AbstractJin Qinian and Min Zhong [10] considered an iteratively regularized Gauss–Newton method in Banach spaces to find a stable approximate solution of the nonlinear ill-posed operator equation. They have considered a Morozov-type stopping rule (Rule 1) as one of the criterion to stop the iterations and studied the convergence analysis of the method. However, no error estimates have been obtained for this case. In this paper, we consider a modified variant of the method, namely, the simplified Gauss–Newton method under both an a priori as well as a Morozov-type stopping rule. In both cases, we obtain order optimal error estimates under Hölder-type approximate source conditions. An example of a parameter identification problem for which the method can be implemented is discussed in the paper.

Iterative regularization method for an abstract ill-posed generalized elliptic equation

2020 ◽  
pp. 2150069
Author(s):  
Sassane Roumaissa ◽  
Boussetila Nadjib ◽  
Rebbani Faouzia ◽  
Benrabah Abderafik
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Regularization Method ◽  
Boundary Data ◽  
Identification Problem ◽  
Infinite Domain ◽  
Iterative Regularization ◽  
Convergence Results ◽  
Ill Posed ◽  
Well Posed

A preconditioning version of the Kozlov–Maz’ya iteration method for the stable identification of missing boundary data is presented for an ill-posed problem governed by generalized elliptic equations. The ill-posed data identification problem is reformulated as a sequence of well-posed fractional elliptic equations in infinite domain. Moreover, some convergence results are established. Finally, numerical results are included showing the accuracy and efficiency of the proposed method.

Simplified Iteratively Regularized Gauss–Newton Method in Banach Spaces Under a General Source Condition

2020 ◽  
Vol 20 (2) ◽  
pp. 321-341
Author(s):  
Pallavi Mahale ◽  
Sharad Kumar Dixit
Keyword(s):  
Newton Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Identification Problem ◽  
Stopping Rule ◽  
Optimal Error Estimates ◽  
Source Condition ◽  
Parameter Identification Problem ◽  
Ill Posed ◽  
General Source

AbstractIn this paper, we consider a simplified iteratively regularized Gauss–Newton method in a Banach space setting under a general source condition. We will obtain order-optimal error estimates both for an a priori stopping rule and for a Morozov-type stopping rule together with a posteriori choice of the regularization parameter. An advantage of a general source condition is that it provides a unified setting for the error analysis which can be applied to the cases of both severely and mildly ill-posed problems. We will give a numerical example of a parameter identification problem to discuss the performance of the method.

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