scholarly journals Determination of a space-dependent force function in the one-dimensional wave equation

2014 ◽  
Vol 12 (1) ◽  
Author(s):  
S. O. Hussein ◽  
D. Lesnic
Keyword(s):  
Input Data ◽  
Regularization Parameter ◽  
Stable Solution ◽  
Force Function ◽  
Tikhonov Regularization Method ◽  
Vibrating String ◽  
Ill Posed ◽  
Curve Method ◽  
The One

<p class="p1">The determination of an unknown spacewice dependent force function acting on a vibrating string from over-specied Cauchy boundary data is investigated numerically using the boundary element method (BEM) combined with a regularized method of separating variables. This linear inverse problem is ill-posed since small errors in the input data cause large errors in the output force solution. Consequently, when the input data is contaminated with noise we use the Tikhonov regularization method in order to obtain a stable solution. The choice of the regularization parameter is based on the L-curve method. Numerical results show that the solution is accurate for exact data and stable for noisy data.</p>

scholarly journals ABOUT REGULARIZATION OF SEVERELY ILL-POSED PROBLEMS BY STANDARD TIKHONOV'S METHOD WITH THE BALANCING PRINCIPLE

2014 ◽  
Vol 19 (2) ◽  
pp. 199-215 ◽  
Author(s):  
Sergei G. Solodky ◽  
Ganna L. Myleiko
Keyword(s):  
Input Data ◽  
Regularization Parameter ◽  
Stable Solution ◽  
Order Of Accuracy ◽  
Tikhonov Method ◽  
Ill Posed ◽  
Balancing Principle

In the present paper for a stable solution of severely ill-posed problems with perturbed input data, the standard Tikhonov method is applied, and the regularization parameter is chosen according to balancing principle. We establish that the approach provides the order of accuracy on the class of problems under consideration.

scholarly journals A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1685-1697
Author(s):  
Zhenyu Zhao ◽  
Lei You ◽  
Zehong Meng
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
Tikhonov Regularization Method ◽  
Hermite Expansion ◽  
The Cauchy Problem ◽  
Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

The Regularization Method Based on Complex Collinearity Diagnostics and Metrics

2013 ◽  
Vol 416-417 ◽  
pp. 1393-1398
Author(s):  
Chao Zhong Ma ◽  
Yong Wei Gu ◽  
Ji Fu ◽  
Yuan Lu Du ◽  
Qing Ming Gui
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Measurement Data ◽  
Tikhonov Regularization Method ◽  
Parameter Determination ◽  
Selection Methods ◽  
Ill Posed ◽  
Collinearity Diagnostics ◽  
Regularization Matrix

In a large number of measurement data processing, the ill-posed problem is widespread. For such problems, this paper introduces the solution of ill-posed problem of the unity of expression and Tikhonov regularization method, and then to re-collinearity diagnostics and metrics based on proposed based on complex collinearity diagnostics and the metric regularization method is given regularization matrix selection methods and regularization parameter determination formulas. Finally, it uses a simulation example to verify the effectiveness of the method.

A-optimal design method to determine the regularization parameter of coseismic slip distribution inversion

2020 ◽  
Vol 221 (1) ◽  
pp. 440-450
Author(s):  
Leyang Wang ◽  
Wangwang Gu
Keyword(s):  
Optimal Design ◽  
Regularization Parameter ◽  
Design Method ◽  
Slip Distribution ◽  
Coseismic Slip ◽  
Seismic Slip ◽  
Regularization Parameters ◽  
Curve Method ◽  
Optimal Design Method

ABSTRACT The key to the inversion of a coseismic slip distribution is to determine the regularization parameters. In view of the determination of regularization parameters in seismic slip distribution inversion, the A-optimal design method is proposed in this paper. The L-curve method and A-optimal design method are used to design simulation experiments, and the inversion results show that the A-optimal design method is superior to the L-curve method in determining the regularization parameters. These two methods are also used to determine the regularization parameters of the L'Aquila and Lushan earthquake slip distribution inversions, and the results are consistent with those of other research conducted at home and abroad. Compared with the L-curve method, the A-optimal design method has the advantages of a high accuracy that does not rely on the data fitting accuracy.

scholarly journals INVERSE PROBLEM OF A ONE-DIMENSIONAL MODEL IN MULTILAYER HEAT CONDUCTION

2020 ◽  
Vol 19 (1) ◽  
pp. 42
Author(s):  
G. C. Oliveira ◽  
S. S. Ribeiroa ◽  
G. Guimarães
Keyword(s):  
Inverse Problem ◽  
Measurement Errors ◽  
Input Data ◽  
Temperature Measurements ◽  
Dimensional Model ◽  
One Dimensional ◽  
Ill Posed ◽  
Measurements Errors ◽  
Chip Chip

The inverse problem in conducting heat is related to the determination of the boundary condition, rate of heat generation, or thermophysical properties, using temperature measurements at one or more positions of the solid. The inverse problem in conducting heat is mathematically one of the ill-posed problems, because its solution extremely sensitive to measurement errors. For a well-placed problem the following conditions must be satisfied: the solution must exist, it must be unique and must be stable on small changes of the input data. The objective of the work is to estimate the heat flux generated at the tool-chip-chip interface in a manufacturing process. The term "estimation" is used because in the temperature measurements, errors are always present and these affect the accuracy of the calculation of the heat flow.

scholarly journals Solving ill-posed magnetic inverse problem using a Parameterized Trust-Region Sub-problem

2013 ◽  
Vol 43 (2) ◽  
pp. 99-123 ◽  
Author(s):  
Maha Mohamed Abdelazeem
Keyword(s):  
Weighting Function ◽  
Regularization Parameter ◽  
Stable Solution ◽  
Trust Region ◽  
Real Data ◽  
Linear Constraints ◽  
Singular Value Decomposition Method ◽  
Ill Posed ◽  
Value Decomposition ◽  
Natural Decay

Abstract The aim of this paper is to find a plausible and stable solution for the inverse geophysical magnetic problem. Most of the inverse problems in geophysics are considered as ill-posed ones. This is not necessarily due to complex geological situations, but it may arise because of ill-conditioned kernel matrix. To deal with such ill-conditioned matrix, one may truncate the most ill part as in truncated singular value decomposition method (TSVD). In such a method, the question will be where to truncate? In this paper, for comparison, we first try the adaptive pruning algorithm for the discrete L-curve criterion to estimate the regularization parameter for TSVD method. Linear constraints have been added to the ill-conditioned matrix. The same problem is then solved using a global optimizing and regularizing technique based on Parameterized Trust Region Sub-problem (PTRS). The criteria of such technique are to choose a trusted region of the solutions and then to find the satisfying minimum to the objective function. The ambiguity is controlled mainly by proper choosing the trust region. To overcome the natural decay in kernel with depth, a specific depth weighting function is used. A Matlab-based inversion code is implemented and tested on two synthetic total magnetic fields contaminated with different levels of noise to simulate natural fields. The results of PTRS are compared with those of TSVD with adaptive pruning L-curve. Such a comparison proves the high stability of the PTRS method in dealing with potential field problems. The capability of such technique has been further tested by applying it to real data from Saudi Arabia and Italy.

scholarly journals Splitting the one-Dimensional Wave Equation, Part II: Additional Data are Given by an End Displacement Measurement

2021 ◽  
pp. 233-239
Author(s):  
Shilan Othman Hussein ◽  
Mohammed Sabah Hussein
Keyword(s):  
Wave Equation ◽  
Additional Data ◽  
Numerical Solutions ◽  
Separation Of Variables ◽  
Displacement Measurement ◽  
Tikhonov Regularization Method ◽  
Ill Posed ◽  
The One ◽  
Inverse Force ◽  
Force Problem

     In this research, an unknown space-dependent force function in the wave equation is studied. This is a natural continuation of [1] and chapter 2 of [2] and [3], where the finite difference method (FDM)/boundary element method (BEM), with the separation of variables method, were considered. Additional data are given by the one end displacement measurement. Moreover, it is a continuation of [3], with exchanging the boundary condition, where  are extra data, by the initial condition. This is an ill-posed inverse force problem for linear hyperbolic equation. Therefore, in order to stabilize the solution, a zeroth-order Tikhonov regularization method is provided. To assess the accuracy, the minimum error between exact and numerical solutions for the force is computed for various regularization parameters. Numerical results are presented and a good agreement was obtained for the exact and noisy data.

A Meshless Regularization Method for a Two-Dimensional Two-Phase Linear Inverse Stefan Problem

2013 ◽  
Vol 5 (06) ◽  
pp. 825-845 ◽  
Author(s):  
B. Tomas Johansson ◽  
Daniel Lesnic ◽  
Thomas Reeve
Keyword(s):  
Stefan Problem ◽  
Regularization Method ◽  
Input Data ◽  
Stable Solution ◽  
Higher Dimensions ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Two Dimensional ◽  
Two Phase ◽  
Ill Posed

AbstractIn this paper, a meshless regularization method of fundamental solutions is proposed for a two-dimensional, two-phase linear inverse Stefan problem. The numerical implementation and analysis are challenging since one needs to handle composite materials in higher dimensions. Furthermore, the inverse Stefan problem is ill-posed since small errors in the input data cause large errors in the desired output solution. Therefore, regularization is necessary in order to obtain a stable solution. Numerical results for several benchmark test examples are presented and discussed.

scholarly journals A New Method for Determining Optimal Regularization Parameter in Near-Field Acoustic Holography

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Yue Xiao
Keyword(s):  
Point Source ◽  
Regularization Parameter ◽  
Near Field ◽  
New Method ◽  
Tikhonov Regularization Method ◽  
Acoustic Holography ◽  
Reconstruction Process ◽  
Curve Method ◽  
Univariate Optimization ◽  
Equivalent Sources

Tikhonov regularization method is effective in stabilizing reconstruction process of the near-field acoustic holography (NAH) based on the equivalent source method (ESM), and the selection of the optimal regularization parameter is a key problem that determines the regularization effect. In this work, a new method for determining the optimal regularization parameter is proposed. The transfer matrix relating the source strengths of the equivalent sources to the measured pressures on the hologram surface is augmented by adding a fictitious point source with zero strength. The minimization of the norm of this fictitious point source strength is as the criterion for choosing the optimal regularization parameter since the reconstructed value should tend to zero. The original inverse problem in calculating the source strengths is converted into a univariate optimization problem which is solved by a one-dimensional search technique. Two numerical simulations with a point driven simply supported plate and a pulsating sphere are investigated to validate the performance of the proposed method by comparison with the L-curve method. The results demonstrate that the proposed method can determine the regularization parameter correctly and effectively for the reconstruction in NAH.

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