Approximation of Solution Components for Ill-Posed Problems by the Tikhonov Method with Total Variation

2018 ◽  
Vol 97 (3) ◽  
pp. 266-270 ◽  
Author(s):  
V. V. Vasin ◽  
V. V. Belyaev
Keyword(s):  
Total Variation ◽  
Tikhonov Method ◽  
Ill Posed

scholarly journals COMPLEXITY ESTIMATES FOR SEVERELY ILL-POSED PROBLEMS UNDER A POSTERIORI SELECTION OF REGULARIZATION PARAMETER

2017 ◽  
Vol 22 (3) ◽  
pp. 283-299
Author(s):  
Sergii G. Solodky ◽  
Ganna L. Myleiko ◽  
Evgeniya V. Semenova
Keyword(s):  
Integral Equations ◽  
Regularization Parameter ◽  
Efficient Algorithms ◽  
Optimal Order ◽  
A Posteriori ◽  
Order Of Accuracy ◽  
Tikhonov Method ◽  
Discrete Information ◽  
Ill Posed ◽  
Selection Of

In the article the authors developed two efficient algorithms for solving severely ill-posed problems such as Fredholm’s integral equations. The standard Tikhonov method is applied as a regularization. To select a regularization parameter we employ two different a posteriori rules, namely, discrepancy and balancing principles. It is established that proposed strategies not only achieved optimal order of accuracy on the class of problems under consideration, but also they are economical in the sense of used discrete information.

Blind Inversion of Multidimensional Seismic Data Using Sequential Tikhonov and Total Variation Regularizations (STTVR)

Geophysics ◽  
2021 ◽  
pp. 1-56
Author(s):  
Saber jahanjooy ◽  
Mohammad Ali Riahi ◽  
Hamed Ghanbarnejad Moghanloo
Keyword(s):  
Total Variation ◽  
Tikhonov Regularization ◽  
Seismic Data ◽  
Low Frequency ◽  
Seismic Inversion ◽  
Earth Model ◽  
Rapid Changes ◽  
Nonlinear Relation ◽  
Ill Posed ◽  
Seismic Sections

The acoustic impedance (AI) model is key data for seismic interpretation, usually obtained from its nonlinear relation with seismic reflectivity. Common approaches use initial geological and seismic information to constraint the AI model estimation. When no accurate prior information is available, these approaches may dictate false results at some parts of the model. The regularization of ill-posed underdetermined problems requires some constraints to restrict the possible results. Available seismic inversion methods mostly use Tikhonov or total variation (TV) regularizations with some adjustments. Tikhonov regularization assumes smooth variation in the AI model, and it is incurious about the rapid changes in the model. TV allows rapid changes, and it is more stable in presence of noisy data. In a detailed realistic earth model that AI changes gradually, TV creates a stair-casing effect, which could lead to misinterpretation. This could be avoided by using TV and Tikhonov regularization sequentially in the alternating direction method of multipliers (ADMM) and creating the AI model. The result of implementing the proposed algorithm (STTVR) on 2D synthetic and real seismic sections shows that the smaller details in the lithological variations are accounted for as well as the general trend. STTVR can calculate major AI variations without any additional low-frequency constraints. The temporal and spatial transition of the calculated AI in real seismic data is gradual and close to a real geological setting.

An Inverse Diffraction Problem: Shape Reconstruction

2015 ◽  
Vol 5 (4) ◽  
pp. 342-360 ◽  
Author(s):  
Yanfeng Kong ◽  
Zhenping Li ◽  
Xiangtuan Xiong
Keyword(s):  
Error Estimates ◽  
Diffraction Problem ◽  
Shape Reconstruction ◽  
Numerical Examples ◽  
Tikhonov Method ◽  
Ill Posed ◽  
Tikhonov Regularisation ◽  
Slow Evolution ◽  
Inverse Diffraction

AbstractAn inverse diffraction problem is considered. Both classical Tikhonov regularisation and a slow-evolution-from-the-continuation-boundary (SECB) method are used to solve the ill-posed problem. Regularisation error estimates for the two methods are compared, and the SECB method is seen to be an improvement on the classical Tikhonov method. Two numerical examples demonstrate their feasibility and efficiency.

scholarly journals Identification of Source Term for the Time-Fractional Diffusion-Wave Equation by Fractional Tikhonov Method

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 934
Author(s):  
Le Dinh Long ◽  
Nguyen Hoang Luc ◽  
Yong Zhou ◽  
and Can Nguyen
Keyword(s):  
Source Term ◽  
A Priori ◽  
Fractional Diffusion ◽  
Behavior Changes ◽  
Diffusion Wave ◽  
Tikhonov Method ◽  
A Posteriori Parameter Choice ◽  
Ill Posed ◽  
Well Posed ◽  
Parameter Choice Rules

In this article, we consider an inverse problem to determine an unknown source term in a space-time-fractional diffusion equation. The inverse problems are often ill-posed. By an example, we show that this problem is NOT well-posed in the Hadamard sense, i.e., this problem does not satisfy the last condition-the solution’s behavior changes continuously with the input data. It leads to having a regularization model for this problem. We use the Tikhonov method to solve the problem. In the theoretical results, we also propose a priori and a posteriori parameter choice rules and analyze them.

scholarly journals Full-waveform inversion and joint migration inversion with an automatic directional total variation constraint

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. R175-R183 ◽  
Author(s):  
Shan Qu ◽  
Eric Verschuur ◽  
Yangkang Chen
Keyword(s):  
Total Variation ◽  
Waveform Inversion ◽  
Velocity Model ◽  
Full Waveform Inversion ◽  
L1 Norm ◽  
Reflection Data ◽  
Full Waveform ◽  
Ill Posed ◽  
Laplacian Smoothing ◽  
The Impact

As full-waveform inversion (FWI) is a nonunique and typically ill-posed inversion problem, it needs proper regularization to avoid cycle skipping. To reduce the nonlinearity of FWI, we have developed joint migration inversion (JMI) as an alternative, explaining the reflection data with decoupled velocity and reflectivity parameters. However, the velocity update may also suffer from being trapped in local minima. To optimally include geologic information, we have developed FWI/JMI with directional total variation (TV) as an L1-norm regularization on the velocity. We design the directional TV operator based on the local dip field, instead of ignoring the local structural direction of the subsurface and only using horizontal and vertical gradients in the traditional TV. The local dip field is estimated using plane-wave destruction based on a raw reflectivity model, which is usually calculated from the initial velocity model. With two complex synthetic examples, based on the Marmousi model, we determine that our method is much more effective compared with FWI/JMI without regularization and FWI/JMI with the conventional TV regularization. In the JMI-based example, we also determine that L1 directional TV works better than L2 directional Laplacian smoothing. In addition, by comparing these two examples, it can be seen that the impact of regularization is larger for FWI than for JMI because in JMI the velocity model only explains the propagation effects and, thereby, makes it less sensitive to the details in the velocity model.

scholarly journals Two-Step Newton-Tikhonov Method for Hammerstein-Type Equations: Finite-Dimensional Realization

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Santhosh George ◽  
Monnanda Erappa Shobha
Keyword(s):  
Hilbert Space ◽  
Regularization Parameter ◽  
Real Hilbert Space ◽  
Initial Approximation ◽  
Numerical Illustration ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Tikhonov Method ◽  
Ill Posed ◽  
Balancing Principle

Finite-dimensional realization of a Two-Step Newton-Tikhonov method is considered for obtaining a stable approximate solution to nonlinear ill-posed Hammerstein-type operator equations KF(x)=f. Here F:D(F)⊆X→X is nonlinear monotone operator, K:X→Y is a bounded linear operator, X is a real Hilbert space, and Y is a Hilbert space. The error analysis for this method is done under two general source conditions, the first one involves the operator K and the second one involves the Fréchet derivative of F at an initial approximation x0 of the the solution x̂: balancing principle of Pereverzev and Schock (2005) is employed in choosing the regularization parameter and order optimal error bounds are established. Numerical illustration is given to confirm the reliability of our approach.

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