Compact discrepancy and chi-squared principles for over-determined inverse problems

2017 ◽  
Vol 25 (3) ◽  
Author(s):  
Maxim Pisarenco ◽  
Irwan D. Setija
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Regularization Parameter ◽  
Discrepancy Principle ◽  
Chi Squared

AbstractWe discuss and analyze the classical discrepancy principle and the recently proposed and closely related chi-squared principle for selecting the regularization parameter of an inverse problem. Some properties that deteriorate the performance of these methods for over-determined inverse problems are highlighted. We propose a so-called

The inverse problem of recovering the coefficients of a differential equations on a graph

2020 ◽  
Vol 28 (5) ◽  
pp. 727-738
Author(s):  
Victor Sadovnichii ◽  
Yaudat Talgatovich Sultanaev ◽  
Azamat Akhtyamov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Characteristic Equation ◽  
Arbitrary Number ◽  
New Class ◽  
Finite Set

AbstractWe consider a new class of inverse problems on the recovery of the coefficients of differential equations from a finite set of eigenvalues of a boundary value problem with unseparated boundary conditions. A finite number of eigenvalues is possible only for problems in which the roots of the characteristic equation are multiple. The article describes solutions to such a problem for equations of the second, third, and fourth orders on a graph with three, four, and five edges. The inverse problem with an arbitrary number of edges is solved similarly.

Variational-hemivariational inverse problem for electro-elastic unilateral frictional contact problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Othmane Baiz ◽  
Hicham Benaissa ◽  
Zakaria Faiz ◽  
Driss El Moutawakil
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Contact Problem ◽  
Existence And Uniqueness ◽  
Frictional Contact ◽  
Hemivariational Inequalities ◽  
Frictional Contact Problem ◽  
Uniqueness Of A Solution

AbstractIn the present paper, we study inverse problems for a class of nonlinear hemivariational inequalities. We prove the existence and uniqueness of a solution to inverse problems. Finally, we introduce an inverse problem for an electro-elastic frictional contact problem to illustrate our results.

scholarly journals Adjoint Numerical Method for a Multiphysical Inverse Problem of Two-Phase Well Testing in Petroleum Reservoirs

2021 ◽  
Vol 2090 (1) ◽  
pp. 012139
Author(s):  
OA Shishkina ◽  
I M Indrupskiy
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Objective Function ◽  
Petroleum Reservoirs ◽  
Forward Problem ◽  
Adjoint Equations ◽  
Well Testing ◽  
Adjoint Methods ◽  
Two Phase ◽  
Gradient Calculation

Abstract Inverse problem solution is an integral part of data interpretation for well testing in petroleum reservoirs. In case of two-phase well tests with water injection, forward problem is based on the multiphase flow model in porous media and solved numerically. The inverse problem is based on a misfit or likelihood objective function. Adjoint methods have proved robust and efficient for gradient calculation of the objective function in this type of problems. However, if time-lapse electrical resistivity measurements during the well test are included in the objective function, both the forward and inverse problems become multiphysical, and straightforward application of the adjoint method is problematic. In this paper we present a novel adjoint algorithm for the inverse problems considered. It takes into account the structure of cross dependencies between flow and electrical equations and variables, as well as specifics of the equations (mixed parabolic-hyperbolic for flow and elliptic for electricity), numerical discretizations and grids, and measurements in the inverse problem. Decomposition is proposed for the adjoint problem which makes possible step-wise solution of the electric adjoint equations, like in the forward problem, after which a cross-term is computed and added to the right-hand side of the flow adjoint equations at this timestep. The overall procedure provides accurate gradient calculation for the multiphysical objective function while preserving robustness and efficiency of the adjoint methods. Example cases of the adjoint gradient calculation are presented and compared to straightforward difference-based gradient calculation in terms of accuracy and efficiency.

scholarly journals Bayesian statistics of non-linear inverse problems: example of the magnetotelluric 1-D inverse problem

1994 ◽  
Vol 119 (2) ◽  
pp. 353-368 ◽  
Author(s):  
Pascal Tarits ◽  
Virginie Jouanne ◽  
Michel Menvielle ◽  
Michel Roussignol
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Bayesian Statistics ◽  
Linear Inverse Problems ◽  
Non Linear

Regularization by Denoising for simultaneous source separation

Geophysics ◽  
2021 ◽  
pp. 1-56
Author(s):  
Breno Bahia ◽  
Rongzhi Lin ◽  
Mauricio Sacchi
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Data Processing ◽  
Spectrum Analysis ◽  
Numerical Experiments ◽  
Source Separation ◽  
Singular Spectrum Analysis ◽  
Source Data ◽  
Blended Data ◽  
Simultaneous Source

Denoisers can help solve inverse problems via a recently proposed framework known as regularization by denoising (RED). The RED approach defines the regularization term of the inverse problem via explicit denoising engines. Simultaneous source separation techniques, being themselves a combination of inversion and denoising methods, provide a formidable field to explore RED. We investigate the applicability of RED to simultaneous-source data processing and introduce a deblending algorithm named REDeblending (RDB). The formulation permits developing deblending algorithms where the user can select any denoising engine that satisfies RED conditions. Two popular denoisers are tested, but the method is not limited to them: frequency-wavenumber thresholding and singular spectrum analysis. We offer numerical blended data examples to showcase the performance of RDB via numerical experiments.

scholarly journals Convergence Rate of the Modified Landweber Method for Solving Inverse Potential Problems

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 608
Author(s):  
Pornsarp Pornsawad ◽  
Parada Sungcharoen ◽  
Christine Böckmann
Keyword(s):  
Convergence Rate ◽  
Potential Problem ◽  
Regularization Parameter ◽  
Discrepancy Principle ◽  
Source Condition ◽  
Rate Analysis ◽  
Potential Problems ◽  
Ill Posed ◽  
Landweber Method ◽  
Convergence Rate Analysis

In this paper, we present the convergence rate analysis of the modified Landweber method under logarithmic source condition for nonlinear ill-posed problems. The regularization parameter is chosen according to the discrepancy principle. The reconstructions of the shape of an unknown domain for an inverse potential problem by using the modified Landweber method are exhibited.

scholarly journals The Impact of the Discrepancy Principle on the Tikhonov-Regularized Solutions with Oversmoothing Penalties

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 331
Author(s):  
Bernd Hofmann ◽  
Christopher Hofmann
Keyword(s):  
Case Studies ◽  
Convergence Rates ◽  
Regularization Parameter ◽  
A Priori ◽  
Discrepancy Principle ◽  
Parameter Choice ◽  
Operator Equations ◽  
Analytical Results ◽  
Ill Posed ◽  
The Impact

This paper deals with the Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. One focus is on the application of the discrepancy principle for choosing the regularization parameter and its consequences. Numerical case studies are performed in order to complement analytical results concerning the oversmoothing situation. For example, case studies are presented for exact solutions of Hölder type smoothness with a low Hölder exponent. Moreover, the regularization parameter choice using the discrepancy principle, for which rate results are proven in the oversmoothing case in in reference (Hofmann, B.; Mathé, P. Inverse Probl. 2018, 34, 015007) is compared to Hölder type a priori choices. On the other hand, well-known analytical results on the existence and convergence of regularized solutions are summarized and partially augmented. In particular, a sketch for a novel proof to derive Hölder convergence rates in the case of oversmoothing penalties is given, extending ideas from in reference (Hofmann, B.; Plato, R. ETNA. 2020, 93).

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