scholarly journals A Regularization Process for Electrical Impedance Equation Employing Pseudoanalytic Function Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Cesar Marco Antonio Robles Gonzalez ◽  
Ariana Guadalupe Bucio Ramirez ◽  
Volodymyr Ponomaryov ◽  
Marco Pedro Ramirez Tachiquin
Keyword(s):  
Inverse Problem ◽  
Taylor Series ◽  
Regularization Method ◽  
Electrical Impedance ◽  
Function Theory ◽  
Forward Problem ◽  
Problem Method ◽  
Pseudoanalytic Function ◽  
Ill Posed

The electrical impedance equation is considered an ill-posed problem where the solution to the forward problem is more easy to achieve than the inverse problem. This work tries to improve convergence in the forward problem method, where the Pseudoanalytic Function Theory by means of the Taylor series in formal powers is used, incorporating a regularization method to make a solution more stable and to obtain better convergence. In addition, we include a comparison between the designed algorithms that perform proposed method with and without a regularization process and the autoadjustment parameter for this regularization process.

scholarly journals Regularization of a sideways problem for a time-fractional diffusion equation with nonlinear source

2020 ◽  
Vol 28 (2) ◽  
pp. 211-235
Author(s):  
Tran Bao Ngoc ◽  
Nguyen Huy Tuan ◽  
Mokhtar Kirane
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Regularization Method ◽  
A Priori ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Nonlinear Source ◽  
Numerical Example ◽  
Ill Posed ◽  
Time Fractional Diffusion Equation

AbstractIn this paper, we consider an inverse problem for a time-fractional diffusion equation with a nonlinear source. We prove that the considered problem is ill-posed, i.e., the solution does not depend continuously on the data. The problem is ill-posed in the sense of Hadamard. Under some weak a priori assumptions on the sought solution, we propose a new regularization method for stabilizing the ill-posed problem. We also provide a numerical example to illustrate our results.

scholarly journals Inverse problem of the market demand theory and analytical indices of demand

2019 ◽  
Vol 21 (1) ◽  
pp. 89-110 ◽  
Author(s):  
Vladimir K. Gorbunov ◽  
Alexander G. Lvov
Keyword(s):  
Inverse Problem ◽  
Utility Function ◽  
Regularization Method ◽  
Market Demand ◽  
Index Numbers ◽  
Demand Theory ◽  
Finite Set ◽  
Ulyanovsk Region ◽  
Ill Posed ◽  
Analytical Index

The inverse problem of the market demand's theory is constructing a collective utility function via a trade statistics consisting of a finite set of pairs ``prices-quantities''. The main computational problem here is the solution of the Afriat's inequalities system, which determines the values of the utility function and the Lagrange multiplier on the trade statistics data, which are ``Afriat's numbers''. This inverse problem is ill-posed one because of multiplicity of inequalities system's solutions and also because of their possible inconsistency and instability. A regularization method for this problem is proposed, based on the relaxation of the Afriat's system yielding local Hausdorf continuity of its solution set, and on the use of characteristics of analytical index numbers determined via Afriat's numbers. These characteristics formalized by choice criteria are: optimism, pessimism, objectivity. The results of constructing analytical index numbers for real trade statistics of Ulyanovsk region are presented.

Existence of variational source conditions for nonlinear inverse problems in Banach spaces

2018 ◽  
Vol 26 (2) ◽  
pp. 277-286 ◽  
Author(s):  
Jens Flemming
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Banach Spaces ◽  
Regularization Method ◽  
Multiple Solutions ◽  
Convergence Rates ◽  
Nonlinear Inverse Problems ◽  
Range Type ◽  
Ill Posed ◽  
Tikhonov’S Regularization

AbstractVariational source conditions proved to be useful for deriving convergence rates for Tikhonov’s regularization method and also for other methods. Up to now, such conditions have been verified only for few examples or for situations which can be also handled by classical range-type source conditions. Here we show that for almost every ill-posed inverse problem variational source conditions are satisfied. Whether linear or nonlinear, whether Hilbert or Banach spaces, whether one or multiple solutions, variational source conditions are a universal tool for proving convergence rates.

scholarly journals First Characterization of a New Method for Numerically Solving the Dirichlet Problem of the Two-Dimensional Electrical Impedance Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Marco Pedro Ramirez-Tachiquin ◽  
Cesar Marco Antonio Robles Gonzalez ◽  
Rogelio Adrian Hernandez-Becerril ◽  
Ariana Guadalupe Bucio Ramirez
Keyword(s):  
Electrical Impedance ◽  
Dirichlet Boundary ◽  
Function Theory ◽  
New Method ◽  
Two Dimensional ◽  
Dirichlet Boundary Value Problem ◽  
The Finite Element Method ◽  
Impedance Tomography ◽  
Pseudoanalytic Function

Based upon the elements of the modern pseudoanalytic function theory, we analyze a new method for numerically solving the forward Dirichlet boundary value problem corresponding to the two-dimensional electrical impedance equation. The analysis is performed by introducing interpolating piecewise separable-variables conductivity functions in the unit circle. To warrant the effectiveness of the posed method, we consider several examples of conductivity functions, whose boundary conditions are exact solutions of the electrical impedance equation, performing a brief comparison with the finite element method. Finally, we discuss the possible contributions of these results to the field of the electrical impedance tomography.

Statistical inversion in electrical impedance tomography using mixed total variation and non-convex ℓp regularization prior

2015 ◽  
Vol 23 (5) ◽  
Author(s):  
Thilo Strauss ◽  
Taufiquar Khan
Keyword(s):  
Inverse Problem ◽  
Total Variation ◽  
Electrical Impedance Tomography ◽  
Electrical Impedance ◽  
Bayesian Framework ◽  
Electromagnetic Properties ◽  
Impedance Tomography ◽  
Conductivity Distribution ◽  
Ill Posed

AbstractElectrical impedance tomography (EIT) is a well-known technique to estimate the conductivity distribution γ of a body Ω with unknown electromagnetic properties. EIT is a severely ill-posed inverse problem. In this paper, we formulate the EIT problem in the Bayesian framework using mixed total variation (TV) and non-convex ℓ

scholarly journals On an initial inverse problem for a diffusion equation with a conformable derivative

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Tran Thanh Binh ◽  
Nguyen Hoang Luc ◽  
Donal O’Regan ◽  
Nguyen H. Can
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Regularization Method ◽  
Conformable Derivative ◽  
Parameter Choice ◽  
Backward Problem ◽  
Ill Posed ◽  
Two Parameter ◽  
Parameter Choice Rules ◽  
Regularized Solution

AbstractIn this paper, we consider the initial inverse problem for a diffusion equation with a conformable derivative in a general bounded domain. We show that the backward problem is ill-posed, and we propose a regularizing scheme using a fractional Landweber regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.

scholarly journals Determination of a power density by an entropy regularization method

2005 ◽  
Vol 2005 (2) ◽  
pp. 127-152 ◽  
Author(s):  
Olivier Prot ◽  
Maïtine Bergounioux ◽  
Jean Gabriel Trotignon
Keyword(s):  
Inverse Problem ◽  
Electromagnetic Field ◽  
Electromagnetic Wave ◽  
Power Density ◽  
Regularization Method ◽  
Power Density Distribution ◽  
Numerical Examples ◽  
Finite Dimensional ◽  
Ill Posed

The determination of directional power density distribution of an electromagnetic wave from the electromagnetic field measurement can be expressed as an ill-posed inverse problem. We consider the resolution of this inverse problem via a maximum entropy regularization method. A finite-dimensional algorithm is derived from optimality conditions, and we prove its convergence. A variant of this algorithm is also studied. This second one leads to a solution which maximizes entropy in the probabilistic sense. Some numerical examples are given.

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