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.

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 Determination of conductivity in a heat equation

2000 ◽  
Vol 24 (9) ◽  
pp. 589-594 ◽  
Author(s):  
Ping Wang ◽  
Kewang Zheng
Keyword(s):  
Numerical Simulation ◽  
Inverse Problem ◽  
Approximate Solution ◽  
Heat Equation ◽  
Ill Posed ◽  
Smooth Data

We consider the problem of determining the conductivity in a heat equation from overspecified non-smooth data. It is an ill-posed inverse problem. We apply a regularization approach to define and construct a stable approximate solution. We also conduct numerical simulation to demonstrate the accuracy of our approximation.

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 A high accurate method for solving an inverse problem of the Laplace equation in detection of a Robin coefficient

2022 ◽  
Author(s):  
Abdelhak Hadj
Keyword(s):  
Inverse Problem ◽  
Regularization Method ◽  
Fourier Expansion ◽  
Least Squares Method ◽  
Accurate Method ◽  
Cauchy Data ◽  
Trefftz Method ◽  
Numerical Examples ◽  
Collocation Technique ◽  
Harmonic Equation

Abstract This study This work deals with an inverse problem for the harmonic equation to recover a Robin coefficient on a non-accessible part of a circle from Cauchy data measured on an accessible part of that circle. By assuming that the available data has a Fourier expansion, we adopt the Modified Collocation Trefftz Method (MCTM) to solve this problem. We use the truncation regularization method in combination with the collocation technique to approximate the solution, and the conjugate gradient method to obtain the coefficients, thus completing the missing Cauchy data. We recommend the least squares method to achieve a better stability. Finally, we illustrate the feasibility of this method with numerical examples.

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.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document