scholarly journals INVERSE PROBLEMS FOR A GENERALIZED SUBDIFFUSION EQUATION WITH FINAL OVERDETERMINATION

2019 ◽  
Vol 24 (2) ◽  
pp. 236-262 ◽  
Author(s):  
Jaan Janno ◽  
Nataliia Kinash
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Source Term ◽  
Local Existence ◽  
Reaction Term ◽  
Dependent Component ◽  
Linear Reaction ◽  
Existence And Stability ◽  
Subdiffusion Equation ◽  
Stability Of The Solution

We consider two inverse problems for a generalized subdiffusion equation that use the final overdetermination condition. Firstly, we study a problem of reconstruction of a specific space-dependent component in a source term. We prove existence, uniqueness and stability of the solution to this problem. Based on these results, we consider an inverse problem of identification of a space-dependent coefficient of a linear reaction term. We prove the uniqueness and local existence and stability of the solution to this problem.

scholarly journals INVERSE PROBLEMS FOR A GENERALIZED SUBDIFFUSION EQUATION WITH FINAL OVERDETERMINATION

2019 ◽  
Vol 24 (2) ◽  
pp. 236-262 ◽  
Author(s):  
Jaan Janno ◽  
Nataliia Kinash
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Source Term ◽  
Local Existence ◽  
Reaction Term ◽  
Dependent Component ◽  
Linear Reaction ◽  
Existence And Stability ◽  
Subdiffusion Equation ◽  
Stability Of The Solution

We consider two inverse problems for a generalized subdiffusion equation that use the final overdetermination condition. Firstly, we study a problem of reconstruction of a specific space-dependent component in a source term. We prove existence, uniqueness and stability of the solution to this problem. Based on these results, we consider an inverse problem of identification of a space-dependent coefficient of a linear reaction term. We prove the uniqueness and local existence and stability of the solution to this problem.

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 Numerical method of identification of an unknown source term in a heat equation

2002 ◽  
Vol 8 (2) ◽  
pp. 161-168 ◽  
Author(s):  
Afet Golayoğlu Fatullayev
Keyword(s):  
Inverse Problem ◽  
Heat Equation ◽  
Numerical Method ◽  
Minimization Problem ◽  
Unknown Function ◽  
Source Term ◽  
Numerical Procedure ◽  
Numerical Examples ◽  
Unknown Source

A numerical procedure for an inverse problem of identification of an unknown source in a heat equation is presented. Approach of proposed method is to approximate unknown function by polygons linear pieces which are determined consecutively from the solution of minimization problem based on the overspecified data. Numerical examples are presented.

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 On investigation of the inverse problem for a parabolic integro-differential equation with a variable coefficient of thermal conductivity

2020 ◽  
Vol 30 (4) ◽  
pp. 572-584
Author(s):  
D.K. Durdiev ◽  
J.Z. Nuriddinov
Keyword(s):  
Thermal Conductivity ◽  
Inverse Problem ◽  
Variable Coefficient ◽  
Direct Problem ◽  
Local Existence ◽  
Problem Solution ◽  
Volterra Type ◽  
Integral Term ◽  
Local Existence And Uniqueness ◽  
Coefficient Of Thermal Conductivity

The inverse problem of determining a multidimensional kernel of an integral term depending on a time variable $t$ and $ (n-1)$-dimensional spatial variable $x'=\left(x_1,\ldots, x_ {n-1}\right)$ in the $n$-dimensional heat equation with a variable coefficient of thermal conductivity is investigated. The direct problem is the Cauchy problem for this equation. The integral term has the time convolution form of kernel and direct problem solution. As additional information for solving the inverse problem, the solution of the direct problem on the hyperplane $x_n = 0$ is given. At the beginning, the properties of the solution to the direct problem are studied. For this, the problem is reduced to solving an integral equation of the second kind of Volterra-type and the method of successive approximations is applied to it. Further the stated inverse problem is reduced to two auxiliary problems, in the second one of them an unknown kernel is included in an additional condition outside integral. Then the auxiliary problems are replaced by an equivalent closed system of Volterra-type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the Hölder class of functions, we prove the main result of the article, which is a local existence and uniqueness theorem of the inverse problem solution.

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

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
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