scholarly journals An Approximate Solution for a Class of Ill-Posed Nonhomogeneous Cauchy Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Nihed Teniou ◽  
Salah Djezzar
Keyword(s):  
Nonlocal Problem ◽  
Cauchy Problems ◽  
Convergence Results ◽  
Fixed Sign ◽  
The Cauchy Problem ◽  
Boundary Value Method ◽  
Ill Posed ◽  
The Given ◽  
Differential Operator Equation ◽  
Well Posed

In this paper, we consider a nonhomogeneous differential operator equation of first order u ′ t + A u t = f t . The coefficient operator A is linear unbounded and self-adjoint in a Hilbert space. We assume that the operator does not have a fixed sign. We associate to this equation the initial or final conditions u 0 = Φ  or  u T = Φ . We note that the Cauchy problem is severely ill-posed in the sense that the solution if it exists does not depend continuously on the given data. Using a quasi-boundary value method, we obtain an approximate nonlocal problem depending on a small parameter. We show that regularized problem is well-posed and has a strongly solution. Finally, some convergence results are provided.

scholarly journals Quasi-boundary value method for non-well posed problem for a parabolic equation with integral boundary condition

2001 ◽  
Vol 7 (2) ◽  
pp. 129-145 ◽  
Author(s):  
M. Denche ◽  
K. Bessila
Keyword(s):  
Boundary Condition ◽  
Convergence Rates ◽  
Initial Conditions ◽  
Nonlocal Problem ◽  
Convergence Result ◽  
Integral Boundary Condition ◽  
Integral Boundary ◽  
Boundary Value Method ◽  
Ill Posed ◽  
Well Posed

In this paper we study the problem of control by the initial conditions of the heat equation with an integral boundary condition. This problem is ill-posed. Perturbing the final condition we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates.

scholarly journals Improved Regularization Method for Backward Cauchy Problems Associated with Continuous Spectrum Operator

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
Salah Djezzar ◽  
Nihed Teniou
Keyword(s):  
Continuous Spectrum ◽  
Original Problem ◽  
Convergence Rates ◽  
Nonlocal Problem ◽  
Cauchy Problems ◽  
Final Time ◽  
Unbounded Linear Operator ◽  
Convergence Results ◽  
Small Parameters ◽  
The Given

We consider in this paper an abstract parabolic backward Cauchy problem associated with an unbounded linear operator in a Hilbert space , where the coefficient operator in the equation is an unbounded self-adjoint positive operator which has a continuous spectrum and the data is given at the final time and a solution for is sought. It is well known that this problem is illposed in the sense that the solution (if it exists) does not depend continuously on the given data. The method of regularization used here consists of perturbing both the equation and the final condition to obtain an approximate nonlocal problem depending on two small parameters. We give some estimates for the solution of the regularized problem, and we also show that the modified problem is stable and its solution is an approximation of the exact solution of the original problem. Finally, some other convergence results including some explicit convergence rates are also provided.

scholarly journals To the Problem on Small Motions of the System of Two Viscoelastic Fluids in a Fixed Vessel

2018 ◽  
Vol 64 (3) ◽  
pp. 547-572
Author(s):  
N D Kopachevsky
Keyword(s):  
Hilbert Space ◽  
Hydraulic System ◽  
Operator Pencil ◽  
Incompressible Fluids ◽  
Operator Approach ◽  
The Cauchy Problem ◽  
Generalized Operator ◽  
Initialboundary Value Problem ◽  
Differential Operator Equation ◽  
Well Posed

In this paper, we study the problem of small motions of two Oldroyd viscoelastic incompressible fluids contained in a fixed vessel. By means of the operator approach, we reduce the original initialboundary value problem to the Cauchy problem for a differential operator equation in a Hilbert space and prove the well-posed solvability of the problem on an arbitrary interval of time. We obtain the equation for normal oscillations of the hydraulic system under consideration (Krein generalized operator pencil).

Iterative regularization method for an abstract ill-posed generalized elliptic equation

2020 ◽  
pp. 2150069
Author(s):  
Sassane Roumaissa ◽  
Boussetila Nadjib ◽  
Rebbani Faouzia ◽  
Benrabah Abderafik
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Regularization Method ◽  
Boundary Data ◽  
Identification Problem ◽  
Infinite Domain ◽  
Iterative Regularization ◽  
Convergence Results ◽  
Ill Posed ◽  
Well Posed

A preconditioning version of the Kozlov–Maz’ya iteration method for the stable identification of missing boundary data is presented for an ill-posed problem governed by generalized elliptic equations. The ill-posed data identification problem is reformulated as a sequence of well-posed fractional elliptic equations in infinite domain. Moreover, some convergence results are established. Finally, numerical results are included showing the accuracy and efficiency of the proposed method.

scholarly journals A modified quasi-boundary value method for an abstract ill-posed biparabolic problem

2017 ◽  
Vol 15 (1) ◽  
pp. 1649-1666 ◽  
Author(s):  
Khelili Besma ◽  
Boussetila Nadjib ◽  
Rebbani Faouzia
Keyword(s):  
Original Problem ◽  
Convergence Rates ◽  
Boundary Value ◽  
A Priori ◽  
Numerical Tests ◽  
Convergence Results ◽  
A Priori Bound ◽  
Boundary Value Method ◽  
Ill Posed ◽  
Stable Solutions

Abstract In this paper, we are concerned with the problem of approximating a solution of an ill-posed biparabolic problem in the abstract setting. In order to overcome the instability of the original problem, we propose a modified quasi-boundary value method to construct approximate stable solutions for the original ill-posed boundary value problem. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution. Moreover, numerical tests are presented to illustrate the accuracy and efficiency of this method.

scholarly journals On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates

2020 ◽  
Vol 54 (2) ◽  
pp. 493-529
Author(s):  
Laurent Bourgeois ◽  
Lucas Chesnel
Keyword(s):  
Cauchy Problem ◽  
Finite Elements ◽  
Small Parameter ◽  
Laplace Equation ◽  
Limit Problem ◽  
The Cauchy Problem ◽  
Ill Posed ◽  
Finite Elements Methods ◽  
Regularity Results ◽  
Well Posed

We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameter ε > 0. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly in ε. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due to the particular structure of the regularized problems, classical techniques à la Grisvard do not work and instead, we apply the Kondratiev approach. We describe the procedure in detail to keep track of the dependence in ε in all the estimates. The main originality of this study lies in the fact that the limit problem is ill-posed in any framework.

scholarly journals An Energy Regularization for Cauchy Problems of Laplace Equation in Annulus Domain

2011 ◽  
Vol 9 (4) ◽  
pp. 878-896 ◽  
Author(s):  
Houde Han ◽  
Leevan Ling ◽  
Tomoya Takeuchi
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Regularization Method ◽  
Boundary Data ◽  
Outer Boundary ◽  
Cauchy Problems ◽  
Electrical Field ◽  
Numerical Examples ◽  
The Cauchy Problem ◽  
Ill Posed

AbstractDetecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the pipe. The interior surface of the pipe is inaccessible and the nondestructive detection is solely based on measurements from the outer layer. The Cauchy problem for an elliptic equation is a typical ill-posed problem whose solution does not depend continuously on the boundary data. In this work, we assume that the measurements are available on the whole outer boundary on an annulus domain. By imposing reasonable assumptions, the theoretical goal here is to derive the stabilities of the Cauchy solutions and an energy regularization method. Relationship between the proposed energy regularization method and the Tikhonov regularization with Morozov principle is also given. A novel numerical algorithm is proposed and numerical examples are given.

scholarly journals Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems

2021 ◽  
Author(s):  
Radu Boţ ◽  
Guozhi Dong ◽  
Peter Elbau ◽  
Otmar Scherzer
Keyword(s):  
Linear Operator ◽  
Operator Equation ◽  
Convex Analysis ◽  
Convergence Rates ◽  
Linear Operator Equation ◽  
Optimal Convergence Rates ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Given ◽  
Theoretical Results

AbstractRecently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

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