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 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 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 Simultaneous Determination of the Space-Dependent Source and the Initial Distribution in a Heat Equation by Regularizing Fourier Coefficients of the Given Measurements

2018 ◽  
Vol 2018 ◽  
pp. 1-15
Author(s):  
Shufang Qiu ◽  
Wen Zhang ◽  
Jianmei Peng
Keyword(s):  
Inverse Problem ◽  
Fourier Coefficients ◽  
Convergence Rates ◽  
Initial Distribution ◽  
Stability And Convergence ◽  
Convergence Results ◽  
Regularization Parameters ◽  
Regularization Problem ◽  
The Given

We consider an inverse problem for simultaneously determining the space-dependent source and the initial distribution in heat conduction equation. First, we study the ill-posedness of the inverse problem. Then, we construct a regularization problem to approximate the originally inverse problem and obtain the regularization solutions with their stability and convergence results. Furthermore, convergence rates of the regularized solutions are presented under a prior and a posteriori strategies for selecting regularization parameters. Results of numerical examples show that the proposed regularization method is stable and effective for the considered inverse problem.

scholarly journals Spectral Regularization Methods for an Abstract Ill-Posed Elliptic Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Nadjib Boussetila ◽  
Salim Hamida ◽  
Faouzia Rebbani
Keyword(s):  
Original Problem ◽  
Convergence Rates ◽  
A Priori ◽  
Stable Solution ◽  
Cauchy Data ◽  
Regularization Methods ◽  
Spectral Regularization ◽  
Convergence Results ◽  
A Priori Bound ◽  
Ill Posed

We study an abstract elliptic Cauchy problem associated with an unbounded self-adjoint positive operator which has a continuous spectrum. It is well-known that such a problem is severely ill-posed; that is, the solution does not depend continuously on the Cauchy data. We propose two spectral regularization methods to construct an approximate stable solution to our original problem. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution.

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.

scholarly journals Spectral methods for Langevin dynamics and associated error estimates

2018 ◽  
Vol 52 (3) ◽  
pp. 1051-1083 ◽  
Author(s):  
Julien Roussel ◽  
Gabriel Stoltz
Keyword(s):  
Convergence Rates ◽  
Langevin Dynamics ◽  
Galerkin Methods ◽  
Discrete Level ◽  
Rigidity Matrix ◽  
Poisson Equations ◽  
The Poisson Equation ◽  
Convergence Results ◽  
Abstract Setting ◽  
Tensor Basis

We prove the consistency of Galerkin methods to solve Poisson equations where the differential operator under consideration is hypocoercive. We show in particular how the hypocoercive nature of the generator associated with Langevin dynamics can be used at the discrete level to first prove the invertibility of the rigidity matrix, and next provide error bounds on the approximation of the solution of the Poisson equation. We present general convergence results in an abstract setting, as well as explicit convergence rates for a simple example discretized using a tensor basis. Our theoretical findings are illustrated by numerical simulations.

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 Tracking probabilistic truths: a logic for statistical learning

Synthese ◽  
2021 ◽  
Author(s):  
Alexandru Baltag ◽  
Soroush Rafiee Rad ◽  
Sonja Smets
Keyword(s):  
Statistical Learning ◽  
Bayes Rule ◽  
Revision Theory ◽  
True Probability ◽  
Doxastic Logic ◽  
Statistical Knowledge ◽  
Convergence Results ◽  
Repeated Sampling ◽  
Types Of Information ◽  
The Given

AbstractWe propose a new model for forming and revising beliefs about unknown probabilities. To go beyond what is known with certainty and represent the agent’s beliefs about probability, we consider a plausibility map, associating to each possible distribution a plausibility ranking. Beliefs are defined as in Belief Revision Theory, in terms of truth in the most plausible worlds (or more generally, truth in all the worlds that are plausible enough). We consider two forms of conditioning or belief update, corresponding to the acquisition of two types of information: (1) learning observable evidence obtained by repeated sampling from the unknown distribution; and (2) learning higher-order information about the distribution. The first changes only the plausibility map (via a ‘plausibilistic’ version of Bayes’ Rule), but leaves the given set of possible distributions essentially unchanged; the second rules out some distributions, thus shrinking the set of possibilities, without changing their plausibility ordering.. We look at stability of beliefs under either of these types of learning, defining two related notions (safe belief and statistical knowledge), as well as a measure of the verisimilitude of a given plausibility model. We prove a number of convergence results, showing how our agent’s beliefs track the true probability after repeated sampling, and how she eventually gains in a sense (statistical) knowledge of that true probability. Finally, we sketch the contours of a dynamic doxastic logic for statistical learning.

scholarly journals Rothe time-discretization method for a nonlocal problem arising in thermoelasticity

2005 ◽  
Vol 2005 (1) ◽  
pp. 13-28 ◽  
Author(s):  
Nabil Merazga ◽  
Abdelfatah Bouziani
Keyword(s):  
Continuous Dependence ◽  
Nonlocal Problem ◽  
Time Discretization ◽  
Boundary Integral ◽  
Time Variable ◽  
Discretization Method ◽  
Integral Conditions ◽  
Rothe Method ◽  
Semidiscrete Approximation ◽  
The Given

We investigate a model parabolic mixed problem with purely boundary integral conditions arising in the context of thermoelasticity. Using the Rothe method which is based on a semidiscretization of the given problem with respect to the time variable, the questions of existence, uniqueness, and continuous dependence upon data of a weak solution are proved. Moreover, we establish convergence and derive an error estimate for a semidiscrete approximation.

scholarly journals Identification of gas pipelines hydraulic efficiency coefficients

2019 ◽  
Vol 102 ◽  
pp. 03002
Author(s):  
Andrey A. Belevitin ◽  
Victoria G. Ryzhkova
Keyword(s):  
Nonlinear Optimization ◽  
Optimization Problem ◽  
Original Problem ◽  
Solution Procedure ◽  
Hydraulic Efficiency ◽  
Gas Pipelines ◽  
Problem Statement ◽  
Nonlinear Optimization Problem ◽  
Successive Linear Programming ◽  
The Given

This study investigates the identification of non-measureable parameters of the gas transmission system (gas pipelines hydraulic efficiency coefficients). The problem statement and solution procedure are presented. The original problem is divided into two interrelated components: the nonlinear optimization problem and the temperature calculation. The nonlinear optimization problem is solved using the Successive Linear Programming (SLP) method. The problems of insufficiency of measurements and multiplicity of solutions are described, and appropriate approaches are proposed (introduction of additional subcriteria and uniting gas pipelines into groups). Identification of gas pipelines hydraulic efficiency coefficients for gas transmission systems of various complexity has been performed using the given algorithm.

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