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 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 The Impact of the Discrepancy Principle on the Tikhonov-Regularized Solutions with Oversmoothing Penalties

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 331
Author(s):  
Bernd Hofmann ◽  
Christopher Hofmann
Keyword(s):  
Case Studies ◽  
Convergence Rates ◽  
Regularization Parameter ◽  
A Priori ◽  
Discrepancy Principle ◽  
Parameter Choice ◽  
Operator Equations ◽  
Analytical Results ◽  
Ill Posed ◽  
The Impact

This paper deals with the Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. One focus is on the application of the discrepancy principle for choosing the regularization parameter and its consequences. Numerical case studies are performed in order to complement analytical results concerning the oversmoothing situation. For example, case studies are presented for exact solutions of Hölder type smoothness with a low Hölder exponent. Moreover, the regularization parameter choice using the discrepancy principle, for which rate results are proven in the oversmoothing case in in reference (Hofmann, B.; Mathé, P. Inverse Probl. 2018, 34, 015007) is compared to Hölder type a priori choices. On the other hand, well-known analytical results on the existence and convergence of regularized solutions are summarized and partially augmented. In particular, a sketch for a novel proof to derive Hölder convergence rates in the case of oversmoothing penalties is given, extending ideas from in reference (Hofmann, B.; Plato, R. ETNA. 2020, 93).

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.

On an ill-posed boundary value problem for the Laplace equation in a circular cylinder

2021 ◽  
pp. 35-43
Author(s):  
Evgeniy B. Laneev ◽  
Dmitriy Yu. Bykov ◽  
Anastasia V. Zubarenko ◽  
Olga N. Kulikova ◽  
Darya A. Morozova ◽  
...  
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Circular Cylinder ◽  
Boundary Value Problem ◽  
Laplace Equation ◽  
Boundary Value ◽  
Stable Solution ◽  
Medical Diagnostics ◽  
Cauchy Data ◽  
Ill Posed

In this paper, we consider a mixed problem for the Laplace equation in a region in a circular cylinder. On the lateral surface of a cylidrical region, the homogeneous boundary conditions of the first kind are given. The cylindrical area is bounded on one side by an arbitrary surface on which the Cauchy conditions are set, i.e. a function and its normal derivative are given. The other border of the cylindrical area is free. This problem is ill-posed, and to construct its approximate solution in the case of Cauchy data known with some error it is necessary to use regularizing algorithms. In this paper, the problem is reduced to a Fredholm integral equation of the first kind. Based on the solution of the integral equation, an explicit representation of the exact solution of the problem is obtained in the form of a Fourier series with the eigenfunctions of the first boundary value problem for the Laplace equation in a circle. A stable solution of the integral equation is obtained by the Tikhonov regularization method. The extremal of the Tikhonov functional is considered as an approximate solution. Based on this solution, an approximate solution of the problem in the whole is constructed. The theorem on convergence of the approximate solution of the problem to the exact one as the error in the Cauchy data tends to zero and the regularization parameter is matched with the error in the data is given. The results can be used for mathematical processing of thermal imaging data in medical diagnostics.

scholarly journals Tikhonov regularization with ℓ0-term complementing a~convex penalty: ℓ1-convergence under sparsity constraints

2019 ◽  
Vol 27 (4) ◽  
pp. 575-590 ◽  
Author(s):  
Wei Wang ◽  
Shuai Lu ◽  
Bernd Hofmann ◽  
Jin Cheng
Keyword(s):  
Convergence Rates ◽  
Regularization Parameter ◽  
A Priori ◽  
Convex Functional ◽  
Important Special Case ◽  
Bounded Linear ◽  
Sparsity Constraints ◽  
Ill Posed ◽  
Sparse Solutions ◽  
Special Case

Abstract Measuring the error by an {\ell^{1}} -norm, we analyze under sparsity assumptions an {\ell^{0}} -regularization approach, where the penalty in the Tikhonov functional is complemented by a general stabilizing convex functional. In this context, ill-posed operator equations {Ax=y} with an injective and bounded linear operator A mapping between {\ell^{2}} and a Banach space Y are regularized. For sparse solutions, error estimates as well as linear and sublinear convergence rates are derived based on a variational inequality approach, where the regularization parameter can be chosen either a priori in an appropriate way or a posteriori by the sequential discrepancy principle. To further illustrate the balance between the {\ell^{0}} -term and the complementing convex penalty, the important special case of the {\ell^{2}} -norm square penalty is investigated showing explicit dependence between both terms. Finally, some numerical experiments verify and illustrate the sparsity promoting properties of corresponding regularized solutions.

scholarly journals A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations

2018 ◽  
Vol 26 (5) ◽  
pp. 639-646 ◽  
Author(s):  
Jens Flemming
Keyword(s):  
Banach Spaces ◽  
Convergence Rates ◽  
A Priori ◽  
Linear Equations ◽  
Rate Function ◽  
Bregman Distances ◽  
Error Measures ◽  
Ill Posed ◽  
General Convex ◽  
Linear Operator Equations

Abstract We consider Tikhonov-type variational regularization of ill-posed linear operator equations in Banach spaces with general convex penalty functionals. Upper bounds for certain error measures expressing the distance between exact and regularized solutions, especially for Bregman distances, can be obtained from variational source conditions. We prove that such bounds are optimal in case of twisted Bregman distances, a specific a priori parameter choice, and low regularity of the exact solution, that is, the rate function is also an asymptotic lower bound for the error measure. This result extends existing converse results from Hilbert space settings to Banach spaces without adhering to spectral theory.

Convergence Rates For Lavrentiev-Type Regularization In Hilbert Scales

2008 ◽  
Vol 8 (3) ◽  
pp. 279-293 ◽  
Author(s):  
M.T. NAIR ◽  
U. TAUTENHAHN
Keyword(s):  
Error Bounds ◽  
Convergence Rates ◽  
Noisy Data ◽  
Regularization Methods ◽  
Operator Equations ◽  
Optimal Error ◽  
Regularization Scheme ◽  
Data Regularization ◽  
Ill Posed ◽  
Adjoint Operators

AbstractFor solving linear ill-posed problems with noisy data regularization methods are required. We analyze a simplified regularization scheme in Hilbert scales for operator equations with nonnegative self-adjoint operators. By exploiting the op-erator monotonicity of certain functions, order-optimal error bounds are derived that characterize the accuracy of the regularized approximations. These error bounds have been obtained under general smoothness conditions.

scholarly journals Regularization of a Cauchy problem for the heat equation

2018 ◽  
Vol 1 (T5) ◽  
pp. 184-192
Author(s):  
Au Van Vo ◽  
Tuan Hoang Nguyen
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Heat Equation ◽  
Error Estimates ◽  
A Priori ◽  
Cauchy Data ◽  
Linear Source ◽  
Truncation Method ◽  
Ill Posed ◽  
Regularized Solution

In this paper, we study a Cauchy problem for the heat equation with linear source in the form ut(x,t)= uxx(x,t)+f(x,t), u(L,t)=  φ(t), u(L,t)= Ψ (t), (x,t) ∈ (0,L) ×(0, 2π). This problem is ill-posed in the sense of Hadamard. To regularize the problem, the truncation method is proposed to solve the problem in the presence of noisy Cauchy data φε and Ψε satisfying ‖ φε - φ ‖+‖ Ψε - Ψ ‖ ≤ ε and that fε satisfying ‖ fε(x,. ) - f(x,.) ‖ ≤ ε .  We give some error estimates between the regularized solution and the exact solution under some different a-priori conditions of exact solution.

scholarly journals Two fractional regularization methods for identifying the radiogenic source of the Helium production-diffusion equation

2021 ◽  
Vol 6 (10) ◽  
pp. 11425-11448
Author(s):  
Xuemin Xue ◽  
◽  
Xiangtuan Xiong ◽  
Yuanxiang Zhang ◽  
Keyword(s):  
Diffusion Equation ◽  
Source Term ◽  
Regularization Parameter ◽  
A Priori ◽  
Regularization Methods ◽  
A Posteriori ◽  
Parameter Choice ◽  
Ill Posed ◽  
Parameter Choice Rules ◽  
Regularized Solution

<abstract><p>The predication of the helium diffusion concentration as a function of a source term in diffusion equation is an ill-posed problem. This is called inverse radiogenic source problem. Although some classical regularization methods have been considered for this problem, we propose two new fractional regularization methods for the purpose of reducing the over-smoothing of the classical regularized solution. The corresponding error estimates are proved under the a-priori and the a-posteriori regularization parameter choice rules. Some numerical examples are shown to display the necessarity of the methods.</p></abstract>

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