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 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 Regularization of backward time-fractional parabolic equations by Sobolev-type equations

2020 ◽  
Vol 28 (5) ◽  
pp. 659-676
Author(s):  
Dinh Nho Hào ◽  
Nguyen Van Duc ◽  
Nguyen Van Thang ◽  
Nguyen Trung Thành
Keyword(s):  
Error Estimates ◽  
Parabolic Equations ◽  
Regularization Parameter ◽  
A Priori ◽  
Sobolev Type ◽  
A Posteriori ◽  
Parameter Choice ◽  
Numerical Tests ◽  
Ill Posed ◽  
Parameter Choice Rules

AbstractThe problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

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 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 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.

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