A comparison of error estimates at a point and on a set when solving ill-posed problems

2018 ◽  
Vol 26 (4) ◽  
pp. 541-550
Author(s):  
V. P. Tanana
Keyword(s):  
Exact Solution ◽  
Error Estimate ◽  
Error Estimates ◽  
Smooth Function ◽  
Special Class ◽  
Exact Estimate ◽  
Piecewise Smooth ◽  
Piecewise Smooth Function ◽  
Ill Posed

Abstract The problem of correlating the error estimates at a point and on a correctness class is of interest to many mathematicians. Since the desired solution to a real ill-posed problem is unique, the error estimate obtained on the class becomes crude. In this paper, by assuming that an exact solution is a piecewise smooth function, we prove, for a special class of incorrect problems, that an error estimate at a point is an infinitely small quantity compared with an exact estimate on a correctness set.

scholarly journals Surface temperature determination from Borehole measurements: Regularization and error estimates

1995 ◽  
Vol 18 (3) ◽  
pp. 601-605 ◽  
Author(s):  
Tran Thi Le ◽  
Milagros Navarro
Keyword(s):  
Exact Solution ◽  
Temperature Distribution ◽  
Measurement Error ◽  
Error Estimate ◽  
Surface Temperature ◽  
Error Estimates ◽  
Interior Point ◽  
Temperature Determination ◽  
Ill Posed ◽  
Regularized Solution

The authors consider the problem of determining the temperature distributionu(x,t)on the half-linex=0,t>0, from measurements at an interior point, for allt>0. As is well-known, this is an ill-posed problem Using the Tikhonov method, the authors give a regularized solution, and assuming the (unknown) exact solution is inHα(ℝ),α>0. They give an error estimate of the order1/(1n1/ ϵ )αforϵ →0, whereϵ >0is a bound on the measurement error.

scholarly journals A Wavelet Method for the Cauchy Problem for the Helmholtz Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Fang-Fang Dou ◽  
Chu-Li Fu
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Helmholtz Equation ◽  
Error Estimates ◽  
Fixed Frequency ◽  
Wavelet Method ◽  
Numerical Tests ◽  
The Cauchy Problem ◽  
Ill Posed

We consider a Cauchy problem for the Helmholtz equation at a fixed frequency. The problem is severely ill posed in the sense that the solution (if it exists) does not depend continuously on the data. We present a wavelet method to stabilize the problem. Some error estimates between the exact solution and its approximation are given, and numerical tests verify the efficiency and accuracy of the proposed method.

scholarly journals HP estimation for the Cauchy problem for nonlinear elliptic equation

2018 ◽  
Vol 1 (T5) ◽  
pp. 193-202
Author(s):  
Thang Duc Le
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Elliptic Equation ◽  
Error Estimates ◽  
Bounded Domain ◽  
Nonlinear Elliptic Equation ◽  
Nonlinear Elliptic ◽  
The Cauchy Problem ◽  
Ill Posed ◽  
Regularized Solution

In this paper, we investigate the Cauchy problem for a ND nonlinear elliptic equation in a bounded domain. As we know, the problem is severely ill-posed. We apply the Fourier truncation method to regularize the problem. Error estimates between the regularized solution and the exact solution are established in Hp space under some priori assumptions on the exact solution.

scholarly journals Method of iteration of solving invariant equations with an approximate operator in the case of an arbitrary choice of the regularization parameter

2019 ◽  
Vol 54 (4) ◽  
pp. 408-416
Author(s):  
O. V. Matysik ◽  
V. F. Savchuk
Keyword(s):  
Error Estimate ◽  
Error Estimates ◽  
Regularization Parameter ◽  
A Priori ◽  
Operator Equations ◽  
Mathematical Economics ◽  
Prove Theorem ◽  
A Value ◽  
Ill Posed ◽  
The Right

In the introduction, the object of investigation is indicated – incorrect problems described by first-kind operator equations. The subject of the study is an explicit iterative method for solving first-kind equations. The aim of the paper is to prove the convergence of the proposed method of simple iterations with an alternating step alternately and to obtain error estimates in the original norm of a Hilbert space for the cases of self-conjugated and non self-conjugated problems. The a priori choice of the regularization parameter is studied for a source-like representable solution under the assumption that the operator and the right-hand side of the equation are given approximately. In the main part of the work, the achievement of the stated goal is expressed in four reduced and proved theorems. In Section 1, the first-kind equation is written down and a new explicit method of simple iteration with alternating steps is proposed to solve it. In Section 2, we consider the case of the selfconjugated problem and prove Theorem 1 on the convergence of the method and Theorem 2, in which an error estimate is obtained. To obtain an error estimate, an additional condition is required – the requirement of the source representability of the exact solution. In Section 3, the non-self-conjugated problem is solved, the convergence of the proposed method is proved, which in this case is written differently, and its error estimate is obtained in the case of an a priori choice of the regularization parameter. In sections 2 and 3, the error estimates obtained are optimized, that is, a value is found – the step number of the iteration, in which the error estimate is minimal. Since incorrect problems constantly arise in numerous applications of mathematics, the problem of studying them and constructing methods for their solution is topical. The obtained results can be used in theoretical studies of solution of first-kind operator equations, as well as applied ill-posed problems encountered in dynamics and kinetics, mathematical economics, geophysics, spectroscopy, systems for complete automatic processing and interpretation of experiments, plasma diagnostics, seismic and medicine.

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 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 A Posteriori Error Estimates for the Generalized Overlapping Domain Decomposition Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Hakima Benlarbi ◽  
Ahmed-Salah Chibi
Keyword(s):  
Error Estimate ◽  
Domain Decomposition ◽  
Error Estimates ◽  
Domain Decomposition Method ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Decomposition Methods ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Overlapping Domain Decomposition

A posteriori error estimates for the generalized overlapping domain decomposition method (GODDM) (i.e., with Robin boundary conditions on the interfaces), for second order boundary value problems, are derived. We show that the error estimate in the continuous case depends on the differences of the traces of the subdomain solutions on the interfaces. After discretization of the domain by finite elements we use the techniques of the residuala posteriorierror analysis to get ana posteriorierror estimate for the discrete solutions on subdomains. The results of some numerical experiments are presented to support the theory.

UNIFORM INF–SUP CONDITIONS FOR THE SPECTRAL DISCRETIZATION OF THE STOKES PROBLEM

1999 ◽  
Vol 09 (03) ◽  
pp. 395-414 ◽  
Author(s):  
C. BERNARDI ◽  
Y. MADAY
Keyword(s):  
Error Estimate ◽  
Error Estimates ◽  
Best Approximation ◽  
Stokes Problem ◽  
Optimal Error Estimate ◽  
Optimal Error ◽  
Discretization Parameter ◽  
The Best Approximation ◽  
Discrete Spaces ◽  
Spectral Discretization

In standard spectral discretizations of the Stokes problem, error estimates on the pressure are slightly less accurate than the best approximation estimates, since the constant of the Babuška–Brezzi inf–sup condition is not bounded independently of the discretization parameter. In this paper, we propose two possible discrete spaces for the pressure: for each of them, we prove a uniform inf–sup condition, which leads in particular to an optimal error estimate on the pressure.

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