Approximate Solution of Ill-Posed Equations: Arbitrarily Slow Convergence vs. Superconvergence

1985 ◽  
pp. 234-243 ◽  
Author(s):  
Eberhard Schock
Keyword(s):  
Approximate Solution ◽  
Slow Convergence ◽  
Ill Posed

Numerical methods for the approximate solution of ill-posed problems on compact sets

1995 ◽  
pp. 65-79 ◽  
Author(s):  
A. N. Tikhonov ◽  
A. V. Goncharsky ◽  
V. V. Stepanov ◽  
A. G. Yagola
Keyword(s):  
Approximate Solution ◽  
Numerical Methods ◽  
Compact Sets ◽  
Ill Posed

scholarly journals Determination of conductivity in a heat equation

2000 ◽  
Vol 24 (9) ◽  
pp. 589-594 ◽  
Author(s):  
Ping Wang ◽  
Kewang Zheng
Keyword(s):  
Numerical Simulation ◽  
Inverse Problem ◽  
Approximate Solution ◽  
Heat Equation ◽  
Ill Posed ◽  
Smooth Data

We consider the problem of determining the conductivity in a heat equation from overspecified non-smooth data. It is an ill-posed inverse problem. We apply a regularization approach to define and construct a stable approximate solution. We also conduct numerical simulation to demonstrate the accuracy of our approximation.

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 An optimal order yielding discrepancy principle for simplified regularization of ill-posed problems in Hilbert scales

2003 ◽  
Vol 2003 (39) ◽  
pp. 2487-2499 ◽  
Author(s):  
Santhosh George ◽  
M. Thamban Nair
Keyword(s):  
Approximate Solution ◽  
Hilbert Spaces ◽  
Regularization Parameter ◽  
Optimal Order ◽  
Discrepancy Principle ◽  
Parameter Choice ◽  
Bounded Linear ◽  
Choice Strategy ◽  
Ill Posed ◽  
Parameter Choice Strategy

Recently, Tautenhahn and Hämarik (1999) have considered a monotone rule as a parameter choice strategy for choosing the regularization parameter while considering approximate solution of an ill-posed operator equationTx=y, whereTis a bounded linear operator between Hilbert spaces. Motivated by this, we propose a new discrepancy principle for the simplified regularization, in the setting of Hilbert scales, whenTis a positive and selfadjoint operator. When the datayis known only approximately, our method provides optimal order under certain natural assumptions on the ill-posedness of the equation and smoothness of the solution. The result, in fact, improves an earlier work of the authors (1997).

A fast multilevel iteration method for solving linear ill-posed integral equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hongqi Yang ◽  
Rong Zhang
Keyword(s):  
Approximate Solution ◽  
Linear System ◽  
Integral Equations ◽  
Tikhonov Regularization ◽  
Iteration Method ◽  
Noise Level ◽  
Regularization Parameter ◽  
Ill Posed ◽  
Linear Integral ◽  
Linear Integral Equations

Abstract We propose a new concept of noise level: R ⁢ ( K * ) \mathcal{R}(K^{*}) -noise level for ill-posed linear integral equations in Tikhonov regularization, which extends the range of regularization parameter. This noise level allows us to choose a more suitable regularization parameter. Moreover, we also analyze error estimates of the approximate solution with respect to this noise level. For ill-posed integral equations, finding fast and effective numerical methods is a challenging problem. For this, we formulate a matrix truncated strategy based on multiscale Galerkin method to generate the linear system of Tikhonov regularization for ill-posed linear integral equations, which greatly reduce the computational complexity. To further reduce the computational cost, a fast multilevel iteration method for solving the linear system is established. At the same time, we also prove convergence rates of the approximate solution obtained by this fast method with respect to the R ⁢ ( K * ) \mathcal{R}(K^{*}) -noise level under the balance principle. By numerical results, we show that R ⁢ ( K * ) \mathcal{R}(K^{*}) -noise level is very useful and the proposed method is a fast and effective method, respectively.

scholarly journals Numerical Solution of Eikonal Equation Using Finite Difference Method

2018 ◽  
Vol 15 ◽  
pp. 8174-8184
Author(s):  
Sana'a Abdullah Alotibi
Keyword(s):  
Approximate Solution ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Solution ◽  
Wave Front ◽  
Difference Method ◽  
Eikonal Equation ◽  
Tsunami Wave ◽  
Ill Posed ◽  
The Finite Difference Method

In this paper, a method to calculate tsunami wave front is introduced using the finite difference method to solve the ill-posed problem and to calculate perturbed velocity of the wave front. Comparison between the actual and approximate solution will be proposed in a table form and a graphic form.

scholarly journals Placement Inverse Source Problem under Partition Hyperbolic Equation

2021 ◽  
pp. 1994-1999
Author(s):  
Shilan Othman Hussein
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Finite Difference Method ◽  
Hyperbolic Equation ◽  
Additional Condition ◽  
Inverse Source Problem ◽  
Ill Posed ◽  
The Finite Difference Method ◽  
Regularization Matrix ◽  
Regularized Solution

In this article, the inverse source problem is determined by the partition hyperbolic equation under the left end flux tension of the string, where the extra measurement is considered. The approximate solution is obtained in the form of splitting and applying the finite difference method (FDM). Moreover, this problem is ill-posed, dealing with instability of force after adding noise to the additional condition. To stabilize the solution, the regularization matrix is considered. Consequently, it is proved by error estimates between the regularized solution and the exact solution. The numerical results show that the method is efficient and stable.

Algorithms for the approximate solution of ill-posed problems on special sets

1995 ◽  
pp. 81-96
Author(s):  
A. N. Tikhonov ◽  
A. V. Goncharsky ◽  
V. V. Stepanov ◽  
A. G. Yagola
Keyword(s):  
Approximate Solution ◽  
Ill Posed
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