Generalized Tikhonov-type regularization method for the Cauchy problem of a semi-linear elliptic equation

This paper considers the Cauchy problem of a semi-linear elliptic equation and uses a generalized Tikhonov-type regularization method to overcome its ill-posedness. The existence, uniqueness, and stability for regularized solution are proven. Under an a priori bound assumption for exact solution, we...

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Bibliographic Details
Published in:Numerical algorithms Vol. 81; no. 3; pp. 833 - 851
Main Authors: Zhang, Hongwu, Zhang, Xiaoju
Format: Journal Article
Language:English
Published: New York Springer US 01.07.2019
Springer Nature B.V
Subjects:
Algebra
Algorithms
Cauchy problems
Computer Science
Convergence
Eigenvalues
Exact solutions
Helmholtz equations
Integral equations
Iterative methods
Numeric Computing
Numerical Analysis
Original Paper
Regularization
Regularization methods
Stability
Theory of Computation
Semi-linear elliptic equation
Ill-posed problem
Convergence estimate
Generalized Tikhonov-type regularization method
Cauchy problem
ISSN:1017-1398, 1572-9265
Online Access:Get full text
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Summary:This paper considers the Cauchy problem of a semi-linear elliptic equation and uses a generalized Tikhonov-type regularization method to overcome its ill-posedness. The existence, uniqueness, and stability for regularized solution are proven. Under an a priori bound assumption for exact solution, we derive the convergence estimate of H ö lder type for this method. An application of this method to the Cauchy problem of Helmholtz equation is discussed, and we investigate the stability and convergence estimates for different wave numbers. Finally, an iterative scheme is constructed to calculate the regularization solution, numerical results show that this method is stable and feasible.
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ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-018-0573-4