Finite dimensional realization of a quadratic convergence yielding iterative regularization method for ill-posed equations with monotone operators

2016 ◽  
Vol 273 ◽  
pp. 1041-1050
Author(s):  
Vorkady. S. Shubha ◽  
Santhosh George ◽  
P. Jidesh ◽  
M.E. Shobha
Keyword(s):  
Regularization Method ◽  
Quadratic Convergence ◽  
Monotone Operators ◽  
Iterative Regularization ◽  
Finite Dimensional ◽  
Ill Posed

scholarly journals Heating source localization in a reduced time

2016 ◽  
Vol 26 (3) ◽  
pp. 623-640 ◽  
Author(s):  
Sara Beddiaf ◽  
Laurent Autrique ◽  
Laetitia Perez ◽  
Jean-Claude Jolly
Keyword(s):  
Heat Conduction ◽  
Source Localization ◽  
Conjugate Gradient ◽  
Regularization Method ◽  
Three Dimensional ◽  
Gradient Algorithm ◽  
Iterative Regularization ◽  
Heating Source ◽  
Ill Posed ◽  
Reduced Time

Abstract Inverse three-dimensional heat conduction problems devoted to heating source localization are ill posed. Identification can be performed using an iterative regularization method based on the conjugate gradient algorithm. Such a method is usually implemented off-line, taking into account observations (temperature measurements, for example). However, in a practical context, if the source has to be located as fast as possible (e.g., for diagnosis), the observation horizon has to be reduced. To this end, several configurations are detailed and effects of noisy observations are investigated.

scholarly journals On fractional asymptotical regularization of linear ill-posed problems in hilbert spaces

2019 ◽  
Vol 22 (3) ◽  
pp. 699-721 ◽  
Author(s):  
Ye Zhang ◽  
Bernd Hofmann
Keyword(s):  
Fractional Order ◽  
Hilbert Spaces ◽  
Regularization Method ◽  
Numerical Realization ◽  
Regularization Methods ◽  
Iterative Regularization ◽  
Operator Equations ◽  
Ill Posed ◽  
One Step ◽  
The One

Abstract In this paper, we study a fractional-order variant of the asymptotical regularization method, called Fractional Asymptotical Regularization (FAR), for solving linear ill-posed operator equations in a Hilbert space setting. We assign the method to the general linear regularization schema and prove that under certain smoothness assumptions, FAR with fractional order in the range (1, 2) yields an acceleration with respect to comparable order optimal regularization methods. Based on the one-step Adams-Moulton method, a novel iterative regularization scheme is developed for the numerical realization of FAR. Two numerical examples are given to show the accuracy and the acceleration effect of FAR.

scholarly journals Smoothing and Regularization with Modified Sparse Approximate Inverses

2010 ◽  
Vol 2010 ◽  
pp. 1-16 ◽  
Author(s):  
T. Huckle ◽  
M. Sedlacek
Keyword(s):  
Laplace Operator ◽  
Constant Coefficient ◽  
Regularization Method ◽  
Low Frequency ◽  
Attractive Alternative ◽  
Iterative Regularization ◽  
Dynamic Computation ◽  
Approximate Inverses ◽  
Ill Posed ◽  
Original Information

Sparse approximate inverses which satisfy have shown to be an attractive alternative to classical smoothers like Jacobi or Gauss-Seidel (Tang and Wan; 2000). The static and dynamic computation of a SAI and a SPAI (Grote and Huckle; 1997), respectively, comes along with advantages like inherent parallelism and robustness with equal smoothing properties (Bröker et al.; 2001). Here, we are interested in developing preconditioners that can incorporate probing conditions for improving the approximation relative to high- or low-frequency subspaces. We present analytically derived optimal smoothers for the discretization of the constant-coefficient Laplace operator. On this basis, we introduce probing conditions in the generalized Modified SPAI (MSPAI) approach (Huckle and Kallischko; 2007) which yields efficient smoothers for multigrid. In the second part, we transfer our approach to the domain of ill-posed problems to recover original information from blurred signals. Using the probing facility of MSPAI, we impose the preconditioner to act as approximately zero on the noise subspace. In combination with an iterative regularization method, it thus becomes possible to reconstruct the original information more accurately in many cases. A variety of numerical results demonstrate the usefulness of this approach.

A modified iterative regularization method for ill-posed problems

2017 ◽  
Vol 122 ◽  
pp. 108-128 ◽  
Author(s):  
Xiangtuan Xiong ◽  
Xuemin Xue ◽  
Zhi Qian
Keyword(s):  
Regularization Method ◽  
Iterative Regularization ◽  
Ill Posed

scholarly journals An iterative regularization method for variational inequalities in Hilbert spaces

2020 ◽  
Vol 36 (3) ◽  
pp. 475-482
Author(s):  
HONG-KUN XU ◽  
NAJLA ALTWAIJRY ◽  
SOUHAIL CHEBBI
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Hilbert Spaces ◽  
Regularization Method ◽  
Iterative Solution ◽  
Iterative Regularization ◽  
Lipschitz Continuous ◽  
Ill Posed ◽  
Feasible Solutions ◽  
First Time

We consider an iterative method for regularization of a variational inequality (VI) defined by a Lipschitz continuous monotone operator in the case where the set of feasible solutions is decomposed to the intersection of finitely many closed convex subsets of a Hilbert space. We prove the strong convergence of the sequence generated by our algorithm. It seems that this is the first time in the literature to handle iterative solution of ill-posed VIs in the domain decomposition case.

Iterative regularization method for an abstract ill-posed generalized elliptic equation

2020 ◽  
pp. 2150069
Author(s):  
Sassane Roumaissa ◽  
Boussetila Nadjib ◽  
Rebbani Faouzia ◽  
Benrabah Abderafik
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Regularization Method ◽  
Boundary Data ◽  
Identification Problem ◽  
Infinite Domain ◽  
Iterative Regularization ◽  
Convergence Results ◽  
Ill Posed ◽  
Well Posed

A preconditioning version of the Kozlov–Maz’ya iteration method for the stable identification of missing boundary data is presented for an ill-posed problem governed by generalized elliptic equations. The ill-posed data identification problem is reformulated as a sequence of well-posed fractional elliptic equations in infinite domain. Moreover, some convergence results are established. Finally, numerical results are included showing the accuracy and efficiency of the proposed method.

scholarly journals Determination of a power density by an entropy regularization method

2005 ◽  
Vol 2005 (2) ◽  
pp. 127-152 ◽  
Author(s):  
Olivier Prot ◽  
Maïtine Bergounioux ◽  
Jean Gabriel Trotignon
Keyword(s):  
Inverse Problem ◽  
Electromagnetic Field ◽  
Electromagnetic Wave ◽  
Power Density ◽  
Regularization Method ◽  
Power Density Distribution ◽  
Numerical Examples ◽  
Finite Dimensional ◽  
Ill Posed

The determination of directional power density distribution of an electromagnetic wave from the electromagnetic field measurement can be expressed as an ill-posed inverse problem. We consider the resolution of this inverse problem via a maximum entropy regularization method. A finite-dimensional algorithm is derived from optimality conditions, and we prove its convergence. A variant of this algorithm is also studied. This second one leads to a solution which maximizes entropy in the probabilistic sense. Some numerical examples are given.

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