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An efficient method to set up a Lanczos based preconditioner for discrete ill-posed problems

Shervan Erfani, Ali Tavakoli, Davod Khojasteh Salkuyeh
2013 Applied Mathematical Modelling  
Hosseini, Computing 88(2010), 79-96] presented a Lanczos based preconditioner for discrete ill-posed problems.  ...  Their preconditioner is constructed by using few steps (e.g., k) of the Lanczos bidiagonalization and corresponding computed singular values and right Lanczos vectors.  ...  In [15] , a new regularized preconditioner obtained by k steps of Lanczos bidiagonalization for discrete ill-posed problems has been introduced.  ... 
doi:10.1016/j.apm.2013.03.064 fatcat:qspft3iek5c2tl6xttjzvipcgm

Constraint interface preconditioning for topology optimization problems [article]

Michal Kocvara and Daniel Loghin and James Turner
2015 arXiv   pre-print
The main contribution is the choice of interface preconditioner which allows for the acceleration of the domain decomposition method, leading to performance independent of problem size.  ...  ill-conditioning, strong nonlinearity and size.  ...  By the well-posedness of problems of the form (3.1-3.3) for the case ∂Ω N = ∅, we conclude that the associated first-order optimality conditions are well-posed and yield a Jacobian matrix which is non-singular  ... 
arXiv:1510.04568v1 fatcat:t2htymozvbhg7ccjrglixj6foy

Page 6602 of Mathematical Reviews Vol. , Issue 2001I [page]

2001 Mathematical Reviews  
Both approaches are tested on many challenging problems arising in the field of elasticity and diffusion, which are discretized by finite elements.  ...  Such problems are normally very difficult when iterative methods are used. Therefore, finding a good preconditioner is an important task.  ... 

A Comparison of Preconditioned Krylov Subspace Methods for Large-Scale Nonsymmetric Linear Systems [article]

Aditi Ghai, Cao Lu, Xiangmin Jiao
2018 arXiv   pre-print
BoomerAMG with proper choice of coarsening and interpolation techniques typically converges faster than ML, but both may fail for ill-conditioned or saddle-point problems while multilevel ILU tends to  ...  While implementations of preconditioned KSP methods are usually readily available, it is unclear to users which methods are the best for different classes of problems.  ...  Acknowledgements The authors acknowledge supports by DoD-ARO under contract #W911NF0910306 and by Argonne National Laboratory under Contract DE-AC02-06CH11357 for the SciDAC program funded by the Office  ... 
arXiv:1607.00351v4 fatcat:62uskwa745gkdd7kvpooob47ly

Page 6299 of Mathematical Reviews Vol. , Issue 93k [page]

1993 Mathematical Reviews  
ill-posed problems by means of the L-curve.  ...  When discrete ill-posed problems are analyzed and solved by various numerical regular- ization techniques, a convenient way to display information about the regularized solution is to plot the norm or  ... 

Constraint Interface Preconditioning for Topology Optimization Problems

M. Kočvara, D. Loghin, J. Turner
2016 SIAM Journal on Scientific Computing  
By the well-posedness of problems of the form (3.1)-(3.3) for the case ∂Ω N = ∅, we conclude that the associated first-order optimality conditions are well-posed and yield a Jacobian matrix which is nonsingular  ...  The problem was reformulated in order to allow for well-posed subdomain problems which could be viewed as local Jacobian solves.  ... 
doi:10.1137/140980387 fatcat:aw7za35hhrawbnhetkvusckcki

Page 4316 of Mathematical Reviews Vol. , Issue 2000f [page]

2000 Mathematical Reviews  
For some choices of the function e, this problem is an ill- posed problem in the classical optimisation methods sense, such as the nonlinear least squares.  ...  Chapter eight is the most exhaustive discussion of matrix-based preconditioners that comes to mind.  ... 

A new trust region algorithm for image restoration

Zaiwen WEN
2005 Science in China Series A  
Lanczos method.  ...  In this paper we use the trust region method to deal with the image restoration problem, meanwhile, the trust region subproblem is solved by the truncated Lanczos method and the preconditioned truncated  ...  also partially supported by the Special Innovation Fund for graduate students of CAS.  ... 
doi:10.1360/03ys0178 fatcat:vfpatlwpt5fq7pye66w7sbgl2u

A Trust-Region Approach to the Regularization of Large-Scale Discrete Forms of Ill-Posed Problems

Marielba Rojas, Danny C. Sorensen
2002 SIAM Journal on Scientific Computing  
We consider large-scale least squares problems where the coefficient matrix comes from the discretization of an operator in an ill-posed problem, and the right-hand side contains noise.  ...  The method relies on matrix-vector products only, has low and fixed storage requirements, and can handle the singularities arising in ill-posed problems.  ...  Bill also provided the model seismic problem in section 5.3 and the AVO data set.  ... 
doi:10.1137/s1064827500378167 fatcat:mwabpgtu4fb57bm4xy4elikf6q

Acceleration techniques for regularized Newton methods applied to electromagnetic inverse medium scattering problems

Thorsten Hohage, Stefan Langer
2010 Inverse Problems  
We study the construction and updating of spectral preconditioners for regularized Newton methods and their application to electromagnetic inverse medium scattering problems.  ...  Moreover, we show how a Lepskiĭ-type stopping rule can be implemented efficiently for these methods.  ...  For an overview on iterative regularization methods for nonlinear ill-posed problems we refer to [22] .  ... 
doi:10.1088/0266-5611/26/7/074011 fatcat:ujqcofcxencpzjrjt2luyum2mq

A Robust Preconditioner for High-Contrast Problems [article]

Yuliya Gorb, Daria Kurzanova, Yuri Kuznetsov
2018 arXiv   pre-print
Then a robust preconditioner for the Lanczos method of minimized iterations for solving the derived saddle point problem is proposed.  ...  This paper introduces a procedure by which the discrete system obtained from a linear finite element discretization of the given continuum problem is converted into an equivalent linear system of the saddle  ...  Below, we propose a block-diagonal preconditioner for the Lanczos method employed to solve the problem (3) , and this preconditioner is also singular.  ... 
arXiv:1801.01578v1 fatcat:or5fgqa4inhzzcf22tzy53q5ky

Regularization of inverse problems by an approximate matrix-function technique

Stefano Cipolla, Marco Donatelli, Fabio Durastante
2021 Numerical Algorithms  
AbstractIn this work, we introduce and investigate a class of matrix-free regularization techniques for discrete linear ill-posed problems based on the approximate computation of a special matrix-function  ...  Numerical tests on a gallery of standard benchmark problems are included to prove the efficacy of our approach even for problems affected by a high level of noise.  ...  Introduction Object of this work is the numerical solution of the problem A n,m x m = g n ≡ḡ n + ε, A n,m ∈ R n×m , x m ∈ R m , g n , ε ∈ R n , n,m > 0, (1) obtained from the discretization of an ill-posed  ... 
doi:10.1007/s11075-021-01076-y fatcat:tw4kwgdrtnh2hlhxmbzsc5hcpq

Page 2521 of Mathematical Reviews Vol. , Issue 98D [page]

1998 Mathematical Reviews  
The focus is on discrete ill-posed problems (problems with very ill-conditioned coefficient matrices whose sin- gular values decay gradually), for which the standard truncated TLS (T-TLS) method is ineffective  ...  For certain choices of preconditioners, he obtains upper bounds for the leading term in the perturbation expansion of the eigenvalues arising from O(e) perturbation of the matrix elements.  ... 

Preconditioned Recycling Krylov subspace methods for self-adjoint problems [article]

André Gaul, Nico Schlömer
2015 arXiv   pre-print
Such problems appear, for example, in the Newton process for solving nonlinear equations.  ...  The method is designed to work with arbitrary inner products and arbitrary self-adjoint positive-definite preconditioners whose inverse can be computed with high accuracy.  ...  The authors wish to thank Jörg Liesen for his valuable feedback, Alexander Schlote for providing experimental results with harmonic Ritz vectors and the anonymous referees for their helpful remarks.  ... 
arXiv:1208.0264v4 fatcat:jywlrv4is5dq7ktxwmxe3ihwsm

Singular Value Decomposition Approximation via Kronecker Summations for Imaging Applications [article]

Clarissa Garvey, Chang Meng, James G. Nagy
2018 arXiv   pre-print
We provide theoretical results and numerical experiments to demonstrate the accuracy of our approximation and show how the approximation can be used to solve large scale ill-posed inverse problems, either  ...  as an approximate filtering method, or as a preconditioner to accelerate iterative algorithms.  ...  In this paper we are concerned with computing approximations of large scale linear systems that arise from discretization of ill-posed inverse problems in imaging applications.  ... 
arXiv:1803.11525v2 fatcat:lvvv4j47sfeq3chmxcx66p2uiq
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