#### DMCA

## Analysis of Bounded Variation Penalty Methods for Ill-Posed Problems (1994)

Venue: | INVERSE PROBLEMS |

Citations: | 165 - 1 self |

### Citations

2232 | Nonlinear total variation based noise removal algorithms
- Rudin, Osher, et al.
- 1992
(Show Context)
Citation Context ...he size of the error in the data and J(u) is the Bounded Variation (BV) norm or seminorm of u (see [4] for definitions and background). This is essentially the approach taken by Rudin and Osher et al =-=[8]-=-, [9]. A closely related approach is taken by Dobson and Santosa [3], where the constraint in (1.3) is replaced by the operator equation (1.1). In the application considered in [3], discretizations of... |

697 |
Nonlinear Functional Analysis
- Deimling
- 1980
(Show Context)
Citation Context ...mma1 , dimension ds2, and T is weakly lower semicontinuous, then a solution also exists. In either case, the solution is unique if T is strictly convex. Proof. The following argument is standard (see =-=[2]-=-): Let u n be a minimizing sequence for T ; in other words, T (u n ) ! inf u2L p (\Omega\Gamma T (u) def = T min : (3.3) By hypothesis (3.2), the u n 's are BV-bounded. As a consequence of Theorem 2.6... |

625 |
Minimal Surfaces and Functions of Bounded Variation, BirkhĂ¤user
- Giusti
- 1994
(Show Context)
Citation Context ...tion problem like min u J(u) subject to kAu \Gamma zk 2 = oe 2 ; (1.3) where oe 2 is an estimate of the size of the error in the data and J(u) is the Bounded Variation (BV) norm or seminorm of u (see =-=[4]-=- for definitions and background). This is essentially the approach taken by Rudin and Osher et al [8], [9]. A closely related approach is taken by Dobson and Santosa [3], where the constraint in (1.3)... |

85 |
Applications of Functional Analysis and Operator Theory
- Hutson, Pym, et al.
- 2005
(Show Context)
Citation Context ...j 1=p + CMskuk L 1(\Omega\Gamma j\Omega j 1=p\Gamma1 + CM (j\Omega j 1=p\Gamma1 + C)M = (j\Omega j \Gamma1=d + C)M: Relative weak compactness in dimensions ds2 follows from the Banach-Alaoglu Theorem =-=[6]-=-. The following example shows that the above result is sharp. Example 2.1. Let\Omega = fx 2 R d : jxj ! 2g and u n = n d\Gamma1sn , wheresn (x) = ( 1; if jxjs1 n ; 0; otherwise: Let ! d denote the vol... |

28 |
An image enhancement technique for electrical impedance tomography
- Dobson, Santosa
- 1994
(Show Context)
Citation Context ... as "regularization " methods, since stability was typically obtained by imposing smoothness constraints on the approximate solutions. In many applications, particularly in image processing =-=(see [9], [3]-=-) and parameter identification (see [5]), a serious shortcoming of standard regularization methods is that they do not allow discontinuous solutions. This difficulty can be overcome Department of Math... |

28 |
Identi of discontinuous parameters in flow equations
- Gutman
- 1990
(Show Context)
Citation Context ...ability was typically obtained by imposing smoothness constraints on the approximate solutions. In many applications, particularly in image processing (see [9], [3]) and parameter identification (see =-=[5]-=-), a serious shortcoming of standard regularization methods is that they do not allow discontinuous solutions. This difficulty can be overcome Department of Mathematics, Eastern Montana College, Billi... |

14 | C.R.Vogel,Well-posedness and convergence of some regularization methods for nonlinear ill-posed problems, Inverse Problems 5
- Seidman
- 1989
(Show Context)
Citation Context ... to an underlying exact solution as perturbations in the data and the operator vanish. Results in sections 3-5 (in particular, Theorems 3.1 and 5.1) are similar to those obtained by Seidman and Vogel =-=[11]-=- for ill-posed nonlinear operator equations in a reflexive Banach space setting. The stronger results (in particular, stability in Theorem 3.2) obtained in this paper rely on the linearity of the oper... |

10 |
Funzioni BV e tracce
- Anzellotti, Giaquinta
- 1978
(Show Context)
Citation Context ...ctions S is defined to be BV-bounded if there exists a constant B ? 0 for which kukBVsB for all u 2 S. The relative compactness of BV-bounded sets in L p(\Omega\Gamma follows from the next lemma (see =-=[1]-=- and [4, p. 14]): Lemma 2.5. If u 2 BV(\Omega\Gamma , then there exists a sequence fu n g in C 1 (\Omega\Gamma such that limku n \Gamma uk L p (\Omega\Gamma = 0 and limJ 0 (u n ) = J 0 (u). Theorem 2.... |

6 |
Reconstruction of blocky impedance profile from normal-incidence reflection seismograms which are band-limited and miscalibrated
- Santosa, Symes
- 1988
(Show Context)
Citation Context ... BV functions in a parameter identification setting (where a constraint on J(u) is imposed instead) is the paper by Gutman ([5]). Another closely related approach, which is taken by Santosa and Symes =-=[10]-=- and Vogel [13], is to solve the unconstrained minimization problem min u kAu \Gamma zk 2 + ffJ(u): (1.4) This can be viewed as a penalty method approach to solving the constrained minimization proble... |

6 |
Total variation regularization for ill-posed problems
- Vogel
- 1993
(Show Context)
Citation Context ...n a parameter identification setting (where a constraint on J(u) is imposed instead) is the paper by Gutman ([5]). Another closely related approach, which is taken by Santosa and Symes [10] and Vogel =-=[13]-=-, is to solve the unconstrained minimization problem min u kAu \Gamma zk 2 + ffJ(u): (1.4) This can be viewed as a penalty method approach to solving the constrained minimization problem (1.3). Here t... |

3 |
Denoising and Deblurring Algorithms with Constrained Nonlinear PDE's, submitted to
- Lions, Osher, et al.
(Show Context)
Citation Context ...(to an unperturbed problem)? The goal of this paper is to provide qualitative answers to these questions. The analysis here is substantially different from that of Lions, Osher and Rudin presented in =-=[7]-=-, which deals with artificial time-evolution algorithms to solve the Euler-Lagrange equations (obtained from first order necessary conditions) for a constrained problem similar to (1.3). This paper de... |

2 |
Total Variation Based Restoration of Noisy
- Rudin, Osher, et al.
(Show Context)
Citation Context ...egral operators Au(x) = Z \Omega k(x; y) u(y) dy: (1.2) For example, certain blurring effects in image processing may be described by convolution operators, in which case k(x; y) = k(x \Gamma y). See =-=[9]-=-. Problem (1.1) is ill-posed and discretizations of it are highly ill-conditioned. To deal with ill-posedness, one should apply methods which impose stability while retaining certain desired features ... |