Matrix forms of iterative algorithms to solve large-scale discrete ill-posed problems with an application to image restoration

2018 ◽  
Vol 60 (1-2) ◽  
pp. 113-145
Author(s):  
Fernando Pazos ◽  
Amit Bhaya
Keyword(s):  
Image Restoration ◽  
Large Scale ◽  
Iterative Algorithms ◽  
Ill Posed

scholarly journals Bayesian Image Restoration Using a Large-Scale Total Patch Variation Prior

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Yang Chen ◽  
Weimin Yu ◽  
Yinsheng Li ◽  
Zhou Yang ◽  
Limin Luo ◽  
...  
Keyword(s):  
Image Restoration ◽  
Large Scale ◽  
Computational Cost ◽  
Real Data ◽  
Inverse Image ◽  
Edge Preserving ◽  
Device Architecture ◽  
Mixture Prior ◽  
Ill Posed ◽  
Improved Performance

Edge-preserving Bayesian restorations using nonquadratic priors are often inefficient in restoring continuous variations and tend to produce block artifacts around edges in ill-posed inverse image restorations. To overcome this, we have proposed a spatial adaptive (SA) prior with improved performance. However, this SA prior restoration suffers from high computational cost and the unguaranteed convergence problem. Concerning these issues, this paper proposes a Large-scale Total Patch Variation (LS-TPV) Prior model for Bayesian image restoration. In this model, the prior for each pixel is defined as a singleton conditional probability, which is in a mixture prior form of one patch similarity prior and one weight entropy prior. A joint MAP estimation is thus built to ensure the iteration monotonicity. The intensive calculation of patch distances is greatly alleviated by the parallelization of Compute Unified Device Architecture(CUDA). Experiments with both simulated and real data validate the good performance of the proposed restoration.

Novel Iterative Truncated Total Least Squares Algorithm for Image Restoration

2013 ◽  
Vol 401-403 ◽  
pp. 1397-1400
Author(s):  
Lei Zhang ◽  
Yue Yun Cao ◽  
Zi Chun Yang
Keyword(s):  
Image Restoration ◽  
Least Squares ◽  
Large Scale ◽  
Total Least Squares ◽  
System Matrix ◽  
Scale Problem ◽  
Truncation Parameter ◽  
Lanczos Bidiagonalization ◽  
Ill Posed ◽  
Truncated Total Least Squares

Image restoration is a typical ill-posed inverse problem, which can be solved by a successful total least squares (TLS) method when not only the observation but the system matrix is also contaminated by addition noise. Considering the image restoration is a large-scale problem in general, project the TLS problem onto a subspace defined by a Lanczos bidiagonalization algorithm, and then the Truncated TLS method is applied on the subspace. Therefore, a novel iterative TTLS method, involving appropriate the choice of truncation parameter, is proposed. Finally, an Image reconstruction example is given to illustrate the effectiveness and robustness of proposed algorithm.

scholarly journals Inverse Infiltration Modeling of Dike Covers Made of Dredged Material Using PEST and AMALGAM

Geosciences ◽  
2021 ◽  
Vol 11 (2) ◽  
pp. 41
Author(s):  
Tim Jurisch ◽  
Stefan Cantré ◽  
Fokke Saathoff
Keyword(s):  
Large Scale ◽  
Measurement Data ◽  
Fine Grained ◽  
Dredged Materials ◽  
Measurement Results ◽  
Large Scale Field ◽  
Eigenvalue Ratio ◽  
Ill Posed ◽  
Materials Used ◽  
Installation Technology

A variety of studies recently proved the applicability of different dried, fine-grained dredged materials as replacement material for erosion-resistant sea dike covers. In Rostock, Germany, a large-scale field experiment was conducted, in which different dredged materials were tested with regard to installation technology, stability, turf development, infiltration, and erosion resistance. The infiltration experiments to study the development of a seepage line in the dike body showed unexpected measurement results. Due to the high complexity of the problem, standard geo-hydraulic models proved to be unable to analyze these results. Therefore, different methods of inverse infiltration modeling were applied, such as the parameter estimation tool (PEST) and the AMALGAM algorithm. In the paper, the two approaches are compared and discussed. A sensitivity analysis proved the presumption of a non-linear model behavior for the infiltration problem and the Eigenvalue ratio indicates that the dike infiltration is an ill-posed problem. Although this complicates the inverse modeling (e.g., termination in local minima), parameter sets close to an optimum were found with both the PEST and the AMALGAM algorithms. Together with the field measurement data, this information supports the rating of the effective material properties of the applied dredged materials used as dike cover material.

scholarly journals Alternating Asymmetric Iterative Algorithm Based on Domain Decomposition for 3D Poisson Problem

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 281
Author(s):  
Qiuyan Xu ◽  
Zhiyong Liu
Keyword(s):  
Domain Decomposition ◽  
Iterative Algorithm ◽  
Poisson Equation ◽  
Large Scale ◽  
Linear Equations ◽  
Iterative Algorithms ◽  
Computation Time ◽  
Explicit Method ◽  
Computational Process ◽  
Poisson Problem

Poisson equation is a widely used partial differential equation. It is very important to study its numerical solution. Based on the strategy of domain decomposition, the alternating asymmetric iterative algorithm for 3D Poisson equation is provided. The solution domain is divided into several sub-domains, and eight asymmetric iterative schemes with the relaxation factor for 3D Poisson equation are constructed. When the numbers of iteration are odd or even, the computational process of the presented iterative algorithm are proposed respectively. In the calculation of the inner interfaces, the group explicit method is used, which makes the algorithm to be performed fast and in parallel, and avoids the difficulty of solving large-scale linear equations. Furthermore, the convergence of the algorithm is analyzed theoretically. Finally, by comparing with the numerical experimental results of Jacobi and Gauss Seidel iterative algorithms, it is shown that the alternating asymmetric iterative algorithm based on domain decomposition has shorter computation time, fewer iteration numbers and good parallelism.

A novel modified TRSVD method for large-scale linear discrete ill-posed problems

2020 ◽  
Author(s):  
Xianglan Bai ◽  
Guang-Xin Huang ◽  
Xiao-Jun Lei ◽  
Lothar Reichel ◽  
Feng Yin
Keyword(s):  
Large Scale ◽  
Ill Posed

Improved inversion of geopotential field anomalies for lithospheric investigations

Geophysics ◽  
1988 ◽  
Vol 53 (3) ◽  
pp. 375-385 ◽  
Author(s):  
R. R. B. von Frese ◽  
D. N. Ravat ◽  
W. J. Hinze ◽  
C. A. McGue
Keyword(s):  
Large Scale ◽  
Superposition Principle ◽  
Gravity Anomalies ◽  
Damping Factor ◽  
Stable Model ◽  
Memory Integration ◽  
Unstable Solutions ◽  
Ill Posed ◽  
Anomaly Analysis ◽  
Magnetic And Gravity Anomalies

Instabilities and the large matrices which are common to inversions of regional magnetic and gravity anomalies often complicate the use of efficient least‐squares matrix procedures. Inversion stability profoundly affects anomaly analysis, and hence it must be considered in any application. Wildly varying or unstable solutions are the products of errors in the anomaly observations and the integrated effects of observation spacing, source spacing, elevation differences between sources and observations, geographic coordinate attributes, geomagnetic field attitudes, and other factors which influence the conditioning of inversion. Solution instabilities caused by ill‐posed parameters can be efficiently minimized by ridge regression with a damping factor large enough to stabilize the inversion, but small enough to produce an analytically useful solution. An effective choice for the damping factor is facilitated by plotting damping factors against residuals between observed and modeled anomalies and by then comparing this curve to curves of damping factors plotted against solution variance or the residuals between predicted anomaly maps representing the processing objective (e.g., downward continuation, differential reduction to the radial pole, etc.). To obtain accurate and efficient large‐scale inversions of anomaly data, a procedure based on the superposition principle of potential fields may be used. This method involves successive inversions of residuals between the observations and various stable model fields which can be readily accommodated by available computer memory. Integration of the model fields yields a well‐resolved representation of the observed anomalies corresponding to an integrated model which normally could not be obtained by direct inversion because the memory requirements would be excessive. MAGSAT magnetic anomaly inversions over India demonstrate the utility of these procedures for improving the geologic analysis of potential field anomalies.

scholarly journals A conjugate gradient algorithm and its application in large-scale optimization problems and image restoration

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Gonglin Yuan ◽  
Tingting Li ◽  
Wujie Hu
Keyword(s):  
Image Restoration ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Optimization Problems ◽  
Trust Region ◽  
Conjugate Gradient Algorithm ◽  
Gradient Algorithm ◽  
Smooth Functions ◽  
Nonsmooth Functions ◽  
Large Scale Unconstrained Optimization

Abstract To solve large-scale unconstrained optimization problems, a modified PRP conjugate gradient algorithm is proposed and is found to be interesting because it combines the steepest descent algorithm with the conjugate gradient method and successfully fully utilizes their excellent properties. For smooth functions, the objective algorithm sufficiently utilizes information about the gradient function and the previous direction to determine the next search direction. For nonsmooth functions, a Moreau–Yosida regularization is introduced into the proposed algorithm, which simplifies the process in addressing complex problems. The proposed algorithm has the following characteristics: (i) a sufficient descent feature as well as a trust region trait; (ii) the ability to achieve global convergence; (iii) numerical results for large-scale smooth/nonsmooth functions prove that the proposed algorithm is outstanding compared to other similar optimization methods; (iv) image restoration problems are done to turn out that the given algorithm is successful.

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