A reduced basis Landweber method for the identification of piecewise constant Robin coefficient in an elliptic equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Junjun Hu ◽  
Daijun Jiang
Keyword(s):  
Elliptic Equation ◽  
Computational Time ◽  
High Dimensional ◽  
Robin Problem ◽  
Reduced Basis ◽  
Iterative Regularization ◽  
Piecewise Constant ◽  
Ill Posed ◽  
Landweber Method ◽  
Basis Method

Abstract In this paper, we are concerned with the identification of the piecewise constant Robin coefficient in an elliptic equation. The iterative regularization method is one of the very effective methods for solving this kind of nonlinear ill-posed inverse problems. But it usually requires to solve numerous amounts of forward solutions during the iterative process, which will cost a lot of computational time in high-dimensional spaces. A reduced basis method is considered to reduce the computational time for solving the forward problems, and its error estimate is also studied. Finally, we propose a reduced basis Landweber algorithm to solve the elliptic inverse Robin problem and present several numerical experiments to demonstrate the accuracy and efficiency of the algorithm.

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 NUMERICAL HOMOGENIZATION OF A NONLINEARLY COUPLED ELLIPTIC–PARABOLIC SYSTEM, REDUCED BASIS METHOD, AND APPLICATION TO NUCLEAR WASTE STORAGE

2013 ◽  
Vol 23 (13) ◽  
pp. 2523-2560 ◽  
Author(s):  
ANTOINE GLORIA ◽  
THIERRY GOUDON ◽  
STELLA KRELL
Keyword(s):  
Coupled System ◽  
Slow Variable ◽  
Heterogeneous Porous Media ◽  
Computational Time ◽  
Reduced Basis ◽  
Nuclear Waste Storage ◽  
Waste Storage ◽  
Effective Coefficients ◽  
Basis Method

We consider the homogenization of a coupled system of PDEs describing flows in heterogeneous porous media. Due to the coupling, the effective coefficients always depend on the slow variable, even in the simple case when the porosity is periodic. Therefore the most important part of the computational time for the numerical simulation of such flows is dedicated to the determination of these coefficients. We propose a new numerical algorithm based on Reduced Basis techniques, which significantly improves the computational performances.

scholarly journals Computing Expectiles Using k-Nearest Neighbours Approach

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 645
Author(s):  
Muhammad Farooq ◽  
Sehrish Sarfraz ◽  
Christophe Chesneau ◽  
Mahmood Ul Hassan ◽  
Muhammad Ali Raza ◽  
...  
Keyword(s):  
Computational Cost ◽  
Real Life ◽  
Distance Measures ◽  
Computational Time ◽  
High Dimensional ◽  
Test Error ◽  
Nearest Neighbours ◽  
Comparable Performance ◽  
Asymmetric Least Squares ◽  
Low Computational Cost

Expectiles have gained considerable attention in recent years due to wide applications in many areas. In this study, the k-nearest neighbours approach, together with the asymmetric least squares loss function, called ex-kNN, is proposed for computing expectiles. Firstly, the effect of various distance measures on ex-kNN in terms of test error and computational time is evaluated. It is found that Canberra, Lorentzian, and Soergel distance measures lead to minimum test error, whereas Euclidean, Canberra, and Average of (L1,L∞) lead to a low computational cost. Secondly, the performance of ex-kNN is compared with existing packages er-boost and ex-svm for computing expectiles that are based on nine real life examples. Depending on the nature of data, the ex-kNN showed two to 10 times better performance than er-boost and comparable performance with ex-svm regarding test error. Computationally, the ex-kNN is found two to five times faster than ex-svm and much faster than er-boost, particularly, in the case of high dimensional data.

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.

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