scholarly journals Characterizations of Variational Source Conditions, Converse Results, and Maxisets of Spectral Regularization Methods

2017 ◽  
Vol 55 (2) ◽  
pp. 598-620 ◽  
Author(s):  
Thorsten Hohage ◽  
Frederic Weidling
Keyword(s):  
Regularization Methods ◽  
Spectral Regularization ◽  
Converse Results

Wavelet and spectral regularization methods for a sideways parabolic equation

2005 ◽  
Vol 160 (3) ◽  
pp. 881-908 ◽  
Author(s):  
Chu-Li Fu ◽  
Xiang-Tuan Xiong ◽  
Hong-Fang Li ◽  
You-Bin Zhu
Keyword(s):  
Parabolic Equation ◽  
Regularization Methods ◽  
Spectral Regularization

scholarly journals Spectral Algorithms for Supervised Learning

2008 ◽  
Vol 20 (7) ◽  
pp. 1873-1897 ◽  
Author(s):  
L. Lo Gerfo ◽  
L. Rosasco ◽  
F. Odone ◽  
E. De Vito ◽  
A. Verri
Keyword(s):  
Kernel Methods ◽  
Classification Performance ◽  
Matrix Inversion ◽  
Regularization Methods ◽  
Data Sets ◽  
Inversion Problem ◽  
Spectral Regularization ◽  
Numerical Stabilization ◽  
Ill Posed ◽  
Real World Problems

We discuss how a large class of regularization methods, collectively known as spectral regularization and originally designed for solving ill-posed inverse problems, gives rise to regularized learning algorithms. All of these algorithms are consistent kernel methods that can be easily implemented. The intuition behind their derivation is that the same principle allowing for the numerical stabilization of a matrix inversion problem is crucial to avoid overfitting. The various methods have a common derivation but different computational and theoretical properties. We describe examples of such algorithms, analyze their classification performance on several data sets and discuss their applicability to real-world problems.

scholarly journals Spectral Regularization Methods for an Abstract Ill-Posed Elliptic Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Nadjib Boussetila ◽  
Salim Hamida ◽  
Faouzia Rebbani
Keyword(s):  
Original Problem ◽  
Convergence Rates ◽  
A Priori ◽  
Stable Solution ◽  
Cauchy Data ◽  
Regularization Methods ◽  
Spectral Regularization ◽  
Convergence Results ◽  
A Priori Bound ◽  
Ill Posed

We study an abstract elliptic Cauchy problem associated with an unbounded self-adjoint positive operator which has a continuous spectrum. It is well-known that such a problem is severely ill-posed; that is, the solution does not depend continuously on the Cauchy data. We propose two spectral regularization methods to construct an approximate stable solution to our original problem. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution.

Comparisons of regularization methods and regularization parameter selection methods in sound source identification using inverse boundary element method

2019 ◽  
Vol 67 (3) ◽  
pp. 219-227
Author(s):  
Youhong Xiao ◽  
Qingqing Song ◽  
Shaowei Li ◽  
Guoxue Lv ◽  
Zhenlin Ji
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Transfer Matrix ◽  
Sound Source ◽  
Source Identification ◽  
Regularization Parameter ◽  
Vibration Velocity ◽  
Regularization Methods ◽  
Inverse Boundary ◽  
Element Method

In noise source identification based on the inverse boundary element method (IBEM), the boundary vibration velocity is predicted based on the field pressure through a transfer matrix of the vibration velocity and field pressure established on the Helmholtz integral equation. Because the matrix is often ill-posed, it needs to be regularized before reconstructing the vibration velocity. Two regularization methods and two methods of selecting the regularization parameter are investigated through the simulation analysis of a pulsating sphere. The result of transfer matrix regularization is further verified through the reconstruction of the vibration of an aluminum plate. Additionally, to reduce the large errors at some frequencies in the reconstruction result, increasing the number of measuring points is more effective than reducing the distance between the measurement plane and the sound source.

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