The use of the L-curve in the regularization of discrete ill-posed problems
SIAM Journal on Scientific Computing
An introduction to the mathematical theory of inverse problems
An introduction to the mathematical theory of inverse problems
A general heuristic for choosing the regularization parameter in ill-posed problems
SIAM Journal on Scientific Computing
Manifold Regularization: A Geometric Framework for Learning from Labeled and Unlabeled Examples
The Journal of Machine Learning Research
New model function methods for determining regularization parameters in linear inverse problems
Applied Numerical Mathematics
Wavelet domain image restoration with adaptive edge-preserving regularization
IEEE Transactions on Image Processing
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In this paper, we study the multi-parameter Tikhonov regularization method which adds multiple different penalties to exhibit multi-scale features of the solution. An optimal error bound of the regularization solution is obtained by a priori choice of multiple regularization parameters. Some theoretical results of the regularization solution about the dependence on regularization parameters are presented. Then, an a posteriori parameter choice, i.e., the damped Morozov discrepancy principle, is introduced to determine multiple regularization parameters. Five model functions, i.e., two hyperbolic model functions, a linear model function, an exponential model function and a logarithmic model function, are proposed to solve the damped Morozov discrepancy principle. Furthermore, four efficient model function algorithms are developed for finding reasonable multiple regularization parameters, and their convergence properties are also studied. Numerical results of several examples show that the damped discrepancy principle is competitive with the standard one, and the model function algorithms are efficient for choosing regularization parameters.