The L-curve and its use in the numerical treatment of inverse problems

By ProblemsPer Christian HansenDepartment. 2000

994 citations recorded by the source corpus.

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Citations in the corpus

Accelerated Microstructure Imaging via Convex Optimization (AMICO) from diffusion MRI data (2015, methodology): “Ψ = ∥ ⋅ ∥ 1 is a popular choice to promote sparsity in the FOD (Ramirez-Manzanares et al., 2007; Michailovich et al., 2011; Landman et al., 2012), when in fact Tikhonov regularization (Tikhonov and Arsenin, 1977) is usually adopted to reduce ill-conditioning in the matrix Φ (Jian and Vemuri, 2007;…”
Preprocessing of Vector Magnetograph Data for a Nonlinear Force-Free Magnetic Field Reconstruction (2006, methodology): “…selection of these parameters we proceed in a similar way as it is customary for regularization techniques of inversion problems (e.g. Hansen (2000)) For this purpose we visualize the different solutions Bmin(µ3, µ4) for different µ parameters in a phase space of log(L12), log(L3) and log(L4).”
Comparing parameter choice methods for regularization of ill-posed problems (2011, background): “These effects have been observed in numerical experiments [72, 84].”
A survey of applications of the MFS to inverse problems (2011, methodology): “Alternatively, instead of the Tikhonov regularization method one can use the truncated singular value decomposition method [34,35].”
MRI Estimates of Brain Iron Concentration in Normal Aging Using Quantitative Susceptibility Mapping (2011, methodology): “Selection of regularization parameters λ and β To choose appropriate regularization parameters that balance data consistency and the amount of regularization, the L-curve method was employed (Hansen, 2000).”
On instabilities of deep learning in image reconstruction and the potential costs of AI (2019, background): “Hence, there is an extensive literature on designing stable methods for image reconstruction in inverse problems [1]– [4].”
The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations (2005, unknown): “Several regularization procedures have been developed to solve such ill-conditioned systems, see for example Hansen [43].”
TUnfold, an algorithm for correcting migration effects in high energy physics (2012, methodology): “Several methods to choose the strength of the regularisation are discussed in literature, for example eigenvalue analyses [14], minimisation of correlation coefficients [15], and the L-curve method [16].”