scholarly journals Generalized cross validation for ℓp-ℓq minimization

2021 ◽  
Author(s):  
Alessandro Buccini ◽  
Lothar Reichel
Keyword(s):  
Minimization Problem ◽  
Original Problem ◽  
Cross Validation ◽  
Regularization Parameter ◽  
Relative Importance ◽  
Science And Engineering ◽  
Generalized Cross Validation ◽  
Minimization Problems ◽  
Ill Posed ◽  
Well Posed

AbstractDiscrete ill-posed inverse problems arise in various areas of science and engineering. The presence of noise in the data often makes it difficult to compute an accurate approximate solution. To reduce the sensitivity of the computed solution to the noise, one replaces the original problem by a nearby well-posed minimization problem, whose solution is less sensitive to the noise in the data than the solution of the original problem. This replacement is known as regularization. We consider the situation when the minimization problem consists of a fidelity term, that is defined in terms of a p-norm, and a regularization term, that is defined in terms of a q-norm. We allow 0 < p,q ≤ 2. The relative importance of the fidelity and regularization terms is determined by a regularization parameter. This paper develops an automatic strategy for determining the regularization parameter for these minimization problems. The proposed approach is based on a new application of generalized cross validation. Computed examples illustrate the performance of the method proposed.

On generalized cross validation for stable parameter selection in disease models

2015 ◽  
Vol 23 (5) ◽  
Author(s):  
Alexandra Smirnova ◽  
Maia Martcheva ◽  
Hui Liu
Keyword(s):  
Inverse Problem ◽  
Operator Equation ◽  
Recovery Rate ◽  
Cross Validation ◽  
Regularization Parameter ◽  
Parameter Selection ◽  
Time Dependent ◽  
Disease Models ◽  
Generalized Cross Validation ◽  
Irregular Operator

AbstractIn this paper we study advantages and limitations of the Generalized Cross Validation (GCV) approach for selecting a regularization parameter in the case of a partially stochastic linear irregular operator equation. The research has been motivated by an inverse problem in epidemiology, where the goal was to reconstruct a time dependent treatment recovery rate for

scholarly journals A variational non-linear constrained model for the inversion of FDEM data

2021 ◽  
Author(s):  
Alessandro Buccini ◽  
Patricia Díaz de Alba
Keyword(s):  
Variational Problem ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Real Data ◽  
Measured Data ◽  
Non Invasive ◽  
Non Linear ◽  
Ill Posed ◽  
Invasive Techniques ◽  
Well Posed

Abstract Reconstructing the structure of the soil using non-invasive techniques is a very relevant problem in many scientific fields, like geophysics and archaeology. This can be done, for instance, with the aid of Frequency Domain Electromagnetic (FDEM) induction devices. Inverting FDEM data is a very challenging inverse problem, as the problem is extremely ill-posed, i.e., sensible to the presence of noise in the measured data, and non-linear. Regularization methods substitute the original ill-posed problem with a well-posed one whose solution is an accurate approximation of the desired one. In this paper we develop a regularization method to invert FDEM data. We propose to determine the electrical conductivity of the ground by solving a variational problem. The minimized functional is made up by the sum of two term: the data fitting term ensures that the recovered solution fits the measured data, while the regularization term enforces sparsity on the Laplacian of the solution. The trade-off between the two terms is determined by the regularization parameter. This is achieved by minimizing an $\ell_2-\ell_q$ functional with $0<q\leq 2$. Since the functional we wish to minimize is non-convex, we show that the variational problem admits a solution. Moreover, we prove that, if the regularization parameter is tuned accordingly to the amount of noise present in the data, this model induces a regularization method. Some selected numerical examples on synthetic and real data show the good performances of our proposal.

scholarly journals On tensor GMRES and Golub-Kahan methods via the T-product for color image processing

2021 ◽  
Vol 37 ◽  
pp. 524-543
Author(s):  
Mohamed El Guide ◽  
Alaa El Ichi ◽  
Khalide Jbilou ◽  
Rachid Sadaka
Keyword(s):  
Tikhonov Regularization ◽  
Cross Validation ◽  
Krylov Subspace ◽  
Color Image ◽  
Regularization Parameter ◽  
Color Image Processing ◽  
Regularization Technique ◽  
Image Sequence Processing ◽  
Ill Posed ◽  
Selection Of

The present paper is concerned with developing tensor iterative Krylov subspace methods to solve large multi-linear tensor equations. We use the T-product for two tensors to define tensor tubal global Arnoldi and tensor tubal global Golub-Kahan bidiagonalization algorithms. Furthermore, we illustrate how tensor-based global approaches can be exploited to solve ill-posed problems arising from recovering blurry multichannel (color) images and videos, using the so-called Tikhonov regularization technique, to provide computable approximate regularized solutions. We also review a generalized cross-validation and discrepancy principle type of criterion for the selection of the regularization parameter in the Tikhonov regularization. Applications to image sequence processing are given to demonstrate the efficiency of the algorithms.

A unified treatment for Tikhonov regularization using a general stabilizing operator

2015 ◽  
Vol 13 (02) ◽  
pp. 201-215
Author(s):  
M. T. Nair
Keyword(s):  
Tikhonov Regularization ◽  
Regularization Parameter ◽  
Optimal Error Estimates ◽  
Bounded Linear ◽  
Densely Defined Operator ◽  
Special Cases ◽  
Ill Posed ◽  
Residual Norm ◽  
Well Posed ◽  
Densely Defined

While dealing with the problem of solving an ill-posed operator equation Tx = y, where T : X → Y is a bounded linear operator between Hilbert spaces X and Y, one looks for a stable method for approximating [Formula: see text], a least-residual norm solution which minimizes a seminorm x ↦ ‖Lx‖, where L : D(L) ⊆ X → X is a (possibly unbounded) closed densely defined operator in X. If the operators T and L satisfy a completion condition ‖Tx‖2 + ‖Lx‖2 ≥ γ‖x‖2 for all x ∈ D(L*L) for some constant γ > 0, then Tikhonov regularization is one of the simple and widely used of such procedures in which the regularized solution is obtained by solving a well-posed equation [Formula: see text] where yδ is a noisy data and α > 0 is the regularization parameter to be chosen appropriately. We prescribe a condition on (T, L) which unifies the analysis for ordinary Tikhonov regularization, that is, L = I, and also the case of L = Bs with B being a strictly positive closed densely defined unbounded operator which generates a Hilbert scale {Xt}t>0. Under the new framework, we provide estimates for the best possible worst error and order optimal error estimates for the regularized solutions under certain general source condition which incorporates in its fold many existing results as special cases, by choosing regularization parameter using a Morozov-type discrepancy principle.

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