Geodesic
Gradient descent for Tikhonov functionals with sparsity constraints: theory and numerical comparison of step size rules
Electronic transactions on numerical analysis, Tome 39 (2012), pp. 437-463.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: In this paper, we analyze gradient methods for minimization problems arising in the regularization of nonlinear inverse problems with sparsity constraints. In particular, we study a gradient method based on the subsequent minimization of quadratic approximations in Hilbert spaces, which is motivated by a recently proposed equivalent method in a finite-dimensional setting. We prove convergence of this method employing assumptions on the operator which are different compared to other approaches. We also discuss accelerated gradient methods with step size control and present a numerical comparison of different step size selection criteria for a parameter identification problem for an elliptic partial differential equation.
Classification : 65K10, 46N10, 65M32, 90C48
Keywords: nonlinear inverse problems, sparsity constraints, gradient descent, iterated soft shrinkage, accelerated gradient method 
@article{ETNA_2012__39__a1,
     author = {Lorenz, Dirk A. and Maass, Peter and Muoi, Pham Q.},
     title = {Gradient descent for {Tikhonov} functionals with sparsity constraints: theory and numerical comparison of step size rules},
     journal = {Electronic transactions on numerical analysis},
     pages = {437--463},
     publisher = {mathdoc},
     volume = {39},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2012__39__a1/}
}
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Lorenz, Dirk A.; Maass, Peter; Muoi, Pham Q. Gradient descent for Tikhonov functionals with sparsity constraints: theory and numerical comparison of step size rules. Electronic transactions on numerical analysis, Tome 39 (2012), pp. 437-463. http://geodesic.mathdoc.fr/item/ETNA_2012__39__a1/