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
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},
     year = {2012},
     volume = {39},
     zbl = {1287.65105},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2012__39__a1/}
}
TY  - JOUR
AU  - Lorenz,  Dirk A.
AU  - Maass,  Peter
AU  - Muoi,  Pham Q.
TI  - Gradient descent for Tikhonov functionals with sparsity constraints: theory and numerical comparison of step size rules
JO  - Electronic transactions on numerical analysis
PY  - 2012
SP  - 437
EP  - 463
VL  - 39
UR  - http://geodesic.mathdoc.fr/item/ETNA_2012__39__a1/
LA  - en
ID  - ETNA_2012__39__a1
ER  - 
%0 Journal Article
%A Lorenz,  Dirk A.
%A Maass,  Peter
%A Muoi,  Pham Q.
%T Gradient descent for Tikhonov functionals with sparsity constraints: theory and numerical comparison of step size rules
%J Electronic transactions on numerical analysis
%D 2012
%P 437-463
%V 39
%U http://geodesic.mathdoc.fr/item/ETNA_2012__39__a1/
%G en
%F ETNA_2012__39__a1
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/