TY  - JOUR
A1  - Pornsawad, Pornsarp
A1  - Sungcharoen, Parada
A1  - Böckmann, Christine
T1  - Convergence rate of the modified Landweber method for solving inverse potential problems
JF  - Mathematics : open access journal
N2  - In this paper, we present the convergence rate analysis of the modified Landweber method under logarithmic source condition for nonlinear ill-posed problems. The regularization parameter is chosen according to the discrepancy principle. The reconstructions of the shape of an unknown domain for an inverse potential problem by using the modified Landweber method are exhibited.
KW  - nonlinear operator
KW  - regularization
KW  - modified Landweber method
KW  - discrepancy principle
KW  - logarithmic source condition
Y1  - 2020
U6  - https://doi.org/10.3390/math8040608
SN  - 2227-7390
VL  - 8
IS  - 4
PB  - MDPI
CY  - Basel
ER  - 
TY  - JOUR
A1  - Pornsawad, Pornsarp
A1  - Sapsakul, Nantawan
A1  - Böckmann, Christine
T1  - A modified asymptotical regularization of nonlinear ill-posed problems
JF  - Mathematics
N2  - In this paper, we investigate the continuous version of modified iterative Runge–Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of ∥𝐹(𝑥𝛿(𝑇))−𝑦𝛿∥=𝜏𝛿+ for some 𝛿+>𝛿, and an appropriate source condition. We yield the optimal rate of convergence.
KW  - nonlinear operator
KW  - regularization
KW  - discrepancy principle
KW  - asymptotic method
KW  - optimal rate
Y1  - 2019
U6  - https://doi.org/10.3390/math7050419
SN  - 2227-7390
VL  - 7
PB  - MDPI
CY  - Basel, Schweiz
ET  - 5
ER  - 
TY  - JOUR
A1  - Blanchard, Gilles
A1  - Hoffmann, Marc
A1  - Reiss, Markus
T1  - Optimal adaptation for early stopping in statistical inverse problems
JF  - SIAM/ASA Journal on Uncertainty Quantification
N2  - For linear inverse problems Y = A mu + zeta, it is classical to recover the unknown signal mu by iterative regularization methods ((mu) over cap,(m) = 0,1, . . .) and halt at a data-dependent iteration tau using some stopping rule, typically based on a discrepancy principle, so that the weak (or prediction) squared-error parallel to A((mu) over cap (()(tau)) - mu)parallel to(2) is controlled. In the context of statistical estimation with stochastic noise zeta, we study oracle adaptation (that is, compared to the best possible stopping iteration) in strong squared- error E[parallel to((mu) over cap (()(tau)) - mu)parallel to(2)]. For a residual-based stopping rule oracle adaptation bounds are established for general spectral regularization methods. The proofs use bias and variance transfer techniques from weak prediction error to strong L-2-error, as well as convexity arguments and concentration bounds for the stochastic part. Adaptive early stopping for the Landweber method is studied in further detail and illustrated numerically.
KW  - linear inverse problems
KW  - early stopping
KW  - discrepancy principle
KW  - adaptive estimation
KW  - oracle inequality
KW  - Landweber iteration
Y1  - 2018
U6  - https://doi.org/10.1137/17M1154096
SN  - 2166-2525
VL  - 6
IS  - 3
SP  - 1043
EP  - 1075
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Blanchard, Gilles
A1  - Hoffmann, Marc
A1  - Reiss, Markus
T1  - Early stopping for statistical inverse problems via truncated SVD estimation
JF  - Electronic journal of statistics
N2  - We consider truncated SVD (or spectral cut-off, projection) estimators for a prototypical statistical inverse problem in dimension D. Since calculating the singular value decomposition (SVD) only for the largest singular values is much less costly than the full SVD, our aim is to select a data-driven truncation level (m) over cap is an element of {1, . . . , D} only based on the knowledge of the first (m) over cap singular values and vectors. We analyse in detail whether sequential early stopping rules of this type can preserve statistical optimality. Information-constrained lower bounds and matching upper bounds for a residual based stopping rule are provided, which give a clear picture in which situation optimal sequential adaptation is feasible. Finally, a hybrid two-step approach is proposed which allows for classical oracle inequalities while considerably reducing numerical complexity.
KW  - Linear inverse problems
KW  - truncated SVD
KW  - spectral cut-off
KW  - early stopping
KW  - discrepancy principle
KW  - adaptive estimation
KW  - oracle inequalities
Y1  - 2018
U6  - https://doi.org/10.1214/18-EJS1482
SN  - 1935-7524
VL  - 12
IS  - 2
SP  - 3204
EP  - 3231
PB  - Institute of Mathematical Statistics
CY  - Cleveland
ER  - 
TY  - JOUR
A1  - Böckmann, Christine
A1  - Kammanee, Athassawat
A1  - Braunss, Andreas
T1  - Logarithmic convergence rate of Levenberg-Marquardt method with application to an inverse potential problem
JF  - Journal of inverse and ill-posed problems
N2  - We prove logarithmic convergence rate of the Levenberg-Marquardt method in a Hilbert space if a logarithmic source condition is satisfied. This method is applied to an inverse potential problem. Numerical implementations demonstrate the convergence rate.
KW  - Levenberg-Marquardt method
KW  - inverse potential problems
KW  - logarithmic convergence rate
KW  - discrepancy principle
KW  - logarithmic source condition
Y1  - 2011
U6  - https://doi.org/10.1515/JIIP.2011.034
SN  - 0928-0219
VL  - 19
IS  - 3
SP  - 345
EP  - 367
PB  - De Gruyter
CY  - Berlin
ER  -