TY  - JOUR
A1  - Pornsawad, Pornsarp
A1  - Böckmann, Christine
A1  - Panitsupakamon, Wannapa
T1  - The Levenberg–Marquardt regularization for the backward heat equation with fractional derivative
JF  - Electronic transactions on numerical analysis - ETNA
N2  - The backward heat problem with time-fractional derivative in Caputo's sense is studied. The inverse problem is severely ill-posed in the case when the fractional order is close to unity. A Levenberg-Marquardt method with a new a posteriori stopping rule is investigated. We show that optimal order can be obtained for the proposed method under a Hölder-type source condition. Numerical examples for one and two dimensions are provided.
KW  - ill-posed problems
KW  - time-fractional derivative
KW  - backward heat problem
KW  - Levenberg-Marquardt method
KW  - a posteriori stopping rule
KW  - optimal order
Y1  - 2022
SN  - 978-3-7001-8258-0
U6  - https://doi.org/10.1553/etna_vol57s67
SN  - 1068-9613
VL  - 57
SP  - 67
EP  - 79
PB  - Kent State University
CY  - Kent
ER  - 
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  - GEN
A1  - Pornsawad, Pornsarp
A1  - Sapsakul, Nantawan
A1  - Böckmann, Christine
T1  - A modified asymptotical regularization of nonlinear ill-posed problems
T2  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1335 
KW  - nonlinear operator
KW  - regularization
KW  - discrepancy principle
KW  - asymptotic method
KW  - optimal rate
Y1  - 2019
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-473433
SN  - 1866-8372
IS  - 1335
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  - GEN
A1  - Pornsawad, Pornsarp
A1  - Sungcharoen, Parada
A1  - Böckmann, Christine
T1  - Convergence rate of the modified Landweber method for solving inverse potential problems
T2  - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
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.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1034 
KW  - nonlinear operator
KW  - regularization
KW  - modified Landweber method
KW  - discrepancy principle
KW  - logarithmic source condition
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-471942
SN  - 1866-8372
IS  - 1034
ER  - 
TY  - JOUR
A1  - Pornsawad, Pornsarp
A1  - Böckmann, Christine
T1  - Modified Iterative Runge-Kutta-Type Methods for Nonlinear Ill-Posed Problems
JF  - Numerical functional analysis and optimization : an international journal of rapid publication
N2  - This work is devoted to the convergence analysis of a modified Runge-Kutta-type iterative regularization method for solving nonlinear ill-posed problems under a priori and a posteriori stopping rules. The convergence rate results of the proposed method can be obtained under a Holder-type sourcewise condition if the Frechet derivative is properly scaled and locally Lipschitz continuous. Numerical results are achieved by using the Levenberg-Marquardt, Lobatto, and Radau methods.
KW  - Nonlinear ill-posed problems
KW  - Runge-Kutta methods
KW  - regularization methods
KW  - Holder-type source condition
KW  - stopping rules
Y1  - 2016
U6  - https://doi.org/10.1080/01630563.2016.1219744
SN  - 0163-0563
SN  - 1532-2467
VL  - 37
SP  - 1562
EP  - 1589
PB  - Wiley-VCH
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Pornsawad, Pornsarp
A1  - D'Amico, Giuseppe
A1  - Böckmann, Christine
A1  - Amodeo, Aldo
A1  - Pappalardo, Gelsomina
T1  - Retrieval of aerosol extinction coefficient profiles from Raman lidar data by inversion method
JF  - Applied optics
N2  - We regard the problem of differentiation occurring in the retrieval of aerosol extinction coefficient profiles from inelastic Raman lidar signals by searching for a stable solution of the resulting Volterra integral equation. An algorithm based on a projection method and iterative regularization together with the L-curve method has been performed on synthetic and measured lidar signals. A strategy to choose a suitable range for the integration within the framework of the retrieval of optical properties is proposed here for the first time to our knowledge. The Monte Carlo procedure has been adapted to treat the uncertainty in the retrieval of extinction coefficients.
Y1  - 2012
U6  - https://doi.org/10.1364/AO.51.002035
SN  - 1559-128X
SN  - 2155-3165
VL  - 51
IS  - 12
SP  - 2035
EP  - 2044
PB  - Optical Society of America
CY  - Washington
ER  - 
TY  - THES
A1  - Pornsawad, Pornsarp
T1  - Solution of nonlinear inverse ill-posed problems via Runge-Kutta methods
Y1  - 2010
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Pornsawad, Pornsarp
A1  - Böckmann, Christine
T1  - Modified iterative Runge-Kutta-type methods for nonlinear ill-posed problems
N2  - This work is devoted to the convergence analysis of a modified Runge-Kutta-type iterative regularization method for solving nonlinear ill-posed problems under a priori and a posteriori stopping rules. The convergence rate results of the proposed method can be obtained under Hölder-type source-wise condition if the Fréchet derivative is properly scaled and locally Lipschitz continuous. Numerical results are achieved by using the Levenberg-Marquardt and Radau methods.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 7 
KW  - ill-posed problems
KW  - Runge-Kutta methods
KW  - regularization methods
KW  - Hölder-type source condition
KW  - stopping rules
Y1  - 2014
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-70834
SN  - 2193-6943
VL  - 3
IS  - 7
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  -