Modified Iterative Runge-Kutta-Type Methods for Nonlinear Ill-Posed Problems and Applications Christine Böckmann, Pornsarp Pornsawad, S tefanos S amaras, Moritz Haarig University of Potsdam, Institute of Mathematics, Germany S ilpakorn University, Department of Mathematics, Thailand University of Potsdam and DLR Oberpfaffenhofen, Germany Institute of Troposheric Research Leipzig, Germany New Trends in Parameter Identification for Mathematical Model IMPA, Rio de Janeiro, October 30 th to November 3rd, 2017 Chemnitz Symposium on Inverse Problems 2017 (on Tour in Rio) Acknow ledgment: The research leading t o t hese result s has received funding from t he European Unions S event h Framework Program for research, t echnological development and demonst rat ion under grant agreement no. 289923 – ITaRS and ACTRIS -2.
Introduction ITaRS Proj ect 2
Iterative Regularization Method Looking for: w 0 – minimum-norm solution 3
Convergence of the whole RK-Family ITaRS Proj ect 4
Order Optimality of the first stage RK- Family 5
Example with different steplength τ (relaxation parameter α =1/ τ ) 6
Modified Iterative Runge-Kutta-Type Methods Original RK method (Pornsawad, C.B., 2008,2010) Iteratively regularized Gauss-Newton method (Blaschke et al, 1997, Jin, 2008,2013) Motiviation: this additional term Advantage of this term: Modified Landweber Iteration (S cherzer, 1998) No representation of F‘ by linear bounded operators nessecary Modified iterative RK-Method (Pornsawad, C.B. 2016 ) 7 ( 1.5 )
Assumption A for Convergence 1) Tangential cone condition: 2) Boundedness condition: i) and and S =F‘ ii) 3) Closeness condition: i) a-posteriori PCR: with ii) a-priori PCR: with with 4) S um of relaxation parameters finite: 8
Convergence of the whole modified RK-Family Under a-posteriori stopping criterion with ( 2.6 ) Under a-priori stopping criterion ( 2.8 ) Let Assumption A 1) - 4) be fullfiled. 9
Assumption B for Convergence Rate 1) Derivative is Lipschitz-continuous: 2) S ourcewise representation condition: 3) Relaxation condition: i) and ii) 4) Closeness condition: 10
Convergence Rate of modified RK-Family Theorem: Let Assumpt ion B 1) – 4) fullfiled. where 11
Example comparing different Methods 12
Examples comparing different Methods 13
Conclusion of first Part Advantages: 1) The used modified Runge-Kutta-type methods need less iteration steps as the Gauss-Newton method which also includes an additional term 2) The convergence rate analysis of the modified RK-method can be obtained under less restrictive assumptions on F (wit hout requiring any represent at ion condit ion on F‘ ) for the whole family whereas for the original RK-method only under more restrictive representation conditions and only for the first-stage family it was shown until now. Drawbacks: 1) Due to the additional term the modified RK-method is more time consuming compared to the original RK-method 2) Discrepancy of different assumptions to the specific choice of between Assumptions A and B 14
Application Part 15 Acknowledgment: Figure courtesy of http://www.wmo.int/pages/prog/arep/gaw/aerosol.html
Remote S ensing of Aerosols by LIDAR Optical Radar: Ligth Decetion and Ranging Global distribution of atmospheric aerosols represented as aerosol optical thickness Acknowledgment: Figure courtesy of Stefan Kinne 2016, Personal communication. Source: ftp-proj ects.zmaw.de/ aerocom/ climatology/ MACv2_2015 16 National Technical University of Athens, 6-Wavelength Raman Lidar Acknowledgment: Photo courtesy of Alexandros Papayannis 2015, Alfred Wegener Institute: Research Station in Ny Ålesund on Spitzsbergen. Personal communication Lidar in operation: green laser 532 nm. Source: Photo J. Schmid, AWI
S pheroidal Partical Model: Two-dimensional S ize Distribution Source: Stéphane Reyné, Guillaume Duchateau, Jean-Yves Natoli, Laurent Lamaignère, Laser-induced damage of KDP crystals S phere Oblate S pheroid Prolete S pheorid by 1ω nanosecond pulses: influence of crystal orientation , Opt. Express 17, 21652-21665 (2009); https://www.osapublishing.org/oe/ Aspect ratio: 1 <1 >1 abstract.cfm?uri=oe-17-24-21652 Input Data: 3ß( ≜ ß ∥ ) +2 α +3 δ ( ≜ ß ⊥ ) ß: backscatter coefficient Aspect ratio Radius α : extinction coefficient 17 range range
S imulation Results: Known Refractive Index Input Distribution: Prolate Particles Input Data: 3ß+2 α +3 δ 1% Input data error 10% Input data error 18 S . S amaras, PhD t hesis, Universit y of Pot sdam, 2017. Method: Pade-Iteration-DP Pade-Iteration-LC Tikhonov-DP RK- RK-
S imulation Results: Known Refractive Index Input Distribution: Oblate Particles Input Data: 3ß+2 α +3 δ 1% Input error 10% Input error 19 S . S amaras, PhD t hesis, Universit y of Pot sdam, 2017. Method: Pade-Iteration-DP Pade-Iteration-LC Tikhonov-DP RK- RK-
Comparison of two Methods with heuristic parameter rules = RK– It eration-LC ITaRS Proj ect 20 S . S amaras, PhD t hesis, Universit y of Pot sdam, 2017.
Two-dimensinal Bi-modal Distribution Example Retrieved Distribution: Input Distribution: by RK-Iteration Noiseless Input Data: 3ß+2 α +3 δ 21 S . S amaras, PhD t hesis, Universit y of Pot sdam, 2017.
Measurement Case: S aharan Dust S torm Barbados Acknowledgment:R.R. Draxler, G.D. Rolph, HYS PLIT (HYbrid S ingle-Particle Lagrangian Integrat ed 22 Traj ect ory) model access via NOAA ARL READY websit e (ht t p:/ / www.arl. noaa.gov/ ready/ hysplit 4.html), NOAA Air Resources Laborat ory, S ilver S pring, MD, 2014.
Case S tudy – Barbados: S aharan Dust Aerosol Event, June 2014 Layer: 1.5 – 2.75 km Depolarization is high assuming: non-spherical particles Acknowledgment: Measurement s from Inst it ute of Tropospheric Research Leipzig, Germany. 23 Haarig, M., Ansmann, A., Althausen, D., Klepel, A., Groß, S ., Freudenthaler, V., Toledano, C., Mamouri, R.-E., Farrell, D. A., Prescod, D. A., Marinou, E., Burton, S . P., Gasteiger,J., Engelmann, R., and Baars, H.: Triple-wavelength depolarization-ratio profiling of S aharan dust over Barbados during S ALTRACE in 2013 and 2014, Atmos. Chem. Phys., 17, 10767-10794, https:/ / doi.org/ 10.5194/ acp-17-10767-2017, 2017.
Retrieval Result of 2D S ize Distribution Input Data: 3ß+2 α +3 δ Fine mode Coarse mode 24 Oblates Prolates S pheres S . S amaras, PhD t hesis, Universit y of Pot sdam, 2017.
It is a honour for me to thank Prof. Bernd Hofmann for his very fruitful Chemnitz S ymposium over all the years from 2002, hosting in particular young researchers, most of my PhD students, e.g., Pornsarp Pornsawad on the picture and today my coauthor. 2008 Thanks for your interest ! Obrigado! New Trends in Parameter Identification for Mathematical Model IMPA, Rio de Janeiro, October 30 th to November 3rd, 2017 Chemnitz Symposium on Inverse Problems 2017 (on Tour in Rio) 25 Acknow ledgment: The research leading t o t hese result s has received funding from t he European Unions S event h Framework Program for research, t echnological development and demonst rat ion under grant agreement no. 289923 – ITaRS and ACTRIS -2.
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