Modified Iterative Runge-Kutta-Type Methods for Nonlinear Ill-Posed - - PowerPoint PPT Presentation

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Modified Iterative Runge-Kutta-Type Methods for Nonlinear Ill-Posed - - PowerPoint PPT Presentation

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Modified Iterative Runge-Kutta-Type Methods for Nonlinear Ill-Posed Problems and Applications Christine Bckmann, Pornsarp Pornsawad, S tefanos S amaras, Moritz Haarig University of Potsdam, Institute of Mathematics, Germany S ilpakorn

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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

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.

New Trends in Parameter Identification for Mathematical Model IMPA, Rio de Janeiro, October 30th to November 3rd, 2017 Chemnitz Symposium on Inverse Problems 2017 (on Tour in Rio)

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Introduction

ITaRS Proj ect 2

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Iterative Regularization Method

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Looking for: w0 – minimum-norm solution

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Convergence of the whole RK-Family

ITaRS Proj ect 4

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Order Optimality of the first stage RK- Family

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Example with different steplength τ (relaxation parameter α=1/ τ)

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Modified Iterative Runge-Kutta-Type Methods

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Original RK method (Pornsawad, C.B., 2008,2010) Iteratively regularized Gauss-Newton method (Blaschke et al, 1997, Jin, 2008,2013) Motiviation: this additional term Modified Landweber Iteration (S cherzer, 1998)

Modified iterative RK-Method (Pornsawad, C.B. 2016)

(1.5) Advantage of this term: No representation of F‘ by linear bounded operators nessecary

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Assumption A for Convergence

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1) Tangential cone condition: 2) Boundedness condition:

and

ii) 3) Closeness condition:

with

4) S um of relaxation parameters finite: i) a-posteriori PCR: ii) a-priori PCR:

with with

i)

and S =F‘

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Convergence of the whole modified RK-Family

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Under a-posteriori stopping criterion Under a-priori stopping criterion

(2.6) (2.8)

with

Let Assumption A 1) - 4) be fullfiled.

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Assumption B for Convergence Rate

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1) Derivative is Lipschitz-continuous: 2) S

  • urcewise representation condition:

3) Relaxation condition: i)

and ii)

4) Closeness condition:

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Convergence Rate of modified RK-Family

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Theorem:

Let Assumpt ion B 1) – 4) fullfiled. where

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Example comparing different Methods

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Examples comparing different Methods

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Conclusion of first Part

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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

  • nly 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

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Application Part

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Acknowledgment: Figure courtesy of http://www.wmo.int/pages/prog/arep/gaw/aerosol.html

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Remote S ensing of Aerosols by LIDAR

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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 Alfred Wegener Institute: Research Station in Ny Ålesund on Spitzsbergen. Lidar in operation: green laser 532 nm. Source: Photo J. Schmid, AWI National Technical University of Athens, 6-Wavelength Raman Lidar Acknowledgment: Photo courtesy of Alexandros Papayannis 2015, Personal communication

Optical Radar: Ligth Decetion and Ranging

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S pheroidal Partical Model: Two-dimensional S ize Distribution

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Source: Stéphane Reyné, Guillaume Duchateau, Jean-Yves Natoli, Laurent Lamaignère, Laser-induced damage of KDP crystals by 1ω nanosecond pulses: influence of crystal orientation,

  • Opt. Express 17, 21652-21665 (2009);

https://www.osapublishing.org/oe/ abstract.cfm?uri=oe-17-24-21652

Oblate S pheroid S phere Prolete S pheorid Aspect ratio range Radius range Aspect ratio: 1 <1 >1 Input Data: 3ß(≜ß∥)+2α+3δ(≜ß⊥) ß: backscatter coefficient

α: extinction coefficient

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S imulation Results: Known Refractive Index

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Input Distribution: Prolate Particles Input Data: 3ß+2α+3δ 1% Input data error 10% Input data error Method: Pade-Iteration-DP Pade-Iteration-LC Tikhonov-DP RK- RK-

S . S amaras, PhD t hesis, Universit y of Pot sdam, 2017.

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Input Distribution: Oblate Particles

S imulation Results: Known Refractive Index

Input Data: 3ß+2α+3δ Method: Pade-Iteration-DP Pade-Iteration-LC Tikhonov-DP 1% Input error 10% Input error RK- RK-

S . S amaras, PhD t hesis, Universit y of Pot sdam, 2017.

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Comparison of two Methods with heuristic parameter rules

ITaRS Proj ect 20

S . S amaras, PhD t hesis, Universit y of Pot sdam, 2017. = RK– It eration-LC

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Two-dimensinal Bi-modal Distribution Example

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Input Distribution: Retrieved Distribution: by RK-Iteration Noiseless Input Data: 3ß+2α+3δ

S . S amaras, PhD t hesis, Universit y of Pot sdam, 2017.

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Measurement Case: S aharan Dust S torm

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Barbados

Acknowledgment:R.R. Draxler, G.D. Rolph, HYS PLIT (HYbrid S ingle-Particle Lagrangian Integrat ed 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.

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Case S tudy – Barbados: S aharan Dust Aerosol Event, June 2014

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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.

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.
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Retrieval Result of 2D S ize Distribution

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Oblates Prolates S pheres Input Data: 3ß+2α+3δ Fine mode Coarse mode

S . S amaras, PhD t hesis, Universit y of Pot sdam, 2017.

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Thanks for your interest ! Obrigado!

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New Trends in Parameter Identification for Mathematical Model IMPA, Rio de Janeiro, October 30th 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.

It is a honour for me to thank

  • Prof. Bernd Hofmann

for his very fruitful Chemnitz S ymposium

  • ver 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