1. Indirect imaging of stellar non-radial pulsations Svetlana V. Berdyugina University of Oulu, Finland Institute of Astronomy, ETH Zurich, Switzerland

2. Moletai, August 20052 Overview Inversion methods in astrophysics  Inverse problem  Maximum likelihood method  Regularization Stellar surface imaging  Line profile distortions  Localization of inhomogeneities Imaging of stellar non-radial pulsations  Temperature variations  Velocity field Mode identification  sectoral modes:  symmetric tesseral modes:  antisymmetric tesseral modes:  zonal modes:

3. Moletai, August 20053 1. Inversion methods in astrophysics Inverse problem Maximum likelihood method Regularization  Maximum Entropy  Tikhonov  Spherical harmonics  Occamian approach

4. Moletai, August 20054 Inverse problem Determine true properties of phenomena (objects) from observed effects All problems in astronomy are inverse

5. Moletai, August 20055 Inverse problem Trial-and-error method  Response operator (PSF, model) is known  Direct modeling while assuming various properties of the object Inversion  True inversion:  unstable solution due to noise  ill-posed problem  Parameter estimation: fighting the noise DataObject Response operator

6. Moletai, August 20056 Inverse problem Estimate true properties of phenomena (objects) from observed effects Parameter estimation problem

7. Moletai, August 20057 Maximum likelihood method Probability density function (PDF): Normal distribution: Likelihood function Maximum likelihood

8. Moletai, August 20058 Maximum likelihood method Maximum likelihood Normal distribution Residual minimization

9. Moletai, August 20059 Maximum likelihood method Maximum likelihood solution:  Unique  Unbiased  Minimum variance  UNSTABLE !!! Reduce the overall probability Statistical tests  test  Kolmogorov  Mean information

10. Moletai, August 200510 Maximum likelihood method A multitude of solutions with probability New solution  Biased only within noise level  Stable  NOT UNIQUE !!! Likelihood Solutions

11. Moletai, August 200511 Regularization Provide a unique solution  Invoke additional constraints  Assign special properties of a new solution Maximize the functional Regularized solution is forced to possess properties

12. Moletai, August 200512 Bayesian approach Thomas Bayes (1702-1761)  Posterior and prior probabilities Prior information on the solution Using a priori constraints is the Bayesian approach

13. Moletai, August 200513 Maximum entropy regularization Entropy  In physics: a measure of ”disorder”  In math (Shannon): a measure of “uninformativeness” Maximum entropy method (MEM, Skilling & Bryan, 1984): MEM solution  Largest entropy (within the noise level of data)  Minimum information (minimum correlation)

14. Moletai, August 200514 Tikhonov regularization Tikhonov (1963): Goncharsky et al. (1982): TR solution  Least gradient (within the noise level of data)  Smoothest solution (maximum correlation)

15. Moletai, August 200515 Spherical harmonics regularization Piskunov & Kochukhov (2002): multipole regularization MPR solution  Closest to the spherical harmonics expansion  Can be justified by the physics of a phenomenon Mixed regularization:

16. Moletai, August 200516 Occamian approach William of Occam (1285 --1347):  Occam's Razor: the simplest explanation to any problem is the best explanation Terebizh & Biryukov (1994, 1995):  Simplest solution (within the noise level of data)  No a priori information Fisher information matrix:

17. Moletai, August 200517 Occamian approach Orthogonal transform Principal components Simplest solution  Unique  Stable

18. Moletai, August 200518 Key issues Inverse problem is to estimate true properties of phenomena (objects) from observed effects Maximum likelihood method results in the unique but unstable solution Statistical tests provide a multitude of stable solutions Regularization is needed to choose a unique solution Regularized solution is forced to possess assigned properties MEM solution  minimum correlation between parameters TR solution  maximum correlation between parameters MPR solution  closest to the spherical harmonics expansion OA solution  simplest among statistically acceptable