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

Moletai, August 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:

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

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

Moletai, August 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

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

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

Moletai, August Maximum likelihood method Maximum likelihood Normal distribution Residual minimization

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

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

Moletai, August 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

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

Moletai, August 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)

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

Moletai, August 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:

Moletai, August Occamian approach William of Occam ( ):  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:

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

Moletai, August 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