Monday, March 16 2020
11:00am - 12:00pm
Computational Mathematics Colloquium,Statistics Seminar
CANCELLED: Model-aware learning approaches to data-driven inverse/UQ problems

In the first part of the talk, we present a subspace model-aware regularization technique (SMART) that combines advantages of the classical truncated SVD and Tikhonov regularization. In particular, the SMART approach does not pollute the data-informed modes, and regularizes only less data-informed ones. As a direct consequence, the approach is at least as good as the Tikhonov method for any value of the regularization parameter and it is more accurate than the TSVD (for reasonable regularization parameter). Due to this blending of these two classical methods, SMART is robust with regularization parameter. We show that the SMART approach has an interesting statistical interpretation, that is, it transforms both the data distribution (i.e. the likelihood) and prior distribution (induced by Tikhonov regularization) to the same Gaussian distribution whose covariance matrix is diagonal and diagonal elements are exactly the singular values of a composition of the prior covariance matrix, the forward map, and the noise covariance matrix. In other words, SMART finds the modes that are most equally data-informed and prior-informed and leaves these modes untouched so that the inverse solution receive the best possible (balanced) information from both prior and the data. We will show that SMART is regularization strategy and admissible regularization. To demonstrate and to support our findings, we have presented various results for popular computer vision and imaging problems including deblurring, denosing, and X-ray tomography. We also present the theoretical aspect of SMART methods on infinite dimensional spaces.

The second part of the talk presents our recent work on developing model-aware deep learning approaches for inverse problems. The first approach combines the traditional ROM method and deep learning to learn the parameter-to-observable map, and the second develops an Auto-Inversion (AI) approach using a model-aware autoencoder method to learn the inverse parameter-to-observable map. Various numerical inversion results will be presented to verify the proposed approach.
Speaker:Tan Bui Thanh
Affiliation:Aerospace Engineering, UT Austin
Location:SCB 4017


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