# Robust inversion, dimensionality reduction, and randomized sampling

Mathematical Programming, 134(1):101–125, 2012, 2012

Mathematical Programming, 134(1):101–125, 2012, 2012

We consider a class of inverse problems in which the forward model is the solution operator to linear ODEs or PDEs. This class admits several dimensionality-reduction techniques based on data averaging or sampling, which are especially useful for large-scale problems. We survey these approaches and their connection to stochastic optimization. The data-averaging approach is only viable, however, for a least-squares misfit, which is sensitive to outliers in the data and artifacts unexplained by the forward model. This motivates us to propose a robust formulation based on the Student's t-distribution of the error. We demonstrate how the corresponding penalty function, together with the sampling approach, can obtain good results for a large-scale seismic inverse problem with 50% corrupted data.

```
@article{AravkinFHV:2012,
author = {A. Aravkin and M. P. Friedlander and F. Herrmann and T. van Leeuwen},
title = {Robust inversion, dimensionality reduction, and randomized sampling},
journal = {Mathematical Programming},
volume = 134,
number = 1,
pages = {101-125},
year = {2012}
DOI = {10.1007/s10107-012-0571-6}
}
```