Robust inversion, dimensionality reduction, and randomized sampling

We consider a class of inverse problems inwhich 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 stochas- tic 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.