# Approximate inference for inverse models (Brian Reich and Arvind Saibaba)

**Prerequisites:** Linear algebra, interest in programming.

**Outline:** Gaussian process inverse models are used to extract features of a physical process from spatial image data in fields ranging from remote sensing of deforestation to short-term weather forecasting. Wellestablished methods such as the Kalman filter are infeasible for large spatial datasets because the computational load for a Gaussian process model increases exponentially with the number of observations (1). This project will utilize mathematical and statistical approximation theory to visualize and analyze big data.

**Research objectives: **Extension of the new Krylov subspace approximation method developed by Reich and Saibaba (2) to analyze the enhanced vegetation index using divide-and-conquer methods that split

the data into spatiotemporal blocks and carefully fuse the results to produce coherent maps.

**Outcomes**: Publicly available code implementing their methods and apply their code to produce high-resolution phenology maps and test for changes in the critical aspect of the climate.

- Heaton MJ, Datta A, Finley AO, Furrer R, Guinness J, Guhaniyogi R, Gerber F, Gramacy RB, Hammerling D, Katzfuss M, Lindgren F, Nychka DW, Sun F, Zammit-Mangion A. A Case Study Competition Among Methods for Analyzing Large Spatial Data. J Agric Biol Environ Stat. 2019;24:398-425.
- Majumder S, Guan Y, Reich B, AK S. Kryging: Geostatistical analysis of large-scale datasets using Krylov subspace methods. arXiv. 2020;2012.13133.