Explore recent developments in matrix factorization techniques for discretized differential and integral operators in this 33-minute conference talk from the 2013 SIAM Annual Meeting. Delve into the innovative use of interpolative decomposition for numerically low-rank matrices, and discover efficient algorithms for applying and inverting these operators. Follow the presentation as it covers problem statements, notations, Schur complement, skeletonization, and hierarchical interpolative factorization methods for both differential and integral equations in 2D and 3D contexts. Gain insights into nested dissection, DSRS, and practical examples that demonstrate the application of these advanced mathematical concepts.