The arrival of massive amounts of data from imaging, sensors, computation and the internet brought with it significant challenges for information science. New methods for analysis and manipulation of big data have come from many scientific disciplines. The first focus of this presentation is the application of ideas from PDEs, such as variational principles and numerical diffusion, to image and data analysis. Examples include denoising, segmentation, inpainting and texture extraction for images. The second focus is the development of new ideas in information science, such as soft-thresholding, sparsity and compressed sensing. The subsequent application of these ideas to PDEs and numerical computation is the third focus of this talk. Examples include soft-thresholding in multiscale computation, solutions with compact support and “compressed modes” for PDEs that come from variational principles, and applications to density functional theory.