Explore the impact of sparsity and compressive sensing in science, technology, medicine, imaging, machine learning, and solving multiscale problems in applied partial differential equations through this 59-minute conference talk. Delve into the power of l1 and related optimization solvers, understanding how their unique nature enables rapid problem-solving and error forgiveness in Bregman iterative methods. Learn about simple, fast algorithms and their novel applications, ranging from sparse dynamics for PDE to new regularization paths for logistic regression and support vector machines. Discover how these techniques are revolutionizing optimal data collection and hyperspectral image processing, as presented by Stanley Osher of UCLA at the 2013 SIAM Annual Meeting.