Scientific Computation and Applied & Industrial Mathematics: Curt Da Silva

Many useful and interesting optimization problems can be cast in a convex composite form min_x h(c(x)), where h is a non-smooth but convex function and c is a smooth nonlinear or linear mapping. The non-smoothness of the outer function prevents traditional methods such as the Gauss-Newton method from converging quickly, which is problematic for large scale problems. In this talk, we will explore level set methods, aka the SPGL1 'trick', for solving this class of problem when we can easily project on to the level sets of h(z). The resulting subproblems will be smooth and have simple constraints, which are amenable to smooth optimization methods such as LBFGS. We also use the variable projection technique, which gives us an alternate interpretation as computing the minimal distance between the level set of h and the image of our nonlinear mapping c. We will demonstrate the effectiveness of this technique on a number of convex and non-convex problems, including cosparsity-based compressed sensing for seismic data interpolation, audio signal declipping, robust tensor PCA/completion, and more.