Randomized subspace methods for linear systems and eigenvalue problems

Tall-skinny least-squares problems are among the first classes of problems in numerical linear algebra for which randomization proved to be an effective tool. Such problems arise naturally when subspace methods are used for solving linear systems and eigenvalue problems. The combination of these ideas has led to a range of exciting advances in randomized linear algebra, yielding algorithms that are not only faster than classical ones, but in some cases come with equally strong stability guarantees. In this talk I will discuss various aspects of randomized subspace methods, including (i) fast solution with randomized sketching or subsampling, (ii) backward stability, and (iii) parameter-dependent problems.