The solution of large sparse linear systems is an essential building block in many science and engineering applications. It is also often the main computational bottleneck. For large problems, direct solvers (based
on, e.g., LU or Cholesky factorizations) can require a significant amount of computing resources. By contrast, iterative solvers (e.g., CG and GMRES) can be much more efficient when effective preconditioners are provided. In this talk, I will present a randomized approach to constructing preconditioners for symmetric diagonally dominant matrices that arise from applications in scientific computing, data science, and machine learning. The new method computes an incomplete factorization of a sparse input matrix. It leverages a randomized sampling scheme developed by Spielman and Kyng that prevents excessive fill-in during Gaussian elimination. Numerical experiments demonstrate that the randomized preconditioner outperforms classical deterministic methods and delivers faster convergence, less running time, and better scalability. Finally, I will discuss some exciting research opportunities related to the new method with applications in high-performance computing and machine learning applications.