Algorithms for the numerical solution of partial differential equations often come in two flavors: iterative methods (which build successively better approximations of the solution until convergence to some tolerance is reached) and direct methods (which construct the solution all at once). Historically, iterative methods have been the norm in many settings, as naive direct solvers can be prohibitively expensive for large-scale problems. However, direct solvers offer many advantages, particularly in the case of high-order discretizations—they are robust, insensitive to ill-conditioning, and allow the efficient reuse of matrix factorizations for multiple righthand sides. Recently, fast direct solvers have emerged as an attractive class of direct solvers, leveraging the structure and compressibility of PDEs to compute factorizations of linear systems in quasi-optimal complexity. In this talk, I will show how fast direct solvers shine when used alongside high-order accurate numerical methods, with examples drawn from elliptic and parabolic PDEs in complex geometries, on surfaces, and in close-to-touching fluid suspensions.
About the speaker:
Dan Fortunato is an Associate Research Scientist in the Center for Computational Mathematics and the Center for Computational Biology at the Flatiron Institute. His research focuses on spectral methods, fast methods for PDEs, integral equations, computational fluid & solid mechanics, and multigrid methods. In 2019, Fortunato received the IMA Leslie Fox Prize for Numerical Analysis (Second Prize). In the past, Dan has worked at Walt Disney Animation Studios, Lawrence Berkeley National Laboratory, Wolfram Research, and Apple. He holds a Ph.D. and M.S. in Applied Mathematics from Harvard University, and a B.S. in Mathematics and Computer Science from Tufts University.
