  In this work, we propose a parallel-in-time solver for linear and nonlinear
ordinary differential equations. The approach is based on an efficient
multilevel solver of the Schur complement related to a multilevel time
partition. For linear problems, the scheme leads to a fast direct method. Next,
two different strategies for solving nonlinear ODEs are proposed. First, we
consider a Newton method over the global nonlinear ODE, using the multilevel
Schur complement solver at every nonlinear iteration. Second, we state the
global nonlinear problem in terms of the nonlinear Schur complement (at an
arbitrary level), and perform nonlinear iterations over it. Numerical
experiments show that the proposed schemes are weakly scalable, i.e., we can
efficiently exploit increasing computational resources to solve for more time
steps the same problem.
