Nonlinear parallel-in-time multilevel Schur complement solvers for ordinary differential equations
This work addresses the need for scalable parallel solvers for time-dependent ODEs, offering a method that efficiently uses additional resources for longer time horizons.
The authors propose a parallel-in-time solver for ODEs using a multilevel Schur complement approach, achieving weak scalability where increasing computational resources efficiently solve more time steps.
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.