# Projects

## Below you will find information on some current projects at LBNL involving PFASST

## Parallel-in-time geometric integrators

Many physics problems are characterized by specific qualitative features or invariants that are important in long-term dynamics. Traditional numerical integrators such as Euler or Runge-Kutta methods often have difficulty in preserving these important invariants in the solution. We are developing integrators based on Magnus-type expansions which are able to preserve this important geometric structure. The example on the right shows the non-linear dynamics of a 1-D periodic 11-particle Toda flow projected onto a circle obtained using a fourth-order Magnus integrator. The solution can be represented by a matrix whose eigenvalues are time invariant, and the Magnus integrator preserves this invariant. The goal of this project is to provide time parallelization for the real-time time-dependent density functional theory package in the NWChem software framework.

## Parallel in time for parabolic optimal control problems

Gradient based methods for PDE optimal control problems are extremely computationally expensive due to the need to fully solve the forward PDE and an adjoint equation every optimization iteration. We are therefore investigating the use of the PFASST algorithm to parallelize both forward and adjoint solution steps. The figure on the left shows the space-time evolution of the solution to a prototypical reaction-diffusion equation computed with the PFASST algorithm applied to both forward and adjoint solves. Since PFASST is iterative, space-time initial conditions for each PDE solve can use the solution from the previous optimization iteration leading to additional computational savings. One motivating application for this project is the optimal placement and signal for pacemakers. See this preprint for more details

## PARALLEL-In-TIme INTEGRATion FOR ATMOSPHERIC FLOWS

The accurate modeling of atmospheric processes over long periods of time is computationally expensive. The high-order integration of the PDEs often requires a very large number of time steps, even when the terms accounting for the propagation of fast atmospheric waves are treated implicitly. Therefore, parallel-in-time schemes are attractive to compute multiple time steps concurrently and reduce the time-to-solution. We propose a multi-level parallel-in-time integration method combining PFASST with the Spherical Harmonics to solve the shallow-water equations on the sphere. We test our parallel-in-time method using nonlinear shallow-water problems capturing the horizontal features of realistic atmospheric flows. We show that PFASST can resolve the main features of the solution significantly faster than with serial time integration schemes. For this project, our libpfasst library has been linked to SWEET (Shallow Water Equation Environment for Tests).

Vorticity field at different times

(Galewsky et al, 2004)