Parallel-in-time computation for sedimentary landscapes

This proposal is about novel mathematical techniques underpinning
computational stratigraphic models that simulate the formation of
landscapes of sedimentary rock. Sedimentary rocks form after gradual
settling of microscopic particles (formed from minerals, or coming
from plants or animals) that are suspended in ocean water. Over
millions of years, the particles settle on the ocean floor, eventually
condensing into rocks such as shale and limestone. By mathematically
modelling this process on a computer and comparing with geological
data, we can learn about the evolution of our present landscape on
Planet Earth, and we can use it to fill in the gaps between
data. These models have applications in locating carbon capture and
storage sites, and in reconstructions of recent geological history of
coral reefs, for example.

Stratigraphic models simulate the evolution of the sediment over time,
stepping from one moment in time to a later one in the near future,
executing "timesteps" one by one in a sequential manner. Accurate
modelling of the sediment processes requires that these timesteps are
0.1-1 years long. Since sedimentary landscapes form over geological
eras that are millions of years long, this means that we have to
execute millions of timesteps, one after the other. This is
prohibitively long, especially when the models are needed for data
assimilation algorithms that search for unknown properties of past
rock formation processes in the light of data obtained from geological
measurement campaigns. This is because these data assimilation
algorithms have to repeat the simulation many times with varying
parameter values. In these situations, stratigraphic modellers are
forced to use timesteps that are 1000s of years long: this yields
results of insufficient accuracy.

Our goal is to create new mathematical techniques that can make use of
highly parallel supercomputers, leading to much faster simulations and
enabling more sophisticated data assimilation algorithms to be used.
Instead of the sequential one-timestep-at-a-time approach, we will
create new algorithms that solve for all of the timesteps
simultaneously on a large number of computer processors in parallel.
We call this parallel-in-time integration. The algorithms will be
iterative, computing first guesses for the model predictions for each
timestep and then updating them until they are sufficiently
accurate. A good parallel-in-time integration method will only require
a small number of iterations, so that the result of the algorithm is
quicker than sequential computation. Finding a good parallel-in-time
integration method is a mathematical problem, with the number of
iterations being strongly dependent on the structure of the equations
that describe the simulation model. Parallel-in-time approaches have
never been investigated for stratigraphic models. In this project we
will start a new field of numerical analysis research, designing
parallel-in-time integration methods for stratigraphic models and
analysing them using a blend of theoretical analysis and high
performance computational experiments to identify the best path
forward.