Parallel-in-time computation for sedimentary landscapes

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This proposal is about novel mathematical techniques underpinningcomputational stratigraphic models that simulate the formation oflandscapes of sedimentary rock. Sedimentary rocks form after gradualsettling of microscopic particles (formed from minerals, or comingfrom plants or animals) that are suspended in ocean water. Overmillions of years, the particles settle on the ocean floor, eventuallycondensing into rocks such as shale and limestone. By mathematicallymodelling this process on a computer and comparing with geologicaldata, we can learn about the evolution of our present landscape onPlanet Earth, and we can use it to fill in the gaps betweendata. These models have applications in locating carbon capture andstorage sites, and in reconstructions of recent geological history ofcoral 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. Accuratemodelling of the sediment processes requires that these timesteps are0.1-1 years long. Since sedimentary landscapes form over geologicaleras that are millions of years long, this means that we have toexecute millions of timesteps, one after the other. This isprohibitively long, especially when the models are needed for dataassimilation algorithms that search for unknown properties of pastrock formation processes in the light of data obtained from geologicalmeasurement campaigns. This is because these data assimilationalgorithms have to repeat the simulation many times with varyingparameter values. In these situations, stratigraphic modellers areforced to use timesteps that are 1000s of years long: this yieldsresults of insufficient accuracy.Our goal is to create new mathematical techniques that can make use ofhighly parallel supercomputers, leading to much faster simulations andenabling more sophisticated data assimilation algorithms to be used.Instead of the sequential one-timestep-at-a-time approach, we willcreate new algorithms that solve for all of the timestepssimultaneously on a large number of computer processors in parallel.We call this parallel-in-time integration. The algorithms will beiterative, computing first guesses for the model predictions for eachtimestep and then updating them until they are sufficientlyaccurate. A good parallel-in-time integration method will only requirea small number of iterations, so that the result of the algorithm isquicker than sequential computation. Finding a good parallel-in-timeintegration method is a mathematical problem, with the number ofiterations being strongly dependent on the structure of the equationsthat describe the simulation model. Parallel-in-time approaches havenever been investigated for stratigraphic models. In this project wewill start a new field of numerical analysis research, designingparallel-in-time integration methods for stratigraphic models andanalysing them using a blend of theoretical analysis and highperformance computational experiments to identify the best pathforward.