Uncertainty Quantification on Large-Scale Models

Any model of real-world processes is affected by uncertainties due to measurement errors and incomplete knowledge. It is the goal of uncertainty quantification to determine how these affect the reliability of predictions about or inferences from reality.

Unfortunately, each uncertain parameter essentially contributes to the dimensionality of the model. For numerical simulations, treating uncertainties typically means that large numbers of simulation runs are required. Clearly, for models that are computationally challenging themselves, the computational power required easily becomes excessive.

Multilevel Markov Chain Monte Carlo (MLMCMC) alleviates this by shifting most of the effort to cheap-to-compute approximate models, only using the full model in order to correct for the errors introduced in the process.

Tsunami model hierarchy

The key contribution of my PhD project is a massively parallelized version of that method. Markov Chain methods introduce data dependencies, even more so in the multilevel setting. Nevertheless, the proposed method allows for near-optimal scaling across at least thousands of processor cores. My parallelized implementation was accepted in the MIT Uncertainty Quantification library.

Using this method, we could solve a Bayesian inference problem on a challenging Tsunami model on ~3,500 processor cores of SuperMUC-NG at Leibniz supercomputing center.

GenEO: Scalable Solver for Challenging PDE Models

Mathematical models of real-world processes are often formulated in terms of partial differential equations (PDEs). They are essential in numerous fields of science and engineering, but may exhibit significant challenges if they are ill-conditioned.

This is the case when modelling composite carbon-fiber materials in aerospace engineering, since material parameters are extremely heterogeneous and anisotropic. Established industry standard solutions are limited to small-scale simulations in that setting, and do not scale beyond a few dozen processor cores.

The CerTest project aims to improve next-generation aircraft design through advanced simulation techniques. My contribution to dune-composites, a powerful simulation tool developed as part of CerTest, is a highly scalable implementation of the GenEO method. This mathematical method automatically identifies components of the solution space that slow down numerical solvers, directly solves for those in a separate coarse level, and thus retains the solver’s efficiency for the overall problem. Due to its general-purpose architecture, the implementation was accepted into the dune-pdelab module of the DUNE numerics framework.

GenEO weak scaling

My GenEO implementation overcomes the limitations of existing solutions for simulating composite materials, and we could demonstrate its effectiveness simulating large-scale aircraft wing structures on up to ~16,000 processor cores on the ARCHER UK national supercomputer.