For the scientific applications modeled using partial differential equations (PDEs), it is now common to use millions of spatial discretization for deterministic simulations. Incorporating uncertainty characterized by thousands of stochastic discretization increases the problem size and complexity level at which the simulation can necessitate the exascale computational capabilities. Therefore, uncertainty quantification (UQ) of extreme-scale numerical models demands the scalable algorithms employed using high-performance computing (HPC).

As part of my Ph.D. research, I have extended and efficiently implemented scalable domain decomposition (DD) solvers for stochastic PDEs. The in-house development of these solvers package utilizes sophisticated HPC libraries like MPI, PETSc and FEniCS on modern HPC clusters.

Numerical and parallel scalabilities of these solvers are investigated for the diffusion equation having spatially varying diffusion coefficient modeled by a non-Gaussian stochastic process. Scalability with respect to the number of random variables is also focused. Furthermore, these solvers are also integrated with an in-house Bayesian estimation algorithm to tackle large-scale data assimilation problems.