UC San Diego

A plethora of computational techniques have been developed for computing quantities of interest in “multiscale” and “multiphysics” problems combining processes occurring on a broad spatiotemporal range. However, a dearth exists in systematic studies of the impact of random fluctuations on the predictive ability and numerical properties of these algorithms. We consider two nonintrusive approaches for multiphysics problems with random noise: domain decomposition and stochastic collocation. A mass-conserving domain decomposition achieving tight Newton- or Picard-based coupling between linear diffusion equations, one having a Gaussian white-noise source term, reveals that Newton's iteration scales linearly with noise amplitude, while Picard's iteration may scale superlinearly. For a given solution error, fully-converged (``implicit'') coupling is more efficient than single-iteration (``explicit'') coupling at low noise strength; at high noise amplitudes, this remains true provided that the time interval between two subsequent implicit coupling communications is sufficiently long. A similar strategy using Jacobian-free Newton-Krylov iteration to solve a highly nonlinear, multiscale diffusion problem forced by a truncated Gaussian boundary noise shows that ensuring path-wise continuity of the state variable and its flux, as opposed to continuity in the mean, accurately propagates random fluctuations and correctly captures system dynamics. Implicit coupling is more efficient than explicit coupling at all coefficients of variation considered, and domain decomposition with path-wise implicit coupling resolves temporally correlated boundary fluctuations when the correlation time exceeds some multiple of an appropriately defined characteristic diffusion time. Application of stochastic collocation to estimate the energy deposition into a brain tumor via X-ray irradiation with parametric uncertainty reveals that the uncertain parameters' coefficients of variation may be amplified by the problem's nonlinearity to the extent that the predictive uncertainty in the energy deposition almost equals the prediction itself. Algorithm refinement for the Ginzburg-Landau equation (GLE) demonstrates the need for adding a coarse-scale random source term to correctly propagate fine-scale Ising fluctuations throughout the computational domain. A moment-based approach with Gaussian closure enabling direct computation of the state variable's statistical moments is shown to be an accurate, and potentially more efficient, alternative to numerical time integration of the system state. A statistically learned stochastic GLE exhibits optimal predictive capacity at a complexity that may differ from that of standard models in the literature. This approach enables data-driven computation of the coarse-scale noise term's amplitude.