Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection–Diffusion–Reaction Equations

We consider the estimation of parameter-dependent statistics of functional outputs of steady-state convection–diffusion–reaction equations with parametrized random and deterministic inputs in the framework of linear elliptic partial differential equations. For a given value of the deterministic parameter, a stochastic Galerkin finite element (SGFE) method can estimate the statistical moments of interest of a linear output at the cost of solving a single, large, block-structured linear system of equations. We propose a stochastic Galerkin reduced basis (SGRB) method as a means to lower the computational burden when statistical outputs are required for a large number of deterministic parameter queries. Our working assumption is that we have access to the computational resources necessary to set up such a reduced-order model for a spatial-stochastic weak formulation of the parameter-dependent model equations. In this scenario, the complexity of evaluating the SGRB model for a new value of the deterministic parameter only depends on the reduced dimension. To derive an SGRB model, we project the spatial-stochastic weak solution of a parameter-dependent SGFE model onto a reduced basis generated by a proper orthogonal decomposition (POD) of snapshots of SGFE solutions at representative values of the parameter. We propose residual-corrected estimates of the parameter-dependent expectation and variance of linear functional outputs and provide respective computable error bounds. We test the SGRB method numerically for a convection–diffusion–reaction problem, choosing the convective velocity as a deterministic parameter and the parametrized reactivity or diffusivity field as a random input. Compared to a standard reduced basis model embedded in a Monte Carlo sampling procedure, the SGRB model requires a similar number of reduced basis functions to meet a given tolerance requirement. However, only a single run of the SGRB model suffices to estimate a statistical output for a new deterministic parameter value, while the standard reduced basis model must be solved for each Monte Carlo sample.

On the multi-species Boltzmann equation with uncertainty and its stochastic Galerkin approximation

In this paper the nonlinear multi-species Boltzmann equation with random uncertainty coming from the initial data and collision kernel is studied. Well-posedness and long-time behavior – exponential decay to the global equilibrium – of the analytical solution, and spectral gap estimate for the corresponding linearized gPC-based stochastic Galerkin system are obtained, by using and extending the analytical tools provided in [M. Briant and E. S. Daus, Arch. Ration. Mech. Anal., 3, 1367–1443, 2016] for the deterministic problem in the perturbative regime, and in [E. S. Daus, S. Jin and L. Liu, Kinet. Relat. Models, 12, 909–922, 2019] for the single-species problem with uncertainty. The well-posedness result of the sensitivity system presented here has not been obtained so far even for the single-species case.

A test of an alternative approach for uncertainty representation in weather forecasting

<p>Quantification of evolving uncertainties is required for both probabilistic forecasting and data assimilation in weather prediction. In current practice, the ensemble of model simulations is often used as primary tool to describe the required uncertainties. In this work, we explore an alternative approach, so called stochastic Galerkin method which integrates uncertainties forward in time using a spectral approximation in the stochastic space. </p><p>In an idealized two-dimensional model that couples compressible non-hydrostatic Navier-Stokes equations to cloud dynamics, we investigate the propagation of initial uncertainty. The propagation of initial perturbations is followed through time for all model variables during two types of forecasts: the ensemble forecast and stochastic Galerkin forecast. Since model simulations are very expensive in weather forecasting, our hypothesis is that the stochastic Galerkin would provide more accurate and cheaper forecast statistics than the ensemble simulations. Results indicate that uncertainty as represented with mean, standard deviation and evolution of trace through time provides almost identical results if a 10000-member ensemble is used and truncation of stochastic Galerkin is made at ten spectral modes.  However, for coarser approximations,  for example if 50 ensemble members are used or the stochastic Galerkin is truncated at two modes, differences in standard deviations become significant in both approaches.  A series of experiments indicates that differences in performance of the two methods depend on the system state. For example, for stable flows, the stochastic Galerkin outperforms the ensemble of simulations for every truncation and every variable. In very unstable,  turbulent flows the estimate of the mean between the two methods still remains similar. However,  the ensemble of simulations needs more than 100 members (depending on the model variable) and the stochastic Galerkin a truncation with more than five spectral modes, to produce accurate results.</p>