A reduced-basis method for input-output uncertainty propagation in stochastic PDEs

Abstract

Recently there has been a growing interest in quantifying the effects of random inputs in the solution of partial differential equations that arise in a number of areas, including fluid mechanics, elasticity, and wave theory to describe phenomena such as turbulence, random vibrations, flow through porous media, and wave propagation through random media. Monte-Carlo based sampling methods, generalized polynomial chaos and stochastic collocation methods are some of the popular approaches that have been used in the analysis of such problems. This work proposes a non-intrusive reduced-basis method for the rapid and reliable evaluation of the statistics of linear functionals of stochastic PDEs. Our approach is based on constructing a reduced-basis model for the quantity of interest that enables to solve the full problem very efficiently. In particular, we apply a reduced-basis technique to the Hybridizable Discontinuous Galerkin (HDG) approximation of the underlying PDE, which allows for a rapid and accurate evaluation of the input-output relationship represented by a functional of the solution of the PDE. The method has been devised for problems where an affine parametrization of the PDE in terms of the uncertain input parameters may be obtained. This particular structure enables us to seek an offline-online computational strategy to economize the output evaluation. Indeed, the offline stage (performed once) is computationally intensive since its computational complexity depends on the dimension of the underlying high-order discontinuous finite element space. The online stage (performed many times) provides rapid output evaluation with a computational cost which is several orders of magnitude smaller than the computational cost of the HDG approximation. In addition, we incorporate two ingredients to the reduced-basis method. First, we employ the greedy algorithm to drive the sampling in the parameter space, by computing inexpensive bounds of the error in the output on the online stage. These error bounds allow us to detect which samples contribute most to the error, thereby enriching the reduced basis with high-quality basis functions. Furthermore, we develop the reduced basis for not only the primal problem, but also the adjoint problem. This allows us to compute an improved reduced basis output that is crucial in reducing the number of basis functions needed to achieve a prescribed error tolerance. Once the reduced bases have been constructed, we employ Monte-Carlo based sampling methods to perform the uncertainty propagation. The main achievement is that the forward evaluations needed for each Monte-Carlo sample are inexpensive, and therefore statistics of the output can be computed very efficiently. This combined technique renders an uncertainty propagation method that requires a small number of full forward model evaluations and thus greatly reduces the computational burden. We apply our approach to study the heat conduction of the thermal fin under uncertainty from the diffusivity coefficient and the wave propagation generated by a Gaussian source under uncertainty from the propagation medium. We shall also compare our approach to stochastic collocation methods and Monte-Carlo methods to assess the reliability of the computations.