An adaptive sampling approach for solving PDEs with uncertain inputs and evaluating risk

© 2017, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved. When solving partial differential equations (PDEs) with random inputs, it is often computationally inefficient to merely propagate samples of the input probability law (or an approximation thereof) because the input law may not accurately capture the behavior of critical system responses that depend on the PDE solution. To further complicate matters, in many applications it is critical to accurately approximate the “risk” associated with the statistical tails of the system responses, not just the statistical moments. In this paper, we develop an adaptive sampling and local reduced basis method for approximately solving PDEs with random inputs. Our method determines a set of parameter atoms and an associated (implicit) Voronoi partition of the parameter domain on which we build local reduced basis approximations of the PDE solution. In addition, we extend our adaptive sampling approach to accurately compute measures of risk evaluated at quantities of interest that depend on the PDE solution.