Parameter Space: The Final Frontier. Certified Reduced Basis Methods for Real-Time Reliable Solution of Parametrized Partial Differential Equations

摘要

This project is focused on reduced basis approximation methods, associated rigorous and sharp aposteriori error bounds, and offline-online computational strategies for the rapid and reliable solution of parametrized elliptic, parabolic, and more recently hyperbolic partial differential equations relevant to mechanics from the quantum through the meso-scale to the macro-scale. Typical equations and applications of interest include Density Functional Theory for solid state property calculations, the Boltzmann equation for microscale gas flows, the Navier-Stokes equations for natural convection calculations, elasticity for stress intensity factors/brittle failure, and Helmholtz and the wave equation for acoustic waveguide applications. Of particular interest is real-time and robust parameter estimation with application to detection, nondestructive evaluation, adaptive design/optimization, and control. In the online/deployed stage, we can provide results for key engineering outputs in real-time without loss of accuracy or reliability: the outputs provided - in milliseconds (online) - by our approach are provably indistinguishable from the outputs provided - typically in many minutes or even hours - by classical methods.