This project is based on a relatively new “forward technology”: reduced basis approximation and associated a posteriori error bounds for rapid and reliable solution of parametrized partial differential equations (PDEs). The crucial ingredients are Galerkin approximation over a space spanned by “snapshots” on the parametrically induced manifold; rigorous a posteriori error bounds for the field variable and outputs of interest; efficient POD/Greedy selection of quasi-optimal samples; and Offline-Online computational procedures.
The Offline, or pre-processing stage, is very expensive; the Offline stage may typically be associated to a large parallel supercomputer. In contrast, the Online stage – input-output prediction, rigorous error bounds, and visualization for each parameter value of interest – requires minimal FLOPs, memory, and bandwidth; the Online stage may thus be associated to a thin deployed platform. The Online stage is currently implemented on the (Android OS) Nexus One Google smartphone with our Smartphone PDE app.
In this project we consider the development of an “inverse app”: integration of our novel forward methodology with algorithms for in-the-field parameter estimation, individuated product design, and control and optimisation. We are developing the methodology, the implementations on suitable thin platforms, and the proofs-of-concept through simple examples in solid mechanics and structures, heat transfer, acoustics, and fluid flow. Our emphasis is on problems where high-fidelity PDE models of the underlying physics play an important role in the siting, design, visualisation, optimisation, function, and operation of an engineering component or system.