COMPUTATIONAL RESEARCH in BOSTON and BEYOND (CRIBB)

Numerical methods for partial differential equations (PDEs) are a crucial tool in modern science and engineering, but these methods are often highly demanding in terms of computational resources. In many engineering contexts the ability to rapidly and accurately solve PDEs on a portable device (i.e. "in the field") would be extremely valuable, but this is clearly out of reach of classical numerical methods. However, by employing model reduction --- more specifically, the Certified Reduced Basis method --- we are able to make high-fidelity scientific computing available on deployed platforms with limited processor speed and memory.

The Certified Reduced Basis method applies to a class of PDEs of significant engineering interest; it involves a computationally expensive Offline stage in which we generate a reduced model followed by the invocation of a very cheap Online stage in which the reduced model is evaluated in real-time for user-specified parameters. We discuss the realization of this framework via a two-level hierarchy of computational architectures: we employ a TeraGrid supercomputer for the Offline and a Nexus One Android smartphone for the Online. The crucial component of our approach that yields "supercomputing on a smartphone" in a rigorous sense is that we compute a posteriori error bounds for the reduced order model with respect to the high-fidelity solution.

In this talk we emphasize the computational aspects of this hierarchical architecture: parallelization of the Offline stage in space and parameter; efficient implementation of the Online stage for input-output evaluation and visualization on `thin platforms.' We illustrate our methodology with examples and demonstrations from heat transfer, solid mechanics, acoustics, and fluid dynamics.