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James PENN et Masayuki YANO, postdoctoral research associates dans l'équipe d'Anthony T. PATERA, Professor of Mechanical Engineering au Massachusetts Institute of Technology, recipient de la Chaire d'Excellence de la Fondation Sciences Mathématiques, au laboratoire LJLL de l'UPMC, Paris, invités par l'équipe Représentations Musicales - MuTant, présentent en anglais :

"Real-time Data Assimilation: The Parametrized-Background Data-Weak Formulation and Robotic Observation Platform"

We present the Parametrized-Background Data-Weak (PBDW) formulation, an integrated variational data assimilation framework which combines a "model" (partial differential equation) and "data" (M experimental observations) to yield estimates for state in real-time. We first abstract the estimation problem as a variational problem in the presence of unlimited observations. We then consider an approximate solution of the variational problem based on experimentally-realizable limited observations. We provide an associated a priori theory which identifies distinct contributions to reduction in the state error with the number of observations. The theory in addition identifies the optimal test and trial space; in the context of our data-driven approximation, the former is associated with the optimal sensor placement and the latter is informed by the parametric manifold. We develop an efficient offline-online computational strategy in the reduced basis setting in which we invoke real data in real-time.

To automate our real data collection, we design and fabricate a robotic system capable of rapid and accurate positioning of a sensor in three axes---x, y, and z. The system comprises a two-axis, planar actuator for x and y positioning that is magnetically coupled to a single-axis, linear actuator for z positioning. The sensor is attached to the low profile z positioning stage, thus affording three-axis control of its position with minimal invasiveness.

We consider the application of the PBDW method and automated measurement system to problems in acoustics, starting with a three-dimensional acoustic resonator. To collect data, we use a speaker to generate sound at various frequencies inside a raised box and move a microphone to various positions within the box to measure the local pressure. We demonstrate that "data" significantly improves the numerical predictability compared to the "model" alone.

We anticipate that the method may be used in the context of sound spatialization and synthesis to assist the accurate characterization of sound sources and environmental acoustic properties.