Abstract:
We present the model-data weak formulation (MDWF), an integrated variational framework which combines a "model" (partial differential equation) and "data" (M experimental observations) to yield estimates for state and model bias.  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 spaces, which, in the context of MDWF, corresponds to the optimal sensor location and bias-shape spaces, respectively.  We then incorporate certified reduced basis method into the model-data variational formulation. We in particular develop an efficient offline-online computational strategy in the reduced basis setting in which we invoke real data in real-time.  We finally apply the method to real data associated with a (three-dimensional) acoustic resonator to assess the effectiveness of the proposed method.

(This is work in collaboration with Prof. Anthony Patera and Dr. James Penn.)