Fast recursive algorithms for large time-varying mulitdimensional fields

In many applications, the signals of interest are often modeled by partial differential equations (pde) and the measurements are highly sparse. A major challenge in reconstructing these time and spatial dependent fields is the curse of dimensionality. Large spatial domains preclude the direct application of sophisticated signal processing algorithms like the Wiener filter, or the Kalman-Bucy filter (KBf). We propose in this thesis several efficient implementations of the KBf for two dimensional (2D) and for three dimensional (3D) linear time varying fields. We consider two types of measurement programs: scanned measurements as collected by instrumentation on board a satellite and point measurements as obtained by moored or floating buoys. Our implementations use the intrinsic block structure of the underlying dynamical models and the sparseness of the measurements. These representations are grouped for scanned measurements into two categories: the block implementations and the localized block implementations. The block KBF (bKBf) involves no approximations. It is exact. For time varying 2D fields, the bKBf reduces the computational complexity by O(I), where I is the linear dimension of the spatial domain. The localized block KBf (lbKBf) is approximate and reduces by O({dollar}Isp2{dollar}) the computational effort and by O(I) the storage requirements. For 3D time varying fields, the computational gains may be up to O({dollar}Isp3{dollar}) and O({dollar}Isp6{dollar}) for the bKBf and the lbKBf, respectively. The storage demands are reduced by O({dollar}Isb3{dollar}) for the lbKBf. Similar results are obtained for point measurements for which we have the scalar KBf (sKBf) and the localized scalar Kbf (lsKBf).; The practical significance of the KBf implementations is demonstrated through applications in physical oceanography. In particular, we apply our KBf algorithms to assimilate data with two different ocean circulation models: the 2D linear equatorial beta-plane model and the 3D nonlinear stratified layer model, developed by the Naval Atmospheric and Oceanographic Research Laboratory (NAORL). For the nonlinear NAORL model, we perform data assimilation via the lbKBf using dynamic linearization. Simulations for the equatorial Pacific region are performed. The measurements follow the Topex/Poseidon scanning pattern. We illustrate the subjective quality achieved by the data assimilation scheme by generating a movie depicting the time evolution of the ocean surface height. Our experiments show that data assimilation may improve significantly the field estimates over the predictions based solely on the ocean global circulation models.