Non-overlapping domain decomposition methods for computationally efficient estimation of parabolic PDEs with mobile sensors

Abstract

Addressing the perpetual quest for computational reduction and efficiency in the estimation of large scale systems such as those governed by partial differential equations, this paper presents a new approach for such a computational reduction. Considering the state estimation of parabolic PDEs over large spatial domains, especially the ones with unknown dimensions and boundaries, the proposed method decomposes the spatial domain into a smaller region of interest that surrounds the current position of a moving sensor, and a region away from the current position of the sensor. To increase the spatial resolution of the estimation scheme, the inner subdomain is discretized using a refined grid with the remainder of the large domain discretized by a coarse grid, thus resulting in a significant reduction in the computational load. Additionally, by solving the Kalman filter over the smaller subdomain with refined grid over the neighborhood of the sensor, the proposed scheme offers additional reduction in the computation of the filter kernels that correspond to the filter operators.