Model-based solution techniques for the source localization problem

The problem of locating sources in dynamical systems described by partial differential equations (PDEs) is a particular case of the more general class of inverse problems, Source localization problems are often approached using non-model-based techniques. By using a priori knowledge of system dynamics, model-based approaches to this problem can be developed, reducing the number of sensors required to solve the problem. This paper presents three such approaches: off-line numerical computation of the time response data at the sensor(s) from all possible source locations and functions of source strength, spatial and time discretization of the PDE model, and off-line solution of a dual ("forward") PDE problem based on the adjoint system model. In each case, a particular algorithm is presented, and analysis of appropriate sensor placement and the minimal number of sensors required is given. In all three approaches, a minimal amount of on-line processing is required. The relative strengths and shortcomings of the three approaches are discussed and are demonstrated through application to the two-dimensional isotropic heat equation.