Sampling and Reconstruction of Diffusive Fields on Graphs In this paper, the focus is on the reconstruction of a diffusive field and the localization of the underlying driving sources on arbitrary graphs by observing a significantly smaller subset of vertices of the graph uniformly in time. Specifically, we focus on the heat diffusion equation driven by an initial field and an external timeinvariant input. When the underlying driving sources are modeled as an initial field or external input, the sources (hence the diffusive field) can be recovered from the subsampled observations without imposing any bandlimiting or sparsity constraints. When the diffusion is induced by both the initial field and external input, then the field and sources can be recovered from the subsampled observations, however, by imposing bandlimiting constraints on either the initial field or external input. For heat diffusion on graphs, we can compensate for the unobserved vertices with the temporal samples at the observed vertices. If the observations are noiseless, then the recovery is exact. Nonetheless, the developed least squares estimators perform reasonably well with noisy observations. We apply the developed theory for localizing and recovering hot spots on a rectangular metal plate with a cavity.

Modeling and Recovery of Graph Signals and DifferenceBased Signals In this paper, we consider the problem of representing and recovering graph signals with a nonlinear measurement model. We propose a twostage graph signal processing (GSP) framework. First, a GSP representation is obtained by finding the graph filter that best approximates the known measurement function. The new GSP representation enables performing tractable operations over graphs, as well as gaining insights into the signal graphfrequency contents. Then, we formulate the signal recovery problem under the smoothness constraint and derive a regularized leastsquares (LS) estimator, which is obtained by applying the inverse of the approximated graph filter on the nonlinear measurements. In the second part of this paper, we investigate the proposed recovery and representation approach for the special case of graph signals that are influenced by the differences between vertex values only. Differencebased graph signals arise, for example, when modeling power signals as a function of the voltages in electrical networks. We show that any differencebased graph signal corresponds to a filter that lacks the zeroorder filter coefficient, and thus, these signals can be recovered up to a constant by the regularized LS estimator. In our simulations, we show that for the special case of state estimation in power systems the proposed GSP approach outperforms the stateoftheart estimator in terms of total variation.