  Graph Signal Processing (GSP) is an emerging research field that extends the
concepts of digital signal processing to graphs. GSP has numerous applications
in different areas such as sensor networks, machine learning, and image
processing. The sampling and reconstruction of static graph signals have played
a central role in GSP. However, many real-world graph signals are inherently
time-varying and the smoothness of the temporal differences of such graph
signals may be used as a prior assumption. In the current work, we assume that
the temporal differences of graph signals are smooth, and we introduce a novel
algorithm based on the extension of a Sobolev smoothness function for the
reconstruction of time-varying graph signals from discrete samples. We explore
some theoretical aspects of the convergence rate of our Time-varying Graph
signal Reconstruction via Sobolev Smoothness (GraphTRSS) algorithm by studying
the condition number of the Hessian associated with our optimization problem.
Our algorithm has the advantage of converging faster than other methods that
are based on Laplacian operators without requiring expensive eigenvalue
decomposition or matrix inversions. The proposed GraphTRSS is evaluated on
several datasets including two COVID-19 datasets and it has outperformed many
existing state-of-the-art methods for time-varying graph signal reconstruction.
GraphTRSS has also shown excellent performance on two environmental datasets
for the recovery of particulate matter and sea surface temperature signals.
