We propose and investigate two new methods to approximate f(A)b for large, sparse, Hermitian matrices A. Computations of this form play an important role in numerous signal processing and machine learning tasks. The main idea behind both methods is to first estimate the spectral density of A, and then find polynomials of a fixed order that better approximate the function f on areas of the spectrum with a higher density of eigenvalues. Compared to state-of-the-art methods such as the Lanczos method and truncated Chebyshev expansion, the proposed methods tend to provide more accurate approximations of f(A)b at lower polynomial orders, and for matrices A with a large number of distinct interior eigenvalues and a small spectral width.

We investigate an M-channel critically sampled filter bank for graph signals where each of the M filters is supported on a different subband of the graph Laplacian spectrum. We partition the graph vertices such that the mthset comprises a uniqueness set for signals supported on the mthsubband. For analysis, the graph signal is filtered on each subband and downsampled on the corresponding set of vertices. However, the classical synthesis filters are replaced with interpolation operators, circumventing the issue of how to design a downsampling pattern and graph spectral filters to ensure perfect reconstruction for signals that do not reside on bipartite graphs. The resulting transform is critically sampled and graph signals are perfectly reconstructable from their analysis coefficients. We empirically explore the joint vertex-frequency localization of the dictionary atoms and sparsity of the analysis coefficients, as well as the ability of the proposed transform to compress piecewise-smooth graph signals.

This paper considers a single source transmitting data to one or more receivers/users over a shared wireless channel. Due to random fading, the wireless channel conditions vary with time and from user to user. Each user has a buffer to store received packets before they are drained. At each time step, the source determines how much power to use for transmission to each user. The source's objective is to dynamically allocate power in a manner that minimizes total power consumption and packet holding costs, while satisfying strict buffer underflow constraints and a joint power constraint in each slot. The primary application motivating this problem is wireless media streaming. For this application, the buffer underflow constraints prevent the user buffers from emptying, so as to maintain playout quality. In the case of a single user, a state-dependent modified base-stock policy is shown to be optimal with linear power-rate curves, and a state-dependent finite generalized base-stock policy is shown to be optimal with piecewise-linear convex power-rate curves. When certain technical conditions are satisfied, efficient methods to compute the critical numbers that complete the characterizations of the optimal control laws in each of these cases are presented. The structure of the optimal policy for the case of two users is then analyzed.

In this paper, we consider the problem of monitoring soil moisture evolution using a wireless network of in situ sensors. Continuously sampling moisture levels with these sensors incurs high-maintenance and energy consumption costs, which are particularly undesirable for wireless networks. Our main hypothesis is that a sparser set of measurements can meet the monitoring objectives in an energy-efficient manner. The underlying idea is that we can trade off some inaccuracy in estimating soil moisture evolution for a significant reduction in energy consumption. We investigate how to dynamically schedule the sensor measurements so as to balance this tradeoff. Unlike many prior studies on sensor scheduling that make generic assumptions on the statistics of the observed phenomenon, we obtain statistics of soil moisture evolution from a physical model. We formulate the optimal measurement scheduling and estimation problem as a partially observable Markov decision problem (POMDP). We then utilize special features of the problem to approximate the POMDP by a computationally simpler finite-state Markov decision problem (MDP). The result is a scalable, implementable technology that we have tested and validated numerically and in the field.

We have developed a new concept for a smart sensor web technology for measurements of soil moisture that include spaceborne and in-situ assets. The objective of the technology is to enable a guided/adaptive sampling strategy for the in-situ sensor network to meet the measurement validation objectives of the spaceborne sensors with respect to resolution and accuracy. One potential application is the Soil Moisture Active/Passive (SMAP) mission. The science measurements considered are the surface-to-depth profiles of soil moisture estimated from satellite radars and radiometers, with calibration and validation using in-situ sensors. Installing an in-situ network to sample the field for all ranges of variability is impractical. However, a sparser but smarter network can provide the validation estimates by operating in a guided fashion with guidance from its own sparse measurements. The feedback and control take place in the context of a dynamic data assimilation system subject to energy and accuracy constraints. The overall design of the smart sensor web including the control architecture, assimilation framework, and actuation hardware are presented in this paper. We also present results of initial numerical and laboratory demonstrations of the sensor web concept, which includes a small number of soil moisture.