Vladimir Protopopescu

Boundary value problems in abstract kinetic theory

This monograph is intended to be a reasonably self -contained and fairly complete exposition of rigorous results in abstract kinetic theory. Throughout, abstract kinetic equations refer to (an abstract formulation of) equations which describe transport of particles, momentum, energy, or, indeed, any transportable physical quantity. These include the equations of traditional (neutron) transport theory, radiative transfer, and rarefied gas dynamics, as well as a plethora of additional applications in various areas of physics, chemistry, biology and engineering. The mathematical problems addressed within the monograph deal with existence and uniqueness of solutions of initial-boundary value problems, as well as questions of positivity, continuity, growth, stability, explicit representation of solutions, and equivalence of various formulations of the transport equations under consideration. The reader is assumed to have a certain familiarity with elementary aspects of functional analysis, especially basic semigroup theory, and an effort is made to outline any more specialized topics as they are introduced. Over the past several years there has been substantial progress in developing an abstract mathematical framework for treating linear transport problems. The benefits of such an abstract theory are twofold: (i) a mathematically rigorous basis has been established for a variety of problems which were traditionally treated by somewhat heuristic distribution theory methods; and (ii) the results obtained are applicable to a great variety of disparate kinetic processes. Thus, numerous different systems of integrodifferential equations which model a variety of kinetic processes are themselves modelled by an abstract operator equation on a Hilbert (or Banach) space.

Radio interferometric tracking of mobile wireless nodes

Location-awareness is an important requirement for many mobile wireless applications today. When GPS is not applicable because of the required precision and/or the resource constraints on the hardware platform, radio interferometric ranging may offer an alternative. In this paper, we present a technique that enables the precise tracking of multiple wireless nodes simultaneously. It relies on multiple infrastructure nodes deployed at known locations measuring the position of tracked mobile nodes using radio interferometry. In addition to location information, the approach also provides node velocity estimates by measuring the Doppler shift of the interference signal. The performance of the technique is evaluated using a prototype implementation on mote-class wireless sensor nodes. Finally, a possible application scenario of dirty bomb detection in a football stadium is briefly described.

Channel-consistent forewarning of epileptic events from scalp EEG

Phase-space dissimilarity measures (PSDM) have been recently proposed to provide forewarning of impending epileptic events from scalp electroencephalography (EEG) for eventual ambulatory settings. Despite high noise in scalp EEG, PSDM yield consistently superior performance over traditional nonlinear indicators, such as Kolmogorov entropy, Lyapunov exponents, and correlation dimension. However, blind application of PSDM may result in channel inconsistency, whereby multiple datasets from the same patient yield conflicting forewarning indications in the same channel. This paper presents a first attempt to solve this problem.

DETECTING DYNAMICAL CHANGE IN NONLINEAR TIME SERIES

Abstract We present a robust, model-independent technique for measuring changes in the dynamics underlying nonlinear time-serial data. After constructing discrete density distributions of phase-space points on the attractor for time-windowed data sets, we measure the dissimilarity between density distributions via L 1 -distance and χ 2 statistics. The discriminating power of the new measures is first tested on the Lorenz model and then applied to EEG data to detect the transition between non-seizure and epileptic activity. We find a clear superiority of the new measures in comparison to traditional nonlinear measures as discriminators of changing dynamics.

The Impact of Market Clearing Time and Price Signal Delay on the Stability of Electric Power Markets

We generalize a model, proposed by Alvarado, of the electric power market by including the effects of control and communication. To simulate realistic markets, our model issues control signals only at given times and those signals are delayed during transmission. These two effects transform Alvarados continuous system into a hybrid system, with consequential effects. The stability analysis of the new system reveals two important properties. First, there is an upper limit on the market clearing time and the delay of the price signal beyond which the system becomes unstable. Second, there is a counter-intuitive relationship between the market clearing time and price signal delay: when the market clearing time is relatively long, delaying the price signal can improve the markets stability while reducing the communication delay can destabilize the market. This counter-intuitive effect shows that the full impact of information technology on power markets can be significant and difficult to anticipate. Therefore, as markets are designed and regulated, careful attention should be paid to the effects of information technology on the markets dynamic behavior.

Optimal control of boundary habitat hostility for interacting species

We consider boundary control for a parabolic system describing the evolution of two interacting species in a bounded habitat. The control models the hostility of the boundary environment to the maintenance of the species. The objective functional represents the balance between the ecological benefit (modelled by the size of the two populations) and the economic cost of maintaining an ecologically favorable boundary environment (modelled by the boundary friendliness). The unique optimal control is characterized in terms of the solution of the optimality system, which consists of the state system coupled with an adjoint system. Copyright

Solving the dynamical inverse problem for the Schrödinger equation by the boundary control method

We consider the inverse problem of determining the potential in the one-dimensional Schrodinger equation from dynamical boundary observations, which are the range values of the Neumann-to-Dirichlet map. Dynamical boundary data have not been used in the inverse problem for the Schrodinger equation, since the traditional Gelfand–Levitan–Marchenko approach reconstructs the potential from spectral or scattering data. Here we show that one can completely recover the spectral data from the dynamical boundary data. The construction of the spectral data uses new results on exact and spectral controllability for the Schrodinger equation, which we obtain by using the properties of exponential Riesz bases (nonharmonic Fourier series). From the spectral data, we solve the inverse problem using the boundary control method, which—unlike other identification methods based on control and optimization—is consistently linear and, in principle, independent of dimensionality.

Nonlinear statistics reveals stronger ties between ENSO and the tropical hydrological cycle

Cross-spectrum analysis based on linear correlations in the time domain suggested a coupling between large river flows and the El Nino-Southern Oscillation (ENSO) cycle. A nonlinear measure based on mutual information (MI) reveals extrabasinal connections between ENSO and river flows in the tropics and subtropics, that are 20-70% higher than those suggested so far by linear correlations. The enhanced dependence observed for the Nile, Amazon, Congo, Paran{acute a}, and Ganges rivers, which affect large, densely populated regions of the world, has significant impacts on inter-annual river flow predictabilities and, hence, on water resources and agricultural planning.

Empirical Mode Decomposition Technique With Conditional Mutual Information for Denoising Operational Sensor Data

This paper presents a new approach for denoising sensor signals using the Empirical Mode Decomposition (EMD) technique and the Information-theoretic method. The EMD technique is applied to decompose a noisy sensor signal into the so-called intrinsic mode functions (IMFs). These functions are of the same length and in the same time domain as the original signal. Therefore, the EMD technique preserves varying frequency in time. Assuming the given signal is corrupted by high-frequency (HF) Gaussian noise implies that most of the noise should be captured by the first few modes. Therefore, our proposition is to separate the modes into HF and low-frequency (LF) groups. We applied an information-theoretic method, namely, mutual information to determine the cutoff for separating the modes. A denoising procedure is applied only to the HF group using a shrinkage approach. Upon denoising, this group is combined with the original LF group to obtain the overall denoised signal. We illustrate our approach with simulated and real-world cargo radiation data sets. The results are compared to two popular denoising techniques in the literature, namely discrete Fourier transform (DFT) and discrete wavelet transform (DWT). We found that our approach performs better than DWT and DFT in most cases, and comparatively to DWT in some cases in terms of: 1) mean square error; 2) recomputed signal-to-noise ratio; and 3) visual quality of the denoised signals.

Learning algorithms for feedforward networks based on finite samples

We present two classes of convergent algorithms for learning continuous functions and regressions that are approximated by feedforward networks. The first class of algorithms, applicable to networks with unknown weights located only in the output layer, is obtained by utilizing the potential function methods of Aizerman et al. (1970). The second class, applicable to general feedforward networks, is obtained by utilizing the classical Robbins-Monro style stochastic approximation methods (1951). Conditions relating the sample sizes to the error bounds are derived for both classes of algorithms using martingale-type inequalities. For concreteness, the discussion is presented in terms of neural networks, but the results are applicable to general feedforward networks, in particular to wavelet networks. The algorithms can be directly adapted to concept learning problems.