Sparse solution of the Lyapunov equation for large-scale interconnected systems

We consider the problem of computing an approximate banded solution of the continuous-time Lyapunov equation $\underline{A}\underline{X}+\underline{X}\underline{A}^{T}=\underline{P}$, where the coefficient matrices $\underline{A}$ and $\underline{P}$ are large, symmetric banded matrices. The (sparsity) pattern of $\underline{A}$ describes the interconnection structure of a large-scale interconnected system. Recently, it has been shown that the entries of the solution $\underline{X}$ are spatially localized or decaying away from a banded pattern. We show that the decay of the entries of $\underline{X}$ is faster if the condition number of $\underline{A}$ is smaller. By exploiting the decay of entries of $\underline{X}$, we develop two computationally efficient methods for approximating $\underline{X}$ by a banded matrix. For a well-conditioned and sparse banded $\underline{A}$, the computational and memory complexities of the methods scale linearly with the state dimension. We perform extensive numerical experiments that confirm this, and that demonstrate the effectiveness of the developed methods. The methods proposed in this paper can be generalized to (sparsity) patterns of $\underline{A}$ and $\underline{P}$ that are more general than banded matrices. The results of this paper open the possibility for developing computationally efficient methods for approximating the solution of the large-scale Riccati equation by a sparse matrix.Comment: Accepted in Automatica, Final version, 16 pages, 10 figure

Localization for MCMC: sampling high-dimensional posterior distributions with local structure

We investigate how ideas from covariance localization in numerical weather prediction can be used in Markov chain Monte Carlo (MCMC) sampling of high-dimensional posterior distributions arising in Bayesian inverse problems. To localize an inverse problem is to enforce an anticipated "local" structure by (i) neglecting small off-diagonal elements of the prior precision and covariance matrices; and (ii) restricting the influence of observations to their neighborhood. For linear problems we can specify the conditions under which posterior moments of the localized problem are close to those of the original problem. We explain physical interpretations of our assumptions about local structure and discuss the notion of high dimensionality in local problems, which is different from the usual notion of high dimensionality in function space MCMC. The Gibbs sampler is a natural choice of MCMC algorithm for localized inverse problems and we demonstrate that its convergence rate is independent of dimension for localized linear problems. Nonlinear problems can also be tackled efficiently by localization and, as a simple illustration of these ideas, we present a localized Metropolis-within-Gibbs sampler. Several linear and nonlinear numerical examples illustrate localization in the context of MCMC samplers for inverse problems.Comment: 33 pages, 5 figure

Convergence of Fundamental Limitations in Feedback Communication, Estimation, and Feedback Control over Gaussian Channels

In this paper, we establish the connections of the fundamental limitations in feedback communication, estimation, and feedback control over Gaussian channels, from a unifying perspective for information, estimation, and control. The optimal feedback communication system over a Gaussian necessarily employs the Kalman filter (KF) algorithm, and hence can be transformed into an estimation system and a feedback control system over the same channel. This follows that the information rate of the communication system is alternatively given by the decay rate of the Cramer-Rao bound (CRB) of the estimation system and by the Bode integral (BI) of the control system. Furthermore, the optimal tradeoff between the channel input power and information rate in feedback communication is alternatively characterized by the optimal tradeoff between the (causal) one-step prediction mean-square error (MSE) and (anti-causal) smoothing MSE (of an appropriate form) in estimation, and by the optimal tradeoff between the regulated output variance with causal feedback and the disturbance rejection measure (BI or degree of anti-causality) in feedback control. All these optimal tradeoffs have an interpretation as the tradeoff between causality and anti-causality. Utilizing and motivated by these relations, we provide several new results regarding the feedback codes and information theoretic characterization of KF. Finally, the extension of the finite-horizon results to infinite horizon is briefly discussed under specific dimension assumptions (the asymptotic feedback capacity problem is left open in this paper).Comment: Submitted to Transactions on Information Theor

An Efficient Approximation of the Kalman Filter for Multiple Systems Coupled via Low-Dimensional Stochastic Input

We formulate a recursive estimation problem for multiple dynamical systems coupled through a low dimensional stochastic input, and we propose an efficient sub-optimal solution. The suggested approach is an approximation of the Kalman filter that discards the off diagonal entries of the correlation matrix in its "update" step. The time complexity associated with propagating this approximate block-diagonal covariance is linear in the number of systems, compared to the cubic complexity of the full Kalman filter. The stability of the proposed block-diagonal filter and its behavior for a large number of systems are analyzed in some simple cases. It is then examined in the context of electric field estimation in a high-contrast space coronagraph, for which it was designed. The numerical simulations provide encouraging results for the cost-efficiency of the newly suggested filter.Comment: 20 pages, 2 figure

Spin relaxation in a complex environment

We report the study of a model of a two-level system interacting in a non-diagonal way with a complex environment described by Gaussian orthogonal random matrices (GORM). The effect of the interaction on the total spectrum and its consequences on the dynamics of the two-level system are analyzed. We show the existence of a critical value of the interaction, depending on the mean level spacing of the environment, above which the dynamics is self-averaging and closely obey a master equation for the time evolution of the observables of the two-level system. Analytic results are also obtained in the strong coupling regimes. We finally study the equilibrium values of the two-level system population and show under which condition it thermalizes to the environment temperature.Comment: 45 pages, 49 figure

Scaling and interleaving of sub-system Lyapunov exponents for spatio-temporal systems

The computation of the entire Lyapunov spectrum for extended dynamical systems is a very time consuming task. If the system is in a chaotic spatio-temporal regime it is possible to approximately reconstruct the Lyapunov spectrum from the spectrum of a sub-system in a very cost effective way. In this work we present a new rescaling method, which gives a significantly better fit to the original Lyapunov spectrum. It is inspired by the stability analysis of the homogeneous evolution in a one-dimensional coupled map lattice but appears to be equally valid in a much wider range of cases. We evaluate the performance of our rescaling method by comparing it to the conventional rescaling (dividing by the relative sub-system volume) for one and two-dimensional lattices in spatio-temporal chaotic regimes. In doing so we notice that the Lyapunov spectra for consecutive sub-system sizes are interleaved and we discuss the possible ways in which this may arise. Finally, we use the new rescaling to approximate quantities derived from the Lyapunov spectrum (largest Lyapunov exponent, Lyapunov dimension and Kolmogorov-Sinai entropy) finding better convergence as the sub-system size is increased than with conventional rescaling.Comment: 18 pages, double column, REVTeX, 27 embedded postscript figures with psfig. Submitted to Chao

Mapping quantum chemical dynamics problems onto spin-lattice simulators

The accurate computational determination of chemical, materials, biological, and atmospheric properties has critical impact on a wide range of health and environmental problems, but is deeply limited by the computational scaling of quantum-mechanical methods. The complexity of quantum-chemical studies arises from the steep algebraic scaling of electron correlation methods, and the exponential scaling in studying nuclear dynamics and molecular flexibility. To date, efforts to apply quantum hardware to such quantum chemistry problems have focused primarily on electron correlation. Here, we provide a framework which allows for the solution of quantum chemical nuclear dynamics by mapping these to quantum spin-lattice simulators. Using the example case of a short-strong hydrogen bonded system, we construct the Hamiltonian for the nuclear degrees of freedom on a single Born-Oppenheimer surface and show how it can be transformed to a generalized Ising model Hamiltonian. We then demonstrate a method to determine the local fields and spin-spin couplings needed to identically match the molecular and spin-lattice Hamiltonians. We describe a protocol to determine the on-site and inter-site coupling parameters of this Ising Hamiltonian from the Born-Oppenheimer potential and nuclear kinetic energy operator. Our approach represents a paradigm shift in the methods used to study quantum nuclear dynamics, opening the possibility to solve both electronic structure and nuclear dynamics problems using quantum computing systems.Comment: 20 pages including supplementary information, 10 figures tota

Computational Bayesian Methods Applied to Complex Problems in Bio and Astro Statistics

In this dissertation we apply computational Bayesian methods to three distinct problems. In the first chapter, we address the issue of unrealistic covariance matrices used to estimate collision probabilities. We model covariance matrices with a Bayesian Normal-Inverse-Wishart model, which we fit with Gibbs sampling. In the second chapter, we are interested in determining the sample sizes necessary to achieve a particular interval width and establish non-inferiority in the analysis of prevalences using two fallible tests. To this end, we use a third order asymptotic approximation. In the third chapter, we wish to synthesize evidence across multiple domains in measurements taken longitudinally across time, featuring a substantial amount of structurally missing data, and fit the model with Hamiltonian Monte Carlo in a simulation to analyze how estimates of a parameter of interest change across sample sizes

Thouless and relaxation time scales in many-body quantum systems

A major open question in studies of nonequilibrium quantum dynamics is the identification of the time scales involved in the relaxation process of isolated quantum systems that have many interacting particles. We demonstrate that long time scales can be analytically found by analyzing dynamical manifestations of spectral correlations. Using this approach, we show that the Thouless time, $t_{\text{Th}}$, and the relaxation time, $t_{\text{R}}$, increase exponentially with system size. We define $t_{\text{Th}}$ as the time at which the spread of the initial state in the many-body Hilbert space is complete and verify that it agrees with the inverse of the Thouless energy. $t_{\text{Th}}$ marks the point beyond which the dynamics acquire universal features, while relaxation happens later when the evolution reaches a stationary state. In chaotic systems, $t_{\text{Th}}\ll t_{\text{R}}$, while for systems approaching a many-body localized phase, $t_{\text{Th}}\rightarrow t_{\text{R}}$. Our analytical results for $t_{\text{Th}}$ and $t_{\text{R}}$ are obtained for the survival probability, which is a global quantity. We show numerically that the same time scales appear also in the evolution of the spin autocorrelation function, which is an experimental local observable. Our studies are carried out for realistic many-body quantum models. The results are compared with those for random matrices.Comment: Final published version. 14 pages, 8 figure

Disordered statistical physics in low dimensions: extremes, glass transition, and localization

This thesis presents original results in two domains of disordered statistical physics: logarithmic correlated Random Energy Models (logREMs), and localization transitions in long-range random matrices. In the first part devoted to logREMs, we show how to characterise their common properties and model--specific data. Then we develop their replica symmetry breaking treatment, which leads to the freezing scenario of their free energy distribution and the general description of their minima process, in terms of decorated Poisson point process. We also report a series of new applications of the Jack polynomials in the exact predictions of some observables in the circular model and its variants. Finally, we present the recent progress on the exact connection between logREMs and the Liouville conformal field theory. The goal of the second part is to introduce and study a new class of banded random matrices, the broadly distributed class, which is characterid an effective sparseness. We will first study a specific model of the class, the Beta Banded random matrices, inspired by an exact mapping to a recently studied statistical model of long--range first--passage percolation/epidemics dynamics. Using analytical arguments based on the mapping and numerics, we show the existence of localization transitions with mobility edges in the "stretch--exponential" parameter--regime of the statistical models. Then, using a block--diagonalization renormalization approach, we argue that such localization transitions occur generically in the broadly distributed class.Comment: Ph.D. thesis, 154 pages, 47 figure