Computationally efficient approximations of the joint spectral radius

The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts but is notoriously difficult to compute and to approximate. We introduce in this paper a procedure for approximating the joint spectral radius of a finite set of matrices with arbitrary high accuracy. Our approximation procedure is polynomial in the size of the matrices once the number of matrices and the desired accuracy are fixed

Efficient delay-tolerant particle filtering

This paper proposes a novel framework for delay-tolerant particle filtering that is computationally efficient and has limited memory requirements. Within this framework the informativeness of a delayed (out-of-sequence) measurement (OOSM) is estimated using a lightweight procedure and uninformative measurements are immediately discarded. The framework requires the identification of a threshold that separates informative from uninformative; this threshold selection task is formulated as a constrained optimization problem, where the goal is to minimize tracking error whilst controlling the computational requirements. We develop an algorithm that provides an approximate solution for the optimization problem. Simulation experiments provide an example where the proposed framework processes less than 40% of all OOSMs with only a small reduction in tracking accuracy

An Application of Joint Spectral Radius in Power Control Problem for Wireless Communications

Resource management, including power control, is one of the most essential functionalities of any wireless telecommunication system. Various transmitter power-control methods have been developed to deliver a desired quality of service in wireless networks. We consider two of these methods: Distributed Power Control and Distributed Balancing Algorithm schemes. We use the concept of joint spectral radius to come up with conditions for convergence of the transmitted power in these two schemes when the gains on all the communications links are assumed to vary at each time-step.Comment: 4 page

An approximate empirical Bayesian method for large-scale linear-Gaussian inverse problems

We study Bayesian inference methods for solving linear inverse problems, focusing on hierarchical formulations where the prior or the likelihood function depend on unspecified hyperparameters. In practice, these hyperparameters are often determined via an empirical Bayesian method that maximizes the marginal likelihood function, i.e., the probability density of the data conditional on the hyperparameters. Evaluating the marginal likelihood, however, is computationally challenging for large-scale problems. In this work, we present a method to approximately evaluate marginal likelihood functions, based on a low-rank approximation of the update from the prior covariance to the posterior covariance. We show that this approximation is optimal in a minimax sense. Moreover, we provide an efficient algorithm to implement the proposed method, based on a combination of the randomized SVD and a spectral approximation method to compute square roots of the prior covariance matrix. Several numerical examples demonstrate good performance of the proposed method

Approximation of the joint spectral radius using sum of squares

We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for an SOS polynomial that proves simultaneous contractibility of a finite set of matrices. We provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices. Additionally, we present a comparison between our approximation scheme and earlier techniques, including the use of common quadratic Lyapunov functions and a method based on matrix liftings. Theoretical results and numerical investigations show that our approach yields tighter approximations.Comment: 18 pages, 1 figur

Surrogate Accelerated Bayesian Inversion for the Determination of the Thermal Diffusivity of a Material

Determination of the thermal properties of a material is an important task in many scientific and engineering applications. How a material behaves when subjected to high or fluctuating temperatures can be critical to the safety and longevity of a system's essential components. The laser flash experiment is a well-established technique for indirectly measuring the thermal diffusivity, and hence the thermal conductivity, of a material. In previous works, optimization schemes have been used to find estimates of the thermal conductivity and other quantities of interest which best fit a given model to experimental data. Adopting a Bayesian approach allows for prior beliefs about uncertain model inputs to be conditioned on experimental data to determine a posterior distribution, but probing this distribution using sampling techniques such as Markov chain Monte Carlo methods can be incredibly computationally intensive. This difficulty is especially true for forward models consisting of time-dependent partial differential equations. We pose the problem of determining the thermal conductivity of a material via the laser flash experiment as a Bayesian inverse problem in which the laser intensity is also treated as uncertain. We introduce a parametric surrogate model that takes the form of a stochastic Galerkin finite element approximation, also known as a generalized polynomial chaos expansion, and show how it can be used to sample efficiently from the approximate posterior distribution. This approach gives access not only to the sought-after estimate of the thermal conductivity but also important information about its relationship to the laser intensity, and information for uncertainty quantification. We also investigate the effects of the spatial profile of the laser on the estimated posterior distribution for the thermal conductivity