On statistically convergent complex uncertain sequences

In this paper, we extend the study of statistical convergence of complex uncertain sequences in a given uncertainty space. We produce the relation between convergence and statistical convergence in an uncertain environment. We also initiate statistically Cauchy complex uncertain sequence to prove that a complex uncertain sequence is statistically convergent if and only if it is statistically Cauchy. We further characterize a statistically convergent complex uncertain sequence via boundedness and density operator

CONVERGENCE OF COMPLEX UNCERTAIN TRIPLE SEQUENCE VIA METRIC OPERATOR, p-DISTANCE AND COMPLETE CONVERGENCE

In this paper, we have introduced three new types of convergence concepts, namely convergence in p-distance, completely convergence and convergence in metric by considering triple sequences of complex uncertain variable. We have established the interrelationships among these notions and also with the existing ones. In this process, we have proven that the notions of convergence in metric and convergence in almost surely are equivalent in nature. Overall, this study presents a more complete scenario of interconnections between the notions of convergences initiated in different directions

Characterization of matrix transformation of complex uncertain sequences via expected value operator

The aim of this paper is to study the concept of matrix transformation between complex uncertain sequences in mean. The characterization of the matrix transformation has been made by applying the concept of convergence of complex uncertain series. Moreover, in this context, some well-known theorems of real sequence spaces have been established by considering complex uncertain sequence via expected value operator

Sparse Deterministic Approximation of Bayesian Inverse Problems

We present a parametric deterministic formulation of Bayesian inverse problems with input parameter from infinite dimensional, separable Banach spaces. In this formulation, the forward problems are parametric, deterministic elliptic partial differential equations, and the inverse problem is to determine the unknown, parametric deterministic coefficients from noisy observations comprising linear functionals of the solution. We prove a generalized polynomial chaos representation of the posterior density with respect to the prior measure, given noisy observational data. We analyze the sparsity of the posterior density in terms of the summability of the input data's coefficient sequence. To this end, we estimate the fluctuations in the prior. We exhibit sufficient conditions on the prior model in order for approximations of the posterior density to converge at a given algebraic rate, in terms of the number $N$ of unknowns appearing in the parameteric representation of the prior measure. Similar sparsity and approximation results are also exhibited for the solution and covariance of the elliptic partial differential equation under the posterior. These results then form the basis for efficient uncertainty quantification, in the presence of data with noise

Scalable Approach to Uncertainty Quantification and Robust Design of Interconnected Dynamical Systems

Development of robust dynamical systems and networks such as autonomous aircraft systems capable of accomplishing complex missions faces challenges due to the dynamically evolving uncertainties coming from model uncertainties, necessity to operate in a hostile cluttered urban environment, and the distributed and dynamic nature of the communication and computation resources. Model-based robust design is difficult because of the complexity of the hybrid dynamic models including continuous vehicle dynamics, the discrete models of computations and communications, and the size of the problem. We will overview recent advances in methodology and tools to model, analyze, and design robust autonomous aerospace systems operating in uncertain environment, with stress on efficient uncertainty quantification and robust design using the case studies of the mission including model-based target tracking and search, and trajectory planning in uncertain urban environment. To show that the methodology is generally applicable to uncertain dynamical systems, we will also show examples of application of the new methods to efficient uncertainty quantification of energy usage in buildings, and stability assessment of interconnected power networks