arXiv subjects

Raktim Bhattacharya

Publications and source records attributed to Raktim Bhattacharya.

At least 55 records · Page 3Linked to original sources

arXiv2020

Optimal Transport Based Filtering with Nonlinear State Equality Constraints

In this work we propose a framework to address the issue of state dependent nonlinear equality-constrained state estimation using Bayesian filtering. This framework is constructed specifically for a linear approximation of Bayesian filtering that uses the theory of Optimal Transport. As a part of this framework, we present three traditionally-used nonlinear equality constraint-preserving algorithms coupled with the Optimal Transport based filter: the equality-constrained Optimal Transport filter, the projected Optimal Transport filter, and the measurement-augmented Optimal Transport filter. In cases where the nonlinear equality-constraints represent an arbitrary convex manifold, we show that the re-sampling step of Optimal Transport filter, can generate initial samples for filtering, from any probability distribution function defined on this manifold. We show numerical results using our proposed framework.

math.OC↗

arXiv2019

A unified framework to generate optimized compact finite difference schemes

A unified framework to derive optimized compact schemes for a uniform grid is presented. The optimal scheme coefficients are determined analytically by solving an optimization problem to minimize the spectral error subject to equality constraints that ensure specified order of accuracy. A rigorous stability analysis for the optimized schemes is also presented. We analytically prove the relation between order of a derivative and symmetry or skew-symmetry of the optimal coefficients approximating it. We also show that other types of schemes e.g., spatially explicit, and biased finite differences, can be generated as special cases of the framework.

math.NA↗

arXiv2019

Privacy and Utility Aware Data Sharing for Space Situational Awareness from Ensemble and Unscented Kalman Filtering Perspective

In this paper, we present an optimization-based formulation for privacy-utility tradeoff in the Ensemble and Unscented Kalman filtering framework, with a focus on space situational awareness. Privacy and utility are defined in terms of a lower and an upper bound on the state estimation error covariance, respectively. Synthetic sensor noise is used to satisfy these bounds and is determined by solving an optimization problem. Given privacy and utility bounds, we present optimization problem formulations to determine a) the maximum noise for which utility is satisfied or the estimation errors are upper-bounded, b) the minimum noise for which privacy is satisfied or the estimation errors are lower-bounded, c) the optimal noise that satisfies utility constraints and maximizes privacy, and d) the optimal noise that satisfies privacy constraints and maximizes utility. We demonstrate application of these formulations to the tracking of the International Space Station, and highlight the optimal privacy vs utility tradeoff for this dynamical system.

math.OC↗

arXiv2019

Surrogate Modeling of Dynamics From Sparse Data Using Maximum Entropy Basis Functions

In this paper we present a data driven approach for approximating dynamical systems. A dynamics is approximated using basis functions, which are derived from maximization of the information-theoretic entropy, and can be generated directly from the data provided. This approach has advantages over other methods, where a dictionary of basis functions have to be provided by the user, which is non trivial in some applications. We compare the accuracy of the proposed data-driven modeling approach to existing methods in the literature, and demonstrate that for some applications the maximum entropy basis functions provide significantly more accurate models.

math.DS↗

arXiv2019

On Improved Statistical Accuracy of Low-Order Polynomial Chaos Approximations

Polynomial chaos expansion is a popular way to develop surrogate models for stochastic systems with arbitrary random variables. Standard techniques such as Galerkin projection, stochastic collocation, and least squares approximation, are applied to determine polynomial chaos coefficients, which define the surrogate model. Since the surrogate models are developed from a function approximation perspective, there is no reason to expect accuracy of statistics from these models. The statistical moments estimated from the surrogate model may significantly differ from the true moments, especially for lower order approximations. Often arbitrary high orders are required to recover, for example, the second moment. In this paper, we present modifications of standard techniques and determine polynomial chaos coefficients by solving a constrained optimization problem. We present this new approach for algebraic functions and differential equations with random parameters, and demonstrate that the surrogate models from the new approach are able to recover the first two moments exactly.

math.OC↗

arXiv2019

A Unified Approach for Deriving Optimal Finite Differences

A unified approach to derive optimal finite differences is presented which combines three critical elements for numerical performance especially for multi-scale physical problems, namely, order of accuracy, spectral resolution and stability. The resulting mathematical framework reduces to a minimization problem subjected to equality and inequality constraints. We show that the framework can provide analytical results for optimal schemes and their numerical performance including, for example, the type of errors that appear for spectrally optimal schemes. By coupling the problem in this unified framework, one can effectively decouple the requirements for order of accuracy and spectral resolution, for example. Alternatively, we show how the framework exposes the tradeoffs between e.g. accuracy and stability and how this can be used to construct explicit schemes that remain stable with very large time steps. We also show how spectrally optimal schemes only bias odd-order derivatives to remain stable, at the expense of accuracy, while leaving even-order derivatives with symmetric coefficients. Schemes constructed within this framework are tested for diverse model problems with an emphasis on reproducing the physics accurately.

physics.comp-ph↗

arXiv2017

Design of Linear Parameter Varying Quadratic Regulator in Polynomial Chaos Framework

We present a new theoretical framework for designing linear parameter varying controllers in the polynomial chaos framework. We assume the scheduling variable to be random and apply polynomial chaos approach to synthesize the controller for the resulting linear stochastic dynamical system. Two algorithms are presented that minimize the performance objective with respect to the stochastic system. The first algorithm is based on Galerkin projection and the second algorithm is based on stochastic collocation. LPV controllers from both the algorithms are shown to outperform classical LPV designs with respect to regulator design for nonlinear missile system.

math.OC↗

arXiv2016

Convergence Analysis of Asynchronous Consensus in Discrete-time Multi-agent Systems with Fixed Topology

In this paper, we study a convergence condition for asynchronous consensus problems in multi-agent systems. The convergence in this context implies the asynchronous consensus value converges to the synchronous one and is unique. Although it is reported in the literature that the consensus value under asynchronous communications may not coincide with the synchronous consensus value, it has not received much attention. In some applications, the discrepancy between them may result in serious consequences. For such applications it is critical to determine under what conditions the asynchronous consensus value is the same as the synchronous consensus value. We illustrate these issues with a few examples and then provide a condition, which guarantees that the asynchronous consensus value converges to the synchronous one. The validity of the proposed result is verified with simulations.

eess.SY↗

arXiv2015

Asynchronous Parallel Computing Algorithm implemented in 1D Heat Equation with CUDA

In this note, we present the stability as well as performance analysis of asynchronous parallel computing algorithm implemented in 1D heat equation with CUDA. The primary objective of this note lies in dissemination of asynchronous parallel computing algorithm by providing CUDA code for fast and easy implementation. We show that the simulations carried out on nVIDIA GPU device with asynchronous scheme outperforms synchronous parallel computing algorithm. In addition, we also discuss some drawbacks of asynchronous parallel computing algorithms.

cs.DC↗

arXiv2015

On the Convergence Analysis of Asynchronous Distributed Quadratic Programming via Dual Decomposition

In this paper, we analyze the convergence as well as the rate of convergence of asynchronous distributed quadratic programming (QP) with dual decomposition technique. In general, distributed optimization requires synchronization of data at each iteration step due to the interdependency of data. This synchronization latency may incur a large amount of waiting time caused by an idle process during computation. We aim to attack this synchronization penalty in distributed QP problems by implementing asynchronous update of dual variable. The price to pay for adopting asynchronous computing algorithms is unpredictability of the solution, resulting in a tradeoff between speedup and accuracy. Thus, the convergence to an optimal solution is not guaranteed owing to the stochastic behavior of asynchrony. In this paper, we employ the switched system framework as an analysis tool to investigate the convergence of asynchronous distributed QP. This switched system will facilitate analysis on asynchronous distributed QP with dual decomposition, providing necessary and sufficient conditions for the mean square convergence. Also, we provide an analytic expression for the rate of convergence through the switched system, which enables performance analysis of asynchronous algorithms as compared with synchronous case. To verify the validity of the proposed methods, numerical examples are presented with an implementation of asynchronous parallel QP using OpenMP.

math.OC↗

arXiv2015

A Polynomial Chaos Framework for Designing Linear Parameter Varying Control Systems

Here we use polynomial chaos framework to design controllers for linear parameter varying (LPV) dynamical systems. We assume the scheduling variable to be random and use polynomial chaos approach to synthesize the controller for the resulting linear stochastic dynamical system. The stability of the LPV system is formulated as an exponential mean-square (EMS) stability problem. Two algorithms are presented that guarantee EMS stability of the stochastic system and correspond to parameter dependent and independent Lyapunov functions, respectively. LPV controllers from the polynomial chaos based framework is shown to outperform LPV controller from classical design for an example nonlinear system.

eess.SY↗

arXiv2015

Robust State Feedback Control Design with Probabilistic System Parameters

In this paper, a new polynomial chaos based framework for analyzing linear systems with probabilistic parameters is presented. Stability analysis and synthesis of optimal quadratically stabilizing controllers for such systems are presented as convex optimization problems, with exponential mean square stability guarantees. A Monte-Carlo approach for analysis and synthesis is also presented, which is used to benchmark the polynomial chaos based approach. The computational advantage of the polynomial chaos approach is shown with an example based on the design of an optimal EMS-stabilizing controller, for an F-16 aircraft model.

eess.SY↗

arXiv2015

A Switched Dynamical System Framework for Analysis of Massively Parallel Asynchronous Numerical Algorithms

In the near future, massively parallel computing systems will be necessary to solve computation intensive applications. The key bottleneck in massively parallel implementation of numerical algorithms is the synchronization of data across processing elements (PEs) after each iteration, which results in significant idle time. Thus, there is a trend towards relaxing the synchronization and adopting an asynchronous model of computation to reduce idle time. However, it is not clear what is the effect of this relaxation on the stability and accuracy of the numerical algorithm. In this paper we present a new framework to analyze such algorithms. We treat the computation in each PE as a dynamical system and model the asynchrony as stochastic switching. The overall system is then analyzed as a switched dynamical system. However, modeling of massively parallel numerical algorithms as switched dynamical systems results in a very large number of modes, which makes current analysis tools available for such systems computationally intractable. We develop new techniques that circumvent this scalability issue. The framework is presented on a one-dimensional heat equation as a case study and the proposed analysis framework is verified by solving the partial differential equation (PDE) in a $\mathtt{nVIDIA\: Tesla^{\scriptsize{TM}}}$ GPU machine, with asynchronous communication between cores.

eess.SY↗

arXiv2015

Stability Analysis of Large-Scale Distributed Networked Control Systems with Random Communication Delays: A Switched System Approach

In this paper, we consider the stability analysis of large-scale distributed networked control systems with random communication delays between linearly interconnected subsystems. The stability analysis is performed in the Markov jump linear system framework. There have been considerable researches on stability analysis of Markov jump systems, however, these methods are not applicable to large-scale systems because large numbers of subsystems result in an extremely large number of the switching modes. To avoid this scalability issue, we propose a new reduced mode model for stability analysis, which is computationally efficient. We also consider the case in which the transition probabilities for the Markov jump process contain uncertainties. We provide a new method that estimates bounds for uncertain Markov transition probability matrix to guarantee the system stability. The efficiency and the usefulness of the proposed methods are verified through examples.

eess.SY↗

arXiv2014

Optimal Switching Synthesis for Jump Linear Systems with Gaussian initial state uncertainty

This paper provides a method to design an optimal switching sequence for jump linear systems with given Gaussian initial state uncertainty. In the practical perspective, the initial state contains some uncertainties that come from measurement errors or sensor inaccuracies and we assume that the type of this uncertainty has the form of Gaussian distribution. In order to cope with Gaussian initial state uncertainty and to measure the system performance, Wasserstein metric that defines the distance between probability density functions is used. Combining with the receding horizon framework, an optimal switching sequence for jump linear systems can be obtained by minimizing the objective function that is expressed in terms of Wasserstein distance. The proposed optimal switching synthesis also guarantees the mean square stability for jump linear systems. The validations of the proposed methods are verified by examples.

eess.SY↗

arXiv2014

Performance and Robustness Analysis of Stochastic Jump Linear Systems using Wasserstein metric

This paper focuses on the performance and the robustness analysis of stochastic jump linear systems. The state trajectory under stochastic jump process becomes random variables, which brings forth the probability distributions in the system state. Therefore, we need to adopt a proper metric to measure the system performance with respect to stochastic switching. In this perspective, Wasserstein metric that assesses the distance between probability density functions is applied to provide the performance and the robustness analysis. Both the transient and steady-state performance of the systems with given initial state uncertainties can be measured in this framework. Also, we prove that the convergence of this metric implies the mean square stability. Overall, this study provides a unifying framework for the performance and the robustness analysis of general stochastic jump linear systems, but not necessarily Markovian jump process that is commonly used for stochastic switching. The practical usefulness and efficiency of the proposed method are verified through numerical examples.

eess.SY↗

arXiv2014

Mean Square Stability for Stochastic Jump Linear Systems via Optimal Transport

In this note, we provide a unified framework for the mean square stability of stochastic jump linear systems via optimal transport. The Wasserstein metric known as an optimal transport, that assesses the distance between probability density functions enables the stability analysis. Without any assumption on the underlying jump process, this Wasserstein distance guarantees the mean square stability for general stochastic jump linear systems, not necessarily for Markovian jump. The validity of the proposed methods are proved by recovering already-known stability conditions under this framework.

eess.SY↗

arXiv2014

Probabilistic Robustness Analysis of Stochastic Jump Linear Systems

In this paper, we propose a new method to measure the probabilistic robustness of stochastic jump linear system with respect to both the initial state uncertainties and the randomness in switching. Wasserstein distance which defines a metric on the manifold of probability density functions is used as tool for the performance and the stability measures. Starting with Gaussian distribution to represent the initial state uncertainties, the probability density function of the system state evolves into mixture of Gaussian, where the number of Gaussian components grows exponentially. To cope with computational complexity caused by mixture of Gaussian, we prove that there exists an alternative probability density function that preserves exact information in the Wasserstein level. The usefulness and the efficiency of the proposed methods are demonstrated by example.

eess.SY↗