arXiv subjects

Tuhin Sahai

Publications and source records attributed to Tuhin Sahai.

34 records · Page 2Linked to original sources

arXiv2020

Stochastic Optimization using Polynomial Chaos Expansions

Polynomial chaos based methods enable the efficient computation of output variability in the presence of input uncertainty in complex models. Consequently, they have been used extensively for propagating uncertainty through a wide variety of physical systems. These methods have also been employed to build surrogate models for accelerating inverse uncertainty quantification (infer model parameters from data) and construct transport maps. In this work, we explore the use of polynomial chaos based approaches for optimizing functions in the presence of uncertainty. These methods enable the fast propagation of uncertainty through smooth systems. If the dimensionality of the random parameters is low, these methods provide orders of magnitude acceleration over Monte Carlo sampling. We construct a generalized polynomial chaos based methodology for optimizing smooth functions in the presence of random parameters that are drawn from \emph{known} distributions. By expanding the optimization variables using orthogonal polynomials, the stochastic optimization problem reduces to a deterministic one that provides estimates for all moments of the output distribution. Thus, this approach enables one to avoid computationally expensive random sampling based approaches such as Monte Carlo and Quasi-Monte Carlo. In this work, we develop the overall framework, derive error bounds, construct the framework for the inclusion of constraints, analyze various properties of the approach, and demonstrate the proposed technique on illustrative examples.

math.OC↗

arXiv2020

Post Quantum Secure Command and Control of Mobile Agents : Inserting quantum-resistant encryption schemes in the Secure Robot Operating System

The secure command and control (C&C) of mobile agents arises in various settings including unmanned aerial vehicles, single pilot operations in commercial settings, and mobile robots to name a few. As more and more of these applications get integrated into aerospace and defense use cases, the security of the communication channel between the ground station and the mobile agent is of increasing importance. The development of quantum computing devices poses a unique threat to secure communications due to the vulnerability of asymmetric ciphers to Shor's algorithm. Given the active development of new quantum resistant encryption techniques, we report the first integration of post-quantum secure encryption schemes with the robot operating system (ROS) and C&C of mobile agents, in general. We integrate these schemes in the application and network layers, and study the performance of these methods by comparing them to present day security schemes such as the widely used RSA algorithm.

cs.CR↗

arXiv2020

Dynamical Systems Theory and Algorithms for NP-hard Problems

This article surveys the burgeoning area at the intersection of dynamical systems theory and algorithms for NP-hard problems. Traditionally, computational complexity and the analysis of non-deterministic polynomial-time (NP)-hard problems have fallen under the purview of computer science and discrete optimization. However, over the past few years, dynamical systems theory has increasingly been used to construct new algorithms and shed light on the hardness of problem instances. We survey a range of examples that illustrate the use of dynamical systems theory in the context of computational complexity analysis and novel algorithm construction. In particular, we summarize a) a novel approach for clustering graphs using the wave equation partial differential equation, b) invariant manifold computations for the traveling salesman problem, c) novel approaches for building quantum networks of Duffing oscillators to solve the MAX-CUT problem, d) applications of the Koopman operator for analyzing optimization algorithms, and e) the use of dynamical systems theory to analyze computational complexity.

math.OC↗

arXiv2019

Estimating the Density of States of Boolean Satisfiability Problems on Classical and Quantum Computing Platforms

Given a Boolean formula $ϕ(x)$ in conjunctive normal form (CNF), the density of states counts the number of variable assignments that violate exactly $e$ clauses, for all values of $e$. Thus, the density of states is a histogram of the number of unsatisfied clauses over all possible assignments. This computation generalizes both maximum-satisfiability (MAX-SAT) and model counting problems and not only provides insight into the entire solution space, but also yields a measure for the \emph{hardness} of the problem instance. Consequently, in real-world scenarios, this problem is typically infeasible even when using state-of-the-art algorithms. While finding an exact answer to this problem is a computationally intensive task, we propose a novel approach for estimating density of states based on the concentration of measure inequalities. The methodology results in a quadratic unconstrained binary optimization (QUBO), which is particularly amenable to quantum annealing-based solutions. We present the overall approach and compare results from the D-Wave quantum annealer against the best-known classical algorithms such as the Hamze-de Freitas-Selby (HFS) algorithm and satisfiability modulo theory (SMT) solvers.

cs.DM↗

arXiv2019

Continuous Relaxations for the Traveling Salesman Problem

In this work, we aim to explore connections between dynamical systems techniques and combinatorial optimization problems. In particular, we construct heuristic approaches for the traveling salesman problem (TSP) based on embedding the relaxed discrete optimization problem into appropriate manifolds. We explore multiple embedding techniques -- namely, the construction of new dynamical systems on the manifold of orthogonal matrices and associated Procrustes approximations of the TSP cost function. Using these dynamical systems, we analyze the local neighborhood around the optimal TSP solutions (which are equilibria) using computations to approximate the associated \emph{stable manifolds}. We find that these flows frequently converge to undesirable equilibria. However, the solutions of the dynamical systems and the associated Procrustes approximation provide an interesting biasing approach for the popular Lin--Kernighan heuristic which yields fast convergence. The Lin--Kernighan heuristic is typically based on the computation of edges that have a `high probability' of being in the shortest tour, thereby effectively pruning the search space. Our new approach, instead, relies on a natural relaxation of the combinatorial optimization problem to the manifold of orthogonal matrices and the subsequent use of this solution to bias the Lin--Kernighan heuristic. Although the initial cost of computing these edges using the Procrustes solution is higher than existing methods, we find that the Procrustes solution, when coupled with a homotopy computation, contains valuable information regarding the optimal edges. We explore the Procrustes based approach on several TSP instances and find that our approach often requires fewer $k$-opt moves than existing approaches. Broadly, we hope that this work initiates more work in the intersection of dynamical systems theory and combinatorial optimization.

cs.DM↗

arXiv2018

Spectral Complexity of Directed Graphs and Application to Structural Decomposition

We introduce a new measure of complexity (called spectral complexity) for directed graphs. We start with splitting of the directed graph into its recurrent and non-recurrent parts. We define the spectral complexity metric in terms of the spectrum of the recurrence matrix (associated with the reccurent part of the graph) and the Wasserstein distance. We show that the total complexity of the graph can then be defined in terms of the spectral complexity, complexities of individual components and edge weights. The essential property of the spectral complexity metric is that it accounts for directed cycles in the graph. In engineered and software systems, such cycles give rise to sub-system interdependencies and increase risk for unintended consequences through positive feedback loops, instabilities, and infinite execution loops in software. In addition, we present a structural decomposition technique that identifies such cycles using a spectral technique. We show that this decomposition complements the well-known spectral decomposition analysis based on the Fiedler vector. We provide several examples of computation of spectral and total complexities, including the demonstration that the complexity increases monotonically with the average degree of a random graph. We also provide an example of spectral complexity computation for the architecture of a realistic fixed wing aircraft system.

math.SP↗

arXiv2017

A Spectral Assignment Approach for the Graph Isomorphism Problem

In this paper, we propose algorithms for the graph isomorphism (GI) problem that are based on the eigendecompositions of the adjacency matrices. The eigenvalues of isomorphic graphs are identical. However, two graphs $ G_A $ and $ G_B $ can be isospectral but non-isomorphic. We first construct a graph isomorphism testing algorithm for friendly graphs and then extend it to unambiguous graphs. We show that isomorphisms can be detected by solving a linear assignment problem. If the graphs possess repeated eigenvalues, which typically correspond to graph symmetries, finding isomorphisms is much harder. By repeatedly perturbing the adjacency matrices and by using properties of eigenpolytopes, it is possible to break symmetries of the graphs and iteratively assign vertices of $ G_A $ to vertices of $ G_B $, provided that an admissible assignment exists. This heuristic approach can be used to construct a permutation which transforms $ G_A $ into $ G_B $ if the graphs are isomorphic. The methods will be illustrated with several guiding examples.

cs.DM↗

arXiv2015

A Chaotic Dynamical System that Paints

Can a dynamical system paint masterpieces such as Da Vinci's Mona Lisa or Monet's Water Lilies? Moreover, can this dynamical system be chaotic in the sense that although the trajectories are sensitive to initial conditions, the same painting is created every time? Setting aside the creative aspect of painting a picture, in this work, we develop a novel algorithm to reproduce paintings and photographs. Combining ideas from ergodic theory and control theory, we construct a chaotic dynamical system with predetermined statistical properties. If one makes the spatial distribution of colors in the picture the target distribution, akin to a human, the algorithm first captures large scale features and then goes on to refine small scale features. Beyond reproducing paintings, this approach is expected to have a wide variety of applications such as uncertainty quantification, sampling for efficient inference in scalable machine learning for big data, and developing effective strategies for search and rescue. In particular, our preliminary studies demonstrate that this algorithm provides significant acceleration and higher accuracy than competing methods for Markov Chain Monte Carlo (MCMC).

nlin.CD↗

arXiv2014

Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems

Polynomial chaos is a powerful technique for propagating uncertainty through ordinary and partial differential equations. Random variables are expanded in terms of orthogonal polynomials and differential equations are derived for the expansion coefficients. Here we study the structure and dynamics of these differential equations when the original system has Hamiltonian structure, has multiple time scales, or displays chaotic dynamics. In particular, we prove that the differential equations for the expansion coefficients in generalized polynomial chaos expansions of Hamiltonian systems retain the Hamiltonian structure relative to the ensemble average Hamiltonian. We connect this with the volume-preserving property of Hamiltonian flows to show that, for an oscillator with uncertain frequency, a finite expansion must fail at long times, regardless of the order of the expansion. Also, using a two-time scale forced nonlinear oscillator, we show that a polynomial chaos expansion of the time-averaged equations captures uncertainty in the slow evolution of the Poincaré section of the system and that, as the time scale separation increases, the computational advantage of this procedure increases. Finally, using the forced Duffing oscillator as an example, we demonstrate that when the original dynamical system displays chaotic dynamics, the resulting dynamical system from polynomial chaos also displays chaotic dynamics, limiting its applicability.

stat.CO↗

arXiv2012

A Traveling Salesman Learns Bayesian Networks

Structure learning of Bayesian networks is an important problem that arises in numerous machine learning applications. In this work, we present a novel approach for learning the structure of Bayesian networks using the solution of an appropriately constructed traveling salesman problem. In our approach, one computes an optimal ordering (partially ordered set) of random variables using methods for the traveling salesman problem. This ordering significantly reduces the search space for the subsequent greedy optimization that computes the final structure of the Bayesian network. We demonstrate our approach of learning Bayesian networks on real world census and weather datasets. In both cases, we demonstrate that the approach very accurately captures dependencies between random variables. We check the accuracy of the predictions based on independent studies in both application domains.

cs.LG↗

arXiv2012

Uncertainty Quantification in Hybrid Dynamical Systems

Uncertainty quantification (UQ) techniques are frequently used to ascertain output variability in systems with parametric uncertainty. Traditional algorithms for UQ are either system-agnostic and slow (such as Monte Carlo) or fast with stringent assumptions on smoothness (such as polynomial chaos and Quasi-Monte Carlo). In this work, we develop a fast UQ approach for hybrid dynamical systems by extending the polynomial chaos methodology to these systems. To capture discontinuities, we use a wavelet-based Wiener-Haar expansion. We develop a boundary layer approach to propagate uncertainty through separable reset conditions. We also introduce a transport theory based approach for propagating uncertainty through hybrid dynamical systems. Here the expansion yields a set of hyperbolic equations that are solved by integrating along characteristics. The solution of the partial differential equation along the characteristics allows one to quantify uncertainty in hybrid or switching dynamical systems. The above methods are demonstrated on example problems.

stat.CO↗

arXiv2012

Mapping and Reducing the Brain on the Cloud

The emergence of cloud computing has enabled an incredible growth in available hardware resources at very low costs. These resources are being increasingly utilized by corporations for scalable analysis of "big data" problems. In this work, we explore the possibility of using commodity hardware such as Amazon EC2 for performing large scale scientific computation. In particular, we simulate interconnected cortical neurons using MapReduce. We build and model a network of 1000 spiking cortical neurons in Hadoop, an opensource implementation of MapReduce, and present results.

cs.DC↗

arXiv2011

Iterative Methods for Scalable Uncertainty Quantification in Complex Networks

In this paper we address the problem of uncertainty management for robust design, and verification of large dynamic networks whose performance is affected by an equally large number of uncertain parameters. Many such networks (e.g. power, thermal and communication networks) are often composed of weakly interacting subnetworks. We propose intrusive and non-intrusive iterative schemes that exploit such weak interconnections to overcome dimensionality curse associated with traditional uncertainty quantification methods (e.g. generalized Polynomial Chaos, Probabilistic Collocation) and accelerate uncertainty propagation in systems with large number of uncertain parameters. This approach relies on integrating graph theoretic methods and waveform relaxation with generalized Polynomial Chaos, and Probabilistic Collocation, rendering these techniques scalable. We analyze convergence properties of this scheme and illustrate it on several examples.

stat.CO↗

arXiv2011

Hearing the clusters in a graph: A distributed algorithm

We propose a novel distributed algorithm to cluster graphs. The algorithm recovers the solution obtained from spectral clustering without the need for expensive eigenvalue/vector computations. We prove that, by propagating waves through the graph, a local fast Fourier transform yields the local component of every eigenvector of the Laplacian matrix, thus providing clustering information. For large graphs, the proposed algorithm is orders of magnitude faster than random walk based approaches. We prove the equivalence of the proposed algorithm to spectral clustering and derive convergence rates. We demonstrate the benefit of using this decentralized clustering algorithm for community detection in social graphs, accelerating distributed estimation in sensor networks and efficient computation of distributed multi-agent search strategies.

cs.DM↗

arXiv2011

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.

eess.SY↗

arXiv2010

An efficient algorithm for the parallel solution of high-dimensional differential equations

The study of high-dimensional differential equations is challenging and difficult due to the analytical and computational intractability. Here, we improve the speed of waveform relaxation (WR), a method to simulate high-dimensional differential-algebraic equations. This new method termed adaptive waveform relaxation (AWR) is tested on a communication network example. Further we propose different heuristics for computing graph partitions tailored to adaptive waveform relaxation. We find that AWR coupled with appropriate graph partitioning methods provides a speedup by a factor between 3 and 16.

cs.DC↗