Ramakrishna Tipireddy

Basis adaptation in homogeneous chaos spaces

We present a new method for the characterization of subspaces associated with low-dimensional quantities of interest (QoI). The probability density function of these QoI is found to be concentrated around one-dimensional subspaces for which we develop projection operators. Our approach builds on the properties of Gaussian Hilbert spaces and associated tensor product spaces.

Quantifying mixed uncertainties in cyber attacker payoffs

Representation and propagation of uncertainty in cyber attacker payoffs is a key aspect of security games. Past research has primarily focused on representing the defenders beliefs about attacker payoffs as point utility estimates. More recently, within the physical security domain, attacker payoff uncertainties have been represented as Uniform and Gaussian probability distributions, and intervals. Within cyber-settings, continuous probability distributions may still be appropriate for addressing statistical (aleatory) uncertainties where the defender may assume that the attackers payoffs differ over time. However, systematic (epistemic) uncertainties may exist, where the defender may not have sufficient knowledge or there is insufficient information about the attackers payoff generation mechanism. Such epistemic uncertainties are more suitably represented as probability boxes with intervals. In this study, we explore the mathematical treatment of such mixed payoff uncertainties.

Propagating mixed uncertainties in cyber attacker payoffs: Exploration of two-phase Monte Carlo sampling and probability bounds analysis

Cyber-system security on a continual basis against a multitude of adverse events is a challenging undertaking. Cybersystem administrators operating with limited protective resources need to account for uncertainties associated with system behavior and types of attackers targeting a system. These uncertainties may arise due to inherent randomness or incomplete knowledge about system behavior and events affecting the system. As a result, uncertainty quantification of attacker payoff functions within stochastic cybersecurity games is a critical area of research interest. This study focuses on operationalizing a probabilistic framework for quantifying attacker payoffs, within a notional case study, through: (1) representation of uncertain attacker and system-related modeling variables as probability distributions and mathematical intervals, and (2) exploration of uncertainty propagation techniques including two-phase Monte Carlo sampling and probability bounds analysis. Enhanced uncertainty representations of payoffs may contribute to further understanding of dynamics between cyber attackers and defenders and advance the state-of-the-art in proactive cyber-system defense and strategic decision-making.

Agent-centric approach for cybersecurity decision-support with partial observability

Generating automated cyber resilience policies for real-world settings is a challenging research problem that must account for uncertainties in system state over time and dynamics between attackers and defenders. In addition to understanding attacker and defender motives and tools, and identifying “relevant” system and attack data, it is also critical to develop rigorous mathematical formulations representing the defenders decision-support problem under uncertainty. Game-theoretic approaches involving cyber resource allocation optimization with Markov decision processes (MDP) have been previously proposed in the literature. However, as is the case in strategic card games such as poker, research challenges using game-theoretic approaches for practical cyber defense applications include equilibrium solvability, existence, and possible multiplicity. Moreover, mixed uncertainties associated with player payoffs also need to be accounted for within game settings. This paper proposes an agent-centric approach for cybersecurity decision-support with partial system state observability. Multiple partially observable MDP (POMDP) problems are formulated and solved from a cyber defenders perspective, against a fixed attacker type, using synthetic (notional) system and attack parameters estimated from a Monte Carlo based sampling scheme. The agent-centric problem formulation helps address equilibrium related research challenges and represents a step toward automated and dynamic cyber resilience policy generation and implementation.

Basis adaptation and domain decomposition for steady-state partial differential equations with random coefficients

Abstract We present a novel approach for solving steady-state stochastic partial differential equations in high-dimensional random parameter space. The proposed approach combines spatial domain decomposition with basis adaptation for each subdomain. The basis adaptation is used to address the curse of dimensionality by constructing an accurate low-dimensional representation of the stochastic PDE solution (probability density function and/or its leading statistical moments) in each subdomain. Restricting the basis adaptation to a specific subdomain affords finding a locally accurate solution. Then, the solutions from all of the subdomains are stitched together to provide a global solution. We support our construction with numerical experiments for a steady-state diffusion equation with a random spatially dependent coefficient. Our results show that accurate global solutions can be obtained with significantly reduced computational costs.