Multiscale Systems Engineering Research Group

research projects > uncertainty

Reliable Modeling & Simulation under Uncertainties

Aleatory and epistemic uncertainties should be differentiated and quantified separately. Epistemic uncertainty associated with computational models come from various sources such as impreciseness of measurement, lack of knowledge about the physical systems, information loss due to numerical approximation, inconsistency of subjective believes or experts' opinions, etc. This research is to develop an imprecise probability theory with the generalized interval form as well as study rigorously verifiable methods to solve interval constraints with various applications.

1. Generalized Interval Probability

Imprecise probability differentiates aleatory and epistemic uncertainties both qualitatively and quantitatively, which is the alternative to the traditional sensitivity analaysis in probablistic reasoning to model indeterminacy and imprecision. Different representations of imprecise probabilities have been proposed. We proposed a new form of imprecise probability based on generalized or modal intervals, called generalized interval probability, which has a more simplified calculus structure than other forms of imprecise probability for ease of use in engineering and science applications. Generalized interval probability provides a concise representation of the two uncertainty components as an extension of the classical precise probability.

With the separation between proper and improper interval probabilities, focal and non-focal events are differentiated based on the associated modalities and logical semantics. Focal events have the semantics of critical, uncontrollable, specified, etc. in probabilistic analysis, whereas the corresponding non-focal events are complementary, controllable, and derived.

A logic coherence constraint is proposed in the new form. Because of the algebraic simplicity, conditional interval probability can be directly defined based on marginal interval probabilities. A Bayes' rule with generalized intervals allows us to interpret the logic relationship between interval prior and posterior probabilities.

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2. Reliable Stochastic Simulation based on Intervals

In the traditional stochastic simulation, probability distributions with deterministic parameters represent the variability of processes. It assumes that parameters of input distributions are known with certitude. The epistemic uncertainty component is ignored in simulation. Epistemic uncertainty in simulation has different sources. For instance, the parameters of probability distributions may be uncertain when the sample size of data for input analysis is small or when the measurement errors and the quality associated with the collected data cannot be ignored. When data are not available, experts usually give judgments, which are subjective and can be inconsistent. Other contributors of uncertainties include lack of information about the dependency among factors and variables, as well as unknown time dependency of these factors. Uncertainties may also come from the simulation model itself because of partial knowledge about the physical system, model errors caused by approximations, numerical errors from floating-point round-off, etc.

We developed a new discrete-event simulation mechanism based on interval-valued probabilities, which are imprecise probabilities with interval parameters. For example, an interval exponential distribution is represented as exp([a1,a2]) where the parameter is an interval. The goal of reliable simulation is to incorporate uncertainties and support robust decision makings. In this approach, probabilistic distributions represent the aleatory component, whereas intervals capture the epistemic component. Simulation is based on random intervals instead of the traditional real-valued random variates. The outputs of simulations are also intervals that incorporate uncertainty. Reliable kinetic Monte Carlo mechanisms are developed to simulate chemical reactions and transitions. The simultaneous evolution of aleatory and epistemic uncertainties over time is also modeled by generalized stochastic dynamics. Sensitivity analysis is not necessary in those reliable stochastic simulation mechanisms.

3. Quantified Interval Constraint Satisfaction and Its Applications in Searching Design Solution Space, Tolerance Modeling, and Human-Agent Collaboration

Constraints are requirements and specifications in engineering design. We are developing new linear and nonlinear constraint solvers to find solutions that satisfy numerical constraints. At the same time, the solution ensures logic interpretability of design intents, which are expressed by first-order logic alike semantics with universal and existential quantifiers. These problems are called quantified constraint satisfaction problems (QCSPs). The developed constraint solving algorithms enhance the traditional numerical analysis methods by preserving logic interpretation information during computation .

We developed semantics-based design solution search algorithms based on generalized intervals. A semantic tolerance modeling scheme is also developed to allow for embedding more tolerancing intents in specifications with a combination of numerical intervals and logic quantifiers. By differentiating a priori and a posteriori tolerances, the logic relationships among variables can be interpreted, which is useful to verify completeness and soundness of numerical estimations in tolerance analysis.

This new generalized interval based constraint representation and solving process enable computable and interpretable numerical analysis. It can also be applied in many other domains, such as modeling the inexactness of natural languages in human-agent collaboration.

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(Sponsored by U.S. Army Research Laboratory)

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