Gabriel Terejanu

Uncertainty propagation for nonlinear dynamic systems using Gaussian mixture models

A Gaussian-mixture-model approach is proposed for accurate uncertainty propagation through a general nonlinear system. The transition probability density function is approximated by a finite sum of Gaussian density functions for which the parameters (mean and covariance) are propagated using linear propagation theory. Two different approaches are introduced to update the weights of different components of a Gaussian-mixture model for uncertainty propagation through nonlinear system. The first method updates the weights such that they minimize the integral square difference between the true forecast probability density function and its Gaussian-sum approximation. The second method uses the Fokker-Planck-Kohnogorov equation error as feedback to adapt for the amplitude of different Gaussian components while solving a quadratic programming problem. The proposed methods are applied to a variety of problems in the open literature and are argued to be an excellent candidate for higher-dimensional uncertainty-propagation problems.

Unscented Kalman Filter/Smoother for a CBRN puff-based dispersion model

Fixed interval smoothing for systems with nonlinear process and measurement models is studied and applied to the assimilation of sensor data in a Chemical, Biological, Radiological or Nuclear (CBRN) incident scenario. A two-filter smoother that uses a Backward Sigma-Point Information Filter, and also a forward-backward Rauch-Tung-Striebel (RTS) smoothing form are re-derived using the weighted statistical linearization concept. Both methods are derived in the context of the Unscented Kalman Filter. The square root version of the resulting RTS Unscented Kalman Filter / Smoother is applied to a CBRN dispersion puff-based model with variable state dimension, and the data assimilation performance of the method is compared with a Particle Filter implementation.

Data partition methodology for validation of predictive models

In many cases, model validation requires that legacy data be partitioned into calibration and validation sets, but how to do so is a nontrivial and open area of research. We present a systematic procedure to partition the data, adapted from cross-validation and in the context of predictive modeling. By considering all possible partitions, we proceed with post-processing steps to find the optimal partition of the data subject to given constraints. We are concerned here with mathematical models of physical systems whose predictions of a given unobservable quantity of interest are the basis for critical decisions. Thus, the proposed approach addresses two critical issues: (1) that the model be evaluated with respect to its ability to reproduce the data and (2) that the model be highly challenged by the validation set with respect to predictions of the quantity of interest. This framework also relies on the interaction between the experimentalist and/or modeler, who understand the physical system and the limitations of the model; the decision-maker, who understands and can quantify the cost of model failure; and the computational scientists, who strive to determine if the model satisfies both the modelers and decision-makers requirements. The framework is general and may be applied to a wide range of problems. It is illustrated here through an example using generated experiments of a nonlinear one degree-of-freedom oscillator.

Approximate interval method for epistemic uncertainty propagation using Polynomial Chaos and evidence theory

The paper builds upon a recent approach to find the approximate bounds of a real function using Polynomial Chaos expansions. Given a function of random variables with compact support probability distributions, the intuition is to quantify the uncertainty in the response using Polynomial Chaos expansion and discard all the information provided about the randomness of the output and extract only the bounds of its compact support. To solve for the bounding range of polynomials, we transform the Polynomial Chaos expansion in the Bernstein form, and use the range enclosure property of Bernstein polynomials to find the minimum and maximum value of the response. This procedure is used to propagate Dempster-Shafer structures on closed intervals through nonlinear functions and it is applied on an algebraic challenge problem.

Quantitative Acoustic Contrast Tomography Reveals Unique Multiscale Physical Fluctuations during Aflatoxin Synthesis in Aspergillus parasiticus

Fungal pathogens need regulated mechanical and morphological fine-tuning for pushing through substrates to meet their metabolic and functional needs. Currently very little is understood on how coordinated colony level morphomechanical modifications regulate their behavior. This is due to an absence of a method that can simultaneously map, quantify, and correlate global fluctuations in physical properties of the expanding fungal colonies. Here, we show that three-dimensional ultrasonic reflections upon decoding can render acoustic contrast tomographs that contain information on material property and morphology in the same time scale of one important phytopathogen, Aspergillus parasiticus, at multiple length scales. By quantitative analysis of the changes in acoustic signatures collected as the A. parasiticus colony expands with time, we further demonstrate that the pathogen displays unique acoustic signatures during synthesis and release of its hepatocarcinogenic secondary metabolite, aflatoxin, suggesting an involvement of a multiscale morphomechanical reorganization of the colony in this process. Our studies illustrate for the first time, the feasibility of generating in any invading cell population, four-dimensional maps of global physical properties, with minimal physical perturbation of the specimens. Our developed method that we term quantitative acoustic contrast tomography (Q-ACT), provides a novel diagnostic framework for the identification of in-cell molecular factors and discovery of small molecules that may modulate pathogen invasion in a host.

An Information-Theoretic Approach to Optimally Calibrate Approximate Models

With the advancements in modeling and numerical algorithms, the decision supported by modeling and simulation has became more mainstream than ever. Even though the computational power is continually increased, in most engineering applications the problem of optimal design under uncertainty has became prohibitively expensive due to long runtimes of single simulations. The obvious solution is to reduce the complexity of the model by employing di erent assumptions and constructing this way an approximate model. The calibration of these simpler models requires a large number of runs of the complex model, which may still be too expensive and ine cient for the task at hand. In this paper, we study the problem of optimal data collection to e ciently learn the model parameters of an approximate model in the context of Bayesian analysis. The paper emphasizes the in uence of model discrepancy on the calibration of the approximate model and hence the choice of optimal designs. Model discrepancy is modeled using a Gaussian process in this study. The optimal design is obtained as a result of an information theoretic sensitivity analysis. Thus, the preferred design is where the statistical dependence between the model parameters and observables is the highest possible. In this paper, the statistical dependence between random variables is quanti ed by mutual information and estimated using a k-nearest neighbor based approximation. As a model problem, a convective-dispersion model is calibrated to approximate the physics of Burgers’ equation in a limited time domain of interest.

Spacecraft attitude estimation using adaptive gaussian sum filter

This paper is concerned with improving the attitude estimation accuracy by implementing an adaptive Gaussian sum filter where the a posteriori density function is approximated by a sum of Gaussian density functions. Compared to the traditional Gaussian sum filter, this adaptive approach utilizes the Fokker-Planck-Kolmogorov residual minimization to update the weights associated with different components of the Gaussian mixture model. Updating the weights provides an accurate approximation of the a posteriori density function and thus superior estimates. Simulation results show that updating the weights during the propagation stage not only provides better estimates between the observations but also provides superior estimator performance where the measurements are ambiguous.

Uncertainty in Modeling and Simulation

Models and simulations are necessarily approximate representations of real-world systems. There are always uncertainties inherent in the data used to create a model, as well as the behaviors and processes defined within the model itself. It is critical to understand and manage these uncertainties in any decision-making process involving the use of M&S. New approaches are required to gain better fundamental understanding of uncertainty and to realize practical methods to manage them. This chapter highlights key challenges such as unifying uncertainty-related efforts in M&S under a consistent theoretical and philosophical foundation, developing advances in theory and methods both for decision making in the M&S process and for M&S to support decision making, advances to understand and address aggregation issues in M&S, and increased use of knowledge concerning humans as decision makers in M&S activities.

Subsurface pressure profiling: a novel mathematical paradigm for computing colony pressures on substrate during fungal infections

Colony expansion is an essential feature of fungal infections. Although mechanisms that regulate hyphal forces on the substrate during expansion have been reported previously, there is a critical need of a methodology that can compute the pressure profiles exerted by fungi on substrates during expansion; this will facilitate the validation of therapeutic efficacy of novel antifungals. Here, we introduce an analytical decoding method based on Biot’s incremental stress model, which was used to map the pressure distribution from an expanding mycelium of a popular plant pathogen, Aspergillus parasiticus. Using our recently developed Quantitative acoustic contrast tomography (Q-ACT) we detected that the mycelial growth on the solid agar created multiple surface and subsurface wrinkles with varying wavelengths across the depth of substrate that were computable with acousto-ultrasonic waves between 50 MHz–175 MHz. We derive here the fundamental correlation between these wrinkle wavelengths and the pressure distribution on the colony subsurface. Using our correlation we show that A. parasiticus can exert pressure as high as 300 KPa on the surface of a standard agar growth medium. The study provides a novel mathematical foundation for quantifying fungal pressures on substrate during hyphal invasions under normal and pathophysiological growth conditions.

A stacked gaussian process for predicting geographical incidence of aflatoxin with quantified uncertainties

The objective of this paper is to develop a methodology for generating probabilistic risk maps for unobserved quantities of interests such as aflatoxin. Aflatoxin is a naturally occurring carcinogenic and it is a serious global issue and an emerging risk for crop producers. The production of aflatoxin is highly dependent on environmental conditions such temperature and water activity, and it can contaminate grains before harvest or during storage. The focus of this paper is to develop a procedure to account for spatial dependencies and uncertainties in risk calculations, to provide various stakeholders with situational awareness to better understand, communicate, and mitigate the aflatoxin risk before harvest. The proposed probabilistic model is obtained in two stages: the production of aflatoxin with quantified uncertainties is modeled under various temperature and water activity conditions within a controlled environment (wet-lab), and then the predictive aflatoxin model is linked with environmental conditions obtained on a regular basis to generate regional probabilistic risk maps. Since both aflatoxin production and environmental data are modeled using Gaussian processes, the resulted probabilistic model is a stacked Gaussian process, where the environmental Gaussian process model governs the input space of the aflatoxin Gaussian process model. The regional prediction of aflatoxin is obtained by marginalizing over the latent space provided by the environmental variables. The methodology is applied to calculate the aflatoxin levels of corn lands in South Carolina in the drought year 2012, where few field measurements are available for an initial comparison with our aflatoxin predictions.