MIT Uncertainty Quantification Group

We propose a new dynamic algorithm which transports samples from a reference distribution to a target distribution in unit time, given access to the target-to-reference density ratio. Our approach is to seek a sequence of transport maps that push forward the reference along a path given by a geometric mixture of the two densities. We take the maps to be simply parameterized, local, sample-driven optimal transport maps which we identify by approximately solving a root-finding problem formulated using importance weights. When feature functions for the maps are taken to be kernels, we obtain a novel interacting particle system from which we derive finite-particle and mean-field ODEs. In discrete time, we introduce an adaptive algorithm for simulating this interacting particle system which adjusts the ODE time steps based on the quality of the transport, automatically uncovering a good "schedule" for traversing the geometric mixture of densities.

@inproceedings{maurais-dynamictransport-2023,
  author     = "Maurais, A. and Marzouk, Y. M.",
  booktitle  = "NeurIPS 2023 Workshop on Optimal Transport and Machine Learning",
  month      = "12",
  title      = "Adaptive Algorithms for Continuous-Time Transport: Homotopy-Driven Sampling and a New Interacting Particle System",
  year       = "2023",
  abstract   = "We propose a new dynamic algorithm which transports samples from a reference distribution to a target distribution in unit time, given access to the target-to-reference density ratio. Our approach is to seek a sequence of transport maps that push forward the reference along a path given by a geometric mixture of the two densities. We take the maps to be simply parameterized, local, sample-driven optimal transport maps which we identify by approximately solving a root-finding problem formulated using importance weights. When feature functions for the maps are taken to be kernels, we obtain a novel interacting particle system from which we derive finite-particle and mean-field ODEs. In discrete time, we introduce an adaptive algorithm for simulating this interacting particle system which adjusts the ODE time steps based on the quality of the transport, automatically uncovering a good "schedule" for traversing the geometric mixture of densities.",
  keywords   = "optimal transport, homotopy, interacting particle system, mean field ODE, adaptivity"
}

We introduce a new mean-field ODE and corresponding interacting particle systems (IPS) for sampling from an unnormalized target density. The IPS are gradient-free, available in closed form, and only require the ability to sample from a reference density and compute the (unnormalized) target-to-reference density ratio. The mean-field ODE is obtained by solving a Poisson equation for a velocity field that transports samples along the geometric mixture of the two densities, which is the path of a particular Fisher-Rao gradient flow. We employ a RKHS ansatz for the velocity field, which makes the Poisson equation tractable and enables discretization of the resulting mean-field ODE over finite samples. The mean-field ODE can be additionally be derived from a discrete-time perspective as the limit of successive linearizations of the Monge-Ampère equations within a framework known as sample-driven optimal transport. We introduce a stochastic variant of our approach and demonstrate empirically that our IPS can produce high-quality samples from varied target distributions, outperforming comparable gradient-free particle systems and competitive with gradient-based alternatives.

@inproceedings{maurais-kfrflow-2024,
  author     = "Maurais, A. and Marzouk, Y. M.",
  booktitle  = "International Conference on Machine Learning (ICML)",
  editor     = "R Salakhutdinov and Z Kolter and K Heller and A Weller and N Oliver and J Scarlett and F Berkenkamp",
  journal    = "Preprint",
  month      = "7",
  pages      = "35138--35162",
  title      = "Sampling in Unit Time with Kernel Fisher-Rao Flow",
  volume     = "235",
  year       = "2024",
  abstract   = "We introduce a new mean-field ODE and corresponding interacting particle systems (IPS) for sampling from an unnormalized target density. The IPS are gradient-free, available in closed form, and only require the ability to sample from a reference density and compute the (unnormalized) target-to-reference density ratio. The mean-field ODE is obtained by solving a Poisson equation for a velocity field that transports samples along the geometric mixture of the two densities, which is the path of a particular Fisher-Rao gradient flow. We employ a RKHS ansatz for the velocity field, which makes the Poisson equation tractable and enables discretization of the resulting mean-field ODE over finite samples. The mean-field ODE can be additionally be derived from a discrete-time perspective as the limit of successive linearizations of the Monge-Ampère equations within a framework known as sample-driven optimal transport. We introduce a stochastic variant of our approach and demonstrate empirically that our IPS can produce high-quality samples from varied target distributions, outperforming comparable gradient-free particle systems and competitive with gradient-based alternatives.",
  keywords   = "gradient flow;generative modelling;sampling"
}

We propose a novel diffusion map particle system (DMPS) for generative modeling, based on diffusion maps and Laplacian-adjusted Wasserstein gradient descent (LAWGD). Diffusion maps are used to approximate the generator of the Langevin diffusion process from samples, and hence to learn the underlying data-generating manifold. On the other hand, LAWGD enables efficient sampling from the target distribution given a suitable choice of kernel, which we construct here via a spectral approximation of the generator, computed with diffusion maps. Numerical experiments show that our method outperforms others on synthetic datasets, including examples with manifold structure.

@inproceedings{li-diffusionparticles-2023,
  author     = "Li, F. Y. and Marzouk, Y. M.",
  booktitle  = "ICML 2023 Workshop on Structured Probabilistic Inference & Generative Modeling",
  journal    = "Preprint",
  month      = "4",
  title      = "Diffusion map particle systems for generative modeling",
  year       = "2023",
  doi        = "10.48550/arXiv.2304.00200",
  abstract   = "We propose a novel diffusion map particle system (DMPS) for generative modeling, based on diffusion maps and Laplacian-adjusted Wasserstein gradient descent (LAWGD). Diffusion maps are used to approximate the generator of the Langevin diffusion process from samples, and hence to learn the underlying data-generating manifold. On the other hand, LAWGD enables efficient sampling from the target distribution given a suitable choice of kernel, which we construct here via a spectral approximation of the generator, computed with diffusion maps. Numerical experiments show that our method outperforms others on synthetic datasets, including examples with manifold structure.",
  keywords   = "diffusion maps, kernel methods, gradient flows, generative modeling, sampling"
}

We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. The estimator fuses samples from a hierarchy of data sources of differing fidelities and costs for variance reduction while guaranteeing definiteness, in contrast with previous approaches. The new estimator makes covariance estimation tractable in applications where simulation or data collection is expensive; to that end, we develop an optimal sample allocation scheme that minimizes the mean-squared error of the estimator given a fixed budget. Guaranteed definiteness is crucial to metric learning, data assimilation, and other downstream tasks. Evaluations of our approach using data from physical applications (heat conduction, fluid dynamics) demonstrate more accurate metric learning and speedups of more than one order of magnitude compared to benchmarks.

@inproceedings{maurais-covlogeuclidean-2023,
  author     = "Maurais, A. and Alsup, T. and Peherstorfer, B. and Marzouk, Y. M.",
  booktitle  = "Proceedings of the 40th International Conference on Machine Learning",
  editor     = "Krause, Andrea and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan",
  month      = "7",
  pages      = "24214--24235",
  publisher  = "PMLR",
  series     = "Proceedings of Machine Learning Research",
  title      = "Multi-Fidelity Covariance Estimation in the Log-Euclidean Geometry",
  volume     = "202",
  year       = "2023",
  abstract   = "We introduce a multi-fidelity estimator of covariance matrices that employs the log-Euclidean geometry of the symmetric positive-definite manifold. The estimator fuses samples from a hierarchy of data sources of differing fidelities and costs for variance reduction while guaranteeing definiteness, in contrast with previous approaches. The new estimator makes covariance estimation tractable in applications where simulation or data collection is expensive; to that end, we develop an optimal sample allocation scheme that minimizes the mean-squared error of the estimator given a fixed budget. Guaranteed definiteness is crucial to metric learning, data assimilation, and other downstream tasks. Evaluations of our approach using data from physical applications (heat conduction, fluid dynamics) demonstrate more accurate metric learning and speedups of more than one order of magnitude compared to benchmarks.",
  keywords   = "multifidelity, covariance"
}

Transportation of probability measures underlies many core tasks in statistics and machine learning, from variational inference to generative modeling. A typical goal is to represent a target probability measure of interest as the push-forward of a tractable source measure through a learned map. We present a new construction of such a transport map, given the ability to evaluate the score of the target distribution. Specifically, we characterize the map as a zero of an infinite-dimensional score-residual operator and derive a Newton-type method for iteratively constructing such a zero. We prove convergence of these iterations by invoking classical elliptic regularity theory for partial differential equations (PDE) and show that this construction enjoys rapid convergence, under smoothness assumptions on the target score. A key element of our approach is a generalization of the elementary Newton method to infinite-dimensional operators, other forms of which have appeared in nonlinear PDE and in dynamical systems. Our Newton construction, while developed in a functional setting, also suggests new iterative algorithms for approximating transport maps.

@inproceedings{chandramoorthy-newtonmethodtransport-2024,
  author     = "Chandramoorthy, N. and Schaefer, F. and Marzouk, Y. M.",
  booktitle  = "Proceedings of The 27th International Conference on Artificial Intelligence and Statistics",
  journal    = "Artificial Intelligence and Statistics (AISTATS)",
  month      = "5",
  organization = "PMLR",
  pages      = "3349--3357",
  title      = "A score-based operator Newton method for measure transport",
  volume     = "238",
  year       = "2024",
  doi        = "10.48550/arXiv.2305.09792",
  abstract   = "Transportation of probability measures underlies many core tasks in statistics and machine learning, from variational inference to generative modeling. A typical goal is to represent a target probability measure of interest as the push-forward of a tractable source measure through a learned map. We present a new construction of such a transport map, given the ability to evaluate the score of the target distribution. Specifically, we characterize the map as a zero of an infinite-dimensional score-residual operator and derive a Newton-type method for iteratively constructing such a zero. We prove convergence of these iterations by invoking classical elliptic regularity theory for partial differential equations (PDE) and show that this construction enjoys rapid convergence, under smoothness assumptions on the target score. A key element of our approach is a generalization of the elementary Newton method to infinite-dimensional operators, other forms of which have appeared in nonlinear PDE and in dynamical systems. Our Newton construction, while developed in a functional setting, also suggests new iterative algorithms for approximating transport maps.",
  keywords   = "Transportation of measure, score-based modeling, optimal transport, Newton method, KAM iteration, elliptic PDE."
}

We propose a method to detect a low-dimensional subspace where a non-Gaussian target distribution departs from a known reference distribution (e.g., a standard Gaussian). We identify this subspace from gradients of the log-ratio between the target and reference densities, which we call the score ratio. Given only samples from the target distribution, we estimate these gradients via score ratio matching, with a tailored parameterization and a regularization method that expose the low-dimensional structure we seek. We show that our approach outperforms standard score matching for dimension reduction of in-class distributions, and that several benchmark UCI datasets in fact exhibit this type of low dimensionality.

@inproceedings{brennan-dimension-2022,
  author     = "Brennan, M. and Baptista, R. and Marzouk, Y. M.",
  booktitle  = "NeurIPS 2022 Workshop on Score-Based Methods",
  title      = "Dimension reduction via score ratio matching",
  year       = "2022",
  abstract   = "We propose a method to detect a low-dimensional subspace where a non-Gaussian target distribution departs from a known reference distribution (e.g., a standard Gaussian). We identify this subspace from gradients of the log-ratio between the target and reference densities, which we call the score ratio. Given only samples from the target distribution, we estimate these gradients via score ratio matching, with a tailored parameterization and a regularization method that expose the low-dimensional structure we seek. We show that our approach outperforms standard score matching for dimension reduction of in-class distributions, and that several benchmark UCI datasets in fact exhibit this type of low dimensionality."
}

@inproceedings{tewari-deeppriors-2022,
  author     = "Tewari, A. and Wheelock, B. and Clark, J. and Foster, D. and Li, M. T. C. and Marzouk, Y. M.",
  booktitle  = "Second International Meeting for Applied Geoscience & Energy",
  title      = "Subsurface uncertainty quantification with deep geologic priors: A variational Bayesian framework",
  year       = "2022",
  doi        = "10.1190/image2022-3750132.1"
}

@inproceedings{zhang-sampling-2021,
  author     = "Zhang, B. J. and Sahai, T. and Marzouk, Y. M.",
  booktitle  = "NeurIPS 2021 workshop: I (Still) Can't Believe It's Not Better!",
  title      = "Sampling via controlled stochastic dynamical systems",
  year       = "2021"
}

@inproceedings{scarinci-tlseg-2021,
  author     = "Scarinci, A. and Marzouk, Y. M. and Gu, C. and Fehler, M. and bin Waheed, U. and Kaka, S. and Dia, B.",
  booktitle  = "First International Meeting for Applied Geoscience & Energy Expanded Abstracts",
  title      = "Transport Lagrangian misfit measures and velocity model uncertainty in Bayesian moment tensor inversion",
  year       = "2021",
  doi        = "10.1190/segam2021-3594806.1"
}

Robustly estimating the separated flow about an airfoil is critical in the design of any closed-loop controller. Darakananda et al. (Phys. Rev. Fluids, 2018) successfully used an ensemble Kalman filter (EnKF) to sequentially estimate the flow using an inviscid vortex model and distributed surface pressure readings. To tackle challenging inference problems with limited observations, classical localization schemes suppress correlations at long distances. However, these techniques would be harmful in our case due to the existence of physical long-range interactions between vortices and pressure readings. Instead, these interactions are best described as interactions between clusters of variables. This work proposes a systematic procedure to identify these clusters of variables from a nonlinear observation model. By projecting the states and observations onto these new sets of variables, the inference is performed in a low-dimensional subspace of the state and the observations. To perform consistent inference with the nonlinear model, we use the stochastic map filter (SMF): a natural generalization of the EnKF that relies on interpretable nonlinear prior-to-posterior transformations (Spantini et al., arXiv, 2019). We combine the identification of these clusters of variables with the SMF to derive a low-rank nonlinear ensemble filter. This filter is assessed on the response of a translating plate at 20 degrees that undergoes strong and overlapping pulses applied near the leading-edge. Our framework outperforms the EnKF at estimating the surface pressure distribution along the entire plate, with only two pressure sensors (placed at the edges of the plate) for collecting measurements.

@inproceedings{leprovost-lr-2021,
  author     = "Le Provost, M. and Baptista, R. and Marzouk, Y. M. and Eldredge, J.",
  booktitle  = "AIAA Scitech 2021 Forum",
  title      = "A low-rank nonlinear ensemble filter for vortex models of aerodynamic flows",
  year       = "2021",
  doi        = "10.2514/6.2021-1937",
  abstract   = "Robustly estimating the separated flow about an airfoil is critical in the design of any closed-loop controller. Darakananda et al. (Phys. Rev. Fluids, 2018) successfully used an ensemble Kalman filter (EnKF) to sequentially estimate the flow using an inviscid vortex model and distributed surface pressure readings. To tackle challenging inference problems with limited observations, classical localization schemes suppress correlations at long distances. However, these techniques would be harmful in our case due to the existence of physical long-range interactions between vortices and pressure readings. Instead, these interactions are best described as interactions between clusters of variables. This work proposes a systematic procedure to identify these clusters of variables from a nonlinear observation model. By projecting the states and observations onto these new sets of variables, the inference is performed in a low-dimensional subspace of the state and the observations. To perform consistent inference with the nonlinear model, we use the stochastic map filter (SMF): a natural generalization of the EnKF that relies on interpretable nonlinear prior-to-posterior transformations (Spantini et al., arXiv, 2019). We combine the identification of these clusters of variables with the SMF to derive a low-rank nonlinear ensemble filter. This filter is assessed on the response of a translating plate at 20 degrees that undergoes strong and overlapping pulses applied near the leading-edge. Our framework outperforms the EnKF at estimating the surface pressure distribution along the entire plate, with only two pressure sensors (placed at the edges of the plate) for collecting measurements.",
  keywords   = "inviscid vortex model, disturbed separated flow, data assimilation, nonlinear ensemble filter, measure transport, low-rank projections"
}

Lightweight aerial vehicles can be strongly affected by environmental disturbances (gusts). It is important to have tools for estimating their aerodynamic response to such gusts and to other disturbances, such as agile maneuvers and flow actuators. In recent work, a framework has been developed for predicting the state of disturbed aerodynamic flows, without prior knowledge of the disturbance, by assimilating surface pressure measurements into an ensemble of aggregated vortex models. In this framework, a randomized ensemble of inexpensive models are advanced in each step to forecast the flow state, and then corrected in a linear analysis step by assimilating new measurements. Further advances to this Ensemble Kalman filter framework have been made in a recent paper (Le Provost and Eldredge 2020). In the present work, we demonstrate the use of the improved framework on encounters of a two-dimensional flat plate with gusts of various amplitudes. In particular, we explore the dependence of the aerodynamic response on the gust strength and configuration, and show that the estimation framework works well even for very large amplitude disturbances.

@inproceedings{eldredge-gust-2021,
  author     = "Eldredge, J. and Le Provost, M. and Baptista, R. and Marzouk, Y. M.",
  booktitle  = "AIAA Scitech 2021 Forum",
  title      = "Applications of ensemble Kalman filtered vortex modeling to gust--wing interactions",
  year       = "2021",
  doi        = "0.2514/6.2021-1936",
  abstract   = "Lightweight aerial vehicles can be strongly affected by environmental disturbances (gusts). It is important to have tools for estimating their aerodynamic response to such gusts and to other disturbances, such as agile maneuvers and flow actuators. In recent work, a framework has been developed for predicting the state of disturbed aerodynamic flows, without prior knowledge of the disturbance, by assimilating surface pressure measurements into an ensemble of aggregated vortex models. In this framework, a randomized ensemble of inexpensive models are advanced in each step to forecast the flow state, and then corrected in a linear analysis step by assimilating new measurements. Further advances to this Ensemble Kalman filter framework have been made in a recent paper (Le Provost and Eldredge 2020). In the present work, we demonstrate the use of the improved framework on encounters of a two-dimensional flat plate with gusts of various amplitudes. In particular, we explore the dependence of the aerodynamic response on the gust strength and configuration, and show that the estimation framework works well even for very large amplitude disturbances."
}

The Bayesian framework is commonly used to quantify uncertainty in seismic inversion. To perform Bayesian inference, Markov chain Monte Carlo (MCMC) algorithms are regarded as the gold standard technique for sampling from the posterior probability distribution. Consistent MCMC methods have trouble for complex, high-dimensional models, and most methods scale poorly to large datasets, such as those arising in seismic inversion. As an alternative, approximate MCMC methods based on unadjusted Langevin dynamics offer scalability and more rapid sampling at the cost of biased inference. However, when assessing the quality of approximate MCMC samples for characterizing the posterior distribution, most diagnostics fail to account for these biases. In this work, we introduce the kernel Stein discrepancy (KSD) as a diagnostic tool to determine the convergence of MCMC samples for Bayesian seismic inversion. We demonstrate the use of the KSD for measuring sample quality and selecting the optimal Langevin MCMC algorithm for two Gaussian Bayesian inference problems.

@inproceedings{izzatullah-langevin-2020,
  author     = "Izzatullah, M. and Baptista, R. and Mackey, L. and Marzouk, Y. M. and Peter, D.",
  booktitle  = "SEG Technical Program Expanded Abstracts",
  title      = "Bayesian seismic inversion: measuring Langevin MCMC sample quality with kernels",
  year       = "2020",
  doi        = "10.1190/segam2020-3422419.1",
  abstract   = "The Bayesian framework is commonly used to quantify uncertainty in seismic inversion. To perform Bayesian inference, Markov chain Monte Carlo (MCMC) algorithms are regarded as the gold standard technique for sampling from the posterior probability distribution. Consistent MCMC methods have trouble for complex, high-dimensional models, and most methods scale poorly to large datasets, such as those arising in seismic inversion. As an alternative, approximate MCMC methods based on unadjusted Langevin dynamics offer scalability and more rapid sampling at the cost of biased inference. However, when assessing the quality of approximate MCMC samples for characterizing the posterior distribution, most diagnostics fail to account for these biases. In this work, we introduce the kernel Stein discrepancy (KSD) as a diagnostic tool to determine the convergence of MCMC samples for Bayesian seismic inversion. We demonstrate the use of the KSD for measuring sample quality and selecting the optimal Langevin MCMC algorithm for two Gaussian Bayesian inference problems."
}

The design of wind power plants is a complex engineering task that requires accurate numerical simulations that include both computational fluid and structural dynamics. The accurate prediction of the global performance of the plant is therefore based on the possibility to capture the evolving dynamics for several fluid scales, from local eddies affecting the loading on the turbine blades up to large eddies that form the interacting wakes downstream with respect to each rotor. The presence of these complex flow structures and their interactions requires the use of high-fidelity tools and high resolution grids. At the same time the operative conditions of the plant are intrinsically stochastic in their nature and uncertainty quantification techniques are needed to both characterize the sources of uncertainty and propagate them through the numerical codes. The high computational burden of this task is very often prohibitive for a single plant configuration and it is exacerbated in the case of a design process in which an uncertainty quantification propagation is required for each design iteration. In this work we present and discuss the integration of multilevel uncertainty quantification and design strategies that have the potential to drastically reduce the overall cost of an optimization under uncertainty study. The main objective of the multilevel strategy is to combine computational tools with different accuracy and computational cost such that information from a hierarchy of resolutions can be efficiently fused to decrease the overall cost without compromising the overall accuracy. We focus on algorithmic advancements for both the forward uncertainty quantification step and the optimization step as well. We present multilevel estimators for higher moments, particularly the variance which is employed in robust optimization problems and show the advantage compared to standard multilevel estimators. We use simple model problems to describe the different algorithmic components and their features and performance. As a final demonstration we consider a wind plant design problem involving two turbines for which we combine fluid dynamics tools with different numerical accuracy based on Reynolds Averaged Navier-Stokes equations. Finally, we compare the performance of the newly developed multilevel strategies with their single fidelity counterpart.

@inproceedings{menhorn-moments-2020,
  author     = "Menhorn, F. and Geraci, G. and Seidl,, D. and Eldred, M. S. and King, R. and Bungartz, H.-J. and Marzouk, Y. M.",
  booktitle  = "AIAA Scitech 2020 Forum",
  title      = "Higher moment multilevel estimators for optimization under uncertainty applied to wind plant design",
  year       = "2020",
  doi        = "https://doi.org/10.2514/6.2020-1952",
  abstract   = "The design of wind power plants is a complex engineering task that requires accurate numerical simulations that include both computational fluid and structural dynamics. The accurate prediction of the global performance of the plant is therefore based on the possibility to capture the evolving dynamics for several fluid scales, from local eddies affecting the loading on the turbine blades up to large eddies that form the interacting wakes downstream with respect to each rotor. The presence of these complex flow structures and their interactions requires the use of high-fidelity tools and high resolution grids. At the same time the operative conditions of the plant are intrinsically stochastic in their nature and uncertainty quantification techniques are needed to both characterize the sources of uncertainty and propagate them through the numerical codes. The high computational burden of this task is very often prohibitive for a single plant configuration and it is exacerbated in the case of a design process in which an uncertainty quantification propagation is required for each design iteration. In this work we present and discuss the integration of multilevel uncertainty quantification and design strategies that have the potential to drastically reduce the overall cost of an optimization under uncertainty study. The main objective of the multilevel strategy is to combine computational tools with different accuracy and computational cost such that information from a hierarchy of resolutions can be efficiently fused to decrease the overall cost without compromising the overall accuracy. We focus on algorithmic advancements for both the forward uncertainty quantification step and the optimization step as well. We present multilevel estimators for higher moments, particularly the variance which is employed in robust optimization problems and show the advantage compared to standard multilevel estimators. We use simple model problems to describe the different algorithmic components and their features and performance. As a final demonstration we consider a wind plant design problem involving two turbines for which we combine fluid dynamics tools with different numerical accuracy based on Reynolds Averaged Navier-Stokes equations. Finally, we compare the performance of the newly developed multilevel strategies with their single fidelity counterpart."
}

@inproceedings{geraci-hifire-2019,
  author     = "Geraci, G. and Menhorn, F. and Huan, X. and Safta, C. and Marzouk, Y. M. and Najm, H. and Eldred, M.",
  booktitle  = "AIAA Scitech 2019 Forum",
  title      = "Progress in scramjet design optimization under uncertainty using simulations of the HIFiRE Direct Connect Rig",
  year       = "2019",
  doi        = "10.2514/6.2019-0725"
}

We demonstrate an algorithm for efficient rare event sampling in a rotorcraft model. Helicopter design parameters are typically chosen for efficient performance in cruise and hover. At the same time, structural components such as the length of the tail are chosen so that the rotorcraft is stable under perturbation by environmental factors such as noisy wind. In the face of stochastic forcing, however, environmental conditions may still lead to rare accidents despite good engineering design. We adapt a recent dynamic importance sampling algorithm for small-noise diffusions, derived from the theory of large deviations, to efficient sampling of rare events in a model rotorcraft system. The method achieves variance reduction in estimating the probabilities of stall events, and helps identify the dynamics leading to these phenomena.

@inproceedings{conf_32,
  address    = "2018 AIAA SciTech Forum and 20th AIAA Non-Deterministic Approaches Conference",
  author     = "Zhang, B. J. and Min, B.-Y. and Sahai, T. and Marzouk, Y. M.",
  title      = "Rare event simulation of rotorcraft systems",
  year       = "2018",
  abstract   = "We demonstrate an algorithm for efficient rare event sampling in a rotorcraft model. Helicopter design parameters are typically chosen for efficient performance in cruise and hover. At the same time, structural components such as the length of the tail are chosen so that the rotorcraft is stable under perturbation by environmental factors such as noisy wind. In the face of stochastic forcing, however, environmental conditions may still lead to rare accidents despite good engineering design. We adapt a recent dynamic importance sampling algorithm for small-noise diffusions, derived from the theory of large deviations, to efficient sampling of rare events in a model rotorcraft system. The method achieves variance reduction in estimating the probabilities of stall events, and helps identify the dynamics leading to these phenomena."
}

Integration-based Gaussian filters such as unscented, cubature, and Gauss-Hermite filters are effective ways to assimilate data and models within nonlinear systems. Traditionally, these filters have only been applicable for systems with a handful of states due to stability and scalability issues. In this paper, we present a new integration method for scaling quadrature-based filters to higher dimensions. Our approach begins by decomposing the dynamics and observation models into separated, low-rank tensor formats. Once in low-rank tensor format, adaptive integration techniques may be used to efficiently propagate the mean and covariance of the distribution of the system state with computational complexity that is polynomial in dimension and rank. Simulation results are shown on nonlinear chaotic systems with 20 state variables.

@inproceedings{conf_30,
  address    = "56th IEEE Conference on Decision and Control (CDC)",
  author     = "Gorodetsky, A. and Karaman, S. and Marzouk, Y. M.",
  title      = "Low-rank tensor integration for Gaussian filtering of continuous time nonlinear systems",
  year       = "2017",
  abstract   = "Integration-based Gaussian filters such as unscented, cubature, and Gauss-Hermite filters are effective ways to assimilate data and models within nonlinear systems. Traditionally, these filters have only been applicable for systems with a handful of states due to stability and scalability issues. In this paper, we present a new integration method for scaling quadrature-based filters to higher dimensions. Our approach begins by decomposing the dynamics and observation models into separated, low-rank tensor formats. Once in low-rank tensor format, adaptive integration techniques may be used to efficiently propagate the mean and covariance of the distribution of the system state with computational complexity that is polynomial in dimension and rank. Simulation results are shown on nonlinear chaotic systems with 20 state variables."
}

We propose an algorithm for fitting $C^1$ piecewise-Bézier curves to (possibly corrupted) data points on manifolds. The curve is chosen as a compromise between proximity to data points and regularity. We apply our algorithm as an example to fit a curve to a set of low-rank covariance matrices, a task arising in wind field modeling. We show that our algorithm has denoising abilities for this application.

@inproceedings{conf_29,
  address    = "25th European Symposium on Artificial Neural Networks, Computational Intelligence, and Machine Learning (ESANN 2017)",
  author     = "Gousenbourger, P.-Y. and Massart, E. M. and Musolas, A. and Absil, P.-A. and Hendrickx, J. M. and Jacques, L. and Marzouk, Y. M.",
  title      = "Piecewise-Bézier C^1 smoothing on manifolds with application to wind field estimation",
  year       = "2017",
  abstract   = "We propose an algorithm for fitting $C^1$ piecewise-Bézier curves to (possibly corrupted) data points on manifolds. The curve is chosen as a compromise between proximity to data points and regularity. We apply our algorithm as an example to fit a curve to a set of low-rank covariance matrices, a task arising in wind field modeling. We show that our algorithm has denoising abilities for this application."
}

Design of complex engineering systems requires coupled analyses of the multiple disciplines affecting system performance. The coupling among disciplines typically contributes significantly to the computational cost of analyzing the system, and can become particularly burdensome when coupled analyses are embedded within a design or optimization loop. In many cases, disciplines may be weakly coupled, so that some of the coupling or interaction terms can be neglected without significantly impacting the accuracy of the system output. However, typical practice derives such approximations in an ad hoc manner using expert opinion and domain experience. This paper proposes a new approach that formulates an optimization problem to find an optimal model that balances accuracy of the model outputs with the sparsity of the discipline couplings. An adaptive sequential Monte Carlo sampling technique is used to efficiently search the combinatorial model space of different discipline couplings. Finally, an algorithm for optimal model selection is presented and applied to identify the important discipline couplings in a fire detection satellite model and a turbine engine cycle analysis model.

@inproceedings{conf_28,
  address    = "2017 AIAA SciTech Forum and 19th AIAA Non-Deterministic Approaches Conference",
  author     = "Baptista, R. and Marzouk, Y. M. and Willcox, K. and Peherstorfer, B.",
  title      = "Optimal approximations of coupling in multi-disciplinary models",
  year       = "2017",
  abstract   = "Design of complex engineering systems requires coupled analyses of the multiple disciplines affecting system performance. The coupling among disciplines typically contributes significantly to the computational cost of analyzing the system, and can become particularly burdensome when coupled analyses are embedded within a design or optimization loop. In many cases, disciplines may be weakly coupled, so that some of the coupling or interaction terms can be neglected without significantly impacting the accuracy of the system output. However, typical practice derives such approximations in an ad hoc manner using expert opinion and domain experience. This paper proposes a new approach that formulates an optimization problem to find an optimal model that balances accuracy of the model outputs with the sparsity of the discipline couplings. An adaptive sequential Monte Carlo sampling technique is used to efficiently search the combinatorial model space of different discipline couplings. Finally, an algorithm for optimal model selection is presented and applied to identify the important discipline couplings in a fire detection satellite model and a turbine engine cycle analysis model."
}

Measure transport provides a useful tool for characterizing multivariate non-Gaussian target distributions arising in Bayesian inference. The transport approach seeks a parametric map that pushes forward a chosen reference distribution to the target/posterior distribution, through minimization of a certain Kullback–Leibler divergence. Among the distinguishing features of this approach is the availability of a tractable error estimator for posterior approximation, along with the idea that transport can be cast as an infinite dimensional optimization problem whose variations can be evaluated in closed form. We use these ingredients to develop a method for adaptively constructing transport maps–-balancing the complexity of the map representation, approximation error, and computational cost.

@inproceedings{conf_27,
  address    = "NIPS 2016 workshop on Advances in Approximate Bayesian Inference",
  author     = "Bigoni, D. and Spantini, A. and Marzouk, Y. M.",
  title      = "Adaptive construction of measure transports for Bayesian inference",
  year       = "2016",
  abstract   = "Measure transport provides a useful tool for characterizing multivariate non-Gaussian target distributions arising in Bayesian inference. The transport approach seeks a parametric map that pushes forward a chosen reference distribution to the target/posterior distribution, through minimization of a certain Kullback--Leibler divergence. Among the distinguishing features of this approach is the availability of a tractable error estimator for posterior approximation, along with the idea that transport can be cast as an infinite dimensional optimization problem whose variations can be evaluated in closed form. We use these ingredients to develop a method for adaptively constructing transport maps---balancing the complexity of the map representation, approximation error, and computational cost."
}

We describe a variational inference method that approximates an intractable target measure as the pushforward of a tractable distribution (e.g., a Gaussian) through a transport map. We then show how such transport maps can be decomposed–-i.e., factorized into the composition of finitely many low-dimensional maps. We use the notion of decomposable transports to derive new deterministic online algorithms for Bayesian filtering and smoothing in nonlinear/non-Gaussian state-space models with static parameters, and illustrate the theory on a stochastic volatility model.

@inproceedings{conf_26,
  address    = "NIPS 2016 workshop on Advances in Approximate Bayesian Inference",
  author     = "Spantini, A. and Bigoni, D. and Marzouk, Y. M.",
  title      = "Variational inference via decomposable transports: new algorithms for Bayesian filtering and smoothing",
  year       = "2016",
  abstract   = "We describe a variational inference method that approximates an intractable target measure as the pushforward of a tractable distribution (e.g., a Gaussian) through a transport map. We then show how such transport maps can be decomposed---i.e., factorized into the composition of finitely many low-dimensional maps. We use the notion of decomposable transports to derive new deterministic online algorithms for Bayesian filtering and smoothing in nonlinear/non-Gaussian state-space models with static parameters, and illustrate the theory on a stochastic volatility model."
}

Correct-by-design automated construction of control systems has attracted a tremendous amount of attention. However, most existing algorithms for automated construction suffer from the curse of dimensionality, i.e., their run time scales exponentially with increasing dimensionality of the state space. As a result, typically, systems with only a few degrees of freedom are considered. In this paper, we propose a novel algorithm based on the tensor-train decomposition that solves stochastic optimal control problems with syntactically co-safe linear temporal logic specifications. We show that, under certain conditions, the run time of the proposed algorithm scales polynomially with the dimensionality of the state space and the rank of the optimal cost-to-go function. We demonstrate the algorithm in a six-dimensional problem instance involving a simple airplane model. In this example, the proposed algorithm provides up to four orders of computational savings when compared to the standard value iteration algorithm.

@inproceedings{conf_25,
  address    = "55th IEEE Conference on Decision and Control (CDC)",
  author     = "Alora, J. and Gorodetsky, A. and Karaman, S. and Marzouk, Y. M. and Lowry, N.",
  title      = "Automated synthesis of low-rank control systems from sc-LTL specifications using tensor-train decompositions",
  year       = "2016",
  abstract   = "Correct-by-design automated construction of control systems has attracted a tremendous amount of attention. However, most existing algorithms for automated construction suffer from the curse of dimensionality, i.e., their run time scales exponentially with increasing dimensionality of the state space. As a result, typically, systems with only a few degrees of freedom are considered. In this paper, we propose a novel algorithm based on the tensor-train decomposition that solves stochastic optimal control problems with syntactically co-safe linear temporal logic specifications. We show that, under certain conditions, the run time of the proposed algorithm scales polynomially with the dimensionality of the state space and the rank of the optimal cost-to-go function. We demonstrate the algorithm in a six-dimensional problem instance involving a simple airplane model. In this example, the proposed algorithm provides up to four orders of computational savings when compared to the standard value iteration algorithm."
}

Many simulation tools for airfoil analysis and design are based on an integral approximation of the boundary layer. This approximate formulation cannot resolve the full dynamics of boundary layer flows, and hence requires additional models to account for unresolved effects. We introduce a new, data-driven, probabilistic model of these unresolved effects for the incompressible and laminar regime. To construct this model, we apply methods from supervised learning to a dataset containing over 1,550 airfoils. The result is a model that (i) is based on a large dataset of realistic airfoil configurations, and (ii) quantifies the model inadequacy associated with the use of an approximate boundary layer formulation. We also create a stochastic version of the airfoil design tool XFOIL by replacing its original boundary layer model with the probabilistic model developed here. We apply this stochastic version of XFOIL to compute the drag polars of two airfoils at low Reynolds numbers, and compare the results with experimental data.

@inproceedings{conf_24,
  address    = "34th AIAA Applied Aerodynamics Conference, AIAA AVIATION Forum",
  author     = "Marques, A. and Wang, Q. and Marzouk, Y. M.",
  title      = "Data-driven probabilistic boundary layer modeling for airfoil performance prediction",
  year       = "2016",
  abstract   = "Many simulation tools for airfoil analysis and design are based on an integral approximation of the boundary layer. This approximate formulation cannot resolve the full dynamics of boundary layer flows, and hence requires additional models to account for unresolved effects. We introduce a new, data-driven, probabilistic model of these unresolved effects for the incompressible and laminar regime. To construct this model, we apply methods from supervised learning to a dataset containing over 1,550 airfoils. The result is a model that (i) is based on a large dataset of realistic airfoil configurations, and (ii) quantifies the model inadequacy associated with the use of an approximate boundary layer formulation. We also create a stochastic version of the airfoil design tool XFOIL by replacing its original boundary layer model with the probabilistic model developed here. We apply this stochastic version of XFOIL to compute the drag polars of two airfoils at low Reynolds numbers, and compare the results with experimental data."
}

Stochastic optimal control problems frequently arise as motion control problems in the context of robotics. Unfortunately, all existing approaches that guarantee arbitrary precision suffer from the curse of dimensionality: the computational effort invested by the algorithm grows exponentially fast with increasing dimensionality of the state space of the underlying dynamic system governing the robot. In this paper, we propose a novel algorithm that utilizes compressed representations to efficiently solve stochastic optimal control problems with arbitrary precision. The running time of the new algorithms scale linearly with increasing dimensionality of the state space! The running time also depends polynomially on the rank of the value function, a measure that quantifies the intrinsic geometric complexity of the value function, due to the geometry and physics embedded in the problem instance at hand. The new algorithms are based on the recent analysis and algorithms for tensor decomposition, generalizing matrix decomposition algorithms, e.g., the singular value decomposition, to three or more dimensions. In computational experiments, we show the computational effort of the new algorithm also scales linearly with the discretization resolution of the state space. We also demonstrate the new algorithm on a problem involving the perching of an aircraft, represented by a nonlinear non-holonomic longitudinal model with a seven-dimensional state space, the full numerical solution to which was not obtained before. In this example, we estimate that the proposed algorithm runs more than seven orders of magnitude faster than the naive value iteration.

@inproceedings{conf_23,
  address    = "Robotics: Science and Systems (RSS) XI. Rome, Italy.",
  author     = "Gorodetsky, A. and Karaman, S. and Marzouk, Y. M.",
  title      = "Efficient high-dimensional stochastic optimal motion control using tensor-train decomposition",
  year       = "2015",
  abstract   = "Stochastic optimal control problems frequently arise as motion control problems in the context of robotics. Unfortunately, all existing approaches that guarantee arbitrary precision suffer from the curse of dimensionality: the computational effort invested by the algorithm grows exponentially fast with increasing dimensionality of the state space of the underlying dynamic system governing the robot. In this paper, we propose a novel algorithm that utilizes compressed representations to efficiently solve stochastic optimal control problems with arbitrary precision. The running time of the new algorithms scale linearly with increasing dimensionality of the state space! The running time also depends polynomially on the rank of the value function, a measure that quantifies the intrinsic geometric complexity of the value function, due to the geometry and physics embedded in the problem instance at hand. The new algorithms are based on the recent analysis and algorithms for tensor decomposition, generalizing matrix decomposition algorithms, e.g., the singular value decomposition, to three or more dimensions. In computational experiments, we show the computational effort of the new algorithm also scales linearly with the discretization resolution of the state space. We also demonstrate the new algorithm on a problem involving the perching of an aircraft, represented by a nonlinear non-holonomic longitudinal model with a seven-dimensional state space, the full numerical solution to which was not obtained before. In this example, we estimate that the proposed algorithm runs more than seven orders of magnitude faster than the naive value iteration."
}

Optimizing wells placement may help in better understanding subsurface solute transport and detecting contaminant plumes. In this work, we use the ensemble Kalman filter (EnKF) as a data assimilation tool and propose a greedy observational design algorithm to optimally select aquifer wells locations for updating the prior contaminant ensemble. The algorithm is greedy in the sense that it operates sequentially, without taking into account expected future gains. The selection criteria is based on maximizing the information gain that the EnKF carries during the update of the prior uncertainties. We test the efficiency of this algorithm in a synthetic aquifer system where a contaminant plume is set to migrate over a 30 years period across a heterogenous domain.

@inproceedings{conf_22,
  address    = "Dynamic Data-Driven Environmental Systems Science (DyDESS), Cambridge, MA.",
  author     = "Gharamti, M. E. and Marzouk, Y. M. and Huan, X. and Hoteit, I.",
  title      = "A greedy approach for placement of subsurface aquifer wells in an ensemble filtering framework",
  year       = "2015",
  abstract   = "Optimizing wells placement may help in better understanding subsurface solute transport and detecting contaminant plumes. In this work, we use the ensemble Kalman filter (EnKF) as a data assimilation tool and propose a greedy observational design algorithm to optimally select aquifer wells locations for updating the prior contaminant ensemble. The algorithm is greedy in the sense that it operates sequentially, without taking into account expected future gains. The selection criteria is based on maximizing the information gain that the EnKF carries during the update of the prior uncertainties. We test the efficiency of this algorithm in a synthetic aquifer system where a contaminant plume is set to migrate over a 30 years period across a heterogenous domain."
}

@inproceedings{conf_21,
  address    = "First SIGPLAN Workshop on Probabilistic and Approximate Computing (Collocated with PLDI 2014). Edinburgh, UK.",
  author     = "Augustin, F. and Marzouk, Y. M.",
  title      = "Uncertainty quantification in high-performance computing",
  year       = "2014"
}

We present a texton-based, multi-stage Bayesian level set algorithm which we use to segment colony images of hESC and their derivatives. We extend our previous research segmenting stem cells according to multiresolution texture methods to accommodate colonies and tissues with diffuse and varied textures via a filter bank approach similar to the MR8. Texture features computed for test images are classified via comparison with learned sets of class-specific textural primitives, known as textons. Encompassing this texture model is the new Bayesian level set algorithm, which smoothes and regularizes classification similar to level sets but is simpler in its probabilistic implementation. The resulting algorithm accurately and automatically classifies images of pluripotent hESC and trophectoderm colonies for high-content screening applications.

@inproceedings{conf_20,
  address    = "9th IEEE International Symposium on Biomedical Imaging (ISBI). Barcelona, Spain.",
  author     = "Lowry, N. and Mangoubi, R. and Desai, M. and Marzouk, Y. M. and Sammak, P.",
  title      = "Texton-based segmentation and classification of human embryonic stem cell colonies using multi-stage Bayesian level sets",
  year       = "2012",
  abstract   = "We present a texton-based, multi-stage Bayesian level set algorithm which we use to segment colony images of hESC and their derivatives. We extend our previous research segmenting stem cells according to multiresolution texture methods to accommodate colonies and tissues with diffuse and varied textures via a filter bank approach similar to the MR8. Texture features computed for test images are classified via comparison with learned sets of class-specific textural primitives, known as textons. Encompassing this texture model is the new Bayesian level set algorithm, which smoothes and regularizes classification similar to level sets but is simpler in its probabilistic implementation. The resulting algorithm accurately and automatically classifies images of pluripotent hESC and trophectoderm colonies for high-content screening applications."
}

Chemical kinetic systems contain both considerable uncertainty in their rate parameters and dynamics at multiple time scales. The latter feature aids model reduction in the deterministic case, but model reduction under uncertainty raises new challenges. We use computational singular perturbation (CSP) to calculate probabilistic “importance indices” for species-reaction pairs. Distributions of these indices are used to form reduced models that yield predictions within probabilistic bounds determined by the full model, or, alternatively, preserve entire output distributions of the full model.

@inproceedings{conf_19,
  address    = "International Workshop on Model Reduction in Reacting Flow/13th International Conference on Numerical Combustion. Corfu, Greece",
  author     = "Coles, T. and Najm, H. and Marzouk, Y. M.",
  title      = "CSP simplification of chemical kinetic systems under uncertainty",
  year       = "2011",
  abstract   = "Chemical kinetic systems contain both considerable uncertainty in their rate parameters and dynamics at multiple time scales. The latter feature aids model reduction in the deterministic case, but model reduction under uncertainty raises new challenges. We use computational singular perturbation (CSP) to calculate probabilistic “importance indices” for species-reaction pairs. Distributions of these indices are used to form reduced models that yield predictions within probabilistic bounds determined by the full model, or, alternatively, preserve entire output distributions of the full model."
}

@inproceedings{conf_18,
  address    = "IEEE International Symposium on Biomedical Imaging. Chicago, IL",
  author     = "Lowry, N. and Mangoubi, R. and Desai, M. and Marzouk, Y. M. and Sammak, P.",
  title      = "A unified approach to expectation maximization and level set segmentation applied to stem cell and brain MRI images",
  year       = "2011"
}

@inproceedings{conf_17,
  address    = "49th AIAA Aerospace Sciences Meeting. Orlando, FL",
  author     = "Huan, X. and Marzouk, Y. M.",
  title      = "Optimal Bayesian experimental design for combustion kinetics.",
  year       = "2011"
}

@inproceedings{conf_16,
  address    = "Western States Section of the Combustion Institute, Fall Meeting, Irvine, CA",
  author     = "Debusschere, B. and Rhoads, B. and Najm, H. and Marzouk, Y. M. and Goussis, D. and Valorani, M. and Frenklach, M.",
  title      = "Time integration of chemical kinetics with computational singular perturbation and tabulation",
  year       = "2009"
}

@inproceedings{conf_15,
  address    = "Second International Workshop on Model Reduction in Reacting Flows, Notre Dame, IN",
  author     = "Marzouk, Y. M. and Debusschere, B. and Najm, H. and Goussis, D. and Valorani, M. and Frenklach, M.",
  title      = "Time integration of reacting flows with CSP tabulation",
  year       = "2009"
}

no abstract

@inproceedings{conf_14,
  address    = "SciDAC 2008, Seattle, WA.",
  author     = "Najm, H. N. and Safta, C. and Marzouk, Y. M. and Ray, J. and Valorani, M. and Goussis, D.",
  title      = "High-order AMR computations of reacting flow with adaptive reduction of chemical stiffness",
  year       = "2008"
}

@inproceedings{conf_13,
  address    = "Proceedings of the 2008 Joint Statistical Meetings; Denver, CO.",
  author     = "Ray, J. and Marzouk, Y. M.",
  title      = "A Bayesian method for inferring transmission chains in a partially observed epidemic",
  year       = "2008"
}

no abstract

@inproceedings{conf_12,
  address    = "RAND/ASA Conference on Quantitative Methods & Statistical Applications in Defense and National Security., Santa Monica, CA, USA",
  author     = "Ray, J. and Marzouk, Y. M. and Najm, H. N. and Kraus, M. and Fast, P.",
  booktitle  = "Proceedings of the American Statistical Association.",
  title      = "Estimating bioterror attacks from patient data: a Bayesian approach",
  year       = "2006"
}

@inproceedings{conf_11,
  address    = "AIAA paper 2006-1492",
  author     = "Marzouk, Y. M. and Ghoniem, A. F.",
  title      = "Actuating transverse jets via nozzle-edge vortical perturbations",
  year       = "2006"
}

no abstract

@inproceedings{conf_10,
  address    = "Bayesian Inference and Maximum Entropy Methods in Science and Engineering",
  author     = "Marzouk, Y. M. and Najm, H. N. and Rahn, L. A.",
  title      = "Stochastic spectral methods for efficient Bayesian solution of inverse problems",
  year       = "2005"
}

no abstract

@inproceedings{conf_9,
  address    = "Third MIT Conference on Computational Fluid and Solid Mechanics",
  author     = "Speth, R. L. and Marzouk, Y. M. and Ghoniem, A. F.",
  title      = "A quasi-one-dimensional unsteady laminar flame formulation with independent strain rate and curvature",
  year       = "2005"
}

@inproceedings{conf_8,
  address    = "AIAA paper 2005-0308",
  author     = "Marzouk, Y. M. and Ghoniem, A. F.",
  title      = "Simulations of high Reynolds number transverse jets and analysis of the underlying vortical structures",
  year       = "2005"
}

@inproceedings{conf_7,
  address    = "AIAA paper 2005-0291",
  author     = "Wee, D. H. and Marzouk, Y. M. and Ghoniem, A. F.",
  title      = "Lagrangian simulation of a jet in crossflow at finite Reynolds number",
  year       = "2005"
}

@inproceedings{conf_6,
  address    = "AIAA paper 2005-0143",
  author     = "Speth, R. L. and Marzouk, Y. M. and Ghoniem, A. F.",
  title      = "Impact of hydrogen addition on flame response to stretch and curvature",
  year       = "2005"
}

@inproceedings{conf_5,
  address    = "AIAA paper 2004-0096",
  author     = "Marzouk, Y. M. and Ghoniem, A. F.",
  title      = "Vorticity formulation for an actuated jet in crossflow",
  year       = "2004"
}

no abstract

@inproceedings{conf_4,
  address    = "Proceedings of the Second MIT Conference on Computational Fluid and Solid Mechanics",
  author     = "Marzouk, Y. M. and Ghoniem, A. F.",
  title      = "Vorticity generation mechanisms and correct boundary conditions for transverse jet simulation",
  year       = "2003"
}

@inproceedings{conf_3,
  address    = "AIAA paper 2002-1063",
  author     = "Marzouk, Y. M. and Ghoniem, A. F. and Najm, H. N.",
  title      = "Mechanism of streamwise vorticity formation in a transverse jet",
  year       = "2002"
}

no abstract

@inproceedings{conf_2,
  address    = "Proceedings of the Second Mediterranean Combustion Symposium: 372--383",
  author     = "Marzouk, Y. M. and Ghoniem, A. F. and Najm, H. N.",
  title      = "Can a one-dimensional strained flame model flame-vortex interactions?",
  year       = "2002"
}

@inproceedings{conf_1,
  address    = "AIAA paper 2000-0866",
  author     = "Marzouk, Y. M. and Ghoniem, A. F. and Najm, H. N.",
  title      = "A flame embedding model for turbulent combustion simulation",
  year       = "2000"
}