### Preprints — Published Articles

#### Certified dimension reduction in nonlinear Bayesian inverse problems

#### BibTeX entry

@article { title = "Certified dimension reduction in nonlinear Bayesian inverse problems", author = "O. Zahm and T. Cui and K. Law and A. Spantini and Y. M. Marzouk", journal = "Preprint", volume = "", year ="2018", number = "", pages = "", doi = "" }

#### A Stein variational Newton method

#### BibTeX entry

@article { title = "A Stein variational Newton method", author = "G. Detomasso and T. Cui and Y. M. Marzouk and R. Scheichl and A. Spantini", journal = "Preprint", volume = "", year ="2018", number = "", pages = "", doi = "" }

#### Gradient-based dimension reduction of multivariate vector-valued functions

#### BibTeX entry

@article { title = "Gradient-based dimension reduction of multivariate vector-valued functions", author = "O. Zahm and P. Constantine and C. Prieur and Y. M. Marzouk", journal = "Preprint", volume = "", year ="2018", number = "", pages = "", doi = "" }

#### Localization for MCMC: sampling high-dimensional posterior distributions with local structure

#### BibTeX entry

@article { title = "Localization for MCMC: sampling high-dimensional posterior distributions with local structure", author = "M. Morzfeld and X. T. Tong and Y. M. Marzouk", journal = "Preprint", volume = "", year ="2017", number = "", pages = "", doi = "" }

#### A trust region method for derivative-free nonlinear constrained stochastic optimization

#### BibTeX entry

@article { title = "A trust region method for derivative-free nonlinear constrained stochastic optimization", author = "F. Augustin and Y. M. Marzouk", journal = "Preprint", volume = "", year ="2017", number = "", pages = "", doi = "" }

#### Function-train: a continuous analogue of the tensor-train decomposition

#### BibTeX entry

@article { title = "Function-train: a continuous analogue of the tensor-train decomposition", author = "A. Gorodetsky and S. Karaman and Y. M. Marzouk", journal = "Preprint", volume = "", year ="2016", number = "", pages = "", doi = "" }

#### Sequential Bayesian optimal experimental design via approximate dynamic programming

#### BibTeX entry

@article { title = "Sequential Bayesian optimal experimental design via approximate dynamic programming", author = "X. Huan and Y. M. Marzouk", journal = "Preprint", volume = "", year ="2016", number = "", pages = "", doi = "" }

#### Bayesian level sets for image segmentation

#### BibTeX entry

@article { title = "Bayesian level sets for image segmentation", author = "N. Lowry and R. Mangoubi and M. Desai and Y. M. Marzouk and P. Sammak", journal = "Preprint", volume = "", year ="2015", number = "", pages = "", doi = "" }

#### NOWPAC: A provably convergent derivative-free nonlinear optimizer with path-augmented constraints

#### BibTeX entry

@article { title = "NOWPAC: A provably convergent derivative-free nonlinear optimizer with path-augmented constraints", author = "F. Augustin and Y. M. Marzouk", journal = "Preprint", volume = "", year ="2014", number = "", pages = "", doi = "" }

**No articles found matching filter.**

### Published Articles

**in press**(2018)

#### Quantifying kinetic uncertainty in turbulent combustion simulations using active subspaces

#### BibTeX entry

@article { title = "Quantifying kinetic uncertainty in turbulent combustion simulations using active subspaces", author = "W. Ji and Z. Ren and Y. M. Marzouk and C. K. Law", journal = "Proceedings of the Combustion Institute", volume = "in press", year ="2018", number = "", pages = "", doi = "10.1016/j.proci.2018.06.206" }

**6**pp. 645–682 (2018)

#### Fulltext options

`https://doi.org/10.1137/17M1134640`

View on arXiv.org

`http://arxiv.org/abs/1412.5492`

#### Transport map accelerated Markov chain Monte Carlo

#### BibTeX entry

@article { title = "Transport map accelerated Markov chain Monte Carlo", author = "M. Parno and Y. M. Marzouk", journal = "SIAM/ASA Journal on Uncertainty Quantification", volume = "6", year ="2018", number = "2", pages = "645–682", doi = "10.1137/17M1134640" }

**in press**(2018)

#### An approximate empirical Bayesian method for large-scale linear-Gaussian inverse problems

#### BibTeX entry

@article { title = "An approximate empirical Bayesian method for large-scale linear-Gaussian inverse problems", author = "Q. Zhou and W. Liu and J. Li and Y. M. Marzouk", journal = "Inverse Problems", volume = "in press", year ="2018", number = "", pages = "", doi = "10.1088/1361-6420/aac287" }

**in press**(2018)

#### Fulltext options

#### Optimal approximations of coupling in multidisciplinary models

#### BibTeX entry

@article { title = "Optimal approximations of coupling in multidisciplinary models", author = "R. Baptista and Y. M. Marzouk and K. Willcox and B. Peherstorfer", journal = "AIAA Journal", volume = "in press", year ="2018", number = "", pages = "", doi = "" }

**in press**(2018)

#### Conditional classifiers and boosted conditional Gaussian mixture models for novelty detection

#### BibTeX entry

@article { title = "Conditional classifiers and boosted conditional Gaussian mixture models for novelty detection", author = "R. Mohammadi-Ghazi and Y. M. Marzouk and Oral Büyüköztürk", journal = "Pattern Recognition", volume = "in press", year ="2018", number = "", pages = "", doi = "10.1016/j.patcog.2018.03.022" }

**in press**(2018)

#### Multilevel sequential Monte Carlo with dimension-independent likelihood-informed proposals

#### BibTeX entry

@article { title = "Multilevel sequential Monte Carlo with dimension-independent likelihood-informed proposals", author = "A. Beskos and A. Jasra and K. Law and Y. M. Marzouk and Y. Zhou", journal = "SIAM/ASA Journal on Uncertainty Quantification", volume = "in press", year ="2018", number = "", pages = "", doi = "" }

**89**pp. 013504 (2018)

#### Efficient design and verification of diagnostics for impurity transport experiments

#### BibTeX entry

@article { title = "Efficient design and verification of diagnostics for impurity transport experiments", author = "M. Chilenski and M. Greenwald and Y. M. Marzouk and J. Rice and A. White", journal = "Review of Scientific Instruments", volume = "89", year ="2018", number = "", pages = "013504", doi = "10.1063/1.4997251" }

**6**pp. 339–373 (2018)

#### Parallel local approximation MCMC for expensive models

#### BibTeX entry

@article { title = "Parallel local approximation MCMC for expensive models", author = "P. Conrad and A. Davis and Y. M. Marzouk and N. Pillai and A. Smith", journal = "SIAM/ASA Journal on Uncertainty Quantification", volume = "6", year ="2018", number = "1", pages = "339–373", doi = "10.1137/16M1084080" }

**37**pp. 340–377 (2018)

#### Fulltext options

`https://doi.org/10.1177/0278364917753994`

View on arXiv.org

`http://arxiv.org/abs/arXiv:1611.04706`

#### High-dimensional stochastic optimal control using continuous tensor decompositions

#### BibTeX entry

@article { title = "High-dimensional stochastic optimal control using continuous tensor decompositions", author = "A. Gorodetsky and S. Karaman and Y. M. Marzouk", journal = "International Journal of Robotics Research", volume = "37", year ="2018", number = "2-3", pages = "340–377", doi = "10.1177/0278364917753994" }

**45**pp. 1313–1320 (2018)

#### Inferring fault frictional and reservoir hydraulic properties from injection-induced seismicity

#### BibTeX entry

@article { title = "Inferring fault frictional and reservoir hydraulic properties from injection-induced seismicity", author = "J. Jagalur-Mohan and B. Jha and Z. Wang and R. Juanes and Y. M. Marzouk", journal = "Geophysical Research Letters", volume = "45", year ="2018", number = "3", pages = "1313–1320", doi = "10.1002/2017GL075925" }

**in press**(2018)

#### Inference via low-dimensional couplings

#### BibTeX entry

@article { title = "Inference via low-dimensional couplings", author = "A. Spantini and D. Bigoni and Y. M. Marzouk", journal = "Journal of Machine Learning Research", volume = "in press", year ="2018", number = "", pages = "", doi = "" }

**212**pp. 1963–1985 (2018)

#### Fulltext options

`https://academic.oup.com/gji/advance-article/doi/10.1093/gji/ggx517/4689122?guestAccessKey=e2cc1771-4f08-4231-a911-ce515f0a2bcd`

#### Waveform-based Bayesian full moment tensor inversion and uncertainty determination for induced seismicity in an oil/gas field

#### BibTeX entry

@article { title = "Waveform-based Bayesian full moment tensor inversion and uncertainty determination for induced seismicity in an oil/gas field", author = "C. Gu and Y. M. Marzouk and M. N. Toksöz", journal = "Geophysical Journal International", volume = "212", year ="2018", number = "1", pages = "1963–1985", doi = "10.1093/gji/ggx517" }

**56**pp. 482–496 (2018)

#### Data-driven integral boundary layer modeling for airfoil performance prediction in the laminar regime

#### BibTeX entry

@article { title = "Data-driven integral boundary layer modeling for airfoil performance prediction in the laminar regime", author = "A. Marques and Q. Wang and Y. M. Marzouk", journal = "AIAA Journal", volume = "56", year ="2018", number = "2", pages = "482–496", doi = "10.2514/1.J055877" }

**190**pp. 146–157 (2018)

#### Shared low-dimensional subspaces for propagating kinetic uncertainty to multiple outputs

#### BibTeX entry

@article { title = "Shared low-dimensional subspaces for propagating kinetic uncertainty to multiple outputs", author = "W. Ji and J. Wang and O. Zahm and Y. M. Marzouk and B. Yang and Z. Ren and C. K. Law", journal = "Combustion and Flame", volume = "190", year ="2018", number = "", pages = "146–157", doi = "10.1016/j.combustflame.2017.11.021" }

**57**pp. 126013 (2017)

#### Experimentally testing the dependence of momentum transport on second derivatives using Gaussian process regression

#### BibTeX entry

@article { title = "Experimentally testing the dependence of momentum transport on second derivatives using Gaussian process regression", author = "M. Chilenski and M. Greenwald and A. Hubbard and J. Hughes and J. Lee and Y. M. Marzouk and J. Rice and A. White", journal = "Nuclear Fusion", volume = "57", year ="2017", number = "12", pages = "126013", doi = "10.1088/1741-4326/aa8387" }

**39**pp. S167–S196 (2017)

#### Goal-oriented optimal approximations of Bayesian linear inverse problems

#### BibTeX entry

@article { title = "Goal-oriented optimal approximations of Bayesian linear inverse problems", author = "A. Spantini and T. Cui and K. Willcox and L. Tenorio and Y. M. Marzouk", journal = "SIAM Journal on Scientific Computing", volume = "39", year ="2017", number = "5", pages = "S167–S196", doi = "10.1137/16M1082123" }

**39**pp. S140–S166 (2017)

#### Bayesian inverse problems with l1 priors: a randomize-then-optimize approach

#### BibTeX entry

@article { title = "Bayesian inverse problems with l1 priors: a randomize-then-optimize approach", author = "Z. Wang and J. M. Bardsley and A. Solonen and T. Cui and Y. M. Marzouk", journal = "SIAM Journal on Scientific Computing", volume = "39", year ="2017", number = "5", pages = "S140–S166", doi = "10.1137/16M1080938" }

**4**pp. 1160–1190 (2016)

#### Fulltext options

`http://dx.doi.org/10.1137/15M1032478`

View on arXiv.org

`http://arxiv.org/abs/1507.07024`

#### A multiscale strategy for Bayesian inference using transport maps

#### BibTeX entry

@article { title = "A multiscale strategy for Bayesian inference using transport maps", author = "M. Parno and T. Moselhy and Y. M. Marzouk", journal = "SIAM/ASA Journal on Uncertainty Quantification", volume = "4", year ="2016", number = "1", pages = "1160–1190", doi = "10.1137/15M1032478" }

**38**pp. A2405–A2439 (2016)

#### Spectral tensor-train decomposition

#### BibTeX entry

@article { title = "Spectral tensor-train decomposition", author = "D. Bigoni and A. Engsig-Karup and Y. M. Marzouk", journal = "SIAM Journal on Scientific Computing", volume = "38", year ="2016", number = "4", pages = "A2405–A2439", doi = "10.1137/15M1036919" }

**315**pp. 363–387 (2016)

#### Scalable posterior approximations for large-scale Bayesian inverse problems via likelihood-informed parameter and state reduction

#### BibTeX entry

@article { title = "Scalable posterior approximations for large-scale Bayesian inverse problems via likelihood-informed parameter and state reduction", author = "T. Cui and Y. M. Marzouk and K. Willcox", journal = "Journal of Computational Physics", volume = "315", year ="2016", number = "", pages = "363–387", doi = "doi:10.1016/j.jcp.2016.03.055" }

**4**pp. 796–828 (2016)

#### Mercer kernels and integrated variance experimental design: connections between Gaussian process regression and polynomial approximation

#### BibTeX entry

@article { title = "Mercer kernels and integrated variance experimental design: connections between Gaussian process regression and polynomial approximation", author = "A. Gorodetsky and Y. M. Marzouk", journal = "SIAM/ASA Journal on Uncertainty Quantification", volume = "4", year ="2016", number = "1", pages = "796–828", doi = "10.1137/15M1017119" }

**32**pp. 045003 (2016)

#### On dimension reduction in Gaussian filters

#### BibTeX entry

@article { title = "On dimension reduction in Gaussian filters", author = "A. Solonen and T. Cui and J. Hakkarainen and Y. M. Marzouk", journal = "Inverse Problems", volume = "32", year ="2016", number = "4", pages = "045003", doi = "10.1088/0266-5611/32/4/045003" }

**300**pp. 490–509 (2016)

#### Fulltext options

#### Multifidelity importance sampling

#### BibTeX entry

@article { title = "Multifidelity importance sampling", author = "B. Peherstorfer and T. Cui and Y. M. Marzouk and K. Willcox", journal = "Computer Methods in Applied Mechanics and Engineering", volume = "300", year ="2016", number = "", pages = "490–509", doi = "http://dx.doi.org/10.1016/j.cma.2015.12.002" }

**304**pp. 109–137 (2016)

#### Dimension-independent likelihood-informed MCMC

#### BibTeX entry

@article { title = "Dimension-independent likelihood-informed MCMC", author = "T. Cui and K. J. H. Law and Y. M. Marzouk", journal = "Journal of Computational Physics", volume = "304", year ="2016", number = "", pages = "109–137", doi = "doi:10.1016/j.jcp.2015.10.008" }

**111**pp. 1591–1607 (2016)

#### Accelerating asymptotically exact MCMC for computationally intensive models via local approximations

#### BibTeX entry

@article { title = "Accelerating asymptotically exact MCMC for computationally intensive models via local approximations", author = "P. Conrad and Y. M. Marzouk and N. Pillai and A. Smith", journal = "Journal of the American Statistical Association", volume = "111", year ="2016", number = "516", pages = "1591–1607", doi = "10.1080/01621459.2015.1096787" }

**37**pp. A2451–A2487 (2015)

#### Optimal low-rank approximations of Bayesian linear inverse problems

#### BibTeX entry

@article { title = "Optimal low-rank approximations of Bayesian linear inverse problems", author = "A. Spantini and A. Solonen and T. Cui and J. Martin and L. Tenorio and Y. M. Marzouk", journal = "SIAM Journal on Scientific Computing", volume = "37", year ="2015", number = "6", pages = "A2451–A2487", doi = "10.1137/140977308" }

**23**pp. 4242–4254 (2015)

#### Uncertainty quantification of silicon photonic devices with correlated and non-Gaussian random parameters

#### BibTeX entry

@article { title = "Uncertainty quantification of silicon photonic devices with correlated and non-Gaussian random parameters", author = "T. Weng and Z. Zhang and Z. Su and Y. M. Marzouk and A. Melloni and L. Daniel", journal = "Optics Express", volume = "23", year ="2015", number = "4", pages = "4242–4254", doi = "10.1364/OE.23.004242" }

**104**pp. 313–329 (2015)

#### A new network approach to Bayesian inference in partial differential equations

#### BibTeX entry

@article { title = "A new network approach to Bayesian inference in partial differential equations", author = "D. Kohler and Y. M. Marzouk and J. Müller and U. Wever", journal = "International Journal for Numerical Methods in Engineering", volume = "104", year ="2015", number = "5", pages = "313–329", doi = "10.1002/nme.4928" }

**55**pp. 023012 (2015)

#### Improved profile fitting and quantification of uncertainty in experimental measurements of impurity transport coefficients using Gaussian process regression

#### BibTeX entry

@article { title = "Improved profile fitting and quantification of uncertainty in experimental measurements of impurity transport coefficients using Gaussian process regression", author = "M. A. Chilenski and M. Greenwald and Y. M. Marzouk and N. T. Howard and A. E. White and J. E. Rice and J. R. Walk", journal = "Nuclear Fusion", volume = "55", year ="2015", number = "2", pages = "023012", doi = "10.1088/0029-5515/55/2/023012" }

**23**pp. 015009 (2015)

#### Bayesian inference of substrate properties from film behavior

#### BibTeX entry

@article { title = "Bayesian inference of substrate properties from film behavior", author = "R. Aggarwal and M. Demkowicz and Y. M. Marzouk", journal = "Modelling Simul. Mater. Sci. Eng. ", volume = "23", year ="2015", number = "", pages = "015009", doi = "http://dx.doi.org/10.1088/0965-0393/23/1/015009" }

**123**pp. 170–190 (2015)

#### Bayesian inference of chemical kinetic models from proposed reactions

#### BibTeX entry

@article { title = "Bayesian inference of chemical kinetic models from proposed reactions", author = "N. Galagali and Y. M. Marzouk", journal = "Chemical Engineering Science", volume = "123", year ="2015", number = "", pages = "170–190", doi = "doi:10.1016/j.ces.2014.10.030" }

**102**pp. 966–990 (2015)

#### Data-driven model reduction for the Bayesian solution of inverse problems

#### BibTeX entry

@article { title = "Data-driven model reduction for the Bayesian solution of inverse problems", author = "T. Cui and Y. M. Marzouk and K. Willcox", journal = "International Journal for Numerical Methods in Engineering", volume = "102", year ="2015", number = "5", pages = "966–990", doi = "doi:10.1002/nme.4748" }

**29**pp. 114015 (2014)

#### Likelihood-informed dimension reduction for nonlinear inverse problems

#### BibTeX entry

@article { title = "Likelihood-informed dimension reduction for nonlinear inverse problems", author = "T. Cui and J. Martin and Y. M. Marzouk and A. Solonen and A. Spantini", journal = "Inverse Problems", volume = "29", year ="2014", number = "", pages = "114015", doi = "doi:10.1088/0266-5611/30/11/114015" }

**36**pp. A2584–A2610 (2014)

#### Efficient localization of discontinuities in complex computational simulations

#### BibTeX entry

@article { title = "Efficient localization of discontinuities in complex computational simulations", author = "A. Gorodetsky and Y. M. Marzouk", journal = "SIAM Journal on Scientific Computing", volume = "36", year ="2014", number = "6", pages = "A2584–A2610", doi = "" }

**36**pp. A1163–A1186 (2014)

#### Adaptive construction of surrogates for the Bayesian solution of inverse problems

#### BibTeX entry

@article { title = "Adaptive construction of surrogates for the Bayesian solution of inverse problems", author = "J. Li and Y. M. Marzouk", journal = "SIAM Journal on Scientific Computing", volume = "36", year ="2014", number = "3", pages = "A1163–A1186", doi = "" }

**4**pp. 479–510 (2014)

#### Gradient-based stochastic optimization methods in Bayesian experimental design

This paper develops gradient-based stochastic optimization methods for the design of experiments on a continuous parameter space. Given a Monte Carlo estimator of expected information gain, we use infinitesimal perturbation analysis to derive gradients of this estimator. We are then able to formulate two gradient-based stochastic optimization approaches: (i) Robbins-Monro stochastic approximation, and (ii) sample average approximation combined with a deterministic quasi-Newton method. A polynomial chaos approximation of the forward model accelerates objective and gradient evaluations in both cases. We discuss the implementation of these optimization methods, then conduct an empirical comparison of their performance. To demonstrate design in a nonlinear setting with partial differential equation forward models, we use the problem of sensor placement for source inversion. Numerical results yield useful guidelines on the choice of algorithm and sample sizes, assess the impact of estimator bias, and quantify tradeoffs of computational cost versus solution quality and robustness.

#### BibTeX entry

@article { title = "Gradient-based stochastic optimization methods in Bayesian experimental design", author = "X. Huan and Y. M. Marzouk", journal = "International Journal for Uncertainty Quantification", volume = "4", year ="2014", number = "6", pages = "479–510", doi = "10.1615/Int.J.UncertaintyQuantification.2014006730" }

**17**pp. 899-911 (2013)

#### A priori testing of sparse adaptive polynomial chaos expansions using an ocean general circulation model database

#### BibTeX entry

@article { title = "A priori testing of sparse adaptive polynomial chaos expansions using an ocean general circulation model database", author = "J. Winokur and P. Conrad and I. Sraj and O. Knio and A. Srinivasan and W. C. Thacker and Y. M. Marzouk and M. Iskandarani,", journal = "Computational Geosciences", volume = "17", year ="2013", number = "6", pages = "899-911", doi = "10.1007/s10596-013-9361-3" }

**35**pp. A2643–A2670 (2013)

#### Adaptive Smolyak pseudospectral approximations

#### BibTeX entry

@article { title = "Adaptive Smolyak pseudospectral approximations", author = "P. Conrad and Y. M. Marzouk", journal = "SIAM Journal on Scientific Computing", volume = "35", year ="2013", number = "6", pages = "A2643–A2670", doi = "10.1137/120890715" }

**7**pp. 81–105 (2013)

#### Bayesian inverse problems with Monte Carlo forward models

The full application of Bayesian inference to inverse problems requires exploration of a posterior distribution that typically does not possess a standard form. In this context, Markov chain Monte Carlo (MCMC) methods are often used. These methods require many evaluations of a computationally intensive forward model to produce the equivalent of one independent sample from the posterior. We consider applications in which approximate forward models at multiple resolution levels are available, each endowed with a probabilistic error estimate. These situations occur, for example, when the forward model involves Monte Carlo integration. We present a novel MCMC method called $MC^3$ that uses low-resolution forward models to approximate draws from a posterior distribution built with the high-resolution forward model. The acceptance ratio is estimated with some statistical error; then a confidence interval for the true acceptance ratio is found, and acceptance is performed correctly with some confidence. The high-resolution models are rarely run and a significant speed up is achieved.

Our multiple-resolution forward models themselves are built around a new importance sampling scheme that allows Monte Carlo forward models to be used efficiently in inverse problems. The method is used to solve an inverse transport problem that finds applications in atmospheric remote sensing. We present a path-recycling methodology to efficiently vary parameters in the transport equation. The forward transport equation is solved by a Monte Carlo method that is amenable to the use of $MC^3$ to solve the inverse transport problem using a Bayesian formalism.

#### BibTeX entry

@article { title = "Bayesian inverse problems with Monte Carlo forward models", author = "G. Bal and I. Langmore and Y. M. Marzouk", journal = "Inverse Problems and Imaging", volume = "7", year ="2013", number = "1", pages = "81–105", doi = "10.3934/ipi.2013.7.81 " }

**232**pp. 288–317 (2013)

#### Simulation-based optimal Bayesian experimental design for nonlinear systems

The optimal selection of experimental conditions is essential to maximizing the value of data for inference and prediction, particularly in situations where experiments are time-consuming and expensive to conduct. We propose a general mathematical framework and an algorithmic approach for optimal experimental design with nonlinear simulation-based models; in particular, we focus on finding sets of experiments that provide the most information about targeted sets of parameters.

Our framework employs a Bayesian statistical setting, which provides a foundation for inference from noisy, indirect, and incomplete data, and a natural mechanism for incorporating heterogeneous sources of information. An objective function is constructed from information theoretic measures, reflecting expected information gain from proposed combinations of experiments. Polynomial chaos approximations and a two-stage Monte Carlo sampling method are used to evaluate the expected information gain. Stochastic approximation algorithms are then used to make optimization feasible in computationally intensive and high-dimensional settings. These algorithms are demonstrated on model problems and on nonlinear parameter inference problems arising in detailed combustion kinetics.

#### BibTeX entry

@article { title = "Simulation-based optimal Bayesian experimental design for nonlinear systems", author = "X. Huan and Y. M. Marzouk", journal = "Journal of Computational Physics", volume = "232", year ="2013", number = "1", pages = "288–317", doi = "10.1016/j.jcp.2012.08.013" }

**231**pp. 7815–7850 (2012)

#### Bayesian inference with optimal maps

We present a new approach to Bayesian inference that entirely avoids Markov chain simulation, by constructing a map that pushes forward the prior measure to the posterior measure. Existence and uniqueness of a suitable measure-preserving map is established by formulating the problem in the context of optimal transport theory. We discuss various means of explicitly parameterizing the map and computing it efficiently through solution of an optimization problem, exploiting gradient information from the forward model when possible. The resulting algorithm overcomes many of the computational bottlenecks associated with Markov chain Monte Carlo. Advantages of a map-based representation of the posterior include analytical expressions for posterior moments and the ability to generate arbitrary numbers of independent posterior samples without additional likelihood evaluations or forward solves. The optimization approach also provides clear convergence criteria for posterior approximation and facilitates model selection through automatic evaluation of the marginal likelihood. We demonstrate the accuracy and efficiency of the approach on nonlinear inverse problems of varying dimension, involving the inference of parameters appearing in ordinary and partial differential equations.

#### BibTeX entry

@article { title = "Bayesian inference with optimal maps", author = "T. A. Moselhy and Y. M. Marzouk", journal = "Journal of Computational Physics", volume = "231", year ="2012", number = "23", pages = "7815–7850", doi = "10.1016/j.jcp.2012.07.022" }

**44**pp. 1–19 (2012)

#### Bayesian reconstruction of binary media with unresolved fine-scale spatial structures

#### BibTeX entry

@article { title = "Bayesian reconstruction of binary media with unresolved fine-scale spatial structures", author = "J. Ray and S. McKenna and B. van Bloemen Waanders and Y. M. Marzouk", journal = "Advances in Water Resources", volume = "44", year ="2012", number = "", pages = "1–19", doi = "10.1016/j.advwatres.2012.04.009" }

**231**pp. 6401–6418 (2012)

#### Sequential data assimilation with multiple models

#### BibTeX entry

@article { title = "Sequential data assimilation with multiple models", author = "A. Narayan and Y. M. Marzouk and D. Xiu", journal = "Journal of Computational Physics", volume = "231", year ="2012", number = "19", pages = "6401–6418", doi = "10.1016/j.jcp.2012.06.002" }

**231**pp. 2180-2198 (2012)

#### Data-free inference of the joint distribution of uncertain model parameters

A critical problem in accurately estimating uncertainty in model predictions is the lack of details in the literature on the correlation (or full joint distribution) of uncertain model parameters. In this paper we describe a framework and a class of algorithms for analyzing such “missing data” problems in the setting of Bayesian statistics. The analysis focuses on the family of posterior distributions consistent with given statistics (e.g. nominal values, confidence intervals). The combining of consistent distributions is addressed via techniques from the opinion pooling literature. The developed approach allows subsequent propagation of uncertainty in model inputs consistent with reported statistics, in the absence of data.

#### BibTeX entry

@article { title = "Data-free inference of the joint distribution of uncertain model parameters", author = "R. D. Berry and H. N. Najm and B. J. Debusschere and Y. M. Marzouk and H. Adalsteinsson", journal = "Journal of Computational Physics", volume = "231", year ="2012", number = "5", pages = "2180-2198", doi = "10.1016/j.jcp.2011.10.031" }

**16**pp. 173-198 (2011)

#### Computational singular perturbation with non-parametric tabulation of slow manifolds for time integration of stiff chemical kinetics

This paper presents a novel tabulation strategy for the adaptive numerical integration of chemical kinetics using the computational singular perturbation (CSP) method. The strategy stores and reuses CSP quantities required to filter out fast dissipative processes, resulting in a non-stiff chemical source term. In particular, non-parametric regression on low-dimensional slow invariant manifolds (SIMs) in the chemical state space is used to approximate the CSP vectors spanning the fast chemical subspace and the associated fast chemical time-scales. The relevant manifold and its dimension varies depending on the local number of exhausted modes at every location in the chemical state space. Multiple manifolds are therefore tabulated, corresponding to different numbers of exhausted modes (dimensions) and associated radical species. Non-parametric representations are inherently adaptive, and rely on efficient approximate-nearest-neighbor queries. As the CSP information is only a function of the non-radical species in the system and has relatively small gradients in the chemical state space, tabulation occurs in a lower-dimensional state space and at a relatively coarse level, thereby improving scalability to larger chemical mechanisms. The approach is demonstrated on the simulation of homogeneous constant pressure H2–air and CH4–air ignition, over a range of initial conditions. For CH4–air, results are shown that outperform direct implicit integration of the stiff chemical kinetics while maintaining good accuracy.

#### BibTeX entry

@article { title = "Computational singular perturbation with non-parametric tabulation of slow manifolds for time integration of stiff chemical kinetics", author = "B. J. Debusschere and Y. M. Marzouk and H. N. Najm and B. Rhoads and D. A. Goussis and M. Valorani", journal = "Combustion Theory and Modeling", volume = "16", year ="2011", number = "1", pages = "173-198", doi = "10.1080/13647830.2011.596575" }

**34**pp. 617-626 (2011)

#### Truncated multi-Gaussian fields and effective conductance of binary media

Truncated Gaussian fields provide a flexible model for defining binary media with dispersed (as opposed to layered) inclusions. General properties of excursion sets on these truncated fields are coupled with a distance-based upscaling algorithm and approximations of point process theory to develop an estimation approach for effective conductivity in two-dimensions. Estimation of effective conductivity is derived directly from knowledge of the kernel size used to create the multiGaussian field, defined as the full-width at half maximum (FWHM), the truncation threshold and conductance values of the two modes. Therefore, instantiation of the multiGaussian field is not necessary for estimation of the effective conductance. The critical component of the effective medium approximation developed here is the mean distance between high conductivity inclusions. This mean distance is characterized as a function of the FWHM, the truncation threshold and the ratio of the two modal conductivities. Sensitivity of the resulting effective conductivity to this mean distance is examined for two levels of contrast in the modal conductances and different FWHM sizes. Results demonstrate that the FWHM is a robust measure of mean travel distance in the background medium. The resulting effective conductivities are accurate when compared to numerical results and results obtained from effective media theory, distance-based upscaling and numerical simulation.

#### BibTeX entry

@article { title = "Truncated multi-Gaussian fields and effective conductance of binary media", author = "S. McKenna and J. Ray and Y. M. Marzouk and B. van Bloemen Waanders", journal = "Advances in Water Resources", volume = "34", year ="2011", number = "5", pages = "617-626", doi = "10.1016/j.advwatres.2011.02.011" }

**676**pp. 461-490 (2011)

#### Contributions of the wall boundary layer to the formation of the counter-rotating vortex pair in transverse jets.

Using high-resolution 3-D vortex simulations, this study seeks a mechanistic understanding of vorticity dynamics in transverse jets at a finite Reynolds number. A full no-slip boundary condition, rigorously formulated in terms of vorticity generation along the channel wall, captures unsteady interactions between the wall boundary layer and the jet – in particular, the separation of the wall boundary layer and its transport into the interior. For comparison, we also implement a reduced boundary condition that suppresses the separation of the wall boundary layer away from the jet nozzle. By contrasting results obtained with these two boundary conditions, we characterize near-field vortical structures formed as the wall boundary layer separates on the backside of the jet. Using various Eulerian and Lagrangian diagnostics, it is demonstrated that several near-wall vortical structures are formed as the wall boundary layer separates. The counter-rotating vortex pair, manifested by the presence of vortices aligned with the jet trajectory, is initiated closer to the jet exit. Moreover tornado-like wall-normal vortices originate from the separation of spanwise vorticity in the wall boundary layer at the side of the jet and from the entrainment of streamwise wall vortices in the recirculation zone on the lee side. These tornado-like vortices are absent in the case where separation is suppressed. Tornado-like vortices merge with counter-rotating vorticity originating in the jet shear layer, significantly increasing wall-normal circulation and causing deeper jet penetration into the crossflow stream.

#### BibTeX entry

@article { title = "Contributions of the wall boundary layer to the formation of the counter-rotating vortex pair in transverse jets.", author = "F. Schlegel and D. Wee and Y. M. Marzouk and A. F. Ghoniem", journal = "Journal of Fluid Mechanics", volume = "676", year ="2011", number = "1", pages = "461-490", doi = "10.1017/jfm.2011.59" }

**9**pp. 486-512 (2011)

#### Bayesian inference of atomic diffusivity in a binary Ni/Al system based on molecular dynamics

This work focuses on characterizing the integral features of atomic diffusion in Ni/Al nanolaminates based on molecular dynamics (MD) computations. Attention is focused on the simplified problem of extracting the diffusivity, D, in an isothermal system at high temperature. To this end, a mixing measure theory is developed that relies on analyzing the moments of the cumulative distribution functions (CDFs) of the constituents. The mixing measures obtained from replica simulations are exploited in a Bayesian inference framework, based on contrasting these measures with corresponding moments of a dimensionless concentration evolving according to a Fickian process. The noise inherent in the MD simulations is described as a Gaussian process, and this hypothesis is verified both a priori and using a posterior predictive check. Computed values of D for an initially unmixed system rapidly heated to 1500 K are found to be consistent with experimental correlation for diffusion of Ni into molten Al. On the contrary, large discrepancies with experimental predictions are observed when D is estimated based on large-time mean-square displacement (MSD) analysis, and when it is evaluated using the Arrhenius correlation calibrated against experimental measurements of self-propagating front velocities. Implications are finally drawn regarding extension of the present work and potential refinement of continuum modeling approaches.

#### BibTeX entry

@article { title = "Bayesian inference of atomic diffusivity in a binary Ni/Al system based on molecular dynamics", author = "F. Rizzi and M. Salloum and Y. M. Marzouk and R. Xu and M. L. Falk and T. P. Weihs and G. Fritz and O. M. Knio", journal = "SIAM Multiscale Modeling and Simulation", volume = "9", year ="2011", number = "1", pages = "486-512", doi = "10.1137/10080590X" }

**30**pp. 101-126 (2010)

#### A Bayesian approach for estimating bioterror attacks from patient data

Terrorist attacks using an aerosolized pathogen have gained credibility as a national security concern after the anthrax attacks of 2001. Inferring some important details of the attack quickly, for example, the number of people infected, the time of infection, and a representative dose received can be crucial to planning a medical response. We use a Bayesian approach, based on a short time series of diagnosed patients, to estimate a joint probability density for these parameters. We first test the formulation with idealized cases and then apply it to realistic scenarios, including the Sverdlovsk anthrax outbreak of 1979. We also use simulated outbreaks to explore the impact of model error, as when the model used for generating simulated epidemic curves does not match the model subsequently used to characterize the attack. We find that in all cases except for the smallest attacks (fewer than 100 infected people), 3–5 days of data are sufficient to characterize the outbreak to a specificity that is useful for directing an emergency response.

#### BibTeX entry

@article { title = "A Bayesian approach for estimating bioterror attacks from patient data", author = "J. Ray and Y. M. Marzouk and H. N. Najm", journal = "Statistics in Medicine", volume = "30", year ="2010", number = "2", pages = "101-126", doi = "10.1002/sim.4090" }

**31**pp. 2510-2527 (2009)

#### Convergence characteristics and computational cost of two algebraic kernels in vortex methods with a tree-code algorithm

We study the convergence characteristics of two algebraic kernels used in vortex calculations: the Rosenhead–Moore kernel, which is a low-order kernel, and the Winckelmans–Leonard kernel, which is a high-order kernel. To facilitate the study, a method of evaluating particle-cluster interactions is introduced for the Winckelmans–Leonard kernel. The method is based on Taylor series expansion in Cartesian coordinates, as initially proposed by Lindsay and Krasny [J. Comput. Phys., 172 (2001), pp. 879–907] for the Rosenhead–Moore kernel. A recurrence relation for the Taylor coefficients of the Winckelmans–Leonard kernel is derived by separating the kernel into two parts, and an error estimate is obtained to ensure adaptive error control. The recurrence relation is incorporated into a tree-code to evaluate vorticity-induced velocity. Next, comparison of convergence is made while utilizing the tree-code. Both algebraic kernels lead to convergence, but the Winckelmans–Leonard kernel exhibits a superior convergence rate. The combined desingularization and discretization error from the Winckelmans–Leonard kernel is an order of magnitude smaller than that from the Rosenhead–Moore kernel at a typical resolution. Simulations of vortex rings are performed using the two algebraic kernels in order to compare their performance in a practical setting. In particular, numerical simulations of the side-by-side collision of two identical vortex rings suggest that the three-dimensional evolution of vorticity at finite resolution can be greatly affected by the choice of the kernel. We find that the Winckelmans–Leonard kernel is able to perform the same task with a much smaller number of vortex elements than the Rosenhead–Moore kernel, greatly reducing the overall computational cost.

#### BibTeX entry

@article { title = "Convergence characteristics and computational cost of two algebraic kernels in vortex methods with a tree-code algorithm", author = "D. Wee and Y. M. Marzouk and F. Schlegel and A. F. Ghoniem", journal = "SIAM Journal on Scientific Computing", volume = "31", year ="2009", number = "4", pages = "2510-2527", doi = "10.1137/080726872" }

**228**pp. 1862-1902 (2009)

#### Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems

We consider a Bayesian approach to nonlinear inverse problems in which the unknown quantity is a spatial or temporal field, endowed with a hierarchical Gaussian process prior. Computational challenges in this construction arise from the need for repeated evaluations of the forward model (e.g., in the context of Markov chain Monte Carlo) and are compounded by high dimensionality of the posterior. We address these challenges by introducing truncated Karhunen–Loève expansions, based on the prior distribution, to efficiently parameterize the unknown field and to specify a stochastic forward problem whose solution captures that of the deterministic forward model over the support of the prior. We seek a solution of this problem using Galerkin projection on a polynomial chaos basis, and use the solution to construct a reduced-dimensionality surrogate posterior density that is inexpensive to evaluate. We demonstrate the formulation on a transient diffusion equation with prescribed source terms, inferring the spatially-varying diffusivity of the medium from limited and noisy data.

#### BibTeX entry

@article { title = "Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems", author = "Y. M. Marzouk and H. N. Najm", journal = "Journal of Computational Physics", volume = "228", year ="2009", number = "6", pages = "1862-1902", doi = "10.1016/j.jcp.2008.11.024" }

**6**pp. 826-847 (2009)

#### A stochastic collocation approach to Bayesian inference in inverse problems

We present an efficient numerical strategy for the Bayesian solution of inverse problems. Stochastic collocation methods, based on generalized polynomial chaos (gPC), are used to construct a polynomial approximation of the forward solution over the support of the prior distribution. This approximation then defines a surrogate posterior probability density that can be evaluated repeatedly at minimal computational cost. The ability to simulate a large number of samples from the posterior distribution results in very accurate estimates of the inverse solution and its associated uncertainty. Combined with high accuracy of the gPC-based forward solver, the new algorithm can provide great efficiency in practical applications. A rigorous error analysis of the algorithm is conducted, where we establish convergence of the approximate posterior to the true posterior and obtain an estimate of the convergence rate. It is proved that fast (exponential) convergence of the gPC forward solution yields similarly fast (exponential) convergence of the posterior. The numerical strategy and the predicted convergence rates are then demonstrated on nonlinear inverse problems ofvarying smoothness and dimension.

#### BibTeX entry

@article { title = "A stochastic collocation approach to Bayesian inference in inverse problems", author = "Y. M. Marzouk and D. Xiu", journal = "Communications in Computational Physics", volume = "6", year ="2009", number = "1", pages = "826-847", doi = "DOI:prism/16" }

**80**pp. 789-814 (2009)

#### Uncertainty quantification in chemical systems

We demonstrate the use of multiwavelet spectral polynomial chaos techniques for uncertainty quantification in non-isothermal ignition of a methane–air system. We employ Bayesian inference for identifying the probabilistic representation of the uncertain parameters and propagate this uncertainty through the ignition process. We analyze the time evolution of moments and probability density functions of the solution. We also examine the role and significance of dependence among the uncertain parameters. We finish with a discussion of the role of non-linearity and the performance of the algorithm.

#### BibTeX entry

@article { title = "Uncertainty quantification in chemical systems", author = "H. N. Najm and B. J. Debusschere and Y. M. Marzouk and S. Widmer and O. LeMaître", journal = "International Journal for Numerical Methods in Engineering", volume = "80", year ="2009", number = "6-7", pages = "789-814", doi = "10.1002/nme.2551" }

**6**pp. 2283-2297 (2009)

#### Bayesian inference of spectral expansions for predictability assessment in stochastic reaction networks

Stochastic reaction networks modeled as jump Markov processes serve as the main mathematical representation of biochemical phenomena in cells, particularly when the relevant molecule count is low, causing deterministic macroscale chemical reaction models to fail. Further, as there is mainly empirical knowledge about the rate parameters, parametric uncertainty analysis becomes very important. The conventional predictability tools for deterministic systems do not readily generalize to the stochastic setting. We use spectral polynomial chaos expansions to represent stochastic processes. Bayesian inference techniques with Markov chain Monte Carlo are used to find the best spectral representation of the system state, taking into account not only intrinsic stochastic noise but also parametric uncertainties. A likelihood-based adaptive domain decomposition is introduced and applied, in particular, for the cases when the parameter range includes deterministic bifurcations. We show that the adaptive multidomain polynomial chaos representation captures the correct system behavior for a benchmark bistable Schlögl model for a wide range of parameter variations.

#### BibTeX entry

@article { title = "Bayesian inference of spectral expansions for predictability assessment in stochastic reaction networks", author = "K. Sargsyan and B. J. Debusschere and H. N. Najm and Y. M. Marzouk", journal = "Journal of Computational and Theoretical Nanoscience", volume = "6", year ="2009", number = "10", pages = "2283-2297", doi = "10.1166/jctn.2009.1285" }

**224**pp. 560-586 (2007)

#### Stochastic spectral methods for efficient Bayesian solution of inverse problems

We present a reformulation of the Bayesian approach to inverse problems, that seeks to accelerate Bayesian inference by using polynomial chaos (PC) expansions to represent random variables. Evaluation of integrals over the unknown parameter space is recast, more efficiently, as Monte Carlo sampling of the random variables underlying the PC expansion. We evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion. The accuracy of posterior estimates is examined with respect to the order of the PC representation, the choice of PC basis, and the decomposition of the support of the prior. The computational cost of the new scheme shows significant gains over direct sampling.

#### BibTeX entry

@article { title = "Stochastic spectral methods for efficient Bayesian solution of inverse problems", author = "Y. M. Marzouk and H. N. Najm and L. A. Rahn,", journal = "Journal of Computational Physics", volume = "224", year ="2007", number = "2", pages = "560-586", doi = "10.1016/j.jcp.2006.10.010" }

**575**pp. 267-305 (2007)

#### Vorticity structure and evolution in a transverse jet

Transverse jets arise in many applications, including propulsion, effluent dispersion, oil field flows, and V/STOL aerodynamics. This study seeks a fundamental, mechanistic understanding of the structure and evolution of vorticity in the transverse jet. We develop a high-resolution three-dimensional vortex simulation of the transverse jet at large Reynolds number and consider jet-to-crossflow velocity ratios r ranging from 5 to 10. A new formulation of vorticity-flux boundary conditions accounts for the interaction of channel wall vorticity with the jet flow immediately around the orifice. We demonstrate that the nascent jet shear layer contains not only azimuthal vorticity generated in the jet pipe, but wall-normal and azimuthal perturbations resulting from the jet–crossflow interaction. This formulation also yields analytical expressions for vortex lines in the near field as a function of $r$.

Transformation of the cylindrical shear layer emanating from the orifice begins with axial elongation of its lee side to form sections of counter-rotating vorticity aligned with the jet trajectory. Periodic roll-up of the shear layer accompanies this deformation, creating complementary vortex arcs on the lee and windward sides of the jet. Counter-rotating vorticity then drives lee-side roll-ups in the windward direction, along the normal to the jet trajectory. Azimuthal vortex arcs of alternating sign thus approach each other on the windward boundary of the jet. Accordingly, initially planar material rings on the shear layer fold completely and assume an interlocking structure that persists for several diameters above the jet exit. Though the near field of the jet is dominated by deformation and periodic roll-up of the shear layer, the resulting counter-rotating vorticity is a pronounced feature of the mean field; in turn, the mean counter-rotation exerts a substantial influence on the deformation of the shear layer. Following the pronounced bending of the trajectory into the crossflow, we observe a sudden breakdown of near-field vortical structures into a dense distribution of smaller scales. Spatial filtering of this region reveals the persistence of counter-rotating streamwise vorticity initiated in the near field.

#### BibTeX entry

@article { title = "Vorticity structure and evolution in a transverse jet", author = "Y. M. Marzouk and A. F. Ghoniem", journal = "Journal of Fluid Mechanics", volume = "575", year ="2007", number = "1", pages = "267-305", doi = "10.1017/S0022112006004411" }

**207**pp. 493-528 (2005)

#### Fulltext options

#### K-means clustering for optimal partitioning and dynamic load balancing of parallel hierarchical N-body simulations

A number of complex physical problems can be approached through N-body simulation, from fluid flow at high Reynolds number to gravitational astrophysics and molecular dynamics. In all these applications, direct summation is prohibitively expensive for large N and thus hierarchical methods are employed for fast summation. This work introduces new algorithms, based on k-means clustering, for partitioning parallel hierarchical N-body interactions. We demonstrate that the number of particle–cluster interactions and the order at which they are performed are directly affected by partition geometry. Weighted k-means partitions minimize the sum of clusters’ second moments and create well-localized domains, and thus reduce the computational cost of N-body approximations by enabling the use of lower-order approximations and fewer cells.

We also introduce compatible techniques for dynamic load balancing, including adaptive scaling of cluster volumes and adaptive redistribution of cluster centroids. We demonstrate the performance of these algorithms by constructing a parallel treecode for vortex particle simulations, based on the serial variable-order Cartesian code developed by Lindsay and Krasny [Journal of Computational Physics 172 (2) (2001) 879–907]. The method is applied to vortex simulations of a transverse jet. Results show outstanding parallel efficiencies even at high concurrencies, with velocity evaluation errors maintained at or below their serial values; on a realistic distribution of 1.2 million vortex particles, we observe a parallel efficiency of 98% on 1024 processors. Excellent load balance is achieved even in the face of several obstacles, such as an irregular, time-evolving particle distribution containing a range of length scales and the continual introduction of new vortex particles throughout the domain. Moreover, results suggest that k-means yields a more efficient partition of the domain than a global oct-tree.

#### BibTeX entry

@article { title = "K-means clustering for optimal partitioning and dynamic load balancing of parallel hierarchical N-body simulations", author = "Y. M. Marzouk and A. F. Ghoniem", journal = "Journal of Computational Physics", volume = "207", year ="2005", number = "2", pages = "493-528", doi = "10.1016/j.jcp.2005.01.021" }

**41**pp. 641-652 (2003)

#### Toward a flame embedding model for turbulent combustion simulation

#### BibTeX entry

@article { title = "Toward a flame embedding model for turbulent combustion simulation", author = "Y. M. Marzouk and A. F. Ghoniem and H. N. Najm", journal = "AIAA Journal", volume = "41", year ="2003", number = "4", pages = "641-652", doi = "" }

**28**pp. 1859-1866 (2000)

#### Fulltext options

#### Dynamic response of strained premixed flames to equivalence ratio gradients

Premixed flames encounter gradients of mixture equivalence ratio in stratified charge engines, lean premixed gas-turbine engines, and a variety of other applications. In cases for which the scales—spatial or temporal—of fuel concentration gradients in the reactants are comparable to flame scales, changes in burning rate, flammability limits, and flame structure have been observed. This paper uses an unsteady strained flame in the stagnation point configuration to examine the effect of temporal gradients on combustion in a premixed methane/air mixture. An inexact Newton backtracking method, coupled with a preconditioned Krylov subspace iterative solver, was used to improve the efficiency of the numerical solution and expand its domain of convergence in the presence of detailed chemistry.

Results indicate that equivalence ratio variations with timescales lower than 10 ms have significant effects on the burning process, including reaction zone broadening, burning rate enhancement, and extension of the flammability limit toward learner mixtures. While the temperature of a flame processing a stoichiometric-to-lean equivalence ratio gradient decreased slightly within the front side of the reaction zone, radical concentrations remained elevated over the entire flame structure. These characteristics are linked to a feature reminiscent of “back-supported” flames—flames in which a stream of products resulting from burning at higher equivalence ratio is continuously supplied to lower equivalence ratio reactants. The relevant feature is the establishment of a positive temperature gradient on the products side of the flame which maintains the temperature high enough and the radical concentration sufficient to sustain combustion there. Unsteadiness in equivalence ratio produces similar gradients within the flame structure, thus compensating for the change in temperature at the leading edge of the reaction zone and accounting for an observed “flame inertia”. For sufficiently large equivalence ratio gradients, a flame starting in a stoichiometric mixture can burn through a very lean one by taking advantage of this mechanism.

#### BibTeX entry

@article { title = "Dynamic response of strained premixed flames to equivalence ratio gradients", author = "Y. M. Marzouk and A. F. Ghoniem and H. N. Najm", journal = "Proceedings of the Combustion Institute", volume = "28", year ="2000", number = "2", pages = "1859-1866", doi = "10.1016/S0082-0784(00)80589-5" }

**25**pp. 401-408 (1998)

#### Asymmetric autocorrelation function to resolve directional ambiguity in PIV images

Autocorrelation of a double-exposed image, unlike cross-correlation between two images, produces a correlation function that is symmetric about the origin. Thus, while it is possible to calculate the speed and direction of tracer particles in a particle image velocimetry (PIV) image using autocorrelation, it is impossible to tell whether the velocity is in the positive or negative direction. This ambiguity can be resolved by spatially shifting one exposure relative to the next or labeling exposures with color or polarization, but the complexity and limitations of these methods can be prohibitive. It is, however, possible to resolve the sign of the velocity from a triple-exposed image using unequal time intervals between exposures.

Triple-exposed images, like double-exposed images, correlate symmetrically about zero. The directional ambiguity, however, can be resolved by calculating the probability that the three exposures occur in a specific temporal order; that is, by assuming that the correlation has a specific sign and testing to see if the assumption is correct. Traditional spectral and statistical correlation techniques are unable to accomplish this. Presented herein is a computationally efficient asymmetric correlation function that is able to differentiate the temporal order of triple exposed images. Included is a discussion of the limitations of this function and of difficulties in experimental implementation.

#### BibTeX entry

@article { title = "Asymmetric autocorrelation function to resolve directional ambiguity in PIV images", author = "Y. M. Marzouk and D. P. Hart", journal = "Experiments in Fluids", volume = "25", year ="1998", number = "5-6", pages = "401-408", doi = "10.1007/s003480050247" }

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**May 2018**

Congratulations to Andrew Davis for successfully defending his PhD thesis!

**February 2018**

Congratulations to Xun Huan for accepting a tenure-track position at the University of Michigan!

**September 2017**

Congratulations to Olivier Zahm for accepting a permanent position at INRIA in Grenoble!

**August 2017**

Congratulations to Alessio Spantini for successfully defending his PhD thesis, "On the low-dimensional structure of Bayesian inference!"

**June 2017**

Congratulations to Rebecca Morrison for accepting a tenure-track position at the University of Colorado Boulder!

**June 2017**

Congratulations to Alex Gorodetsky for accepting a tenure-track position at the University of Michigan!

**Feb 1, 2017**

Congratulations to Ricardo Baptista and Ben Zhang for passing their PhD qualifying exams!

**Sep 2, 2016**

Congratulations to Alex Gorodetsky for successfully defending his PhD thesis!