@article{gamboa_gremaud_klein_lagnoux_2022, title={Global sensitivity analysis: A novel generation of mighty estimators based on rank statistics}, volume={28}, ISSN={["1573-9759"]}, DOI={10.3150/21-BEJ1421}, abstractNote={We propose a new statistical estimation framework for a large family of global sensitivity analysis methods. Our approach is based on rank statistics and uses an empirical correlation coefficient recently introduced by Sourav Chatterjee. We show how to apply this approach to compute not only the Cramer-von-Mises indices, which are directly related to Chatterjee's notion of correlation, but also Sobol indices at any order, higher-order moment indices, and Shapley effects. We establish consistency of the resulting estimators and demonstrate their numerical efficiency, especially for small sample sizes.}, number={4}, journal={BERNOULLI}, author={Gamboa, Fabrice and Gremaud, Pierre and Klein, Thierry and Lagnoux, Agnes}, year={2022}, month={Nov}, pages={2345–2374} }
 @article{gilmore_hart_geddes_olsen_mehlsen_gremaud_olufsen_2021, title={Classification of orthostatic intolerance through data analytics}, volume={59}, ISSN={["1741-0444"]}, DOI={10.1007/s11517-021-02314-0}, abstractNote={Imbalance in the autonomic nervous system can lead to orthostatic intolerance manifested by dizziness, lightheadedness, and a sudden loss of consciousness (syncope); these are common conditions, but they are challenging to diagnose correctly. Uncertainties about the triggering mechanisms and the underlying pathophysiology have led to variations in their classification. This study uses machine learning to categorize patients with orthostatic intolerance. We use random forest classification trees to identify a small number of markers in blood pressure, and heart rate time-series data measured during head-up tilt to (a) distinguish patients with a single pathology and (b) examine data from patients with a mixed pathophysiology. Next, we use Kmeans to cluster the markers representing the time-series data. We apply the proposed method analyzing clinical data from 186 subjects identified as control or suffering from one of four conditions: postural orthostatic tachycardia (POTS), cardioinhibition, vasodepression, and mixed cardioinhibition and vasodepression. Classification results confirm the use of supervised machine learning. We were able to categorize more than 95% of patients with a single condition and were able to subgroup all patients with mixed cardioinhibitory and vasodepressor syncope. Clustering results confirm the disease groups and identify two distinct subgroups within the control and mixed groups. The proposed study demonstrates how to use machine learning to discover structure in blood pressure and heart rate time-series data. The methodology is used in classification of patients with orthostatic intolerance. Diagnosing orthostatic intolerance is challenging, and full characterization of the pathophysiological mechanisms remains a topic of ongoing research. This study provides a step toward leveraging machine learning to assist clinicians and researchers in addressing these challenges.}, number={3}, journal={MEDICAL & BIOLOGICAL ENGINEERING & COMPUTING}, author={Gilmore, Steven and Hart, Joseph and Geddes, Justen and Olsen, Christian H. and Mehlsen, Jesper and Gremaud, Pierre and Olufsen, Mette S.}, year={2021}, month={Mar}, pages={621–632} }
 @article{merritt_alexanderian_gremaud_2021, title={MULTISCALE GLOBAL SENSITIVITY ANALYSIS FOR STOCHASTIC CHEMICAL SYSTEMS}, volume={19}, ISSN={["1540-3467"]}, DOI={10.1137/20M1323989}, abstractNote={Sensitivity analysis is routinely performed on simplified surrogate models as the cost of such analysis on the original model may be prohibitive. Little is known in general about the induced bias on the sensitivity results. Within the framework of chemical kinetics, we provide a full justification of the above approach in the case of variance based methods provided the surrogate model results from the original one through the thermodynamic limit. We also provide illustrative numerical examples in context of a Michaelis--Menten system and a biochemical reaction network describing a genetic oscillator.}, number={1}, journal={MULTISCALE MODELING & SIMULATION}, author={Merritt, Michael and Alexanderian, Alen and Gremaud, Pierre A.}, year={2021}, pages={440–459} }
 @article{alexanderian_gremaud_smith_2020, title={Variance-based sensitivity analysis for time-dependent processes}, volume={196}, ISSN={["1879-0836"]}, DOI={10.1016/j.ress.2019.106722}, abstractNote={The global sensitivity analysis of time-dependent processes requires history-aware approaches. We develop for that purpose a variance-based method that leverages the correlation structure of the problems under study and employs surrogate models to accelerate the computations. The errors resulting from fixing unimportant uncertain parameters to their nominal values are analyzed through a priori estimates. We illustrate our approach on a harmonic oscillator example and on a nonlinear dynamic cholera model.}, journal={RELIABILITY ENGINEERING & SYSTEM SAFETY}, author={Alexanderian, Alen and Gremaud, Pierre A. and Smith, Ralph C.}, year={2020}, month={Apr} }
 @article{williams_brady_gilmore_gremaud_tran_ottesen_mehlsen_olufsen_2019, title={Cardiovascular dynamics during head-up tilt assessed via pulsatile and non-pulsatile models}, volume={79}, ISSN={0303-6812 1432-1416}, url={http://dx.doi.org/10.1007/s00285-019-01386-9}, DOI={10.1007/s00285-019-01386-9}, abstractNote={This study develops non-pulsatile and pulsatile models for the prediction of blood flow and pressure during head-up tilt. This test is used to diagnose potential pathologies within the autonomic control system, which acts to keep the cardiovascular system at homeostasis. We show that mathematical modeling can be used to predict changes in cardiac contractility, vascular resistance, and arterial compliance, quantities that cannot be measured but are useful to assess the system's state. These quantities are predicted as time-varying parameters modeled using piecewise linear splines. Having models with various levels of complexity formulated with a common set of parameters, allows us to combine long-term non-pulsatile simulations with pulsatile simulations on a shorter time-scale. We illustrate results for a representative subject tilted head-up from a supine position to a [Formula: see text] angle. The tilt is maintained for 5 min before the subject is tilted back down. Results show that if volume data is available for all vascular compartments three parameters can be identified, cardiovascular resistance, vascular compliance, and ventricular contractility, whereas if model predictions are made against arterial pressure and cardiac output data alone, only two parameters can be estimated either resistance and contractility or resistance and compliance.}, number={3}, journal={Journal of Mathematical Biology}, publisher={Springer Science and Business Media LLC}, author={Williams, Nakeya D. and Brady, Renee and Gilmore, Steven and Gremaud, Pierre and Tran, Hien T. and Ottesen, Johnny T. and Mehlsen, Jesper and Olufsen, Mette S.}, year={2019}, month={May}, pages={987–1014} }
 @article{schaeffer_barker_tsuji_gremaud_shearer_gray_2019, title={Constitutive relations for compressible granular flow in the inertial regime}, volume={874}, ISSN={["1469-7645"]}, DOI={10.1017/jfm.2019.476}, abstractNote={Granular flows occur in a wide range of situations of practical interest to industry, in our natural environment and in our everyday lives. This paper focuses on granular flow in the so-called inertial regime, when the rheology is independent of the very large particle stiffness. Such flows have been modelled with the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -rheology, which postulates that the bulk friction coefficient $\unicode[STIX]{x1D707}$ (i.e. the ratio of the shear stress to the pressure) and the solids volume fraction $\unicode[STIX]{x1D719}$ are functions of the inertial number $I$ only. Although the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -rheology has been validated in steady state against both experiments and discrete particle simulations in several different geometries, it has recently been shown that this theory is mathematically ill-posed in time-dependent problems. As a direct result, computations using this rheology may blow up exponentially, with a growth rate that tends to infinity as the discretization length tends to zero, as explicitly demonstrated in this paper for the first time. Such catastrophic instability due to ill-posedness is a common issue when developing new mathematical models and implies that either some important physics is missing or the model has not been properly formulated. In this paper an alternative to the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$ -rheology that does not suffer from such defects is proposed. In the framework of compressible $I$ -dependent rheology (CIDR), new constitutive laws for the inertial regime are introduced; these match the well-established $\unicode[STIX]{x1D707}(I)$ and $\unicode[STIX]{x1D6F7}(I)$ relations in the steady-state limit and at the same time are well-posed for all deformations and all packing densities. Time-dependent numerical solutions of the resultant equations are performed to demonstrate that the new inertial CIDR model leads to numerical convergence towards physically realistic solutions that are supported by discrete element method simulations.}, journal={JOURNAL OF FLUID MECHANICS}, author={Schaeffer, D. G. and Barker, T. and Tsuji, D. and Gremaud, P. and Shearer, M. and Gray, J. M. N. T.}, year={2019}, month={Sep}, pages={926–951} }
 @article{hart_gremaud_david_2019, title={Global Sensitivity Analysis of High-Dimensional Neuroscience Models: An Example of Neurovascular Coupling}, volume={81}, ISSN={["1522-9602"]}, DOI={10.1007/s11538-019-00578-0}, abstractNote={The complexity and size of state-of-the-art cell models have significantly increased in part due to the requirement that these models possess complex cellular functions which are thought—but not necessarily proven—to be important. Modern cell models often involve hundreds of parameters; the values of these parameters come, more often than not, from animal experiments whose relationship to the human physiology is weak with very little information on the errors in these measurements. The concomitant uncertainties in parameter values result in uncertainties in the model outputs or quantities of interest (QoIs). Global sensitivity analysis (GSA) aims at apportioning to individual parameters (or sets of parameters) their relative contribution to output uncertainty thereby introducing a measure of influence or importance of said parameters. New GSA approaches are required to deal with increased model size and complexity; a three-stage methodology consisting of screening (dimension reduction), surrogate modeling, and computing Sobol’ indices, is presented. The methodology is used to analyze a physiologically validated numerical model of neurovascular coupling which possess 160 uncertain parameters. The sensitivity analysis investigates three quantities of interest, the average value of $$\hbox {K}^{+}$$ in the extracellular space, the average volumetric flow rate through the perfusing vessel, and the minimum value of the actin/myosin complex in the smooth muscle cell. GSA provides a measure of the influence of each parameter, for each of the three QoIs, giving insight into areas of possible physiological dysfunction and areas of further investigation.}, number={6}, journal={BULLETIN OF MATHEMATICAL BIOLOGY}, author={Hart, J. L. and Gremaud, P. A. and David, T.}, year={2019}, month={Jun}, pages={1805–1828} }
 @article{hart_gremaud_2019, title={ROBUSTNESS OF THE SOBOL' INDICES TO DISTRIBUTIONAL UNCERTAINTY}, volume={9}, ISSN={["2152-5099"]}, DOI={10.1615/Int.J.UncertaintyQuantification.2019030553}, abstractNote={Global sensitivity analysis (GSA) is used to quantify the influence of uncertain variables in a mathematical model. Prior to performing GSA, the user must specific a probability distribution to model the uncertainty, and possibly statistical dependencies, of the variables. Determining this distribution is challenging in practice as the user has limited and imprecise knowledge of the uncertain variables. This article analyzes the robustness of the Sobol' indices, a commonly used tool in GSA, to changes in the distribution of the uncertain variables. A method for assessing such robustness is developed which requires minimal user specification and no additional evaluations of the model. Theoretical and computational aspects of the method are considered and illustrated through examples.}, number={5}, journal={INTERNATIONAL JOURNAL FOR UNCERTAINTY QUANTIFICATION}, author={Hart, Joseph and Gremaud, Pierre}, year={2019}, pages={453–469} }
 @article{hart_gremaud_2019, title={Robustness of the Sobol' Indices to Marginal Distribution Uncertainty}, volume={7}, ISSN={["2166-2525"]}, DOI={10.1137/18M123387X}, abstractNote={Global sensitivity analysis (GSA) quantifies the influence of uncertain variables in a mathematical model. The Sobol' indices, a commonly used tool in GSA, seek to do this by attributing to each variable its relative contribution to the variance of the model output. In order to compute Sobol' indices, the user must specify a probability distribution for the uncertain variables. This distribution is typically unknown and must be chosen using limited data and/or knowledge. The usefulness of the Sobol' indices depends on their robustness to this distributional uncertainty. This article presents a novel method which uses "optimal perturbations" of the marginal probability density functions to analyze the robustness of the Sobol' indices. The method is illustrated through synthetic examples and a model for contaminant transport.}, number={4}, journal={SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION}, author={Hart, Joseph L. and Gremaud, Pierre A.}, year={2019}, pages={1224–1244} }
 @article{hart_gremaud_2018, title={AN APPROXIMATION THEORETIC PERSPECTIVE OF SOBOL' INDICES WITH DEPENDENT VARIABLES}, volume={8}, ISSN={["2152-5099"]}, DOI={10.1615/Int.J.UncertaintyQuantification.2018026498}, abstractNote={The Sobol' indices are a recognized tool in global sensitivity analysis. When the uncertain variables in a model are statistically independent, the Sobol' indices may be easily interpreted and utilized. However, their interpretation and utility is more challenging with statistically dependent variables. This article develops an approximation theoretic perspective to interpret Sobol' indices in the presence of variable dependencies. The value of this perspective is demonstrated in the context of dimension reduction, a common application of the Sobol' indices. Theoretical analysis and illustrative examples are provided.}, number={6}, journal={INTERNATIONAL JOURNAL FOR UNCERTAINTY QUANTIFICATION}, author={Hart, J. L. and Gremaud, P. A.}, year={2018}, pages={483–493} }
 @article{maltba_gremaud_tartakoysky_2018, title={Nonlocal PDF methods for Langevin equations with colored noise}, volume={367}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2018.04.023}, abstractNote={Langevin equations describe systems driven by internally generated or externally imposed random excitations. If these excitations correspond to Gaussian white noise, it is relatively straightforward to derive a closed form equation for the joint probability density function (PDF) of state variables. Many natural phenomena present however correlated (colored) excitations. For such problems, a full probabilistic characterization through the resolution of a PDF equation can be obtained through two levels of approximations: first, mixed ensemble moments have to be approximated to lead to a closed system of equations and, second, the resulting nonlocal equations should be at least partially localized to ensure computational efficiency. We propose a new semi-local formulation based on a modified large-eddy diffusivity (LED) approach; the formulation retains most of the accuracy of a fully nonlocal approach while presenting the same order of algorithmic complexity as the standard LED approach. The accuracy of the approach is successfully tested against Monte Carlo simulations.}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Maltba, Tyler and Gremaud, Pierre A. and Tartakoysky, Daniel M.}, year={2018}, month={Aug}, pages={87–101} }
 @article{hart_alexanderian_gremaud_2017, title={EFFICIENT COMPUTATION OF SOBOL' INDICES FOR STOCHASTIC MODELS}, volume={39}, ISSN={["1095-7197"]}, DOI={10.1137/16m106193x}, abstractNote={Stochastic models are necessary for the realistic description of an increasing number of applications. The ability to identify influential parameters and variables is critical to a thorough analysis and understanding of the underlying phenomena. We present a new global sensitivity analysis approach for stochastic models, i.e., models with both uncertain parameters and intrinsic stochasticity. Our method relies on an analysis of variance through a generalization of Sobol' indices and on the use of surrogate models. We show how to efficiently compute the statistical properties of the resulting indices and illustrate the effectiveness of our approach by computing first order Sobol' indices for two stochastic models.}, number={4}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Hart, J. L. and Alexanderian, A. and Gremaud, P. A.}, year={2017}, pages={A1514–A1530} }
 @article{collins_gremaud_2016, title={Analysis of a domain decomposition method for linear transport problems on networks}, volume={109}, ISSN={["1873-5460"]}, DOI={10.1016/j.apnum.2016.06.004}, abstractNote={In this paper we analyze the convergence of the domain decomposition method applied to transport problems on networks. In particular, we derive estimates for the number of required iterations for linear problems. These estimates can be used to determine when the implementation of domain decomposition methods would be beneficial for this type of problems.}, journal={APPLIED NUMERICAL MATHEMATICS}, author={Collins, J. B. and Gremaud, P. A.}, year={2016}, month={Nov}, pages={61–72} }
 @article{guan_liang_gremaud_2016, title={Comparison of the Windkessel model and structured-tree model applied to prescribe outflow boundary conditions for a one-dimensional arterial tree model}, volume={49}, ISSN={["1873-2380"]}, DOI={10.1016/j.jbiomech.2016.03.037}, abstractNote={One-dimensional (1D) modeling is a widely adopted approach for studying wave propagation phenomena in the arterial system. Despite the frequent use of the Windkessel (WK) model to prescribe outflow boundary conditions for 1D arterial tree models, it remains unclear to what extent the inherent limitation of the WK model in describing wave propagation in distal vasculatures affect hemodynamic variables simulated at the arterial level. In the present study, a 1D model of the arterial tree was coupled respectively with a WK boundary model and a structured-tree (ST) boundary model, yielding two types of arterial tree models. The effective resistances, compliances and inductances of the WK and ST boundary models were matched to facilitate quantitative comparisons. Obtained results showed that pressure/flow waves simulated by the two models were comparable in the aorta, whereas, their discrepancies increased towards the periphery. Wave analysis revealed that the differences in reflected waves generated by the boundary models were the major sources of pressure wave discrepancies observed in large arteries. Additional simulations performed under aging conditions demonstrated that arterial stiffening with age enlarged the discrepancies, but with the effects being partly counteracted by physiological aortic dilatation with age. These findings suggest that the method adopted for modeling the outflow boundary conditions has considerable influence on the performance of a 1D arterial tree model, with the extent of influence varying with the properties of the arterial system.}, number={9}, journal={JOURNAL OF BIOMECHANICS}, author={Guan, Debao and Liang, Fuyou and Gremaud, Pierre A.}, year={2016}, month={Jun}, pages={1583–1592} }
 @article{hart_novak_saunders_gremaud_2016, title={Transcranial Doppler-Based Surrogates for Cerebral Blood Flow: A Statistical Study}, volume={11}, ISSN={["1932-6203"]}, DOI={10.1371/journal.pone.0165536}, abstractNote={It is commonly assumed that perfusion in a given cerebral territory can be inferred from Blood Flow Velocity (BFV) measurements in the corresponding stem artery. In order to test this hypothesis, we construct a cerebral blood flow (CBF) estimator based on transcranial Doppler (TCD) blood flow velocity and ten other easily available patient characteristics and clinical parameters. A total of 261 measurements were collected from 88 older patients. The estimator is based on local regression (Random Forest). Its performance is analyzed against baseline CBF from 3-D pseudocontinuous arterial spin labeling (pCASL) magnetic resonance imaging (MRI). Patient specific CBF predictions are of poor quality (r = 0.41 and p-value = 4.5 × 10−12); the hypothesis is thus not clearly supported by evidence.}, number={11}, journal={PLOS ONE}, author={Hart, Joseph and Novak, Vera and Saunders, Charles and Gremaud, Pierre A.}, year={2016}, month={Nov} }
 @article{cousins_gremaud_2014, title={Impedance boundary conditions for general transient hemodynamics}, volume={30}, ISSN={["2040-7947"]}, DOI={10.1002/cnm.2658}, abstractNote={SUMMARY}, number={11}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING}, author={Cousins, Will and Gremaud, Pierre A.}, year={2014}, month={Nov}, pages={1294–1313} }
 @article{gremaud_sun_2014, title={Numerical Study of Singularity Formation in Relativistic Euler Flows}, volume={16}, ISSN={["1991-7120"]}, DOI={10.4208/cicp.221212.300114a}, abstractNote={Abstract}, number={2}, journal={COMMUNICATIONS IN COMPUTATIONAL PHYSICS}, author={Gremaud, Pierre A. and Sun, Yi}, year={2014}, month={Aug}, pages={348–364} }
 @article{cousins_gremaud_tartakovsky_2013, title={A NEW PHYSIOLOGICAL BOUNDARY CONDITION FOR HEMODYNAMICS}, volume={73}, ISSN={["1095-712X"]}, DOI={10.1137/120895470}, abstractNote={We propose a new physiologically-based outflow boundary condition for hemodynamics under general transient regimes. This is in contrast to previous studies that impose restrictions of temporal periodicity. The new condition is analyzed and its numerical implementation is discussed in detail. We show that existing impedance boundary conditions can be viewed as numerical approximations of the new condition. Our study provides a partial justification for using some of these existing conditions beyond the periodic problems for which they were designed. Moreover, the new condition has better stability properties. The theoretical results are illustrated by numerical experiments pertaining to cerebral blood flow.}, number={3}, journal={SIAM JOURNAL ON APPLIED MATHEMATICS}, author={Cousins, Will and Gremaud, Pierre A. and Tartakovsky, Daniel M.}, year={2013}, pages={1203–1223} }
 @article{gordon-wright_gremaud_2013, title={Granular Free Surfaces}, volume={55}, ISSN={["1095-7200"]}, DOI={10.1137/090747944}, abstractNote={We explore the notion of angle of repose for granular materials. As an illustration, we solve free surface problems for centrifuged granular piles. These have recently been considered as an affordable and simple way to experiment with powders and dust in reduced gravity environments such as on the Moon or on Mars.}, number={1}, journal={SIAM REVIEW}, author={Gordon-Wright, Rachael and Gremaud, Pierre A.}, year={2013}, pages={168–184} }
 @article{cousins_gremaud_2012, title={Boundary conditions for hemodynamics: The structured tree revisited}, volume={231}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2012.04.038}, abstractNote={The structured tree boundary condition is a physiologically-based outflow boundary condition used in hemodynamics. We propose an alternative derivation that is considerably simpler than the original one and yields similar, but not identical, results. We analyze the sensitivity of this boundary condition to its parameters and discuss its domain of validity. Several implementation issues are discussed and tested in the case of arterial flow in the Circle of Willis. Additionally, we compare results obtained from the structured tree boundary condition to the Windkessel boundary condition and measured data.}, number={18}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Cousins, W. and Gremaud, P. A.}, year={2012}, month={Jul}, pages={6086–6096} }
 @article{collins_gremaud_2011, title={A simple model for laser drilling}, volume={81}, ISSN={["1872-7166"]}, DOI={10.1016/j.matcom.2010.07.010}, abstractNote={A simple mathematical model of laser drilling is proposed. Assuming axi-symmetry of the process around the axis of the laser beam, a one-dimensional formulation is obtained after cross-sectional averaging. The novelty of the approach relies on the fact that even after dimension reduction, the shape of the hole can still be described. The model is derived, implemented and validated for drilling using lasers with intensities in the GW/cm2 range and microsecond pulses.}, number={8}, journal={MATHEMATICS AND COMPUTERS IN SIMULATION}, author={Collins, Jeb and Gremaud, Pierre}, year={2011}, month={Apr}, pages={1541–1552} }
 @article{shearer_gremaud_kleiner_2009, title={Periodic motion of a mass-spring system}, volume={74}, ISSN={["0272-4960"]}, DOI={10.1093/imamat/hxp032}, abstractNote={The equations of planar motion of a mass attached to two anchored massless springs form a symmetric Hamiltonian system. The system has a single dimensionless parameter L, corresponding to the spacing between the anchors. For L > 1, there is a stable equilibrium at which the springs are in tension and lie on a line, but for L < 1, this equilibrium has both springs in compression and is unstable. However, there are then two stable equilibria at which both springs carry no force. Oscillations are studied in both regimes, but more systematically in the tension case, where techniques of bifurcation theory, numerical approximation and numerical simulation are used to explore the rich variety of periodic solutions.}, number={6}, journal={IMA JOURNAL OF APPLIED MATHEMATICS}, author={Shearer, Michael and Gremaud, Pierre and Kleiner, Kristoph}, year={2009}, month={Dec}, pages={807–826} }
 @article{devault_gremaud_novak_olufsen_vernieres_zhao_2008, title={BLOOD FLOW IN THE CIRCLE OF WILLIS: MODELING AND CALIBRATION}, volume={7}, ISSN={["1540-3467"]}, DOI={10.1137/07070231X}, abstractNote={A numerical model based on one-dimensional balance laws and ad hoc zero-dimensional boundary conditions is tested against experimental data. The study concentrates on the circle of Willis, a vital subnetwork of the cerebral vasculature. The main goal is to obtain efficient and reliable numerical tools with predictive capabilities. The flow is assumed to obey the Navier-Stokes equations, while the mechanical reactions of the arterial walls follow a viscoelastic model. Like many previous studies, a dimension reduction is performed through averaging. Unlike most previous work, the resulting model is both calibrated and validated against in vivo data, more precisely transcranial Doppler data of cerebral blood velocity. The network considered has three inflow vessels and six outflow vessels. Inflow conditions come from the data, while outflow conditions are modeled. Parameters in the outflow conditions are calibrated using a subset of the data through ensemble Kalman filtering techniques. The rest of the data is used for validation. The results demonstrate the viability of the proposed approach.}, number={2}, journal={MULTISCALE MODELING & SIMULATION}, author={Devault, Kristen and Gremaud, Pierre A. and Novak, Vera and Olufsen, Mette S. and Vernieres, Guillaume and Zhao, Peng}, year={2008}, pages={888–909} }
 @article{gremaud_kuster_li_2007, title={A study of numerical methods for the level set approach}, volume={57}, ISSN={["0168-9274"]}, DOI={10.1016/j.apnum.2006.07.022}, abstractNote={The computation of moving curves by the level set method typically requires reinitializations of the underlying level set function. Two types of reinitialization methods are studied: a high order “PDE” approach and a second order Fast Marching method. Issues related to the efficiency and implementation of both types of methods are discussed, with emphasis on the tube/narrow band implementation and accuracy considerations. The methods are also tested and compared. Fast Marching reinitialization schemes are faster but limited to second order, PDE based reinitialization schemes can easily be made more accurate but are slower, even with a tube/narrow band implementation.}, number={5-7}, journal={APPLIED NUMERICAL MATHEMATICS}, author={Gremaud, Pierre A. and Kuster, Christopher M. and Li, Zhilin}, year={2007}, pages={837–846} }
 @article{kuster_gremaud_touzani_2007, title={Fast numerical methods for Bernoulli free boundary problems}, volume={29}, ISSN={["1064-8275"]}, DOI={10.1137/06065444X}, abstractNote={The numerical solution of the free boundary Bernoulli problem is addressed. An iterative method based on a level-set formulation and boundary element method is proposed. Issues related to the implementation, the accuracy, and the generality of the method are discussed. The efficiency of the approach is illustrated by numerical results.}, number={2}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Kuster, Christopher M. and Gremaud, Pierre A. and Touzani, Rachid}, year={2007}, pages={622–634} }
 @article{devault_gremaud_jenssen_2007, title={Numerical investigation of cavitation in multidimensional compressible flows}, volume={67}, ISSN={["1095-712X"]}, DOI={10.1137/060652713}, abstractNote={The compressible Navier–Stokes equations for an ideal polytropic gas are considered in ${R}^n$, $n = 2,3$. The question of possible vacuum formation, an open theoretical problem, is investigated numerically using highly accurate computational methods. The flow is assumed to be symmetric about the origin with a purely radial velocity field. The numerical results indicate that there are weak solutions to the Navier–Stokes system in two and three space dimensions, which display formation of vacuum when the initial data are discontinuous and sufficiently large. The initial density is constant, while the initial velocity field is symmetric, points radially away from the origin, and belongs to $H^s_{loc}$ for all $s < n/2$. In addition, in the one-dimensional case, the numerical solutions are in agreement with known theoretical results.}, number={6}, journal={SIAM JOURNAL ON APPLIED MATHEMATICS}, author={Devault, Kristen J. and Gremaud, Pierre A. and Jenssen, Helge Kristian}, year={2007}, pages={1675–1692} }
 @article{wambaugh_behringer_matthews_gremaud_2007, title={Response to perturbations for granular flow in a hopper}, volume={76}, ISSN={["1550-2376"]}, DOI={10.1103/physreve.76.051303}, abstractNote={We experimentally investigate the response to perturbations of circular symmetry for dense granular flow inside a three-dimensional right-conical hopper. These experiments consist of particle tracking velocimetry for the flow at the outer boundary of the hopper. We are able to test commonly used constitutive relations and observe granular flow phenomena that we can model numerically. Unperturbed conical hopper flow has been described as a radial velocity field with no azimuthal component. Guided by numerical models based upon continuum descriptions, we find experimental evidence for secondary, azimuthal circulation in response to perturbation of the symmetry with respect to gravity by tilting. For small perturbations we can discriminate between constitutive relations, based upon the agreement between the numerical predictions they produce and our experimental results. We find that the secondary circulation can be suppressed as wall friction is varied, also in agreement with numerical predictions. For large tilt angles we observe the abrupt onset of circulation for parameters where circulation was previously suppressed. Finally, we observe that for large tilt angles the fluctuations in velocity grow, independent of the onset of circulation.}, number={5}, journal={PHYSICAL REVIEW E}, author={Wambaugh, John F. and Behringer, Robert P. and Matthews, John V. and Gremaud, Pierre A.}, year={2007}, month={Nov} }
 @article{gremaud_kuster_2006, title={Computational study of fast methods for the eikonal equation}, volume={27}, ISSN={["1064-8275"]}, DOI={10.1137/040605655}, abstractNote={A computational study of the fast marching and the fast sweeping methods for the eikonal equation is given. It is stressed that both algorithms should be considered as "direct" (as opposed to iterative) methods. On realistic grids, fast sweeping is faster than fast marching for problems with simple geometry. For strongly nonuniform problems and/or complex geometry, the situation may be reversed. Finally, fully second order generalizations of methods of this type for problems with obstacles are proposed and implemented.}, number={6}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Gremaud, PA and Kuster, CM}, year={2006}, pages={1803–1816} }
 @article{gremaud_matthews_schaeffer_2006, title={On the computation of steady hopper flows III: Model comparisons}, volume={219}, ISSN={["0021-9991"]}, DOI={10.1016/j.jcp.2006.03.032}, abstractNote={Gravity flows of granular materials through hoppers are considered. For hoppers shaped as general nonaxisymmetric cones, i.e., “pyramids”, the flow inherits some simplified features from the geometry: similarity solutions can be constructed. Using two different plasticity laws, namely Matsuoka–Nakai and von Mises, those solutions are obtained by solving first-order nonlinear partial differential algebraic systems for stresses, velocities, and a plasticity function. A pseudospectral discretization is applied to both models and the resulting flow fields are examined. Some similarities are found, but important differences appear, especially with regard to velocities near the wall and normal wall stresses. Preliminary comparisons with recent experiments [J.F. Wambaugh, R.P. Behringer, Asymmetry-induced circulation in granular hopper flows, in: R. Garcia-Rojo, H.J. Herrmann, S. McNamara (Eds.), Powders and Grains, 2005, pp. 915–918] based on the present results indicate that for slow granular flows the lesser known Matsuoka–Nakai plasticity law yields better results than more common models based on a von Mises criterion.}, number={1}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Gremaud, Pierre A. and Matthews, John V. and Schaeffer, David G.}, year={2006}, month={Nov}, pages={443–454} }
 @article{gremaud_2005, title={Numerical issues in plasticity models for granular materials}, volume={139}, ISSN={["0377-0273"]}, DOI={10.1016/j.jvolgeores.2004.06.018}, abstractNote={Friction plays a fundamental role in the mechanics of granular materials. Two problems are considered: (i) heap formation and (ii) granular flow. Both problems admit closely related mathematical models. In each case, analytical and numerical difficulties are discussed. Efficient and reliable numerical methods are proposed and implemented. The results are illustrated by several computational experiments.}, number={1-2}, journal={JOURNAL OF VOLCANOLOGY AND GEOTHERMAL RESEARCH}, author={Gremaud, PA}, year={2005}, month={Jan}, pages={23–31} }
 @article{gremaud_matthews_m o'malley_2004, title={On the computation of steady Hopper flows II: von Mises materials in various geometries}, volume={200}, ISSN={["1090-2716"]}, DOI={10.1016/j.jcp.2004.04.021}, abstractNote={Similarity solutions are constructed for the flow of granular materials through hoppers. Unlike previous work, the present approach applies to nonaxisymmetric containers. The model involves ten unknowns (stresses, velocity, and plasticity function) determined by nine nonlinear first order partial differential equations together with a quadratic algebraic constraint (yield condition). A pseudospectral discretization is applied; the resulting problem is solved with a trust region method. The important role of the hopper geometry on the flow is illustrated by several numerical experiments of industrial relevance.}, number={2}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Gremaud, PA and Matthews, JV and M O'Malley}, year={2004}, month={Nov}, pages={639–653} }
 @article{ahmed_buckingham_gremaud_hauck_kuster_prodanovic_royal_silantyev_2004, title={Volume determination for bulk materials in bunkers}, volume={61}, ISSN={["0029-5981"]}, DOI={10.1002/nme.1144}, abstractNote={Abstract}, number={13}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING}, author={Ahmed, SA and Buckingham, R and Gremaud, PA and Hauck, CD and Kuster, CM and Prodanovic, M and Royal, TA and Silantyev, V}, year={2004}, month={Dec}, pages={2239–2249} }
 @article{coffey_gremaud_2003, title={Numerical simulation of aerated powder consolidation}, volume={38}, ISSN={["0020-7462"]}, DOI={10.1016/S0020-7462(02)00063-X}, abstractNote={When a fine powder is dumped into a silo, the gas trapped by the particles will slowly escape by diffusing through the material. The corresponding uneven gas pressure distribution creates a body force that is taken into account through Darcy's law. By using spatial averaging, the formulation, even though essentially one-dimensional in space, includes effects due the geometry of the container. An efficient and robust numerical scheme based on a differential algebraic equation formulation is proposed and implemented. Various computational results are presented and discussed to establish the validity of the approach.}, number={8}, journal={INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS}, author={Coffey, KA and Gremaud, PA}, year={2003}, month={Oct}, pages={1185–1194} }
 @article{gremaud_matthews_schaeffer_2003, title={Secondary circulation in granular flow through nonaxisymmetric hoppers}, volume={64}, ISSN={["1095-712X"]}, DOI={10.1137/S0036139903415124}, abstractNote={Jenike's radial solution, widely used in the design of materials-handling equipment, is a similarity solution of steady-state continuum equations for the flow under gravity of granular material through an infinite, right-circular cone. In this paper we study how the geometry of the hopper influences this solution. Using perturbation theory, we compute a first-order correction to the (steady-state) velocity resulting from a small change in hopper geometry, either distortion of the cross section or tilting away from vertical. Unlike for the Jenike solution, all three components of the correction velocity are nonzero; i.e., there is secondary circulation in the perturbed flow.}, number={2}, journal={SIAM JOURNAL ON APPLIED MATHEMATICS}, author={Gremaud, PA and Matthews, JV and Schaeffer, DG}, year={2003}, pages={583–600} }
 @article{gremaud_kelley_royal_coffey_2001, title={On a powder consolidation problem}, volume={62}, DOI={10.1137/s0036139900368479}, abstractNote={The problem of the consolidation of an aerated fine powder under gravity is considered. The industrial relevance of the problem is discussed and a mathematical model is introduced. The mathematical structure is that of a coupled system for three unknowns, pressure, stress, and height of the powder in the (axisymmetric) bunker containing it. The system itself consists of a parabolic PDE, an ODE, and an integral equation determining a free boundary corresponding to the height of the powder. Existence and uniqueness of a solution is established. A numerical method based on a formulation of the semidiscretized problem as an index 1 DAE is proposed and implemented. The feasibility of the approach is illustrated by computational results.}, number={1}, journal={SIAM Journal on Applied Mathematics}, author={Gremaud, P. A. and Kelley, C. T. and Royal, T. A. and Coffey, K. A.}, year={2001}, pages={1–20} }
 @article{gremaud_matthews_2001, title={On the computation of steady hopper flows I. Stress determination for Coulomb materials}, volume={166}, ISSN={["1090-2716"]}, DOI={10.1006/jcph.2000.6641}, abstractNote={Abstract The problem of determining the steady state flow of granular materials in silos under the action of gravity is considered. In the case of Mohr–Coulomb materials, the stress equations correspond to a system of hyperbolic conservation laws with source terms and boundary conditions. A higher order discontinuous Galerkin method is proposed and implemented for the numerical resolution of those equations. The efficiency of the approach is illustrated by the computation of the stress fields induced in silos with sharp changes of the wall angle.}, number={1}, journal={JOURNAL OF COMPUTATIONAL PHYSICS}, author={Gremaud, PA and Matthews, JV}, year={2001}, month={Jan}, pages={63–83} }
 @article{gremaud_schaeffer_shearer_2000, title={Numerical determination of flow corrective inserts for granular materials in conical hoppers}, volume={35}, ISSN={["0020-7462"]}, DOI={10.1016/S0020-7462(99)00064-5}, abstractNote={The flow of granular materials in hoppers is studied. In industrial applications, inserts of various sizes and shapes are often used to improve the flow properties and get rid of undesirable effects such as material sticking to the walls, funnel flow, arching, etc. We study the case of inverted conical inserts in conical hoppers. In spite of the complexity of the phenomenon as observed in practice, existing methods assume in general a radial structure of the stress and velocity fields. A new numerical approach to the problem of designing “optimal” inserts is proposed and tested. It allows for non-purely radial solutions. General comments about the overall approach and its relationship with experiments are offered.}, number={5}, journal={INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS}, author={Gremaud, PA and Schaeffer, DG and Shearer, M}, year={2000}, month={Sep}, pages={869–882} }
 @article{gremaud_ide_1999, title={Computation of nonclassical solutions to Hamilton-Jacobi problems}, volume={21}, ISSN={["1064-8275"]}, DOI={10.1137/S1064827597327668}, abstractNote={This paper is devoted to the construction of numerical methods for the approximation of nonclassical solutions to multidimensional Hamilton--Jacobi equations for both scalar and vectorial problems. Recent theoretical results have yielded existence of solutions in many cases for which the usual viscosity approach was ill-suited or not applicable. The selection criterion used here is based on a viscoelasticity/capillarity approach, common in solid mechanics. Numerical methods adapted to this framework are built. Consistency of the model equation with the given selection criterion is essential. It is achieved here through the use of high-order finite difference schemes. By considering applications to potential well problems, the convergence of the methods are investigated.}, number={2}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Gremaud, PA and Ide, NR}, year={1999}, month={Oct}, pages={502–521} }
 @article{cockburn_gremaud_yang_1998, title={A priori error estimates for numerical methods for scalar conservation laws - Part III: Multidimensional flux-splitting monotone schemes on non-Cartesian grids}, volume={35}, ISSN={["0036-1429"]}, DOI={10.1137/S0036142997316165}, abstractNote={This paper is the third of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed. In this paper, we consider multidimensional flux-splitting monotone schemes defined on non-Cartesian grids. We identify those schemes which are consistent and prove that the $L^\infty(0,T;L^1(\Bbb{R}^d))$-norm of the error goes to zero as $(\Delta x)^{1/2}$ when the discretization parameter $\Delta x$ goes to zero. Moreover, we show that nonconsistent schemes can converge at optimal rates of $(\Delta x)^{1/2}$ because (i) the conservation form of the schemes and (ii) the so-called consistency of the numerical fluxes allow the regularity properties of the approximate solution to compensate for their lack of consistency.}, number={5}, journal={SIAM JOURNAL ON NUMERICAL ANALYSIS}, author={Cockburn, B and Gremaud, PA and Yang, JXR}, year={1998}, month={Oct}, pages={1775–1803} }
 @article{gremaud_1997, title={Numerical simulation of pattern formation in grain flows}, volume={84}, ISSN={["0096-3003"]}, DOI={10.1016/S0096-3003(96)00084-7}, abstractNote={We present numerical computations for the flow of granular materials in hoppers. The algorithm is based on a cellular automaton approach and takes into account various microscopic effects, including “orientational effects”. The formation and the dynamics of density waves, recently discovered in physical experiments, is observed and analyzed computationally.}, number={2-3}, journal={APPLIED MATHEMATICS AND COMPUTATION}, author={Gremaud, PA}, year={1997}, month={Jul}, pages={145–162} }
 @article{cockburn_gremaud_1997, title={a priori error estimates for numerical methods for scalar conservation laws .2. Flux-splitting monotone schemes on irregular Cartesian grids}, volume={66}, ISSN={["1088-6842"]}, DOI={10.1090/S0025-5718-97-00838-7}, abstractNote={This paper is the second of a series in which a general theory of a priori error estimates for scalar conservation laws is constructed. In this paper, we focus on how the lack of consistency introduced by the nonuniformity of the grids influences the convergence of flux-splitting monotone schemes to the entropy solution. We obtain the optimal rate of convergence of (Δx) 1/2 in L∞(L 1 ) for consistent schemes in arbitrary grids without the use of any regularity property of the approximate solution. We then extend this result to less consistent schemes, called p-consistent schemes, and prove that they converge to the entropy solution with the rate of (Δx) min{1/2,P} in L∞(L 1 ); again, no regularity property of the approximate solution is used. Finally, we propose a new explanation of the fact that even inconsistent schemes converge with the rate of (Δx) 1/2 in L∞(L 1 ). We show that this well-known supraconvergence phenomenon takes place because the consistency of the numerical flux and the fact that the scheme is written in conservation form allows the regularity properties of its approximate solution (total variation boundedness) to compensate for its lack of consistency; the nonlinear nature of the problem does not play any role in this mechanism. All the above results hold in the multidimensional case, provided the grids are Cartesian products of one-dimensional nonuniform grids.}, number={218}, journal={MATHEMATICS OF COMPUTATION}, author={Cockburn, B and Gremaud, PA}, year={1997}, month={Apr}, pages={547–572} }
 @article{cockburn_gremaud_1996, title={Error estimates for finite element methods for scalar conservation laws}, volume={33}, ISSN={["0036-1429"]}, DOI={10.1137/0733028}, abstractNote={In this paper, new a posteriors error estimates for the shock-capturing streamline diffusion (SCSD) method and the shock-capturing discontinuous galerkin (SCDG) method for scalar conservation laws are obtained. These estimates are then used to prove that the SCSD method and the SCDG method converge to the entropy solution with a rate of at least $h^{{1 / 8}} $ and $h^{{1 / 8}} $, respectively, in the $L^\infty (L^1 )$-norm. The triangulations are made of general acute simplices and the approximate solution is taken to be piecewise a polynomial of degree k. The result is independent of the dimension of the space.}, number={2}, journal={SIAM JOURNAL ON NUMERICAL ANALYSIS}, author={Cockburn, B and Gremaud, PA}, year={1996}, month={Apr}, pages={522–554} }
 @article{gremaud_1994, title={NUMERICAL-ANALYSIS OF A NONCONVEX VARIATIONAL PROBLEM RELATED TO SOLID-SOLID PHASE-TRANSITIONS}, volume={31}, ISSN={["0036-1429"]}, DOI={10.1137/0731006}, abstractNote={The description of equilibria of shape memory alloys or other ordered materials gives rise to nonconvex variational problems. In this paper, a two-dimensional model of such materials is studied. Due to the fact that the corresponding functional has two symmetry-related (martensitic) energy wells, the numerical approximation of the deformation gradient does not converge, but tends to oscillate between the two wells, as the size of the mesh is refined. These oscillations may be interpreted in terms of microstructures. Using a nonconforming $\mathcal{P}_1 $ finite element, an estimate is given for the rate of convergence of the probability for the approximated deformation to have its gradient “near” one of the two (martensitic) wells.}, number={1}, journal={SIAM JOURNAL ON NUMERICAL ANALYSIS}, author={GREMAUD, PA}, year={1994}, month={Feb}, pages={111–127} }
 @article{amiez_gremaud_1992, title={ERROR-ESTIMATES FOR EULER FORWARD SCHEME RELATED TO 2-PHASE STEFAN-PROBLEMS}, volume={26}, ISSN={["1290-3841"]}, DOI={10.1051/m2an/1992260203651}, abstractNote={— In this paper, we establish error estimâtes related to the approximation of bidimensional Stefan problems with for eed convection in the fluid phase. An enthalpy formulation of these problems is used. The considered discretization is based on Euler forward finite dijferences in time and C° piecewise linear finite éléments in space combined with a masslumping procedure. The proposed scheme is therefore easy to implement. Under some restrictions relative to the finite element mesh and to stability and non-degeneracy conditions, we prove a L-rate of convergence for température and a H~-rate for enthalpy both of order Résumé. — Cette étude a pour objectif d établir des estimations d'erreurs concernant V approximation de problèmes bidimensionnels de type Stefan en présence de convection forcée dans la phase liquide. Une formulation en enthalpie de ces problèmes est considérée. Le schéma proposé repose sur V utilisation d'une méthode d'Euler progressif dans la variable temporelle et d'éléments finis continus par morceaux en espace combinée avec une procédure de masslumping. Un tel algorithme est donc d'implantation aisée. Sous certaines restrictions relatives à la triangulation et sous des conditions de stabilité et de non-dégénérescence, nous démontrons des estimations d'erreur en norme L pour la température et en norme H~ pour la variable enthalpie toutes deux d'ordre k.}, number={2}, journal={ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE}, author={AMIEZ, G and GREMAUD, PA}, year={1992}, pages={365–383} }