@article{alexanderian_hart_stevens_2023, title={A new perspective on parameter study of optimization problems}, volume={140}, ISSN={["1873-5452"]}, DOI={10.1016/j.aml.2022.108548}, abstractNote={We provide a new perspective on the study of parameterized optimization problems. Our approach combines methods for post-optimal sensitivity analysis and ordinary differential equations to quantify the uncertainty in the minimizer due to uncertain parameters in the optimization problem. We illustrate the proposed approach with a simple analytic example and an inverse problem governed by an advection diffusion equation.}, journal={APPLIED MATHEMATICS LETTERS}, author={Alexanderian, Alen and Hart, Joseph and Stevens, Mason}, year={2023}, month={Jun} }
 @article{stevens_sunseri_alexanderian_2022, title={Hyper-differential sensitivity analysis for inverse problems governed by ODEs with application to COVID-19 modeling}, volume={351}, ISSN={["1879-3134"]}, DOI={10.1016/j.mbs.2022.108887}, abstractNote={We consider inverse problems governed by systems of ordinary differential equations (ODEs) that contain uncertain parameters in addition to the parameters being estimated. In such problems, which are common in applications, it is important to understand the sensitivity of the solution of the inverse problem to the uncertain model parameters. It is also of interest to understand the sensitivity of the inverse problem solution to different types of measurements or parameters describing the experimental setup. Hyper-differential sensitivity analysis (HDSA) is a sensitivity analysis approach that provides tools for such tasks. We extend existing HDSA methods by developing methods for quantifying the uncertainty in the estimated parameters. Specifically, we propose a linear approximation to the solution of the inverse problem that allows efficiently approximating the statistical properties of the estimated parameters. We also explore the use of this linear model for approximate global sensitivity analysis. As a driving application, we consider an inverse problem governed by a COVID-19 model. We present comprehensive computational studies that examine the sensitivity of this inverse problem to several uncertain model parameters and different types of measurement data. Our results also demonstrate the effectiveness of the linear approximation model for uncertainty quantification in inverse problems and for parameter screening.}, journal={MATHEMATICAL BIOSCIENCES}, author={Stevens, Mason and Sunseri, Isaac and Alexanderian, Alen}, year={2022}, month={Sep} }
 @article{dudley_saibaba_alexanderian_2022, title={MONTE CARLO ESTIMATORS FOR THE SCHATTEN p-NORM OF SYMMETRIC POSITIVE SEMIDEFINITE MATRICES}, volume={55}, ISSN={["1068-9613"]}, url={http://dx.doi.org/10.1553/etna_vol55s213}, DOI={10.1553/etna_vol55s213}, abstractNote={We present numerical methods for computing the Schatten $p$-norm of positive semi-definite matrices. Our motivation stems from uncertainty quantification and optimal experimental design for inverse problems, where the Schatten $p$-norm defines a design criterion known as the P-optimal criterion. Computing the Schatten $p$-norm of high-dimensional matrices is computationally expensive. We propose a matrix-free method to estimate the Schatten $p$-norm using a Monte Carlo estimator and derive convergence results and error estimates for the estimator. To efficiently compute the Schatten $p$-norm for non-integer and large values of $p$, we use an estimator using a Chebyshev polynomial approximation and extend our convergence and error analysis to this setting as well. We demonstrate the performance of our proposed estimators on several test matrices and through an application to optimal experimental design of a model inverse problem.}, journal={ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS}, publisher={Osterreichische Akademie der Wissenschaften, Verlag}, author={DUDLEY, E. T. H. A. N. and SAIBABA, A. R. V. I. N. D. K. and ALEXANDERIAN, A. L. E. N.}, year={2022}, pages={213–241} }
 @article{white_alexanderian_yousefian_karbalaeisadegh_bekele-maxwell_kasali_banks_talmant_grimal_muller_2022, title={Using ultrasonic attenuation in cortical bone to infer distributions on pore size}, volume={109}, ISSN={["1872-8480"]}, DOI={10.1016/j.apm.2022.05.024}, abstractNote={In this work we infer the underlying distribution on pore radius in human cortical bone samples using ultrasonic attenuation data. We first discuss how to formulate polydisperse attenuation models using a probabilistic approach and the Waterman Truell model for scattering attenuation. We then compare the Independent Scattering Approximation and the higher-order Waterman Truell models’ forward predictions for total attenuation in polydisperse samples. Following this, we formulate an inverse problem under the Prohorov Metric Framework coupled with variational regularization to stabilize this inverse problem. We then use experimental attenuation data taken from human cadaver samples and solve inverse problems resulting in nonparametric estimates of the probability density function on pore radius. We compare these estimates to the “true” microstructure of the bone samples determined via microCT imaging. We find that our methodology allows us to reliably estimate the underlying microstructure of the bone from attenuation data.}, journal={APPLIED MATHEMATICAL MODELLING}, author={White, R. D. and Alexanderian, A. and Yousefian, O. and Karbalaeisadegh, Y. and Bekele-Maxwell, K. and Kasali, A. and Banks, H. T. and Talmant, M. and Grimal, Q. and Muller, M.}, year={2022}, month={Sep}, pages={819–832} }
 @article{randall_randolph_alexanderian_olufsen_2021, title={Global sensitivity analysis informed model reduction and selection applied to a Valsalva maneuver model}, volume={526}, ISSN={["1095-8541"]}, DOI={10.1016/j.jtbi.2021.110759}, abstractNote={In this study, we develop a methodology for model reduction and selection informed by global sensitivity analysis (GSA) methods. We apply these techniques to a control model that takes systolic blood pressure and thoracic tissue pressure data as inputs and predicts heart rate in response to the Valsalva maneuver (VM). The study compares four GSA methods based on Sobol’ indices (SIs) quantifying the parameter influence on the difference between the model output and the heart rate data. The GSA methods include standard scalar SIs determining the average parameter influence over the time interval studied and three time-varying methods analyzing how parameter influence changes over time. The time-varying methods include a new technique, termed limited-memory SIs, predicting parameter influence using a moving window approach. Using the limited-memory SIs, we perform model reduction and selection to analyze the necessity of modeling both the aortic and carotid baroreceptor regions in response to the VM. We compare the original model to systematically reduced models including (i) the aortic and carotid regions, (ii) the aortic region only, and (iii) the carotid region only. Model selection is done quantitatively using the Akaike and Bayesian Information Criteria and qualitatively by comparing the neurological predictions. Results show that it is necessary to incorporate both the aortic and carotid regions to model the VM.}, journal={JOURNAL OF THEORETICAL BIOLOGY}, author={Randall, E. Benjamin and Randolph, Nicholas Z. and Alexanderian, Alen and Olufsen, Mette S.}, year={2021}, month={Oct} }
 @article{white_yousefian_banks_alexanderian_mueller_2021, title={Inferring pore radius and density from ultrasonic attenuation using physics-based modeling}, volume={149}, ISSN={["1520-8524"]}, DOI={10.1121/10.0003213}, abstractNote={This work proposes the use of two physics-based models for wave attenuation to infer the microstructure of cortical bone-like structures. One model for ultrasound attenuation in porous media is based on the independent scattering approximation (ISA) and the other model is based on the Waterman Truell (WT) approximation. The microstructural parameters of interest are pore radius and pore density. Attenuation data are simulated for three-dimensional structures mimicking cortical bone using the finite-difference time domain package SimSonic. These simulated structures have fixed sized pores (monodisperse), allowing fine-tuned control of the microstructural parameters. Structures with pore radii ranging from 50 to 100  μm and densities ranging from 20 to 50 pores/mm3 are generated in which only the attenuation due to scattering is considered. From here, an inverse problem is formulated and solved, calibrating the models to the simulated data and producing estimates of pore radius and density. The estimated microstructural parameters closely match the values used to simulate the data, validating the use of both the ISA and WT approximations to model ultrasonic wave attenuation in heterogeneous structures mimicking cortical bone. Furthermore, this illustrates the effectiveness of both models in inferring pore radius and density solely from ultrasonic attenuation data.}, number={1}, journal={JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA}, author={White, R. D. and Yousefian, O. and Banks, H. T. and Alexanderian, A. and Mueller, M.}, year={2021}, month={Jan}, pages={340–347} }
 @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_petra_stadler_sunseri_2021, title={Optimal Design of Large-scale Bayesian Linear Inverse Problems Under Reducible Model Uncertainty: Good to Know What You Don't Know}, volume={9}, ISSN={["2166-2525"]}, DOI={10.1137/20M1347292}, abstractNote={We consider optimal design of infinite-dimensional Bayesian linear inverse problems governed by partial differential equations that contain secondary reducible model uncertainties, in addition to the uncertainty in the inversion parameters. By reducible uncertainties we refer to parametric uncertainties that can be reduced through parameter inference. We seek experimental designs that minimize the posterior uncertainty in the primary parameters, while accounting for the uncertainty in secondary parameters. We accomplish this by deriving a marginalized A-optimality criterion and developing an efficient computational approach for its optimization. We illustrate our approach for estimating an uncertain time-dependent source in a contaminant transport model with an uncertain initial state as secondary uncertainty. Our results indicate that accounting for additional model uncertainty in the experimental design process is crucial.}, number={1}, journal={SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION}, author={Alexanderian, Alen and Petra, Noemi and Stadler, Georg and Sunseri, Isaac}, year={2021}, pages={163–184} }
 @misc{alexanderian_2021, title={Optimal experimental design for infinite-dimensional Bayesian inverse problems governed by PDEs: a review}, volume={37}, ISSN={["1361-6420"]}, DOI={10.1088/1361-6420/abe10c}, abstractNote={We present a review of methods for optimal experimental design (OED) for Bayesian inverse problems governed by partial differential equations with infinite-dimensional parameters. The focus is on problems where one seeks to optimize the placement of measurement points, at which data are collected, such that the uncertainty in the estimated parameters is minimized. We present the mathematical foundations of OED in this context and survey the computational methods for the class of OED problems under study. We also outline some directions for future research in this area.}, number={4}, journal={INVERSE PROBLEMS}, author={Alexanderian, Alen}, year={2021}, month={Apr} }
 @article{cleaves_alexanderian_saad_2021, title={Structure exploiting methods for fast uncertainty quantification in multiphase flow through heterogeneous media}, ISSN={["1573-1499"]}, DOI={10.1007/s10596-021-10085-8}, abstractNote={We present a computational framework for dimension reduction and surrogate modeling to accelerate uncertainty quantification in computationally intensive models with high-dimensional inputs and function-valued outputs. Our driving application is multiphase flow in saturated-unsaturated porous media in the context of radioactive waste storage. For fast input dimension reduction, we utilize an approximate global sensitivity measure, for function-valued outputs, motivated by ideas from the active subspace methods. The proposed approach does not require expensive gradient computations. We generate an efficient surrogate model by combining a truncated Karhunen-Loéve (KL) expansion of the output with polynomial chaos expansions, for the output KL modes, constructed in the reduced parameter space. We demonstrate the effectiveness of the proposed surrogate modeling approach with a comprehensive set of numerical experiments, where we consider a number of function-valued (temporally or spatially distributed) QoIs.}, journal={COMPUTATIONAL GEOSCIENCES}, author={Cleaves, Helen and Alexanderian, Alen and Saad, Bilal}, year={2021}, month={Sep} }
 @article{guy_alexanderian_yu_2020, title={A Distributed Active Subspace Method for Scalable Surrogate Modeling of Function Valued Outputs}, volume={85}, ISSN={["1573-7691"]}, DOI={10.1007/s10915-020-01346-2}, abstractNote={We present a distributed active subspace method for training surrogate models of complex physical processes with high-dimensional inputs and function valued outputs. Specifically, we represent the model output with a truncated Karhunen–Loève (KL) expansion, screen the structure of the input space with respect to each KL mode via the active subspace method, and finally form an overall surrogate model of the output by combining surrogates of individual output KL modes. To ensure scalable computation of the gradients of the output KL modes, needed in active subspace discovery, we rely on adjoint-based gradient computation. The proposed method combines benefits of active subspace methods for input dimension reduction and KL expansions used for spectral representation of the output field. We provide a mathematical framework for the proposed method and conduct an error analysis of the mixed KL active subspace approach. Specifically, we provide an error estimate that quantifies errors due to active subspace projection and truncated KL expansion of the output. We demonstrate the numerical performance of the surrogate modeling approach with an application example from biotransport.}, number={2}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, author={Guy, Hayley and Alexanderian, Alen and Yu, Meilin}, year={2020}, month={Oct} }
 @article{sunseri_hart_bloemen waanders_alexanderian_2020, title={Hyper-differential sensitivity analysis for inverse problems constrained by partial differential equations}, volume={36}, ISSN={["1361-6420"]}, DOI={10.1088/1361-6420/abaf63}, abstractNote={High fidelity models used in many science and engineering applications couple multiple physical states and parameters. Inverse problems arise when a model parameter cannot be determined directly, but rather is estimated using (typically sparse and noisy) measurements of the states. The data is usually not sufficient to simultaneously inform all of the parameters. Consequently, the governing model typically contains parameters which are uncertain but must be specified for a complete model characterization necessary to invert for the parameters of interest. We refer to the combination of the additional model parameters (those which are not inverted for) and the measured data states as the ‘complementary parameters’. We seek to quantify the relative importance of these complementary parameters to the solution of the inverse problem. To address this, we present a framework based on hyper-differential sensitivity analysis (HDSA). HDSA computes the derivative of the solution of an inverse problem with respect to complementary parameters. We present a mathematical framework for HDSA in large-scale PDE-constrained inverse problems and show how HDSA can be interpreted to give insight about the inverse problem. We demonstrate the effectiveness of the method on an inverse problem by estimating a permeability field, using pressure and concentration measurements, in a porous medium flow application with uncertainty in the boundary conditions, source injection, and diffusion coefficient.}, number={12}, journal={INVERSE PROBLEMS}, author={Sunseri, Isaac and Hart, Joseph and Bloemen Waanders, Bart and Alexanderian, Alen}, year={2020}, month={Dec} }
 @article{koval_alexanderian_stadler_2020, title={Optimal experimental design under irreducible uncertainty for linear inverse problems governed by PDEs}, volume={36}, ISSN={["1361-6420"]}, DOI={10.1088/1361-6420/ab89c5}, abstractNote={We present a method for computing A-optimal sensor placements for infinite-dimensional Bayesian linear inverse problems governed by PDEs with irreducible model uncertainties. Here, irreducible uncertainties refers to uncertainties in the model that exist in addition to the parameters in the inverse problem, and that cannot be reduced through observations. Specifically, given a statistical distribution for the model uncertainties, we compute the optimal design that minimizes the expected value of the posterior covariance trace. The expected value is discretized using Monte Carlo leading to an objective function consisting of a sum of trace operators and a binary-inducing penalty. Minimization of this objective requires a large number of PDE solves in each step. To make this problem computationally tractable, we construct a composite low-rank basis using a randomized range finder algorithm to eliminate forward and adjoint PDE solves. We also present a novel formulation of the A-optimal design objective that requires the trace of an operator in the observation rather than the parameter space. The binary structure is enforced using a weighted regularized ℓ0-sparsification approach. We present numerical results for inference of the initial condition in a subsurface flow problem with inherent uncertainty in the flow fields and in the initial times.}, number={7}, journal={INVERSE PROBLEMS}, author={Koval, Karina and Alexanderian, Alen and Stadler, Georg}, year={2020}, month={Jul} }
 @article{herman_alexanderian_saibaba_2020, title={RANDOMIZATION AND REWEIGHTED l(1)-MINIMIZATION FOR A-OPTIMAL DESIGN OF LINEAR INVERSE PROBLEMS}, volume={42}, ISSN={["1095-7197"]}, DOI={10.1137/19M1267362}, abstractNote={We consider optimal design of PDE-based Bayesian linear inverse problems with infinite-dimensional parameters. We focus on the A-optimal design criterion, defined as the average posterior variance ...}, number={3}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Herman, Elizabeth and Alexanderian, Alen and Saibaba, Arvind K.}, year={2020}, pages={A1714–A1740} }
 @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{vohra_alexanderian_guy_mahadevan_2019, title={Active subspace-based dimension reduction for chemical kinetics applications with epistemic uncertainty}, volume={204}, ISSN={["1556-2921"]}, DOI={10.1016/j.combustflame.2019.03.006}, abstractNote={We focus on an efficient approach for quantification of uncertainty in complex chemical reaction networks with a large number of uncertain parameters and input conditions. Parameter dimension reduction is accomplished by computing an active subspace that predominantly captures the variability in the quantity of interest (QoI). In the present work, we compute the active subspace for a H2/O2 mechanism that involves 19 chemical reactions, using an efficient iterative strategy. The active subspace is first computed for a 19-parameter problem wherein only the uncertainty in the pre-exponents of the individual reaction rates is considered. This is followed by the analysis of a 36-dimensional case wherein the activation energies and initial conditions are also considered uncertain. In both cases, a 1-dimensional active subspace is observed to capture the uncertainty in the QoI, which indicates enormous potential for efficient statistical analysis of complex chemical systems. In addition, we explore links between active subspaces and global sensitivity analysis, and exploit these links for identification of key contributors to the variability in the model response.}, journal={COMBUSTION AND FLAME}, author={Vohra, Manav and Alexanderian, Alen and Guy, Hayley and Mahadevan, Sankaran}, year={2019}, month={Jun}, pages={152–161} }
 @article{cleaves_alexanderian_guy_smith_yu_2019, title={DERIVATIVE-BASED GLOBAL SENSITIVITY ANALYSIS FOR MODELS WITH HIGH-DIMENSIONAL INPUTS AND FUNCTIONAL OUTPUTS}, volume={41}, ISSN={["1095-7197"]}, DOI={10.1137/19M1243518}, abstractNote={We present a framework for derivative-based global sensitivity analysis (GSA) for models with high-dimensional input parameters and functional outputs. We combine ideas from derivative-based GSA, random field representation via Karhunen--Lo\`{e}ve expansions, and adjoint-based gradient computation to provide a scalable computational framework for computing the proposed derivative-based GSA measures. We illustrate the strategy for a nonlinear ODE model of cholera epidemics and for elliptic PDEs with application examples from geosciences and biotransport.}, number={6}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Cleaves, Helen L. and Alexanderian, Alen and Guy, Hayley and Smith, Ralph C. and Yu, Meilin}, year={2019}, pages={A3524–A3551} }
 @article{saibaba_bardsley_brown_alexanderian_2019, title={Efficient Marginalization-Based MCMC Methods for Hierarchical Bayesian Inverse Problems}, volume={7}, ISSN={["2166-2525"]}, DOI={10.1137/18M1220625}, abstractNote={Hierarchical models in Bayesian inverse problems are characterized by an assumed prior probability distribution for the unknown state and measurement error precision, and hyper-priors for the prior parameters. Combining these probability models using Bayes' law often yields a posterior distribution that cannot be sampled from directly, even for a linear model with Gaussian measurement error and Gaussian prior. Gibbs sampling can be used to sample from the posterior, but problems arise when the dimension of the state is large. This is because the Gaussian sample required for each iteration can be prohibitively expensive to compute, and because the statistical efficiency of the Markov chain degrades as the dimension of the state increases. The latter problem can be mitigated using marginalization-based techniques, but these can be computationally prohibitive as well. In this paper, we combine the low-rank techniques of Brown, Saibaba, and Vallelian (2018) with the marginalization approach of Rue and Held (2005). We consider two variants of this approach: delayed acceptance and pseudo-marginalization. We provide a detailed analysis of the acceptance rates and computational costs associated with our proposed algorithms, and compare their performances on two numerical test cases---image deblurring and inverse heat equation.}, number={3}, journal={SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION}, author={Saibaba, Arvind K. and Bardsley, Johnathan and Brown, D. Andrew and Alexanderian, Alen}, year={2019}, pages={1105–1131} }
 @article{alexanderian_reese_smith_yu_2019, title={Model Input and Output Dimension Reduction Using Karhunen-Loeve Expansions With Application to Biotransport}, volume={5}, ISSN={["2332-9025"]}, DOI={10.1115/1.4044317}, abstractNote={Abstract We consider biotransport in tumors with uncertain heterogeneous material properties. Specifically, we focus on the elliptic partial differential equation (PDE) modeling the pressure field inside the tumor. The permeability field is modeled as a log-Gaussian random field with a prespecified covariance function. We numerically explore dimension reduction of the input parameter and model output. Specifically, truncated Karhunen–Loève (KL) expansions are used to decompose the log-permeability field, as well as the resulting random pressure field. We find that although very high-dimensional representations are needed to accurately represent the permeability field, especially in presence of small correlation lengths, the pressure field is not sensitive to high-order KL terms of the input parameter. Moreover, we find that the pressure field itself can be represented accurately using a KL expansion with a small number of terms. These observations are used to guide a reduced-order modeling approach to accelerate computational studies of biotransport in tumors.}, number={4}, journal={ASCE-ASME JOURNAL OF RISK AND UNCERTAINTY IN ENGINEERING SYSTEMS PART B-MECHANICAL ENGINEERING}, author={Alexanderian, Alen and Reese, William and Smith, Ralph C. and Yu, Meilin}, year={2019}, month={Dec} }
 @article{vohra_alexanderian_safta_mahadevan_2019, title={Sensitivity-Driven Adaptive Construction of Reduced-space Surrogates}, volume={79}, ISSN={["1573-7691"]}, DOI={10.1007/s10915-018-0894-4}, abstractNote={Surrogate modeling has become a critical component of scientific computing in situations involving expensive model evaluations. However, training a surrogate model can be remarkably challenging and even computationally prohibitive in the case of intensive simulations and large-dimensional systems. We develop a systematic approach for surrogate model construction in reduced input parameter spaces. A sparse set of model evaluations in the original input space is used to approximate derivative based global sensitivity measures (DGSMs) for individual uncertain inputs of the model. An iterative screening procedure is developed that exploits DGSM estimates in order to identify the unimportant inputs. The screening procedure forms an integral part of an overall framework for adaptive construction of a surrogate in the reduced space. The framework is tested for computational efficiency through an initial implementation in simple test cases such as the classic Borehole function, and a semilinear elliptic PDE with a random source function. The framework is then deployed for a realistic application from chemical kinetics, where we study the ignition delay in an $$\hbox {H}_2{/}\hbox {O}_2$$ reaction mechanism with 19 and 33 uncertain rate-controlling parameters. It is observed that significant computational gains can be attained by constructing accurate low-dimensional surrogates using the proposed framework.}, number={2}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, author={Vohra, Manav and Alexanderian, Alen and Safta, Cosmin and Mahadevan, Sankaran}, year={2019}, month={May}, pages={1335–1359} }
 @article{alexanderian_saibaba_2018, title={EFFICIENT D-OPTIMAL DESIGN OF EXPERIMENTS FOR INFINITE-DIMENSIONAL BAYESIAN LINEAR INVERSE PROBLEMS}, volume={40}, ISSN={["1095-7197"]}, DOI={10.1137/17M115712X}, abstractNote={We develop a computational framework for D-optimal experimental design for PDE-based Bayesian linear inverse problems with infinite-dimensional parameters. We follow a formulation of the experimental design problem that remains valid in the infinite-dimensional limit. The optimal design is obtained by solving an optimization problem that involves repeated evaluation of the log-determinant of high-dimensional operators along with their derivatives. Forming and manipulating these operators is computationally prohibitive for large-scale problems. Our methods exploit the low-rank structure in the inverse problem in three different ways, yielding efficient algorithms. Our main approach is to use randomized estimators for computing the D-optimal criterion, its derivative, as well as the Kullback--Leibler divergence from posterior to prior. Two other alternatives are proposed based on a low-rank approximation of the prior-preconditioned data misfit Hessian, and a fixed low-rank approximation of the prior-preconditioned forward operator. Detailed error analysis is provided for each of the methods, and their effectiveness is demonstrated on a model sensor placement problem for initial state reconstruction in a time-dependent advection-diffusion equation in two space dimensions.}, number={5}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Alexanderian, Alen and Saibaba, Arvind K.}, year={2018}, pages={A2956–A2985} }
 @article{attia_alexanderian_saibaba_2018, title={Goal-oriented optimal design of experiments for large-scale Bayesian linear inverse problems}, volume={34}, ISSN={["1361-6420"]}, DOI={10.1088/1361-6420/aad210}, abstractNote={We develop a framework for goal-oriented optimal design of experiments (GOODE) for large-scale Bayesian linear inverse problems governed by PDEs. This framework differs from classical Bayesian optimal design of experiments (ODE) in the following sense: we seek experimental designs that minimize the posterior uncertainty in the experiment end-goal, e.g. a quantity of interest (QoI), rather than the estimated parameter itself. This is suitable for scenarios in which the solution of an inverse problem is an intermediate step and the estimated parameter is then used to compute a QoI. In such problems, a GOODE approach has two benefits: the designs can avoid wastage of experimental resources by a targeted collection of data, and the resulting design criteria are computationally easier to evaluate due to the often low-dimensionality of the QoIs. We present two modified design criteria, A-GOODE and D-GOODE, which are natural analogues of classical Bayesian A- and D-optimal criteria. We analyze the connections to other ODE criteria, and provide interpretations for the GOODE criteria by using tools from information theory. Then, we develop an efficient gradient-based optimization framework for solving the GOODE optimization problems. Additionally, we present comprehensive numerical experiments testing the various aspects of the presented approach. The driving application is the optimal placement of sensors to identify the source of contaminants in a diffusion and transport problem. We enforce sparsity of the sensor placements using an -norm penalty approach, and propose a practical strategy for specifying the associated penalty parameter.}, number={9}, journal={INVERSE PROBLEMS}, author={Attia, Ahmed and Alexanderian, Alen and Saibaba, Arvind K.}, year={2018}, month={Sep} }
 @article{saad_alexanderian_prudhomme_knio_2018, title={Probabilistic modeling and global sensitivity analysis for CO2 storage in geological formations: a spectral approach}, volume={53}, ISSN={["1872-8480"]}, DOI={10.1016/j.apm.2017.09.016}, abstractNote={This work focuses on the simulation of CO2 storage in deep underground formations under uncertainty and seeks to understand the impact of uncertainties in reservoir properties on CO2 leakage. To simulate the process, a non-isothermal two-phase two-component flow system with equilibrium phase exchange is used. Since model evaluations are computationally intensive, instead of traditional Monte Carlo methods, we rely on polynomial chaos (PC) expansions for representation of the stochastic model response. A non-intrusive approach is used to determine the PC coefficients. We establish the accuracy of the PC representations within a reasonable error threshold through systematic convergence studies. In addition to characterizing the distributions of model observables, we compute probabilities of excess CO2 leakage. Moreover, we consider the injection rate as a design parameter and compute an optimum injection rate that ensures that the risk of excess pressure buildup at the leaky well remains below acceptable levels. We also provide a comprehensive analysis of sensitivities of CO2 leakage, where we compute the contributions of the random parameters, and their interactions, to the variance by computing first, second, and total order Sobol’ indices.}, journal={APPLIED MATHEMATICAL MODELLING}, author={Saad, Bilal M. and Alexanderian, Alen and Prudhomme, Serge and Knio, Omar M.}, year={2018}, month={Jan}, pages={584–601} }
 @article{crestel_alexanderian_stadler_ghattas_2017, title={A-optimal encoding weights for nonlinear inverse problems, with application to the Helmholtz inverse problem}, volume={33}, ISSN={["1361-6420"]}, DOI={10.1088/1361-6420/aa6d8e}, abstractNote={The computational cost of solving an inverse problem governed by PDEs, using multiple experiments, increases linearly with the number of experiments. A recently proposed method to decrease this cost uses only a small number of random linear combinations of all experiments for solving the inverse problem. This approach applies to inverse problems where the PDE solution depends linearly on the right-hand side function that models the experiment. As this method is stochastic in essence, the quality of the obtained reconstructions can vary, in particular when only a small number of combinations are used. We develop a Bayesian formulation for the definition and computation of encoding weights that lead to a parameter reconstruction with the least uncertainty. We call these weights A-optimal encoding weights. Our framework applies to inverse problems where the governing PDE is nonlinear with respect to the inversion parameter field. We formulate the problem in infinite dimensions and follow the optimize-then-discretize approach, devoting special attention to the discretization and the choice of numerical methods in order to achieve a computational cost that is independent of the parameter discretization. We elaborate our method for a Helmholtz inverse problem, and derive the adjoint-based expressions for the gradient of the objective function of the optimization problem for finding the A-optimal encoding weights. The proposed method is potentially attractive for real-time monitoring applications, where one can invest the effort to compute optimal weights offline, to later solve an inverse problem repeatedly, over time, at a fraction of the initial cost.}, number={7}, journal={INVERSE PROBLEMS}, author={Crestel, Benjamin and Alexanderian, Alen and Stadler, Georg and Ghattas, Omar}, year={2017}, month={Jul} }
 @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{alexanderian_zhu_salloum_ma_yu_2017, title={Investigation of Biotransport in a Tumor With Uncertain Material Properties Using a Nonintrusive Spectral Uncertainty Quantification Method}, volume={139}, ISSN={["1528-8951"]}, DOI={10.1115/1.4037102}, abstractNote={In this study, statistical models are developed for modeling uncertain heterogeneous permeability and porosity in tumors, and the resulting uncertainties in pressure and velocity fields during an intratumoral injection are quantified using a nonintrusive spectral uncertainty quantification (UQ) method. Specifically, the uncertain permeability is modeled as a log-Gaussian random field, represented using a truncated Karhunen-Lòeve (KL) expansion, and the uncertain porosity is modeled as a log-normal random variable. The efficacy of the developed statistical models is validated by simulating the concentration fields with permeability and porosity of different uncertainty levels. The irregularity in the concentration field bears reasonable visual agreement with that in MicroCT images from experiments. The pressure and velocity fields are represented using polynomial chaos (PC) expansions to enable efficient computation of their statistical properties. The coefficients in the PC expansion are computed using a nonintrusive spectral projection method with the Smolyak sparse quadrature. The developed UQ approach is then used to quantify the uncertainties in the random pressure and velocity fields. A global sensitivity analysis is also performed to assess the contribution of individual KL modes of the log-permeability field to the total variance of the pressure field. It is demonstrated that the developed UQ approach can effectively quantify the flow uncertainties induced by uncertain material properties of the tumor.}, number={9}, journal={JOURNAL OF BIOMECHANICAL ENGINEERING-TRANSACTIONS OF THE ASME}, author={Alexanderian, Alen and Zhu, Liang and Salloum, Maher and Ma, Ronghui and Yu, Meilin}, year={2017}, month={Sep} }
 @article{alexanderian_petra_stadler_ghattas_2017, title={Mean-Variance Risk-Averse Optimal Control of Systems Governed by PDEs with Random Parameter Fields Using Quadratic Approximations}, volume={5}, ISSN={["2166-2525"]}, DOI={10.1137/16m106306x}, abstractNote={We present a method for optimal control of systems governed by partial differential equations (PDEs) with uncertain parameter fields. We consider an objective function that involves the mean and variance of the control objective, leading to a risk-averse optimal control problem. To make the problem tractable, we invoke a quadratic Taylor series approximation of the control objective with respect to the uncertain parameter. This enables deriving explicit expressions for the mean and variance of the control objective in terms of its gradients and Hessians with respect to the uncertain parameter. The risk-averse optimal control problem is then formulated as a PDE-constrained optimization problem with constraints given by the forward and adjoint PDEs defining these gradients and Hessians. The expressions for the mean and variance of the control objective under the quadratic approximation involve the trace of the (preconditioned) Hessian and are thus prohibitive to evaluate. To address this, we employ trace estimators that only require a modest number of Hessian-vector products. We illustrate our approach with two problems: the control of a semilinear elliptic PDE with an uncertain boundary source term, and the control of a linear elliptic PDE with an uncertain coefficient field. For the latter problem, we derive adjoint-based expressions for efficient computation of the gradient of the risk-averse objective with respect to the controls. Our method ensures that the cost of computing the risk-averse objective and its gradient with respect to the control, measured in the number of PDE solves, is independent of the (discretized) parameter and control dimensions, and depends only on the number of random vectors employed in the trace estimation. Finally, we present a comprehensive numerical study of an optimal control problem for fluid flow in a porous medium with uncertain permeability field.}, number={1}, journal={SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION}, author={Alexanderian, Alen and Petra, Noemi and Stadler, Georg and Ghattas, Omar}, year={2017}, pages={1166–1192} }
 @article{saibaba_alexanderian_ipsen_2017, title={Randomized matrix-free trace and log-determinant estimators}, volume={137}, ISSN={["0945-3245"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85017115710&partnerID=MN8TOARS}, DOI={10.1007/s00211-017-0880-z}, abstractNote={We present randomized algorithms for estimating the trace and determinant of Hermitian positive semi-definite matrices. The algorithms are based on subspace iteration, and access the matrix only through matrix vector products. We analyse the error due to randomization, for starting guesses whose elements are Gaussian or Rademacher random variables. The analysis is cleanly separated into a structural (deterministic) part followed by a probabilistic part. Our absolute bounds for the expectation and concentration of the estimators are non-asymptotic and informative even for matrices of low dimension. For the trace estimators, we also present asymptotic bounds on the number of samples (columns of the starting guess) required to achieve a user-specified relative error. Numerical experiments illustrate the performance of the estimators and the tightness of the bounds on low-dimensional matrices, and on a challenging application in uncertainty quantification arising from Bayesian optimal experimental design.}, number={2}, journal={NUMERISCHE MATHEMATIK}, author={Saibaba, Arvind K. and Alexanderian, Alen and Ipsen, Ilse C. F.}, year={2017}, month={Oct}, pages={353–395} }
 @article{michoski_alexanderian_paillet_kubatko_dawson_2017, title={Stability of Nonlinear Convection-Diffusion-Reaction Systems in Discontinuous Galerkin Methods}, volume={70}, ISSN={["1573-7691"]}, DOI={10.1007/s10915-016-0256-z}, number={2}, journal={JOURNAL OF SCIENTIFIC COMPUTING}, author={Michoski, C. and Alexanderian, A. and Paillet, C. and Kubatko, E. J. and Dawson, C.}, year={2017}, month={Feb}, pages={516–550} }
 @article{alexanderian_petra_stadler_ghattas_2016, title={A Fast and Scalable Method for A-Optimal Design of Experiments for Infinite-dimensional Bayesian Nonlinear Inverse Problems}, volume={38}, ISSN={1064-8275 1095-7197}, url={http://dx.doi.org/10.1137/140992564}, DOI={10.1137/140992564}, abstractNote={We address the problem of optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by PDEs. The goal is to find a placement of sensors, at which experimental data are collected, so as to minimize the uncertainty in the inferred parameter field. We formulate the OED objective function by generalizing the classical A-optimal experimental design criterion using the expected value of the trace of the posterior covariance. We seek a method that solves the OED problem at a cost (measured in the number of forward PDE solves) that is independent of both the parameter and sensor dimensions. To facilitate this, we construct a Gaussian approximation to the posterior at the maximum a posteriori probability (MAP) point, and use the resulting covariance operator to define the OED objective function. We use randomized trace estimation to compute the trace of this (implicitly defined) covariance operator. The resulting OED problem includes as constraints the PDEs characterizing the MAP point, and the PDEs describing the action of the covariance operator to vectors. The sparsity of the sensor configurations is controlled using sparsifying penalty functions. We elaborate our OED method for the problem of determining the sensor placement to best infer the coefficient of an elliptic PDE. Adjoint methods are used to compute the gradient of the PDE-constrained OED objective function. We provide numerical results for inference of the permeability field in a porous medium flow problem, and demonstrate that the number of PDE solves required for the evaluation of the OED objective function and its gradient is essentially independent of both the parameter and sensor dimensions. The number of quasi-Newton iterations for computing an OED also exhibits the same dimension invariance properties.}, number={1}, journal={SIAM Journal on Scientific Computing}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Alexanderian, Alen and Petra, Noemi and Stadler, Georg and Ghattas, Omar}, year={2016}, month={Jan}, pages={A243–A272} }
 @article{alexanderian_gloor_ghattas_2016, title={On Bayesian A- and D-Optimal Experimental Designs in Infinite Dimensions}, volume={11}, ISSN={["1936-0975"]}, DOI={10.1214/15-ba969}, abstractNote={We consider Bayesian linear inverse problems in infinite-dimensional separable Hilbert spaces, with a Gaussian prior measure and additive Gaussian noise model, and provide an extension of the concept of Bayesian D-optimality to the infinite-dimensional case. To this end, we derive the infinite-dimensional version of the expression for the Kullback-Leibler divergence from the posterior measure to the prior measure, which is subsequently used to derive the expression for the expected information gain. We also study the notion of Bayesian A-optimality in the infinite-dimensional setting, and extend the well known (in the finite-dimensional case) equivalence of the Bayes risk of the MAP estimator with the trace of the posterior covariance, for the Gaussian linear case, to the infinite-dimensional Hilbert space case.}, number={3}, journal={BAYESIAN ANALYSIS}, author={Alexanderian, Alen and Gloor, Philip J. and Ghattas, Omar}, year={2016}, month={Sep}, pages={671–695} }
 @article{alexanderian_2015, title={EXPOSITORY PAPER: A PRIMER ON HOMOGENIZATION OF ELLIPTIC PDES WITH STATIONARY AND ERGODIC RANDOM COEFFICIENT FUNCTIONS}, volume={45}, ISSN={["1945-3795"]}, DOI={10.1216/rmj-2015-45-3-703}, abstractNote={We study the problem of characterizing the effective (homogenized) properties of materials whose diffusive properties are modeled with random fields. Focusing on elliptic PDEs with stationary and ergodic random coefficient functions, we provide a gentle introduction to the mathematical theory of homogenization of random media. We also present numerical examples to elucidate the theoretical concepts and results.}, number={3}, journal={ROCKY MOUNTAIN JOURNAL OF MATHEMATICS}, author={Alexanderian, Alen}, year={2015}, pages={703–735} }
 @article{alexanderian_petra_stadler_ghattas_2014, title={A-Optimal Design of Experiments for Infinite-Dimensional Bayesian Linear Inverse Problems with Regularized l_0-Sparsification}, volume={36}, ISSN={1064-8275 1095-7197}, url={http://dx.doi.org/10.1137/130933381}, DOI={10.1137/130933381}, abstractNote={We present an efficient method for computing A-optimal experimental designs for infinite-dimensional Bayesian linear inverse problems governed by partial differential equations (PDEs). Specifically, we address the problem of optimizing the location of sensors (at which observational data are collected) to minimize the uncertainty in the parameters estimated by solving the inverse problem, where the uncertainty is expressed by the trace of the posterior covariance. Computing optimal experimental designs (OEDs) is particularly challenging for inverse problems governed by computationally expensive PDE models with infinite-dimensional (or, after discretization, high-dimensional) parameters. To alleviate the computational cost, we exploit the problem structure and build a low-rank approximation of the parameter-to-observable map, preconditioned with the square root of the prior covariance operator. This relieves our method from expensive PDE solves when evaluating the optimal experimental design objective function and its derivatives. Moreover, we employ a randomized trace estimator for efficient evaluation of the OED objective function. We control the sparsity of the sensor configuration by employing a sequence of penalty functions that successively approximate the $\ell_0$-"norm"; this results in binary designs that characterize optimal sensor locations. We present numerical results for inference of the initial condition from spatio-temporal observations in a time-dependent advection-diffusion problem in two and three space dimensions. We find that an optimal design can be computed at a cost, measured in number of forward PDE solves, that is independent of the parameter and sensor dimensions. We demonstrate numerically that $\ell_0$-sparsified experimental designs obtained via a continuation method outperform $\ell_1$-sparsified designs.}, number={5}, journal={SIAM Journal on Scientific Computing}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Alexanderian, Alen and Petra, Noemi and Stadler, Georg and Ghattas, Omar}, year={2014}, month={Jan}, pages={A2122–A2148} }
 @article{sraj_iskandarani_srinivasan_thacker_winokur_alexanderian_lee_chen_knio_2013, title={Bayesian Inference of Drag Parameters Using AXBT Data from Typhoon Fanapi}, volume={141}, ISSN={0027-0644 1520-0493}, url={http://dx.doi.org/10.1175/mwr-d-12-00228.1}, DOI={10.1175/mwr-d-12-00228.1}, abstractNote={AbstractThe authors introduce a three-parameter characterization of the wind speed dependence of the drag coefficient and apply a Bayesian formalism to infer values for these parameters from airborne expendable bathythermograph (AXBT) temperature data obtained during Typhoon Fanapi. One parameter is a multiplicative factor that amplifies or attenuates the drag coefficient for all wind speeds, the second is the maximum wind speed at which drag coefficient saturation occurs, and the third is the drag coefficient's rate of change with increasing wind speed after saturation. Bayesian inference provides optimal estimates of the parameters as well as a non-Gaussian probability distribution characterizing the uncertainty of these estimates. The efficiency of this approach stems from the use of adaptive polynomial expansions to build an inexpensive surrogate for the high-resolution numerical model that couples simulated winds to the oceanic temperature data, dramatically reducing the computational burden of the M...}, number={7}, journal={Monthly Weather Review}, publisher={American Meteorological Society}, author={Sraj, Ihab and Iskandarani, Mohamed and Srinivasan, Ashwanth and Thacker, W. Carlisle and Winokur, Justin and Alexanderian, Alen and Lee, Chia-Ying and Chen, Shuyi S. and Knio, Omar M.}, year={2013}, month={Jul}, pages={2347–2367} }
 @article{alexanderian_2013, title={On spectral methods for variance based sensitivity analysis}, volume={10}, ISSN={1549-5787}, url={http://dx.doi.org/10.1214/13-ps219}, DOI={10.1214/13-ps219}, abstractNote={Consider a mathematical model with a finite number of random parameters. 
Variance based sensitivity analysis provides a framework to characterize 
the contribution of the individual parameters to the total variance of 
the model response. We consider the spectral methods for variance based 
sensitivity analysis which 
utilize representations of square integrable random variables in a 
generalized polynomial chaos basis. 
Taking a measure theoretic point of view, we 
provide a rigorous and at the same time intuitive perspective on the 
spectral methods for variance based sensitivity analysis. 
Moreover, we discuss approximation errors incurred by fixing 
inessential random 
parameters, when approximating functions with generalized polynomial 
chaos expansions.}, number={0}, journal={Probability Surveys}, publisher={Institute of Mathematical Statistics}, author={Alexanderian, Alen}, year={2013}, pages={51–68} }
 @article{alexanderian_rizzi_rathinam_le maître_knio_2013, title={Preconditioned Bayesian Regression for Stochastic Chemical Kinetics}, volume={58}, ISSN={0885-7474 1573-7691}, url={http://dx.doi.org/10.1007/s10915-013-9745-5}, DOI={10.1007/s10915-013-9745-5}, number={3}, journal={Journal of Scientific Computing}, publisher={Springer Science and Business Media LLC}, author={Alexanderian, Alen and Rizzi, Francesco and Rathinam, Muruhan and Le Maître, Olivier P. and Knio, Omar M.}, year={2013}, month={Jul}, pages={592–626} }
 @article{alexanderian_winokur_sraj_srinivasan_iskandarani_thacker_knio_2012, title={Global sensitivity analysis in an ocean general circulation model: a sparse spectral projection approach}, volume={16}, ISSN={1420-0597 1573-1499}, url={http://dx.doi.org/10.1007/s10596-012-9286-2}, DOI={10.1007/s10596-012-9286-2}, number={3}, journal={Computational Geosciences}, publisher={Springer Science and Business Media LLC}, author={Alexanderian, Alen and Winokur, Justin and Sraj, Ihab and Srinivasan, Ashwanth and Iskandarani, Mohamed and Thacker, William C. and Knio, Omar M.}, year={2012}, month={Mar}, pages={757–778} }
 @article{alexanderian_rathinam_rostamian_2012, title={Homogenization, Symmetry, and Periodization in Diffusive Random Media}, volume={32}, ISSN={0252-9602}, url={http://dx.doi.org/10.1016/s0252-9602(12)60008-3}, DOI={10.1016/s0252-9602(12)60008-3}, number={1}, journal={Acta Mathematica Scientia}, publisher={Elsevier BV}, author={Alexanderian, Alen and Rathinam, Muruhan and Rostamian, Rouben}, year={2012}, month={Jan}, pages={129–154} }
 @article{salloum_alexanderian_le maître_najm_knio_2012, title={Simplified CSP analysis of a stiff stochastic ODE system}, volume={217-220}, ISSN={0045-7825}, url={http://dx.doi.org/10.1016/j.cma.2012.01.001}, DOI={10.1016/j.cma.2012.01.001}, abstractNote={We develop a simplified computational singular perturbation (CSP) analysis of a stochastic dynamical system. We focus on the case of parametric uncertainty, and rely on polynomial chaos (PC) representations to quantify its impact. We restrict our attention to a system that exhibits distinct timescales, and that tends to a deterministic steady state irrespective of the random inputs. A detailed analysis of eigenvalues and eigenvectors of the stochastic system Jacobian is conducted, which provides a relationship between the PC representation of the stochastic Jacobian and the Jacobian of the Galerkin form of the stochastic system. The analysis is then used to guide the application of a simplified CSP formalism that is based on relating the slow and fast manifolds of the uncertain system to those of a nominal deterministic system. Two approaches are specifically developed with the resulting simplified CSP framework. The first uses the stochastic eigenvectors of the uncertain system as CSP vectors, whereas the second uses the eigenvectors of the nominal system as CSP vectors. Numerical experiments are conducted to demonstrate the results of the stochastic eigenvalue and eigenvector analysis, and illustrate the effectiveness of the simplified CSP algorithms in addressing the stiffness of the system dynamics.}, journal={Computer Methods in Applied Mechanics and Engineering}, publisher={Elsevier BV}, author={Salloum, Maher and Alexanderian, Alen and Le Maître, Olivier P. and Najm, Habib N. and Knio, Omar M.}, year={2012}, month={Apr}, pages={121–138} }
 @article{alexanderian_gobbert_fister_gaff_lenhart_schaefer_2011, title={An age-structured model for the spread of epidemic cholera: Analysis and simulation}, volume={12}, ISSN={1468-1218}, url={http://dx.doi.org/10.1016/j.nonrwa.2011.06.009}, DOI={10.1016/j.nonrwa.2011.06.009}, abstractNote={Abstract Occasional outbreaks of cholera epidemics across the world demonstrate that the disease continues to pose a public health threat. Traditional models for the spread of infectious diseases are based on systems of ordinary differential equations. Since disease dynamics such as vaccine efficacy and the risk for contracting cholera depend on the age of the humans, an age-structured model offers additional insights and the possibility of studying the effects of treatment options. The model investigated is given as a system of hyperbolic (first-order) partial differential equations in combination with ordinary differential equations. First, using a representation from the method of characteristics and a fixed point argument, we prove the existence and uniqueness of a solution to our nonlinear system. Then we present a finite difference approximation to the model and study the effect of high and low rates of shedding of cholera vibrios on the dynamics of the spread of the disease. The simulations demonstrate the explosive nature of cholera outbreaks that is observed in reality. The contrast of results for high and low rates of shedding of vibrios suggest a possible underlying cause for this effect.}, number={6}, journal={Nonlinear Analysis: Real World Applications}, publisher={Elsevier BV}, author={Alexanderian, Alen and Gobbert, Matthias K. and Fister, K. Renee and Gaff, Holly and Lenhart, Suzanne and Schaefer, Elsa}, year={2011}, month={Dec}, pages={3483–3498} }
 @article{alexanderian_le maître_najm_iskandarani_knio_2011, title={Multiscale Stochastic Preconditioners in Non-intrusive Spectral Projection}, volume={50}, ISSN={0885-7474 1573-7691}, url={http://dx.doi.org/10.1007/s10915-011-9486-2}, DOI={10.1007/s10915-011-9486-2}, number={2}, journal={Journal of Scientific Computing}, publisher={Springer Science and Business Media LLC}, author={Alexanderian, Alen and Le Maître, Oliver P. and Najm, Habib N. and Iskandarani, Mohamed and Knio, Omar M.}, year={2011}, month={Apr}, pages={306–340} }
 @article{alexanderian_rathinam_rostamian_2010, title={Irreducibility of a Symmetry Group Implies Isotropy}, volume={102}, ISSN={0374-3535 1573-2681}, url={http://dx.doi.org/10.1007/s10659-010-9268-3}, DOI={10.1007/s10659-010-9268-3}, number={2}, journal={Journal of Elasticity}, publisher={Springer Science and Business Media LLC}, author={Alexanderian, Alen and Rathinam, Muruhan and Rostamian, Rouben}, year={2010}, month={Aug}, pages={151–174} }
 @article{muller_fleck_dimitoglou_caplins_amadigwe_ortiz_wamsler_alexanderian_hughitt_ireland_2009, title={JHelioviewer: Visualizing Large Sets of Solar Images Using JPEG 2000}, volume={11}, ISSN={1521-9615}, url={http://dx.doi.org/10.1109/mcse.2009.142}, DOI={10.1109/mcse.2009.142}, abstractNote={All disciplines that work with image data-from astrophysics to medical research and historic preservation-increasingly require efficient ways to browse and inspect large sets of high-resolution images. Based on the JPEG 2000 image-compression standard, the JHelioviewer solar image visualization tool lets users browse petabyte-scale image archives as well as locate and manipulate specific data sets.}, number={5}, journal={Computing in Science & Engineering}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={Muller, D. and Fleck, B. and Dimitoglou, G. and Caplins, B.W. and Amadigwe, D.E. and Ortiz, J.P.G. and Wamsler, B. and Alexanderian, A. and Hughitt, V.K. and Ireland, J.}, year={2009}, month={Sep}, pages={38–47} }