Society is impacted in myriad ways by the behavior of complex physical systems. The impact of weather prediction, climate change, pollution, and engineering materials are just a few examples.

To dive deeper into this impact, explore Don’s research and papers.

**Taking an Integrated Approach**

A characteristic of multiphysics systems is that it is possible to observe certain aspects of behavior at some scales experimentally and to construct well-validated process models of some component behavior at some scales. But we cannot directly observe or model all important behaviors and interactions.

Thus, the study of multiphysics systems faces significant challenges that require, at a minimum, integrating all available data and models.

## The long-term goal of my research program is to develop powerful computational and statistical tools to draw inferences about complex multiphysics systems, predict their behavior, and quantify uncertainties in the results.

**Research Tools**

The tools used in Don’s research include:

– Statistics

– Measure & Probability Theory

– Differential Geometry

– Differential Equations

– Functional Analysis

– Stochastic Modeling

– Numerical Analysis

**Research Foci**

**Stochastic Inverse Problems**

The stochastic inverse problem for a random vector is the problem of inferring information about unobservable characteristics of a complex system from observed data from experiments. This provides a way to understand and predict system behavior so is core to science and engineering. Estep and collaborators have developed a powerful general measure theoretic framework for solving the stochastic inverse problem.

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A measure-theoretic computational method for inverse sensitivity problems I: Method and analysis, J. Breidt, T. Butler and D. Estep, SIAM Journal on Numerical Analysis 49 (2011), 1836-1859

A computational measure theoretic approach to inverse sensitivity problems II: A posteriori error analysis, T. Butler, D. Estep and J. Sandelin, SIAM Journal on Numerical Analysis, 50 (2012), 22-45

A numerical method for solving a stochastic inverse problem for parameters, T. Butler and D. Estep, Annals of Nuclear Energy, 2012, 86-94, 10.1016/j.anucene.2012.05.016

A measure-theoretic computational method for inverse sensitivity problems III: Multiple quantities of interest, T. Butler, D. Estep, S. Tavener, C. Dawson, J. Westerink, SIAM ASA Journal on Uncertainty Quantification, 2 (2014), 174-202

Definition and solution of a stochastic inverse problem for the Manning’s n parameter field in hydrodynamic models, T. Butler, L. Graham, D. Estep, C. Dawson, and J.J. Westerink, Advances in Water Resources, 78 (2015), 60–79

Parameter estimation and prediction for groundwater contamination based on measure theory, T. Butler, C. Dawson, D. Estep, S. Mattis, V. Vesselinov, Water Resources Research, 52 (2015), 7808-7629

A stochastic inverse problem for multiscale models, N. Panda, T. Butler, D. Estep, L. Graham, and C. Dawson, Journal for Multiscale Computational Engineering, 15 (2017), 265-283

Learning quantities of interest from dynamical systems for data-consistent inversion, S. Mattis, K.R. Steffen, T. Butler, C.N. Dawson, and D. Estep, Computer Methods in Applied Mechanics and Engineering, 2021, accepted.

**Forward Stochastic Sensitivity Analysis and Uncertainty Quantification for Differential Equations**

**Forward Stochastic Sensitivity Analysis and Uncertainty Quantification for Differential Equations**

A powerful approach to predict the behavior of a complex system subject to uncertainty and random effects is to allow for random parameters in a mathematical model. Estep and collaborators have developed a systematic approach to estimating the effect of uncertainty in parameters in mathematical models.

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Fast and reliable methods for determining the evolution of uncertain parameters in differential equations, D. Estep and D. Neckels, Journal on Computational Physics 213 (2006), 530-556

Fast methods for determining the evolution of uncertain parameters in reaction-diffusion equations, D. Estep and D. Neckels, Computer Methods in Applied Mechanics and Engineering 196 (2007), 3967 – 3979

Nonparametric density estimation for randomly perturbed elliptic problems II: Applications and adaptive modeling, D. Estep, A. Malqvist, S. Tavener, International Journal for Numerical Methods in Engineering 80 (2009), 846-867

Nonparametric density estimation for randomly perturbed elliptic problems I: Computational methods, a posteriori analysis, and adaptive error control, D. Estep, A. Malqvist, and S. Tavener, SIAM Journal on Scientific Computing 31 (2009), 2935-2959

Nonparametric density estimation for randomly perturbed elliptic problems III: Convergence, complexity, and generalizations, D. Estep, M. Holst, A. Malqvist, Journal of Applied Mathematics and Computing 38 (2012), 367-387

Uncertainty quantification for approximate p-quantiles for physical models with stochastic inputs, D. Elfverson, D. Estep, F. Hellman, A. Malqvist, SIAM ASA Journal on Uncertainty Quantification, 2 (2014), 826–850

A posteriori error estimation for a cut cell finite volume method with uncertain interface location, J. B. Collins, D. Estep, and S. Tavener, International Journal of Uncertainty Quantification, 5 (2015), 415-432

Exploration of efficient reduced-order modeling and a posteriori error estimation, J. H. Chaudhry, D. Estep, M. Gunzburger, International Journal on Numerical Methods for Engineering, 111 (2016), 102-122

Efficient distribution estimation and uncertainty quantification for elliptic problems on domains with stochastic boundaries, J. H. Chaudhry, N. Burch, D. Estep, SIAM/ASA Journal on Uncertainty Quantification, 6 (2018), 1127-1150

Error estimation and uncertainty quantification for first time to a threshold value, J. H. Chaudhry, D. Estep, Z. Stevens, S. Tavener, BIT Numerical Mathematics, 2020, https://doi.org/10.1007/s10543-020-00825-0, 33 pages.

A posteriori error analysis for Schwarz overlapping domain decomposition methods, J. H. Chaudhry, D. Estep, S. Tavener, BIT Numerical Mathematics, 2021, 0.1007/s10543-021-00864-1.

**A Posteriori Error Analysis and Efficient Control of Discretization Error**

**A Posteriori Error Analysis and Efficient Control of Discretization Error**Science and engineering depends heavily on the approximate solution of complex mathematical models using computer computations. But the complexity and scale of typical models means that numerical approximation error is always significant to the point of affecting scientific conclusions. Estep and collaborators have developed a powerful, comprehensive approach to computing accurate estimates of numerical errors in approximate computer solutions of complex models.

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Global error control for the continuous Galerkin finite element method for ordinary differential equations, D. Estep and D. French, RAIRO Modélisation Mathématique et Analyse Numérique 28 (1994), 815-852

A posteriori error bounds and global error control for approximations of ordinary differential equations, D. Estep, SIAM Journal on Numerical Analysis 32 (1995), 1-48

Introduction to adaptive methods for differential equations, K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Acta Numerica (1995), 105-158.

h-adaptive boundary element schemes, C. Carstensen, D. Estep and E. Stephan, Computational Mechanics 15 (1995), 372-383

Introduction to computational methods for differential equations, K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, in Theory of Numerics for Ordinary and Partial Differential Equations, M. Ainsworth, J. Levesley, W. A. Light, and M. Marletta, eds, Oxford University Press, New York, 1995

Accurate parallel integration of large sparse systems of differential equations, D. Estep

and R. Williams, Mathematical Models and Methods in Applied Sciences 6 (1996), 535-568

Error estimation for numerical differential equations, D. Estep, S. Verduyn Lunel and R. Williams, Invited Article, IEEE Antenna and Propagation Magazine 38 (1996), 71-76

Adaptive methods for reaction diffusion problems, D. Estep, M. Larson and R. Williams, Proceedings of the 12’th Annual Review of Progress in Applied Computational Electromagnetics, 1996, 611-618

Computational error estimation and adaptive mesh refinement for a finite element solution of launch vehicle trajectory problems, D. Estep, D. Hodges and M. Warner, SIAM Journal on Scientific Computing 21 (2000), 1609-1631 (electronic)

The solution of a launch vehicle trajectory problem by an adaptive finite element method, D. Estep, D. H. Hodges, M. Warner, Computer Methods in Applied Mechanics and Engineering, 190 (2001), 4677-4690

Accounting for stability: a posteriori estimates based on residuals and variational analysis, D. Estep, M. Holst, D. Mikulencak, Communications in Numerical Methods in Engineering 18 (2002), 15-30

Generalized Green’s functions and the effective domain of influence, D. Estep, M. Holst, and M. Larson, SIAM Journal on Scientific Computing 26 (2005), 1314-1339

A posteriori – a priori analysis of multiscale operator splitting, D. Estep, V. Ginting, D. Ropp, J. Shadid, and S. Tavener, SIAM Journal on Numerical Analysis 46 (2008), 1116-1146

A posteriori error estimation of approximate boundary fluxes, T. Wildey, S. Tavener, and D. Estep, Communications in Numerical Methods in Engineering, 24 (2008), 421-434

A posteriori analysis and improved accuracy for an operator decomposition solution of a conjugate heat transfer problem, D. Estep, S. Tavener, T. Wildey, SIAM Journal on Numerical Analysis, 46 (2008), 2068-2089

A posteriori error analysis of multiscale operator decomposition methods for multiphysics models, D. Estep, V. Carey, V. Ginting, S. Tavener, T. Wildey, Journal of Physics: Conference Series 125 (2008), 1-16

A posteriori analysis and adaptive error control for multiscale operator decomposition methods for coupled elliptic systems I: Triangular systems, V. Carey, D. Estep, and S. Tavener, SIAM Journal on Numerical Analysis 47 (2009), 740-761

A posteriori error analysis for a transient conjugate heat transfer problem, D. Estep, S. Tavener, T. Wildey, Finite Elements in Analysis and Design 45 (2009), 263-271

A posteriori error analysis of a cell-centered finite volume method for semilinear elliptic problems, D. Estep, M. Pernice, D. Pham, S. Tavener, H. Wang, Journal of Computational and Applied Mathematics 233 (2009), 459-472

A posteriori error estimation and adaptive mesh refinement for a multi-discretization operator decomposition approach to fluid-solid heat transfer, D. Estep, S. Tavener, T. Wildey, Journal of Computational Physics 229 (2010), 4143-4158

Blockwise adaptivity for time dependent problems based on coarse scale adjoint solutions, V. Carey, D. Estep, A. Johansson, M. Larson, and S. Tavener, SIAM Journal on Scientific Computing 32 (2010), 2121-2145

A posteriori error analysis for a cut cell finite volume method, D. Estep, S. Tavener, M. Pernice and H. Wang, Computer Methods in Applied Mechanics and Engineering 200 (2011), 2768-2781

A posteriori analysis of multirate numerical method for ordinary differential equations, D. Estep, V. Ginting, S. Tavener, 2012, Computer Methods in Applied Mechanics and Engineering, 223-224 (2012), 10-27

Adaptive error control for an elliptic optimization problem, Applicable Analysis, D. Estep and S. Lee, 2012, DOI:10.1080/00036811.2012.683785, 1-15

A posteriori analysis of an iterative multi-discretization method for reaction-diffusion systems, D. Estep. V. Ginting, J. Hameed, and S. Tavener, Computer Methods in Applied Mechanics and Engineering, 267 (2013), 1-22

A posteriori analysis and adaptive error control for operator decomposition solution of coupled semilinear elliptic systems, V. Carey, D. Estep, S. Tavener, International Journal of Numerical Methods in Engineering 94 (2013), 826-849

A-posteriori error estimates for mixed finite element and finite volume methods for problems coupled through a boundary with non-matching grids, T. Arbogast, D. Estep, B. Sheehan, and S. Tavener, IMA Journal of Numerical Analysis, 24 (2013), 1625-1653

A posteriori error estimation for the Lax-Wendroff finite difference scheme, J. B. Collins, D. Estep, and S. Tavener, Journal of Computational and Applied Mathematics 263C (2014), 299-311

A posteriori error analysis of IMEX time integration schemes for advection-diffusion-reaction equations, J. Chaudry, D. Estep, V. Ginting, J. Shadid, and S. Tavener, Computer Methods in Applied Mechanics and Engineering, 285 (2014), 730-751

A posteriori error analysis for ﬁnite element methods with projection operators as applied to explicit time integration techniques, J. Collins, D. Estep and S. Tavener, BIT Numerical Mathematics, 55 (2015) 1017-1042

A posteriori error estimates for mixed finite element and finite volume methods for parabolic problems coupled through a boundary with non-matching discretizations, T. Arbogast, D. Estep, B. Sheehan, and S. Tavener, SIAM ASA Journal on Uncertainty Quantification, 3 (2015), 169-198

Adaptive finite element solution of multiscale PDE-ODE systems, A. Johansson, J. H. Chaudhry, V. Carey, D. Estep, V. Ginting, M. Larson, and S. Tavener, Computer Methods in Applied Mechanics and Engineering, 287 (2015), 150–171

A posteriori analysis for iterative solvers for non-autonomous evolution problems, J. H. Chaudry, D. Estep, V. Ginting, and S. Tavener, SIAM ASA Journal on Uncertainty Quantification, 3 (2015), 434-459

A posteriori error analysis of two stage computation methods with application to efficient resource allocation and the Parareal Algorithm, J. H. Chaudhry, D. Estep, S. Tavener, V. Carey, and J. Sandelin, SIAM Journal on Numerical Analysis, 54 (2016), 2729-3122

**Modeling of Large Communication Networks**

**Modeling of Large Communication Networks**Direct simulation of large communication networks is very important for designing networks to achieve desired performance. But simulating large networks on computers is prohibitively expensive in terms of computational time. Estep and collaborators have devised continuum models of complex communication networks that can be simulated very quickly.

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Continuum modeling of large networks, E. Chong, D. Estep, and J. Hannig, International Journal of Numerical Modeling: Electronic Networks, Devices, and Fields, 21 (2008), 169-186

Continuum modeling and control of large mobile networks, Y. Zhang, E. K. P. Chong, J. Hannig, and D. Estep, Proceedings of the 49th Annual Allerton Conference on Communication, Control and Computing, Illinois, 2011, 1670-1677

Analysis of routing protocols and interference-limited communication in large networks via continuum modeling, N. Burch, E. Chong, D. Estep, J. Hannig, Journal of Engineering Mathematics, 79 (2013), 183-199

Continuum modeling and control of large nonuniform wireless networks via nonlinear partial differential equations, Y. Zhang, E. Chong, J. Hannig, and D. Estep, Abstract and Applied Analysis 16 (2013), doi:10.1155/2013/262581, 1-16

Approximating extremely large networks via continuum limits, Y. Zhang, E. Chong, J. Hannig, and D. Estep, IEEE Access, 1 (2013), 577-595

**A Posteriori Error Estimation, Uncertainty Quantification and Adaptive Computation for Electromagnetic Scattering**

**A Posteriori Error Estimation, Uncertainty Quantification and Adaptive Computation for Electromagnetic Scattering**Estep and collaborators have applied the framework for a posteriori error estimation and uncertainty quantifications to the computer simulation of electromagnetic scattering off objects in order to determine material properties and geometry. They have also developed innovative adaptive methods for efficient computations.

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Adjoint methods for uncertainty quantification in applied computational electromagnetics: FEM scattering examples, C. Key, A. Smull, B. M. Notaros, D. Estep, and T. Butler, Invited Paper, Special Issue Advanced Computational Electromagnetic Methodologies and Techniques, ACES Journal, February 2019, ISBN: 978-0-9960078-8-7

A posteriori error estimation and adaptive discretization refinement using adjoint methods in CEM: A study with a one-dimensional higher-order FEM scattering example, C. Key, A. Smull, D. Estep, T. Butler, and B. M. Notaros, IEEE Transactions on Antennas and Propagation 68 (2020), 3791-3806.

Adjoint-based accelerated adaptive refinement in frequency domain 3-D finite element method scattering problems, J. Harmon, C. Key, D. Estep, T. Butler, and B. M. Notaros, IEEE Transactions on Antennas and Propagation, 69 (2020), 940-949.

Adjoint sensitivity analysis for uncertain material parameters in frequency-domain 3-D FEM, J. Harmon, C. Key, D. Estep, T. Butler, and B. M. Notaros, IEEE Transactions on Antennas and Propagation, 69 (2021), 6669-6679.

Adjoint methods for uncertainty quantification in applied computational electromagnetics: FEM scattering examples, C. Key, A. Smull, B. M. Notaros, D. Estep, and T. Butler, Proceedings of the 2018 International Applied Computational Electromagnetics Society (ACES) Symposium – ACES2018, March 25–29, 2018, Denver, Colorado, USA

A posteriori element-wise error quantification for FEM solvers using higher order basis functions, C. Key, A. Smull, D. Estep, T. Butler, and B. M. Notaros, Proceedings of the 2018 IEEE International Symposium on Antennas and Propagation, July 8–13, 2018, Boston, MA, USA, pp. 1319–1320

Applications of adjoint solutions for predicting and analyzing numerical error of forward solutions based on higher order finite element modeling, B. M. Notaros, C. Key, A. Smull, D. Estep, and T. Butler, Proceedings of the 14th International Workshop on Finite Elements for Microwave Engineering – FEM2018, September 10-14, 2018, Cartagena de Indias, Colombia, pp. 3–4

Adjoint-based a posteriori error estimation and its applications in CEM: DHO FEM techniques and 3D scattering problems, J. Harmon, C. Key, B. Troksa, T. Butler, D. Estep, and B. M. Notaros, Proc. 2019 USNC-URSI National Radio Science Meeting, January 9-12, 2019, Boulder, Colorado

Adjoint-based uncertainty quantification in frequency-domain double higher-order FEM, J. Harmon, C. Key, B. M. Notaros, D. Estep, and T. Butler, Proceedings of the 2019 International Applied Computational Electromagnetics Society (ACES) Symposium – ACES2019, April 15–19, 2019, Miami, Florida, USA

Overview of some advances in higher order frequency-domain CEM techniques, B. M. Notaros, S. B. Manic, C. Key, J. Harmon, and D. Estep, Invited Paper, Special Session Advances in Frequency-Domain CEM Techniques and Applications, 21st International Conference on Electromagnetics in Advanced Applications – ICEAA 2019, September 9-13, 2019, Granada, Spain

Error estimation and uncertainty quantification based on adjoint methods in computational electromagnetics, B. M. Notaros, J. Harmon, C. Key, D. Estep, and T. Butler, Invited Paper, Special Session Applications of Machine/Deep Learning and Uncertainty Quantification Techniques in Computational Electromagnetics, 2019 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting – AP-S/URSI 2019, July 7–12, 2019, Atlanta, GA

**Analysis of Numerical Solutions of Dynamical Systems**

**Analysis of Numerical Solutions of Dynamical Systems**Computer simulation of trajectories in dynamical systems is a key scientific tool for inference and prediction. However, numerical approximation introduces error that can have a significant effect on the behavior of simulated trajectories. The analysis of such effects is technically difficult because standard approaches are too crude to provide meaningful information. Estep and collaborators have carried out analysis of the effects of numerical approximation on several classes of dynamical systems.

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Boundedness of dispersive difference schemes, D. Estep, M. Loss, and J. Rauch, Mathematics of Computation 55 (1990), 55-87

Some stability aspects of schemes for the adaptive integration of stiff initial value problems, L. Dieci and D. Estep, SIAM Journal on Scientific and Statistical Computing 12 (1991), 1284-1303

An analysis of numerical approximations of metastable solutions of the bistable equation, D. Estep, Nonlinearity 7 (1994), 1445-1462

A normal form analysis of dispersion in numerical schemes for the linear Korteweg-deVries equation, D. Estep, Applicable Analysis 52 (1994), 53-68

The rate of error growth in Hamiltonian-conserving integrators, D. Estep and A. Stuart, Zeitschrift für Angewandte Mathematik und Physik 46 (1995), 407-418

A modified equation for dispersive difference schemes, D. Estep, Applied Numerical Mathematics 17 (1995), 299-309

Analysis of shear layers in a fluid with temperature-dependent viscosity, D. Estep, S. Verduyn Lunel, and R. Williams, Journal on Computational Physics 173 (2001), 17-60.

The dynamical behavior of the discontinuous Galerkin method and related difference schemes, D. Estep and A. Stuart, Mathematics of Computation 71 (2002), 1075-1103

The formation of shear layers in a fluid with temperature-dependent viscosity, D. Estep, S. Verduyn Lunel, and R. Williams, Equadiff 03, International Conference on Differential Equations, Hasselt 2003, World Scientific, Singapore, 2004

**Miscellaneous Projects**

**Miscellaneous Projects**Most of the following projects arose from collaborations with friends.

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The discontinuous Galerkin method for semilinear parabolic problems, D. Estep and S. Larsson, RAIRO Modélisation Mathématique et Analyse Numérique 27 (1993), 35-54

The computability of the Lorenz system, D. Estep and C. Johnson, Mathematical Models and Methods in Applied Sciences 8 (1998), 1277-1305

Using Krylov-subspace iterations in discontinuous Galerkin methods for nonlinear reaction-diffusion systems, D. Estep and R. Freund, in Lecture Notes in Computational Science and Engineering 11, B. Cockburn, G. E. Karniadakis, C. -W. Shu, Eds, Springer-Verlag, New York, 2000, 327-336.

The nonlinear power method, S. Eastman and D. Estep, Applicable Analysis 86 (2007), 1303 – 1314

Introducing FACETS, the Framework Application for Core-Edge Transport Simulations, J. R. Cary, J. Candy, R. H. Cohen, S. Krasheninnikov, D. C. McCune, D. J. Estep, J. Larson, A. D. Malony, P. H. Worley, J. A. Carlsson, A. H. Hakim, P. Hamill, S. Kruger, S. Muzsala, A. Pletzer, S. Shasharina, D. Wade-Stein, N. Wang, L. McInnes, T. Wildey, T. Casper, L. Diachin, T. Epperly, T. D. Rognlien, M. R. Fahey, J. A. Kuehn, A. Morris, S. Shende, E. Feibush, G. W. Hammett, K. Indireshkumar, C. Ludescher, L. Randerson, D. Stotler, A. Yu Pigarov, .P Bonoli, C. S. Chang, D. A. D’Ippolito, P. Colella, D. E. Keyes, R. Bramley, J. R. Myra, Journal of Physics: Conference Series 78 (2007), 1-6

First results from core-edge parallel composition in the FACETS project, J. R. Cary, J. Candy, R. H. Cohen, S. Krasheninnikov, D. C. McCune, D. J. Estep, J. Larson, A. D. Malony, P. H. Worley, J. A. Carlsson, A. H. Hakim, P. Hamill, S. Kruger, M. Mia, S. Muzsala, A. Pletzer, S. Shasharina, D. Wade-Stein, N. Wang, S. Balay, L. McInnes, H. Zhang, T. Casper, L. Diachin, T. Epperly, T. D. Rognlien, M. R. Fahey, J. Cobb, A. Morris, S. Shende, G. W. Hammett, K. Indireshkumar, D. Stotler, A. Yu Pigarovd, Journal of Physics: Conference Series 125 (2008), 1-5

Analysis of the sensitivity properties of a model of vector-borne bubonic plague, M. Buzby, D. Neckels, M. Antolin, and D. Estep, Royal Society Journal Interface, 5 (2008), 1099-1107

Optimal design and directional leverage with applications in differential equation models, N. Burch, D. Estep, and J. Hoeting, Metrika, DOI: 10.1007/s00184-011-0358-4, 2011

Viscoelastic effects during loading play an integral role in soft tissue mechanics, K. Troyer, D. Estep, and C. Puttlitz, Acta Biomaterialia 8 (2012), 234-244

Multiphysics simulations: Challenges and opportunities, D. E. Keyes, L. C. McInnes, C. Woodward, W. Gropp, E. Myra, M. Pernice, J. Bell, J. Brown, A. Clo, J. Connors, E. Constantinescu, D. Estep, K. Evans, C. Farhat, A. Hakim, G. Hammond, G. Hansen, J. Hill, T. Isaac, X. Jiao, K. Jordan, D. Kaushik, E. Kaxiras, A. Koniges, K. Lee, A. Lott, Q. Lu, J. Magerlein, R. Maxwell, M. McCourt, M. Mehl, R. Pawlowski, A. Peters Randles, D. Reynolds, B. Riviere, U. Ruede, T. Scheibe, J. Shadid, B. Sheehan, M. Shephard, A. Siegel, B. Smith, X. Tang, C. Wilson, and B. Wohlmuth, International Journal of High Performance Computing Applications 27 (2013)

The interaction of iteration error and stability for linear partial differential equations coupled through an interface, B. Sheehan, D. Estep, S. Tavener, J. Cary, S. Kruger, A. Hakim, A. Pletzer, J. Carlsson, and S. Vadlamani, Advances in Mathematical Physics, 2015, 13 pages, doi:10.1155/2015/787198

On a perturbation method for stochastic parabolic PDE, D. Estep and P. Polyakov, Communications in Mathematics and Statistics: 3 (2015), 215-226

The Canadian Statistical Sciences Institute 2003-2022, M. Thompson, N. Reid and D. Estep, Canadian Journal of Statistics, https://doi.org/10.1002/cjs.11716, 2022.