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Papers by Donald Estep

Research paper thumbnail of A Posteriori Analysis and Adaptive Error Control for Operator Decomposition of Coupled Elliptic Systems. Part I: Triangular Systems

In this paper, we perform an a posteriori error analysis of a multiscale operator decomposition f... more In this paper, we perform an a posteriori error analysis of a multiscale operator decomposition finite element method for the solution of a system of coupled elliptic problems. The goal is to compute accurate error estimates that account for the effects arising from multiscale discretization via operator decomposition. Our approach to error estimation is based on a well known a posteriori analysis involving variational analysis, residuals and the generalized Green's function. Our method utilizes adjoint problems to deal with several new features arising from the multiscale operator decomposition. In Part I of this paper, we focus on the propagation of errors arising from the solution of one component to another and the transfer of information between different representations of solution components. We also devise an adaptive discretization strategy based on the error estimates that specifically controls the effects arising from operator decomposition. In Part II of this paper, we address issues related to the iterative solution of a fully coupled nonlinear system.

Research paper thumbnail of A Posteriori Analysis for Iterative Solvers for Nonautonomous Evolution Problems

SIAM/ASA Journal on Uncertainty Quantification, 2015

Research paper thumbnail of Global Error Control For The Continuous Galerkin Finite Element Method For Ordinary Differential Equations

ESAIM Mathematical Modelling and Numerical Analysis

We analyze a continuous Galerkin nite element method for the integration of initial value problem... more We analyze a continuous Galerkin nite element method for the integration of initial value problems in ordinary di erential equations. We derive quasioptimal a priori and a posteriori error bounds. We use these results to construct a rigorous and robust theory of global error control. We conclude by exhibiting the properties of the error control in a series of numerical experiments.

Research paper thumbnail of Using Krylov-Subspace Iterations in Discontinuous Galerkin Methods for Nonlinear Reaction-Diffusion Systems

Research paper thumbnail of A POSTERIORI ANALYSIS OF A MULTIRATE NUMERICAL METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we analyze a multirate time integration method for systems of ordinary differentia... more In this paper, we analyze a multirate time integration method for systems of ordinary differential equations that present significantly different scales within the components of the model. We interpret the multirate method as a multiscale operator decomposition method and use this formulation to conduct both an a priori error analysis and a hybrid a priori - a posteriori error analysis.

Research paper thumbnail of Department of Mathematics

Research paper thumbnail of METRICS AND BENCHMARKS FOR VERIFICATION AND VALIDATION

Research paper thumbnail of DECISION OPTIMIZATION, DATA ASSIMILATION, CONTROL UNDER UNCERTAINTY

Research paper thumbnail of A NON-INTRUSIVE TECHNIQUE FOR INVERSE SENSITIVITY ANALYSIS

Research paper thumbnail of A POSTERIORI ERROR ANALYSIS OF TWO STAGE COMPUTATION METHODS WITH APPLICATION TO EFFICIENT RESOURCE ALLOCATION AND THE PARAREAL ALGORITHM

Research paper thumbnail of CRPC-TR93383 November 1993

Research paper thumbnail of Nathanial J. Burch

Research paper thumbnail of The 2nd Biannual BGCE Student Prize

Research paper thumbnail of A Postetiori Error Control for Multiphysics Systems, invited

Research paper thumbnail of Adaptive error control for an elliptic optimization problem

ABSTRACT In this paper we study optimization of a quantity of interest of a solution of an ellipt... more ABSTRACT In this paper we study optimization of a quantity of interest of a solution of an elliptic problem, with respect to parameters in the data using a gradient search algorithm. We used the generalized Green's function as an efficient way to compute the gradient. We analyze the effect of numerical error on a gradient search, and develop an efficient way to control these errors using a posteriori error analysis. Specifically, we devise an adaptive algorithm to refine and unrefine the finite element mesh at each step in the descent search algorithm. We give basic examples and apply this technique to a model of a healing wound.

Research paper thumbnail of Quantifying uncertainty of land classification within the advanced circulation (adcirc) model I: A computational framework

Research paper thumbnail of A posteriori error analysis for a cut cell nite volume method

We describe a hybrid modeling-discretization numerical method for approximating the solution of a... more We describe a hybrid modeling-discretization numerical method for approximating the solution of an elliptic problem with a discontinuous diffusion coefficient that is suited for cut-cell problems in which the discontinuity interface is not resolved by the mesh. The method is inspired by the well-known Ghost Fluid Method. We carry out an a posteriori error analysis for the numerical solution for

Research paper thumbnail of Definition and solution of a stochastic inverse problem for the Manning’s n parameter field in hydrodynamic models

Advances in Water Resources, 2015

The uncertainty in spatially heterogeneous Manning's n fields is quantified using a novel... more The uncertainty in spatially heterogeneous Manning's n fields is quantified using a novel formulation and numerical solution of stochastic inverse problems for physics-based models. The uncertainty is quantified in terms of a probability measure and the physics-based model considered here is the state-of-the-art ADCIRC model although the presented methodology applies to other hydrodynamic models. An accessible overview of the formulation and solution of the stochastic inverse problem in a mathematically rigorous framework based on measure theory is presented. Technical details that arise in practice by applying the framework to determine the Manning's n parameter field in a shallow water equation model used for coastal hydrodynamics are presented and an efficient computational algorithm and open source software package are developed. A new notion of "condition" for the stochastic inverse problem is defined and analyzed as it relates to the computation of probabilities. This notion of condition is investigated to determine effective output quantities of interest of maximum water elevations to use for the inverse problem for the Manning's n parameter and the effect on model predictions is analyzed.

Research paper thumbnail of The rate of error growth in Hamiltonian-conserving integrators

ZAMP Zeitschrift f�r angewandte Mathematik und Physik, 1995

Research paper thumbnail of A Computational Measure Theoretic Approach to Inverse Sensitivity Problems II: A Posteriori Error Analysis

SIAM Journal on Numerical Analysis, 2012

In part one of this paper [4], we develop and analyze a numerical method to solve a probabilistic... more In part one of this paper [4], we develop and analyze a numerical method to solve a probabilistic inverse sensitivity analysis problem for a given map assuming that the map in question can be evaluated exactly. In this paper, we treat the situation in which the output of the map is determined implicitly and is difficult and/or expensive to evaluate, e.g requiring the solution of a differential equation, and hence the output of the map is approximated numerically. The main goal is an a posteriori error estimate that can be used to evaluate the accuracy of the computed distribution solving the inverse problem taking into account all sources of statistical and numerical deterministic errors. We present a general analysis for the method and then apply the analysis to the case of a map determined by the solution of an initial value problem.

Research paper thumbnail of A Posteriori Analysis and Adaptive Error Control for Operator Decomposition of Coupled Elliptic Systems. Part I: Triangular Systems

In this paper, we perform an a posteriori error analysis of a multiscale operator decomposition f... more In this paper, we perform an a posteriori error analysis of a multiscale operator decomposition finite element method for the solution of a system of coupled elliptic problems. The goal is to compute accurate error estimates that account for the effects arising from multiscale discretization via operator decomposition. Our approach to error estimation is based on a well known a posteriori analysis involving variational analysis, residuals and the generalized Green's function. Our method utilizes adjoint problems to deal with several new features arising from the multiscale operator decomposition. In Part I of this paper, we focus on the propagation of errors arising from the solution of one component to another and the transfer of information between different representations of solution components. We also devise an adaptive discretization strategy based on the error estimates that specifically controls the effects arising from operator decomposition. In Part II of this paper, we address issues related to the iterative solution of a fully coupled nonlinear system.

Research paper thumbnail of A Posteriori Analysis for Iterative Solvers for Nonautonomous Evolution Problems

SIAM/ASA Journal on Uncertainty Quantification, 2015

Research paper thumbnail of Global Error Control For The Continuous Galerkin Finite Element Method For Ordinary Differential Equations

ESAIM Mathematical Modelling and Numerical Analysis

We analyze a continuous Galerkin nite element method for the integration of initial value problem... more We analyze a continuous Galerkin nite element method for the integration of initial value problems in ordinary di erential equations. We derive quasioptimal a priori and a posteriori error bounds. We use these results to construct a rigorous and robust theory of global error control. We conclude by exhibiting the properties of the error control in a series of numerical experiments.

Research paper thumbnail of Using Krylov-Subspace Iterations in Discontinuous Galerkin Methods for Nonlinear Reaction-Diffusion Systems

Research paper thumbnail of A POSTERIORI ANALYSIS OF A MULTIRATE NUMERICAL METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we analyze a multirate time integration method for systems of ordinary differentia... more In this paper, we analyze a multirate time integration method for systems of ordinary differential equations that present significantly different scales within the components of the model. We interpret the multirate method as a multiscale operator decomposition method and use this formulation to conduct both an a priori error analysis and a hybrid a priori - a posteriori error analysis.

Research paper thumbnail of Department of Mathematics

Research paper thumbnail of METRICS AND BENCHMARKS FOR VERIFICATION AND VALIDATION

Research paper thumbnail of DECISION OPTIMIZATION, DATA ASSIMILATION, CONTROL UNDER UNCERTAINTY

Research paper thumbnail of A NON-INTRUSIVE TECHNIQUE FOR INVERSE SENSITIVITY ANALYSIS

Research paper thumbnail of A POSTERIORI ERROR ANALYSIS OF TWO STAGE COMPUTATION METHODS WITH APPLICATION TO EFFICIENT RESOURCE ALLOCATION AND THE PARAREAL ALGORITHM

Research paper thumbnail of CRPC-TR93383 November 1993

Research paper thumbnail of Nathanial J. Burch

Research paper thumbnail of The 2nd Biannual BGCE Student Prize

Research paper thumbnail of A Postetiori Error Control for Multiphysics Systems, invited

Research paper thumbnail of Adaptive error control for an elliptic optimization problem

ABSTRACT In this paper we study optimization of a quantity of interest of a solution of an ellipt... more ABSTRACT In this paper we study optimization of a quantity of interest of a solution of an elliptic problem, with respect to parameters in the data using a gradient search algorithm. We used the generalized Green's function as an efficient way to compute the gradient. We analyze the effect of numerical error on a gradient search, and develop an efficient way to control these errors using a posteriori error analysis. Specifically, we devise an adaptive algorithm to refine and unrefine the finite element mesh at each step in the descent search algorithm. We give basic examples and apply this technique to a model of a healing wound.

Research paper thumbnail of Quantifying uncertainty of land classification within the advanced circulation (adcirc) model I: A computational framework

Research paper thumbnail of A posteriori error analysis for a cut cell nite volume method

We describe a hybrid modeling-discretization numerical method for approximating the solution of a... more We describe a hybrid modeling-discretization numerical method for approximating the solution of an elliptic problem with a discontinuous diffusion coefficient that is suited for cut-cell problems in which the discontinuity interface is not resolved by the mesh. The method is inspired by the well-known Ghost Fluid Method. We carry out an a posteriori error analysis for the numerical solution for

Research paper thumbnail of Definition and solution of a stochastic inverse problem for the Manning’s n parameter field in hydrodynamic models

Advances in Water Resources, 2015

The uncertainty in spatially heterogeneous Manning's n fields is quantified using a novel... more The uncertainty in spatially heterogeneous Manning's n fields is quantified using a novel formulation and numerical solution of stochastic inverse problems for physics-based models. The uncertainty is quantified in terms of a probability measure and the physics-based model considered here is the state-of-the-art ADCIRC model although the presented methodology applies to other hydrodynamic models. An accessible overview of the formulation and solution of the stochastic inverse problem in a mathematically rigorous framework based on measure theory is presented. Technical details that arise in practice by applying the framework to determine the Manning's n parameter field in a shallow water equation model used for coastal hydrodynamics are presented and an efficient computational algorithm and open source software package are developed. A new notion of "condition" for the stochastic inverse problem is defined and analyzed as it relates to the computation of probabilities. This notion of condition is investigated to determine effective output quantities of interest of maximum water elevations to use for the inverse problem for the Manning's n parameter and the effect on model predictions is analyzed.

Research paper thumbnail of The rate of error growth in Hamiltonian-conserving integrators

ZAMP Zeitschrift f�r angewandte Mathematik und Physik, 1995

Research paper thumbnail of A Computational Measure Theoretic Approach to Inverse Sensitivity Problems II: A Posteriori Error Analysis

SIAM Journal on Numerical Analysis, 2012

In part one of this paper [4], we develop and analyze a numerical method to solve a probabilistic... more In part one of this paper [4], we develop and analyze a numerical method to solve a probabilistic inverse sensitivity analysis problem for a given map assuming that the map in question can be evaluated exactly. In this paper, we treat the situation in which the output of the map is determined implicitly and is difficult and/or expensive to evaluate, e.g requiring the solution of a differential equation, and hence the output of the map is approximated numerically. The main goal is an a posteriori error estimate that can be used to evaluate the accuracy of the computed distribution solving the inverse problem taking into account all sources of statistical and numerical deterministic errors. We present a general analysis for the method and then apply the analysis to the case of a map determined by the solution of an initial value problem.

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