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59
Adaptive finite element approximation for distributed elliptic optimal control problems
 SIAM J. Control Optim
, 2002
"... Abstract. In this paper, sharp a posteriori error estimators are derived for a class of distributed elliptic optimal control problems. These error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems and are implemented in the adaptive approach ..."
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Cited by 48 (8 self)
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Abstract. In this paper, sharp a posteriori error estimators are derived for a class of distributed elliptic optimal control problems. These error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems and are implemented in the adaptive approach. Our numerical results indicate that the sharp error estimators work satisfactorily in guiding the mesh adjustments and can save substantial computational work.
Adaptive spacetime finite element methods for parabolic optimization problems
 SIAM J. Control Optim
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A framework for the adaptive finite element solution of large inverse problems. I. Basic techniques
, 2004
"... Abstract. Since problems involving the estimation of distributed coefficients in partial differential equations are numerically very challenging, efficient methods are indispensable. In this paper, we will introduce a framework for the efficient solution of such problems. This comprises the use of a ..."
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Cited by 24 (7 self)
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Abstract. Since problems involving the estimation of distributed coefficients in partial differential equations are numerically very challenging, efficient methods are indispensable. In this paper, we will introduce a framework for the efficient solution of such problems. This comprises the use of adaptive finite element schemes, solvers for the large linear systems arising from discretization, and methods to treat additional information in the form of inequality constraints on the parameter to be recovered. The methods to be developed will be based on an allatonce approach, in which the inverse problem is solved through a Lagrangian formulation. The main feature of the paper is the use of a continuous (function space) setting to formulate algorithms, in order to allow for discretizations that are adaptively refined as nonlinear iterations proceed. This entails that steps such as the description of a Newton step or a line search are first formulated on continuous functions and only then evaluated for discrete functions. On the other hand, this approach avoids the dependence of finite dimensional norms on the mesh size, making individual steps of the algorithm comparable even if they used differently refined meshes. Numerical examples will demonstrate the applicability and efficiency of the method for problems with several million unknowns and more than 10,000 parameters. Key words. Adaptive finite elements, inverse problems, Newton method on function spaces. AMS subject classifications. 65N21,65K10,35R30,49M15,65N50 1. Introduction. Parameter
An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints
, 2006
"... We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residualtype a posteriori error estimator that consists of e ..."
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Cited by 22 (8 self)
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We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residualtype a posteriori error estimator that consists of edge and element residuals. Since we do not assume any regularity of the data of the problem, the error analysis further invokes data oscillations. We prove reliability and efficiency of the error estimator and provide a bulk criterion for mesh refinement that also takes into account data oscillations and is realized by a greedy algorithm. A detailed documentation of numerical results for selected test problems illustrates the convergence of the adaptive finite element method.
A posteriori error estimation and global error control for ordinary differential equations by the adjoint method
 SIAM J. Sci. Comput
, 2002
"... Abstract. In this paper we propose a general method for a posteriori error estimation in the solution of initial value problems in ordinary differential equations (ODEs). With the help of adjoint sensitivity software, this method can be implemented efficiently. It provides a condition estimate for t ..."
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Cited by 21 (2 self)
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Abstract. In this paper we propose a general method for a posteriori error estimation in the solution of initial value problems in ordinary differential equations (ODEs). With the help of adjoint sensitivity software, this method can be implemented efficiently. It provides a condition estimate for the ODE system. We also propose an algorithm for global error control, based on the condition of the system and the perturbation due to the numerical approximation. Key words. adjoint method, ordinary differential equation, global error control
Adaptive finite element methods for the solution of inverse problems
 in optical tomography, Inverse Problems, accepted
, 2008
"... in optical tomography ..."
Gradient recovery type a posteriori error estimates for finite flement approximations on irregular meshes
 Computer Methods in Applied Mechanics and Engineering 190
, 2001
"... Abstract. In this paper, we present a recovery type a posteriori error estimate and the superconvergence analysis for the finite element approximation of the distributed convex optimal control problems governed by integraldifferential equations. We provide the recovery type a posteriori error estima ..."
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Cited by 17 (1 self)
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Abstract. In this paper, we present a recovery type a posteriori error estimate and the superconvergence analysis for the finite element approximation of the distributed convex optimal control problems governed by integraldifferential equations. We provide the recovery type a posteriori error estimates for both the control and the state approximation, which is equivalent to the exact error generally. Under some strong conditions, it is not only equivalent, but also asymptotically exact. 1.
A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints
 COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
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Adaptive finite volume methods for distributed nonsmooth parameter identification
, 2007
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Computing Bounds for Linear Functionals of Exact Weak Solutions to Poisson's Equation
 SIAM J. Numer. Anal
"... We present a method for Poisson's equation that computes guaranteed upper and lower bounds for the values of linear functional outputs of the exact weak solution of the infinitedimensional continuum problem. The method results from exploiting the Lagrangian saddle point property engendered by r ..."
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Cited by 11 (3 self)
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We present a method for Poisson's equation that computes guaranteed upper and lower bounds for the values of linear functional outputs of the exact weak solution of the infinitedimensional continuum problem. The method results from exploiting the Lagrangian saddle point property engendered by recasting the output problem as a constrained minimization problem. Localization is acheived by Lagrangian relaxation and the bounds are computed by appeal to a local dual problem. The proposed method computes approximate Lagrange multipliers using traditional finite element approximations to calculate a primal and an adjoint solution along with well known hybridization techniques to calculate interelement continuity multipliers. The computed bounds hold uniformly for any level of refinement, and in the asymptotic convergence regime of the finite element method, the bound gap decreases at twice the rate of the energy norm measure of the error in the finite element solution. Given a finite element solution and its output adjoint solution, the method can be used to provide a certificate of precision for the output with an asymptotic complexity that is linear in the number of elements in the finite element discretization.