Results 1 
8 of
8
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 ..."
Abstract

Cited by 14 (7 self)
 Add to MetaCart
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
Local Convergence Of A PrimalDual Method For Degenerate Nonlinear Programming
 MATHEMATICS AND COMPUTER SCIENCE DIVISION, ARGONNE NATIONAL LABORATORY, ARGONNE
, 2000
"... In recent work, the local convergence behavior of pathfollowing interiorpoint methods and sequential quadratic programming methods for nonlinear programming has been investigated for the case in which the active constraint gradients may fail to be linearly independent at the solution, but the Mang ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
In recent work, the local convergence behavior of pathfollowing interiorpoint methods and sequential quadratic programming methods for nonlinear programming has been investigated for the case in which the active constraint gradients may fail to be linearly independent at the solution, but the MangasarianFromovitz constraint qualication is satisfied. In this paper, we describe a stabilization of the primaldual interiorpoint approach that ensures rapid local convergence under these conditions without enforcing the usual centrality condition associated with pathfollowing methods. The stabilization takes the form of perturbations to the coefficient matrix in the step equations that vanish as the iterates converge to the solution.
Optimal Control of Unsteady Compressible Viscous Flows
 Inter. J. Num. Meth. Fluids
, 2002
"... This paper presents the formulation and numerical solution of a class of optimal boundary control problems governed by the unsteady twodimensional compressible Navier Stokes equations. Fundamental issues including the choice of the control space and the associated regularization term in the objec ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
This paper presents the formulation and numerical solution of a class of optimal boundary control problems governed by the unsteady twodimensional compressible Navier Stokes equations. Fundamental issues including the choice of the control space and the associated regularization term in the objective function, as well as issues in the gradient computation via the adjoint equation method are discussed. Numerical results are presented for a model problem consisting of two counterrotating viscous vortices above an infinite wall which, due to the selfinduced velocity field, propagate downward and interact with the wall. The wall boundary control is the temporal and spatial distribution of wallnormal velocity. Optimal controls for objective functions that target kinetic energy, heat transfer, and wall shear stress are presented along with the influence of control regularization for each case
Towards AdjointBased Methods for Aeroacoustic Control
 AIAA PAPER
, 2001
"... This paper is concerned with the numerical solution of optimal boundary control problems governed by the unsteady twodimensional compressible NavierStokes equations. Specifically, results are presented for a model problem consisting of two counterrotating viscous vortices above an infinite wall w ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
This paper is concerned with the numerical solution of optimal boundary control problems governed by the unsteady twodimensional compressible NavierStokes equations. Specifically, results are presented for a model problem consisting of two counterrotating viscous vortices above an infinite wall which, due to the selfinduced velocity field, propagate downward and interact with the wall. The wall boundary control is the temporal and spatial distribution of wallnormal velocity which is used to minimize cost functionals of interest. The motivation for this work is onblade control of aeroacoustic noise generated by bladevortex interaction. We discuss some problem formulation issues, especially the choice of regularization terms; we outline our adjoint computations; and we present results from optimal control calculations using two different objectives and different control regularizations.
Vector reduction/transformation operators
 ACM Transactions on Mathematical Software
, 2004
"... Development of flexible linear algebra interfaces is an increasingly critical issue. Efficient and expressive interfaces are well established for some linear algebra abstractions, but not for vectors. Vectors differ from other abstractions in the diversity of necessary operations, sometimes requirin ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Development of flexible linear algebra interfaces is an increasingly critical issue. Efficient and expressive interfaces are well established for some linear algebra abstractions, but not for vectors. Vectors differ from other abstractions in the diversity of necessary operations, sometimes requiring dozens for a given algorithm (e.g. interiorpoint methods for optimization). We discuss a new approach based on operator objects that are transported to the underlying data by the linear algebra library implementation, allowing developers of abstract numerical algorithms to easily extend the functionality regardless of computer architecture, application or data locality/organization. Numerical experiments demonstrate efficient implementation.
Local Analysis of a New Multipliers Method
 European Journal of Operational Research (special volume on Continuous Optimization
"... In this paper we introduce a penalty function and a corresponding multipliers method for the solution of a class of nonlinear programming problems where the equality constraints have a particular structure. The class models optimal control and engineering design problems with bounds on the state and ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
In this paper we introduce a penalty function and a corresponding multipliers method for the solution of a class of nonlinear programming problems where the equality constraints have a particular structure. The class models optimal control and engineering design problems with bounds on the state and control variables and has wide applicability. The multipliers method updates multipliers corresponding to inequality constraints (maintaining their nonnegativity) instead of dealing with multipliers associated with equality constraints. The basic local convergence properties of the method are proved and a dual framework is introduced. We also analyze the properties of the penalized problem related with the penalty function. Keywords. Nonlinear programming, optimal control, state constraints, penalty function, multipliers method, augmented Lagrangian. AMS subject classications. 49M37, 90C06, 90C30 1
A Generalized Trust Region SQP Algorithm for Equality Constrained Optimization
, 2003
"... We introduce and analyze a class of generalized trust region sequential quadratic programming (GTRSQP) algorithms for equality constrained optimization. Unlike in standard trust region SQP (TRSQP) algorithms, the optimization subproblems arising in our GTRSQP algorithm can be generated from models o ..."
Abstract
 Add to MetaCart
We introduce and analyze a class of generalized trust region sequential quadratic programming (GTRSQP) algorithms for equality constrained optimization. Unlike in standard trust region SQP (TRSQP) algorithms, the optimization subproblems arising in our GTRSQP algorithm can be generated from models of the objective and constraint functions that are not necessarily based on Taylor approximations. The need for such generalizations is motivated by optimal control problems for which model problems can be generated using, e.g., different discretizations. Several existing TRSQP algorithms are special cases of our GTRSQP algorithm. Our first order global convergence result for the GTRSQP algorithm applied to TRSQP allows one to relax the condition that the socalled tangential step lies in the nullspace of the linearized constraints. The application of the GTRSQP algorithm to an optimal control problem governed by Burgers equation is discussed.
INVERSE PROBLEMS
, 2008
"... doi:10.1088/02665611/24/3/034011 Adaptive finite element methods for the solution of inverse problems in optical tomography ..."
Abstract
 Add to MetaCart
doi:10.1088/02665611/24/3/034011 Adaptive finite element methods for the solution of inverse problems in optical tomography