Nonlinear functional analysis and its applications
Nonlinear functional analysis and its applications
Introduction to numerical linear algebra and optimisation
Introduction to numerical linear algebra and optimisation
Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
Preconditioners for indefinite systems arising in optimization
SIAM Journal on Matrix Analysis and Applications
An Embedding of Domains Approach in Free Boundary Problems andOptimal Design
SIAM Journal on Control and Optimization
Augmented Lagrangian--SQP Methods for Nonlinear OptimalControl Problems of Tracking Type
SIAM Journal on Control and Optimization
Stable multiscale bases and local error estimation for elliptic problems
Applied Numerical Mathematics - Special issue on multilevel methods
Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems
SIAM Journal on Numerical Analysis
Least-squares methods for optimal control
Proceedings of the second world congress on Nonlinear analysts: part 3
Applied Mathematics and Computation
Composite wavelet bases for operator equations
Mathematics of Computation
On the Lagrange--Newton--SQP Method for the Optimal Control of Semilinear Parabolic Equations
SIAM Journal on Control and Optimization
Inexact Preconditioned Conjugate Gradient Method with Inner-Outer Iteration
SIAM Journal on Scientific Computing
Computational Optimization and Applications
An SQP method for the optimal control of large-scale dynamical systems
Journal of Computational and Applied Mathematics - Special issue on SQP-based direct discretization methods for practical optimal control problems
Adaptive wavelet methods for elliptic operator equations: convergence rates
Mathematics of Computation
Wavelet methods for PDEs — some recent developments
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. VII: partial differential equations
Adaptive Wavelet Schemes for Elliptic Problems---Implementation and Numerical Experiments
SIAM Journal on Scientific Computing
Adaptive Finite Element Methods for Optimal Control of Partial Differential Equations: Basic Concept
SIAM Journal on Control and Optimization
Wavelet Least Squares Methods for Boundary Value Problems
SIAM Journal on Numerical Analysis
Fast iterative solution of elliptic control problems in wavelet discretization
Journal of Computational and Applied Mathematics
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In this paper, wavelet techniques are employed for the fast numerical solution of a control problem governed by an elliptic boundary value problem with boundary control. A quadratic cost functional involving natural norms of the state and the control is to be minimized. Firstly the constraint, the elliptic boundary value problem, is formulated in an appropriate weak form that allows to handle varying boundary conditions explicitly: the boundary conditions are treated by Lagrange multipliers, leading to a saddle point problem. This is combined with a fictitious domain approach in order to cover also more complicated boundaries.Deviating from standard approaches, we then use (biorthogonal) wavelets to derive an equivalent infinite discretized control problem which involves only ℓ2-norms and -operators. Classical methods from optimization yield the corresponding optimality conditions in terms of two weakly coupled (still infinite) saddle point problems for which a unique solution exists. For deriving finite-dimensional systems which are uniformly invertible, stability of the discretizations has to be ensured. This together with the ℓ2-setting circumvents the problem of preconditioning: all operators have uniformly bounded condition numbers independent of the discretization.In order to numerically solve the resulting (finite-dimensional) linear system of the weakly coupled saddle point problems, a fully iterative method is proposed which can be viewed as an inexact gradient scheme. It consists of a gradient algorithm as an outer iteration which alternatingly picks the two saddle point problems, and an inner iteration to solve each of the saddle point problems, exemplified in terms of the Uzawa algorithm. It is proved here that this strategy converges, provided that the inner systems are solved sufficiently well. Moreover, since the system matrix is well-conditioned, it is shown that in combination with a nested iteration strategy this iteration is asymptotically optimal in the sense that it provides the solution on discretization level J with an overall amount of arithmetic operations that is proportional to the number of unknows NJ on that level.Finally, numerical results are provided.