Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets
SIAM Journal on Mathematical Analysis
Element-by-Element Construction of Wavelets Satisfying Stability and Moment Conditions
SIAM Journal on Numerical Analysis
Inexact Preconditioned Conjugate Gradient Method with Inner-Outer Iteration
SIAM Journal on Scientific Computing
Multigrid optimization in applications
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 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 Saddle Point Problems in Optimal Control Based on Wavelets
Computational Optimization and Applications
Adaptive solution of partial differential equations in multiwavelet bases
Journal of Computational Physics
Multigrid methods for parabolic distributed optimal control problems
Journal of Computational and Applied Mathematics
Adaptive Wavelet Schemes for Nonlinear Variational Problems
SIAM Journal on Numerical Analysis
Inexact Krylov Subspace Methods for Linear Systems
SIAM Journal on Matrix Analysis and Applications
Adaptive Wavelet Methods for Linear-Quadratic Elliptic Control Problems: Convergence Rates
SIAM Journal on Control and Optimization
IEEE Transactions on Image Processing
A Posteriori Error Estimates Including Algebraic Error and Stopping Criteria for Iterative Solvers
SIAM Journal on Scientific Computing
SIAM Journal on Control and Optimization
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We investigate wavelet methods for the efficient numerical solution of a class of control problems constrained by a linear elliptic boundary value problem where the cost functional may contain fractional Sobolev norms of the control and the state. Starting point is the formulation of the infinite-dimensional control problem in terms of (boundary-adapted biorthogonal spline-) wavelets, involving only l2 norms of wavelet expansion coefficients (where different norms are realized by a diagonal scaling together with a Riesz map) and constraints in form of an l2 isomorphism. The coupled system of equations resulting from optimization is solved by an inexact conjugate gradient (CG) method for the control, which involves the approximate inversion of the primal and the adjoint operator using again CG iterations. Starting from a coarse discretization level, we use nested iteration to solve the coupled system on successively finer uniform discretizations up to discretization error accuracy on each level. The resulting inexact CG scheme is a 'fast solver': it is of asymptotic optimal complexity in the sense that the overall computational effort to compute the solution up to discretization error on the finest grid is proportional to the number of unknowns on that grid, a consequence of grid-independent condition numbers of the linear operators in wavelet coordinates.In the numerical examples we study the choice of different norms and the regularization parameter in the cost functional and their effect on the solution. Moreover, for different situations the performance of the fully iterative inexact CG scheme is investigated, confirming the theoretical results.