A general Hilbert space approach to framelets.
A method solving optimal control problems.
A note on variational discretization of elliptic Neumann boundary control
A parabolic quasi-variational inequality arising in hydraulics
A priori error estimates for a state-constrained elliptic optimal control problem
We examine an elliptic optimal control problem with control and state constraints in ℝ3. An improved error estimate of 𝒪(hs) with 3/4 ≤ s ≤ 1 − ε is proven for a discretisation involving piecewise constant functions for the control and piecewise linear for the state. The derived order of convergence is illustrated by a numerical example.
A priori error estimates for a state-constrained elliptic optimal control problem
We examine an elliptic optimal control problem with control and state constraints in ℝ3. An improved error estimate of 𝒪(hs) with 3/4 ≤ s ≤ 1 − ε is proven for a discretisation involving piecewise constant functions for the control and piecewise linear for the state. The derived order of convergence is illustrated by a numerical example.
A priori error estimates for a state-constrained elliptic optimal control problem
We examine an elliptic optimal control problem with control and state constraints in ℝ3. An improved error estimate of 𝒪(hs) with 3/4 ≤ s ≤ 1 − ε is proven for a discretisation involving piecewise constant functions for the control and piecewise linear for the state. The derived order of convergence is illustrated by a numerical example.
A priori error estimates for finite element discretizations of a shape optimization problem
In this paper we consider a model shape optimization problem. The state variable solves an elliptic equation on a domain with one part of the boundary described as the graph of a control function. We prove higher regularity of the control and develop a priori error analysis for the finite element discretization of the shape optimization problem under consideration. The derived a priori error estimates are illustrated on two numerical examples.
A refined Newton’s mesh independence principle for a class of optimal shape design problems
Shape optimization is described by finding the geometry of a structure which is optimal in the sense of a minimized cost function with respect to certain constraints. A Newton’s mesh independence principle was very efficiently used to solve a certain class of optimal design problems in [6]. Here motivated by optimization considerations we show that under the same computational cost an even finer mesh independence principle can be given.
A set oriented approach to global optimal control
We describe an algorithm for computing the value function for “all source, single destination” discrete-time nonlinear optimal control problems together with approximations of associated globally optimal control strategies. The method is based on a set oriented approach for the discretization of the problem in combination with graph-theoretic techniques. The central idea is that a discretization of phase space of the given problem leads to an (all source, single destination) shortest path problem...
A set oriented approach to global optimal control
We describe an algorithm for computing the value function for “all source, single destination” discrete-time nonlinear optimal control problems together with approximations of associated globally optimal control strategies. The method is based on a set oriented approach for the discretization of the problem in combination with graph-theoretic techniques. The central idea is that a discretization of phase space of the given problem leads to an (all source, single destination) shortest path...
Adaptive finite element method for shape optimization
We examine shape optimization problems in the context of inexact sequential quadratic programming. Inexactness is a consequence of using adaptive finite element methods (AFEM) to approximate the state and adjoint equations (via the dual weighted residual method), update the boundary, and compute the geometric functional. We present a novel algorithm that equidistributes the errors due to shape optimization and discretization, thereby leading to coarse resolution in the early stages and fine resolution...
Adaptive Newton-like method for shape optimization
All-at-once preconditioning in PDE-constrained optimization
The optimization of functions subject to partial differential equations (PDE) plays an important role in many areas of science and industry. In this paper we introduce the basic concepts of PDE-constrained optimization and show how the all-at-once approach will lead to linear systems in saddle point form. We will discuss implementation details and different boundary conditions. We then show how these system can be solved efficiently and discuss methods and preconditioners also in the case when bound...
An approximation to discrete optimal feedback controls.
An iterative method for solving the generalized system of relaxed cocoercive quasivariational inclusions and fixed point problems of an infinite family of strictly pseudocontractive mappings.
An L2-Error Estimate for an Approximation of the Solution of a Parabolic Variational Inequality.
Analysis and finite element error estimates for the velocity tracking problem for Stokes flows via a penalized formulation
A distributed optimal control problem for evolutionary Stokes flows is studied via a pseudocompressibility formulation. Several results concerning the analysis of the velocity tracking problem are presented. Semidiscrete finite element error estimates for the corresponding optimality system are derived based on estimates for the penalized Stokes problem and the BRR (Brezzi-Rappaz-Raviart) theory. Finally, the convergence of the solutions of the penalized optimality systems as is examined.
Analysis and finite element error estimates for the velocity tracking problem for Stokes flows via a penalized formulation
A distributed optimal control problem for evolutionary Stokes flows is studied via a pseudocompressibility formulation. Several results concerning the analysis of the velocity tracking problem are presented. Semidiscrete finite element error estimates for the corresponding optimality system are derived based on estimates for the penalized Stokes problem and the BRR (Brezzi-Rappaz-Raviart) theory. Finally, the convergence of the solutions of the penalized optimality systems as ε → 0 is examined. ...
Applications of Fixed-Point Methods to Discrete Variational and Quasi-Variational Inequalities.