A Pointwise Quasi-Newton Method for Unconstrained Optimal Control Problems.
A posteriori error analysis for parabolic variational inequalities
Motivated by the pricing of American options for baskets we consider a parabolic variational inequality in a bounded polyhedral domain with a continuous piecewise smooth obstacle. We formulate a fully discrete method by using piecewise linear finite elements in space and the backward Euler method in time. We define an a posteriori error estimator and show that it gives an upper bound for the error in L2(0,T;H1(Ω)). The error estimator is localized in the sense that the size of the elliptic residual...
A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems
We consider the efficient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial differential equations. To this end, we employ the reduced basis method as a low-dimensional surrogate model to solve the optimal control problem and develop a posteriori error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. We show that our approach can be applied to problems involving...
A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD
We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: let an arbitrary admissible control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second-order sufficient optimality condition for the (unknown) locally optimal control, we estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian....
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 Projected Newton Method for Minimization Problems with Nonlinear Inequality Constraints.
A quasi-variational inequality problem arising in the modeling of growing sandpiles
Existence of a solution to the quasi-variational inequality problem arising in a model for sand surface evolution has been an open problem for a long time. Another long-standing open problem concerns determining the dual variable, the flux of sand pouring down the evolving sand surface, which is also of practical interest in a variety of applications of this model. Previously, these problems were solved for the special case in which the inequality is simply variational. Here, we introduce a regularized...
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 Regularized Continuous Projection-Gradient Method of the Fourth Order
A self-adaptive method for solving a system of nonlinear variational inequalities.
A semi-smooth Newton method for solving elliptic equations with gradient constraints
Semi-smooth Newton methods for elliptic equations with gradient constraints are investigated. The one- and multi-dimensional cases are treated separately. Numerical examples illustrate the approach and as well as structural features of the solution.
A semi-smooth Newton method for solving elliptic equations with gradient constraints
Semi-smooth Newton methods for elliptic equations with gradient constraints are investigated. The one- and multi-dimensional cases are treated separately. Numerical examples illustrate the approach and as well as structural features of the solution.
A sensitivity-based extrapolation technique for the numerical solution of state-constrained optimal control problems
Sensitivity analysis (with respect to the regularization parameter) of the solution of a class of regularized state constrained optimal control problems is performed. The theoretical results are then used to establish an extrapolation-based numerical scheme for solving the regularized problem for vanishing regularization parameter. In this context, the extrapolation technique provides excellent initializations along the sequence of reducing regularization parameters. Finally, the favorable numerical behavior...
A sequential iteration algorithm with non-monotoneous behaviour in the method of projections onto convex sets
The method of projections onto convex sets to find a point in the intersection of a finite number of closed convex sets in a Euclidean space, may lead to slow convergence of the constructed sequence when that sequence enters some narrow “corridor” between two or more convex sets. A way to leave such corridor consists in taking a big step at different moments during the iteration, because in that way the monotoneous behaviour that is responsible for the slow convergence may be interrupted. In this...
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...
A variant of Newton's method for generalized equations.