A posteriori error estimation for nonlinear parabolic boundary control

Abstract-We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: Let an arbitrary 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 are able to estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian. We apply this technique to a model reduced optimal control problem obtained by proper orthogonal decomposition (POD). The distance between a (suboptimal) local solution of the reduced problem to a local solution of the original problem is estimated

Functional a posteriori error estimates for time-periodic parabolic optimal control problems

This paper is devoted to the a posteriori error analysis of multiharmonic finite element approximations to distributed optimal control problems with time-periodic state equations of parabolic type. We derive a posteriori estimates of functional type, which are easily computable and provide guaranteed upper bounds for the state and co-state errors as well as for the cost functional. These theoretical results are confirmed by several numerical tests that show high efficiency of the a posteriori error bounds

An a posteriori error analysis of a mixed finite element Galerkin approximation to second order linear parabolic problems

In this article, a posteriori error estimates are derived for a mixed finite element Galerkin approximation to second order linear parabolic initial and boundary value problems. Using mixed elliptic reconstruction method, a posteriori error estimates in $L^\infty(L^2)$ and $L^2(L^2)$-norms with optimal order of convergence for the solution as well as its flux are proved for the semidiscrete scheme. Finally, based on backward Euler method, a completely discrete scheme is analyzed and a posteriori bounds are derived, which improves earlier results on a posteriori estimates for mixed parabolic problems

A posteriori error analysis for the mean curvature flow of graphs

We study the equation describing the motion of a nonparametric surface according to its mean curvature flow. This is a nonlinear nonuniformly parabolic PDE that can be discretized in space via a finite element method. We conduct an aposteriori error analysis of the spatial discretization and derive upper bounds on the error in terms of computable estimators based on local residual indicators. The reliability of the estimators is illustrated with two numerical simulations, one of which treats the case of a singular solution

Functional a posteriori error estimates for parabolic time-periodic boundary value problems

The paper is concerned with parabolic time-periodic boundary value problems which are of theoretical interest and arise in different practical applications. The multiharmonic finite element method is well adapted to this class of parabolic problems. We study properties of multiharmonic approximations and derive guaranteed and fully computable bounds of approximation errors. For this purpose, we use the functional a posteriori error estimation techniques earlier introduced by S. Repin. Numerical tests confirm the efficiency of the a posteriori error bounds derived

Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations

This work derives a residual-based a posteriori error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control inputs. It is shown that quantities that are necessary for the error estimator can be either obtained exactly as the solutions of least-squares problems in a non-intrusive way from data such as initial conditions, control inputs, and high-dimensional solution trajectories or bounded in a probabilistic sense. The computational procedure follows an offline/online decomposition. In the offline (training) phase, the high-dimensional system is judiciously solved in a black-box fashion to generate data and to set up the error estimator. In the online phase, the estimator is used to bound the error of the reduced-model predictions for new initial conditions and new control inputs without recourse to the high-dimensional system. Numerical results demonstrate the workflow of the proposed approach from data to reduced models to certified predictions

Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems

We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto. We derive novel a posteriori estimates for the norms of L∞(0, T; L2(Ω)) and the higher order spaces, L∞(0, T;H1(Ω)) and H1(0, T; L2(Ω)), with optimal orders of convergence