Asymptotic Stability of POD based Model Predictive Control for a semilinear parabolic PDE

In this article a stabilizing feedback control is computed for a semilinear parabolic partial differential equation utilizing a nonlinear model predictive (NMPC) method. In each level of the NMPC algorithm the finite time horizon open loop problem is solved by a reduced-order strategy based on proper orthogonal decomposition (POD). A stability analysis is derived for the combined POD-NMPC algorithm so that the lengths of the finite time horizons are chosen in order to ensure the asymptotic stability of the computed feedback controls. The proposed method is successfully tested by numerical examples

An adaptive certified space-time reduced basis method for nonsmooth parabolic partial differential equations

In this paper, a nonsmooth semilinear parabolic partial differential equation (PDE) is considered. For a reduced basis (RB) approach, a space-time formulation is used to develop a certified a-posteriori error estimator. This error estimator is adopted to the presence of the discrete empirical interpolation method (DEIM) as approximation technique for the nonsmoothness. The separability of the estimated error into an RB and a DEIM part then guides the development of an adaptive RB-DEIM algorithm, combining both offline phases into one. Numerical experiments show the capabilities of this novel approach in comparison with classical RB and RB-DEIM approaches

Suboptimal control of laser surface hardening using proper orthogonal decomposition

Laser surface hardening of steel is formulated in terms of an optimal control problem, where the state equations are a semilinear heat equation and an ordinary differential equation, which describes the evolution of the high temperature phase. The optimal control problem is analyzed and first-order necessary optimality conditions are derived. An error estimate for POD (proper orthogonal decomposition) Galerkin methods for the state system is proved. Finally a strategy to obtain suboptimal controls using POD is developed and validated by computing some numerical examples

Adjoint-based calibration of nonlinear stochastic differential equations

To study the nonlinear properties of complex natural phenomena, the evolution of the quantity of interest can be often represented by systems of coupled nonlinear stochastic differential equations (SDEs). These SDEs typically contain several parameters which have to be chosen carefully to match the experimental data and to validate the effectiveness of the model. In the present paper the calibration of these parameters is described by nonlinear SDE-constrained optimization problems. In the optimize-before-discretize setting a rigorous analysis is carried out to ensure the existence of optimal solutions and to derive necessary first-order optimality conditions. For the numerical solution a Monte-Carlo method is applied using parallelization strategies to compensate for the high computational time. In the numerical examples an Ornstein-Uhlenbeck and a stochastic Prandtl-Tomlinson bath model are considered

Adaptive Parameter Optimization For An Elliptic-Parabolic System Using The Reduced-Basis Method With Hierarchical A-Posteriori Error Analysis

In this paper the authors study a non-linear elliptic-parabolic system, which is motivated by mathematical models for lithium-ion batteries. One state satisfies a parabolic reaction diffusion equation and the other one an elliptic equation. The goal is to determine several scalar parameters in the coupled model in an optimal manner by utilizing a reliable reduced-order approach based on the reduced basis (RB) method. However, the states are coupled through a strongly non-linear function, and this makes the evaluation of online-efficient error estimates difficult. First the well-posedness of the system is proved. Then a Galerkin finite element and RB discretization is described for the coupled system. To certify the RB scheme hierarchical a-posteriori error estimators are utilized in an adaptive trust-region optimization method. Numerical experiments illustrate good approximation properties and efficiencies by using only a relatively small number of reduced bases functions.Comment: 24 pages, 3 figure

Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints

Optimal control problems for the heat equation with pointwise bilateral control-state constraints are considered. A locally superlinearly convergent numerical solution algorithm is proposed and its mesh independence is established. Further, for the efficient numerical solution reduced space and Schur complement based preconditioners are proposed which take into account the active and inactive set structure of the problem. The paper ends by numerical tests illustrating our theoretical findings and comparing the efficiency of the proposed preconditioners

Model Order Reduction by Proper Orthogonal Decomposition

We provide an introduction to POD-MOR with focus on (nonlinear) parametric PDEs and (nonlinear) time-dependent PDEs, and PDE constrained optimization with POD surrogate models as application. We cover the relation of POD and SVD, POD from the infinite-dimensional perspective, reduction of nonlinearities, certification with a priori and a posteriori error estimates, spatial and temporal adaptivity, input dependency of the POD surrogate model, POD basis update strategies in optimal control with surrogate models, and sketch related algorithmic frameworks. The perspective of the method is demonstrated with several numerical examples.Comment: arXiv admin note: substantial text overlap with arXiv:1701.0505