Luise Blank

Introduction to Model Based Optimization of Chemical Processes on Moving Horizons

Dynamic optimization problems are typically quite challenging for large-scale applications. Even more challenging are on-line applications with demanding real-time constraints. This contribution provides a concise introduction into problem formulation and standard numerical techniques commonly found in the context of moving horizon optimization using nonlinear differential algebraic process models.

On the regularization of dynamic data reconciliation problems

Abstract Dynamic data reconciliation problems are discussed from the perspective of the mathematical theory of ill-posed inverse problems. Regularization is of crucial importance to obtain satisfactory estimation quality of the reconciled variables. Usually, some penalty is added to the least-squares objective to achieve a well-posed problem. However, appropriate discretization schemes of the time-continuous problem act themselves as regularization, reducing the need of problem modification. Based on this property, we suggest to refine successively the discretization of the continuous problem starting from a coarse grid, to find a suitable regularization which renders a good compromise between (measurement) data and regularization error in the estimate. In particular, our experience supports the conjecture, that non-equidistant discretization grids offer advantages over uniform grids.

Phase-field Approaches to Structural Topology Optimization

The mean compliance minimization in structural topology optimization is solved with the help of a phase field approach. Two steepest descent approaches based on L2- and H-1-gradient flow dynamics are discussed. The resulting flows are given by Allen-Cahn and Cahn-Hilliard type dynamics coupled to a linear elasticity system. We finally compare numerical results obtained from the two different approaches.

Towards Multiscale Dynamic Data Reconciliation

Although reconciliation of steady-state process data is routinely applied in industrial practice, the theoretical understanding of the problem and its adequate formulation in a dynamic setting is still not mature. Existing formulation approaches are based on stochastic filters, deterministic observers or mathematical programming techniques. In this contribution, we suggest a general problem formulation of dynamic data reconciliation based on the theory of ill-posed problems and their regularizations. It results in a large-scale dynamic optimization problem which requires efficient numerical solution methods in real-time under strict limitations of computational resources. We explore a novel mathematical framework for the discretization of the dynamic optimization problem and the solution of the discretized nonlinear programming problem based on multiscale approximation. The framework attempts to integrate signal processing and optimization finally leading to a fully adaptive and highly efficient numerical treatment which always provides the best possible estimate which is attainable in the allotted time period.

Iterative algorithms for multiscale state estimation, part I: concepts

The objective of the present investigation is to explore the potential of multiscale refinement schemes for the numerical solution of dynamic optimization problems arising in connection with chemical process systems monitoring. State estimation is accomplished by the solution of an appropriately posed least-squares problem. To offer at any instant of time an approximate solution, a hierarchy of successively refined problems is designed using a wavelet-based Galerkin discretization. In order to fully exploit at any stage the approximate solution obtained also for an efficient treatment of the arising linear algebra tasks, we employ iterative solvers. In particular, we will apply a nested iteration scheme to the hierarchy of arising equation systems and adapt the Uzawa algorithm to the present context. Moreover, we show that, using wavelets for the formulation of the problem hierarchy, the largest eigenvalues of the resulting linear systems can be controlled effectively with scaled diagonal preconditioning. Finally, we deduce appropriate stopping criteria and illustrate the characteristics of the solver with a numerical example.

Grid refinement in multiscale dynamic optimization

Abstract In the present work we explore an adaptive discretization scheme for dynamic optimization problems applied to input and state estimation. The proposed method is embedded into a solution methodology where the dynamic optimization problem is approximated by a hierarchy of successively refined finite dimensional problems. Information on the solution of the coarser approximations is used to construct a fully adaptive, problem dependent discretization where the finite dimensional spaces are spanned by biorthogonal wavelets arising from B-splines. We demonstrate exemplarily that the proposed strategy is capable to identify accurate discretization meshes which are more economical than uniform meshes with respect to the ratio of approximation quality vs. number of used trial functions.

Allen-Cahn and Cahn-Hilliard Variational Inequalities Solved with Optimization Techniques

Parabolic variational inequalities of Allen-Cahn and Cahn-Hilliard type are solved using methods involving constrained optimization. Time discrete variants are formulated with the help of Lagrange multipliers for local and non-local equality and inequality constraints. Fully discrete problems resulting from finite element discretizations in space are solved with the help of a primal-dual active set approach. We show several numerical computations also involving systems of parabolic variational inequalities.

Multi-material Phase Field Approach to Structural Topology Optimization

Multi-material structural topology and shape optimization problems are formulated within a phase field approach. First-order conditions are stated and the relation of the necessary conditions to classical shape derivatives are discussed. An efficient numerical method based on an H 1–gradient projection method is introduced and finally several numerical results demonstrate the applicability of the approach.

Multiscale Concepts for Moving Horizon Optimization

In chemical engineering complex dynamic optimization problems formulated on moving horizons have to be solved on-line. In this work, we present a multiscale approach based on wavelets where a hierarchy of successively, adaptively refined problems are constructed. They are solved in the framework of nested iteration as long as the real-time restrictions are fulfilled. To avoid repeated calculations previously gained information is extensively exploited on all levels of the solver when progressing to the next finer discretization and/or to the moved horizon. Moreover, each discrete problem has to be solved only with an accuracy comparable to the current approximation error. Hence, we suggest the use of an iterative solver also for the arising systems of linear equations. To facilitate fast data transfer the necessary signal processing of measurements and setpoint trajectories is organized in the same framework as the treatment of the optimization problems. Moreover, since the original estimation problem is potentially ill-posed we apply the multiscale approach to determine a suitable regularization without a priori knowledge of the noise level.

Preconditioning for Allen-Cahn variational inequalities with non-local constraints

The solution of Allen-Cahn variational inequalities with mass constraints is of interest in many applications. This problem can be solved both in its scalar and vector-valued form as a PDE-constrained optimization problem by means of a primal-dual active set method. At the heart of this method lies the solution of linear systems in saddle point form. In this paper we propose the use of Krylov-subspace solvers and suitable preconditioners for the saddle point systems. Numerical results illustrate the competitiveness of this approach.