Multiobjective Optimization of Non-Smooth PDE-Constrained Problems

Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control - potentially with non-smoothness both on the level of the objectives or in the system dynamics. This results in new challenges such as dealing with expensive models (e.g., governed by partial differential equations (PDEs)) and developing dedicated algorithms handling the non-smoothness. Since in contrast to single-objective optimization, the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in the field of multiobjective optimization of non-smooth PDE-constrained problems. In particular we report on the advances achieved within Project 2 "Multiobjective Optimization of Non-Smooth PDE-Constrained Problems - Switches, State Constraints and Model Order Reduction" of the DFG Priority Programm 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization"

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

Adaptive machine learning-based surrogate modeling to accelerate PDE-constrained optimization in enhanced oil recovery

In this contribution, we develop an efficient surrogate modeling framework for simulation-based optimization of enhanced oil recovery, where we particularly focus on polymer flooding. The computational approach is based on an adaptive training procedure of a neural network that directly approximates an input-output map of the underlying PDE-constrained optimization problem. The training process thereby focuses on the construction of an accurate surrogate model solely related to the optimization path of an outer iterative optimization loop. True evaluations of the objective function are used to finally obtain certified results. Numerical experiments are given to evaluate the accuracy and efficiency of the approach for a heterogeneous five-spot benchmark problem.publishedVersio

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Proximal Galerkin: A structure-preserving finite element method for pointwise bound constraints

The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop scalable, mesh-independent algorithms for optimal design. The paper leads to a derivation of the latent variable proximal point (LVPP) algorithm: an unconditionally stable alternative to the interior point method. LVPP is an infinite-dimensional optimization algorithm that may be viewed as having an adaptive barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of the main benefits of this algorithm is witnessed when analyzing the classical obstacle problem. Therein, we find that the original variational inequality can be replaced by a sequence of semilinear partial differential equations (PDEs) that are readily discretized and solved with, e.g., high-order finite elements. Throughout this work, we arrive at several unexpected contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field density-based topology optimization. The complete latent variable proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis

Adaptive mesh refinement in topology optimization

This dissertation presents developments in stress constrained topology optimization with Adaptive Mesh Refinement (AMR). Regions with stress concentrations dominate the optimized design. As such, we first present an approach to obtain designs with accurately computed stress fields within the context of topology optimization. To achieve this goal, we invoke threshold and AMR operations during the optimization. We do so in an optimal fashion, by applying AMR techniques that use error indicators to refine and coarsen the mesh as needed. In this way, we obtain accurate simulations and greater resolution of the design domain in a computationally efficient manner. We present results in two dimensions to demonstrate the efficacy of our method. The topology optimization community has regularly employed optimization algorithms from the operations research community. However, these algorithms are implemented in the Euclidean space instead of the proper function space where the design, i.e. volume fraction, field resides. In this thesis, we show that, when discretizing the volume fraction field over a non-uniform mesh, algorithms in Euclidean space are mesh dependent. We do so by first explaining the functional analysis tools necessary to understand why convergence is affected by the mesh. Namely, the distinction between derivative and gradient definitions and the role of the mesh dependent inner product. These tools are subsequently used to make the Globally Convergent Method of Moving Asymptotes (GCMMA), a popular optimization algorithm in the topology optimization community, mesh independent. We then benchmark our algorithm with three common problems in topology optimization. High resolution three-dimensional design models optimized for arbitrary cost and constraint functions are absolutely necessary ingredients for the solution of real{world engineering design problems. However, such requirements are non trivial to implement. In this thesis, we address this dilemma by developing a large scale topology optimization framework with AMR. We discuss the need for efficient parallelizable regularization methods that work across different mesh resolutions, iterative solvers and data structures. Furthermore, the optimization algorithm needs to be implemented with the same data structure that is used for the design field. To demonstrate the versatility of our framework, we optimize the designs of a three dimensional stress constrained benchmark L-bracket and a stress-constrained compliant mechanism

Snapshot-Based Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science

A local hybrid surrogate‐based finite element tearing interconnecting dual‐primal method for nonsmooth random partial differential equations

A domain decomposition approach for high‐dimensional random partial differential equations exploiting the localization of random parameters is presented. To obtain high efficiency, surrogate models in multielement representations in the parameter space are constructed locally when possible. The method makes use of a stochastic Galerkin finite element tearing interconnecting dual‐primal formulation of the underlying problem with localized representations of involved input random fields. Each local parameter space associated to a subdomain is explored by a subdivision into regions where either the parametric surrogate accuracy can be trusted or where instead one has to resort to Monte Carlo. A heuristic adaptive algorithm carries out a problem‐dependent hp‐refinement in a stochastic multielement sense, anisotropically enlarging the trusted surrogate region as far as possible. This results in an efficient global parameter to solution sampling scheme making use of local parametric smoothness exploration for the surrogate construction. Adequately structured problems for this scheme occur naturally when uncertainties are defined on subdomains, for example, in a multiphysics setting, or when the Karhunen–Loève expansion of a random field can be localized. The efficiency of the proposed hybrid technique is assessed with numerical benchmark problems illustrating the identification of trusted (possibly higher order) surrogate regions and nontrusted sampling regions