arXiv subjects

Winnifried Wollner

Publications and source records attributed to Winnifried Wollner.

At least 19 recordsLinked to original sources

arXiv2026

Gradient-robustness in optimization subject to stationary Navier-Stokes equations

In this article, we discuss gradient robust discretizations for the simulation of non-linear incompressible Navier-Stokes problem and the optimal control of such flow. We consider several formulations of the flow problem that are equivalent for the continuous non-linear forward problem and compare their gradient robust discretization. We will then discuss the influence of the chosen formulation on the adjoint equations needed for gradient computation in the solution of the optimal control problem.

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arXiv2025

Stabilized SQP Methods in Hilbert Spaces

Based on techniques by (S.J. Wright 1998) for finite-dimensional optimization, we investigate a stabilized sequential quadratic programming method for nonlinear optimization problems in infinite-dimensional Hilbert spaces. The method is shown to achieve fast local convergence even in the absence of a constraint qualification, generalizing the results obtained by (S.J. Wright 1998 and W.W. Hager 1999) in finite dimensions to this broader setting.

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arXiv2025

Fracture propagation by using shape optimization techniques based on outer Riemannian metrics

In this work, we investigate a novel approach for the simulation of two-dimensional, brittle, quasi-static fracture problems based on a shape optimization approach. In contrast to the commonly-used phase-field approach, this proposed approach for investigating fracture paths does not require a specified `length-scale' parameter defining the diffuse interface region nor a level set function. Instead, it interprets the fracture as part of the boundary of the domain and uses shape optimization algorithms to minimize the energy in the system and therefore describes the fracture propagation directly. Embedding the problem of energy minimization in a Riemannian manifold framework formulated on a suitable shape space, together with the choice of an outer Riemannian metric, yields both advantages from an analytical as well as an applied perspective. Furthermore, an eigenvalue decomposition of the strain tensor is used to produce more realistic fracture paths (the so-called strain splitting), which only allows fracture growth from tensile loads. Numerical simulations for the commonly considered single-edge notch tension and shear test are performed and the results are evaluated in comparison to phase-field results.

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arXiv2025

Control in the Coefficients of an Obstacle Problem

In this work, we consider optimality conditions of an optimal control problem governed by an obstacle problem. Here, we focus on introducing a, matrix valued, control variable as the coefficients of the obstacle problem. As it is well known, obstacle problems can be formulated as a complementarity system and consequently the associated solution operator is not Gateaux differentiable. As a consequence, we utilize a regularization approach to obtain optimality conditions as the limit of optimality conditions of a family of regularized problems. Due to the coupling of the controlled coefficient with the gradients of the solution to the obstacle problem, weak convergence arguments can not be applied and we need to argue by $H$-convergence. We show, that, based on initial $H$-convergence, a bootstrapping argument can be utilized to prove strong $L^p$-convergence of the control and thus enable the passage to the limit in the optimality conditions.

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arXiv2025

A~posteriori error analysis for optimization with PDE constraints

We consider finite element solutions to optimization problems, where the state depends on the possibly constrained control through a linear partial differential equation. Basing upon a reduced and rescaled optimality system, we derive a posteriori bounds capturing the approximation of the state, the adjoint state, the control and the observation. The upper and lower bounds show a gap, which grows with decreasing cost or Tikhonov regularization parameter. This growth is mitigated compared to previous results and can be countered by refinement if control and observation involve compact operators. Numerical results illustrate these properties for model problems with distributed and boundary control.

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arXiv2024

Numerical Eigenvalue Optimization by Shape-Variations for Maxwell's Eigenvalue Problem

In this paper we consider the free-form optimization of eigenvalues in electromagnetic systems by means of shape-variations with respect to small deformations. The objective is to optimize a particular eigenvalue to a target value. We introduce the mixed variational formulation of the Maxwell eigenvalue problem introduced by Kikuchi (1987) in function spaces of (H(\operatorname{curl}; Ω)) and (H^1(Ω)). To handle this formulation, suitable transformations of these spaces are utilized, e.g., of Piola-type for the space of (H(\operatorname{curl}; Ω)). This allows for a formulation of the problem on a fixed reference domain together with a domain mapping. Local uniqueness of the solution is obtained by a normalization of the the eigenfunctions. This allows us to derive adjoint formulas for the derivatives of the eigenvalues with respect to domain variations. For the solution of the resulting optimization problem, we develop a particular damped inverse BFGS method that allows for an easy line search procedure while retaining positive definiteness of the inverse Hessian approximation. The infinite dimensional problem is discretized by mixed finite elements and a numerical example shows the efficiency of the proposed approach.

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arXiv2024

Shape optimization of optical microscale inclusions

This paper describes a class of shape optimization problems for optical metamaterials comprised of periodic microscale inclusions composed of a dielectric, low-dimensional material suspended in a non-magnetic bulk dielectric. The shape optimization approach is based on a homogenization theory for time-harmonic Maxwell's equations that describes effective material parameters for the propagation of electromagnetic waves through the metamaterial. The control parameter of the optimization is a deformation field representing the deviation of the microscale geometry from a reference configuration of the cell problem. This allows for describing the homogenized effective permittivity tensor as a function of the deformation field. We show that the underlying deformed cell problem is well-posed and regular. This, in turn, proves that the shape optimization problem is well-posed. In addition, a numerical scheme is formulated that utilizes an adjoint formulation with either gradient descent or BFGS as optimization algorithms. The developed algorithm is tested numerically on a number of prototypical shape optimization problems with a prescribed effective permittivity tensor as the target.

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arXiv2023

Coefficient Control of Variational Inequalities

Within this chapter, we discuss control in the coefficients of an obstacle problem. Utilizing tools from H-convergence, we show existence of optimal solutions. First order necessary optimality conditions are obtained after deriving directional differentiability of the coefficient to solution mapping for the obstacle problem. Further, considering a regularized obstacle problem as a constraint yields a limiting optimality system after proving, strong, convergence of the regularized control and state variables. Numerical examples underline convergence with respect to the regularization. Finally, some numerical experiments highlight the possible extension of the results to coefficient control in phase-field fracture.

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arXiv2023

A Successive Linear Relaxation Method for MINLPs with Multivariate Lipschitz Continuous Nonlinearities

We present a novel method for mixed-integer optimization problems with multivariate and Lipschitz continuous nonlinearities. In particular, we do not assume that the nonlinear constraints are explicitly given but that we can only evaluate them and that we know their global Lipschitz constants. The algorithm is a successive linear relaxation method in which we alternate between solving a master problem, which is a mixed-integer linear relaxation of the original problem, and a subproblem, which is designed to tighten the linear relaxation of the next master problem by using the Lipschitz information about the respective functions. By doing so, we follow the ideas of Schmidt et al. (2018, 2021) and improve the tackling of multivariate constraints. Although multivariate nonlinearities obviously increase modeling capabilities, their incorporation also significantly increases the computational burden of the proposed algorithm. We prove the correctness of our method and also derive a worst-case iteration bound. Finally, we show the generality of the addressed problem class and the proposed method by illustrating that both bilevel optimization problems with nonconvex and quadratic lower levels as well as nonlinear and mixed-integer models of gas transport can be tackled by our method. We provide the necessary theory for both applications and briefly illustrate the outcomes of the new method when applied to these two problems.

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arXiv2022

A new shape optimization approach for fracture propagation

Within this work, we present a novel approach to fracture simulations based on shape optimization techniques. Contrary to widely-used phase-field approaches in literature the proposed method does not require a specified 'length-scale' parameter defining the diffused interface region of the phase-field. We provide the formulation and discuss the used solution approach. We conclude with some numerical comparisons with well-established single-edge notch tension and shear tests.

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arXiv2022

Pressure robust mixed methods for nearly incompressible elasticity

Within the last years pressure robust methods for the discretization of incompressible fluids have been developed. These methods allow the use of standard finite elements for the solution of the problem while simultaneously removing a spurious pressure influence in the approximation error of the velocity of the fluid, or the displacement of an incompressible solid. To this end, reconstruction operators are utilized mapping discretely divergence free functions to divergence free functions. This work shows that the modifications proposed for Stokes equation by Linke (2014) also yield gradient robust methods for nearly incompressible elastic materials without the need to resort to discontinuous finite elements methods as proposed in Fu, Lehrenfeld, Linke, Streckenbach (2021).

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arXiv2022

Pressure-robustness in the context of optimal control

This paper studies the benefits of pressure-robust discretizations in the scope of optimal control of incompressible flows. Gradient forces that may appear in the data can have a negative impact on the accuracy of state and control and can only be correctly balanced if their $L^2$-orthogonality onto discretely divergence-free test functions is restored. Perfectly orthogonal divergence-free discretizations or divergence-free reconstructions of these test functions do the trick and lead to much better analytic a priori estimates that are also validated in numerical examples.

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arXiv2021

Optimality Conditions for Convex Stochastic Optimization Problems in Banach Spaces with Almost Sure State Constraints

We analyze a convex stochastic optimization problem where the state is assumed to belong to the Bochner space of essentially bounded random variables with images in a reflexive and separable Banach space. For this problem, we obtain optimality conditions that are, with an appropriate model, necessary and sufficient. Additionally, the Lagrange multipliers associated with optimality conditions are integrable vector-valued functions and not only measures. A model problem is given demonstrating the application to PDE-constrained optimization under uncertainty with an outlook for further applications.

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arXiv2021

A posteriori estimator for the adaptive solution of a quasi-static fracture phase-field model with irreversibility constraints

Within this article, we develop a residual type a posteriori error estimator for a time discrete quasi-static phase-field fracture model. Particular emphasize is given to the robustness of the error estimator for the variational inequality governing the phase-field evolution with respect to the phase-field regularization parameter $ε$. The article concludes with numerical examples highlighting the performance of the proposed a posteriori error estimators on three standard test cases; the single edge notched tension and shear test as well as the L-shaped panel test.

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arXiv2020

Adaptive and Pressure-Robust Discretization of Incompressible Pressure-Driven Phase-Field Fracture

In this work, we consider pressurized phase-field fracture problems in nearly and fully incompressible materials. To this end, a mixed form for the solid equations is proposed. To enhance the accuracy of the spatial discretization, a residual-type error estimator is developed. Our algorithmic advancements are substantiated with several numerical tests that are inspired from benchmark configurations. Therein, a primal-based formulation is compared to our newly developed mixed phase-field fracture method for Poisson ratios approaching $ν\to 0.5$. Finally, for $ν= 0.5$, we compare the numerical results of the mixed formulation with a pressure robust modification.

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arXiv2020

A Stochastic Gradient Method with Mesh Refinement for PDE Constrained Optimization under Uncertainty

Models incorporating uncertain inputs, such as random forces or material parameters, have been of increasing interest in PDE-constrained optimization. In this paper, we focus on the efficient numerical minimization of a convex and smooth tracking-type functional subject to a linear partial differential equation with random coefficients and box constraints. The approach we take is based on stochastic approximation where, in place of a true gradient, a stochastic gradient is chosen using one sample from a known probability distribution. Feasibility is maintained by performing a projection at each iteration. In the application of this method to PDE-constrained optimization under uncertainty, new challenges arise. We observe the discretization error made by approximating the stochastic gradient using finite elements. Analyzing the interplay between PDE discretization and stochastic error, we develop a mesh refinement strategy coupled with decreasing step sizes. Additionally, we develop a mesh refinement strategy for the modified algorithm using iterate averaging and larger step sizes. The effectiveness of the approach is demonstrated numerically for different random field choices.

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arXiv2020

Adaptive Numerical Simulation of a Phase-field Fracture Model in Mixed Form tested on an L-shaped Specimen with High Poisson Ratios

This work presents a new adaptive approach for the numerical simulation of a phase-field model for fractures in nearly incompressible solids. In order to cope with locking effects, we use a recently proposed mixed form where we have a hydro-static pressure as additional unknown besides the displacement field and the phase-field variable. To fulfill the fracture irreversibility constraint, we consider a formulation as a variational inequality in the phase-field variable. For adaptive mesh refinement, we use a recently developed residual-type a posteriori error estimator for the phase-field variational inequality which is efficient and reliable, and robust with respect to the phase-field regularization parameter. The proposed model and the adaptive error-based refinement strategy are demonstrated by means of numerical tests derived from the L-shaped panel test, originally developed for concrete. Here, the Poisson's ratio is changed from the standard setting towards the incompressible limit $ν\to 0.5$.

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arXiv2020

A One Dimensional Elliptic Distributed Optimal Control Problem with Pointwise Derivative Constraints

We consider a one dimensional elliptic distributed optimal control problem with pointwise constraints on the derivative of the state. By exploiting the variational inequality satisfied by the derivative of the optimal state, we obtain higher regularity for the optimal state under appropriate assumptions on the data. We also solve the optimal control problem as a fourth order variational inequality by a $C^1$ finite element method, and present the error analysis together with numerical results.

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