GAMM2025

Session

S19: Optimisation of differential equations

S19

8 Apr 2025, 08:30

Lecture and Conference Centre

Conveners

S19: Optimisation of differential equations: S19.01

• Philipp Guth
• Daniel Walter

S19: Optimisation of differential equations: S19.02

• Philipp Guth
• Daniel Walter

S19: Optimisation of differential equations: S19.03

• Philipp Guth
• Daniel Walter

S19: Optimisation of differential equations: S19.04

• Daniel Walter
• Philipp Guth

S19: Optimisation of differential equations: S19.05

• Daniel Walter
• Philipp Guth

Description

In this session novel developments devoted to optimization and optimal control problems governed by ordinary or partial differential equations will be discussed. The focus is on theoretical investigations, numerical analysis, algorithmic issues as well as on application.

534. A Riemannian View on PDE-constrained Shape Optimisation

Estefania Loayza Romero

08/04/2025, 08:30

Shape optimization focuses on determining the optimal shape of a domain—such as a boundary or region in space—to minimize or maximize an objective function. This process is typically subject to constraints, often modeled by partial differential equations (PDEs), as the underlying physical phenomena, such as fluid flow, heat conduction, or elasticity, are governed by these equations. Common...

535. A least-squares space-time approach to parabolic shape optimization

Michael Stahl

08/04/2025, 09:10

In this talk we extend the work of Deckelnick et al. (arXiv.2301.08690) to shape optimization with parabolic PDE constraints, where we consider time-dependent as well as time-independent domains. We use the method of mappings and consider steepest descent and Newton-type directions for the solution of the underlying optimization problems. The appearing PDEs for the state and the adjoint are...

537. A combined phase field - sharp interface approach for PDE constrained shape optimization

Michael Hinze

08/04/2025, 09:30

We present a general framework for PDE constrained shape optimization which combines the phase field method with a sharp interface approach based on the method of Lipschitz mappings. On the one hand, phase field methods are very well suited to numerically determine the shape, size and topology of a sought domain, but on the other hand, they have problems to sharpen out domains where they e.g....

538. Incorporating strain decomposition into fracture propagation simulations using shape optimization algorithms

Tim Suchan

08/04/2025, 09:50

A common approach to simulate fracture propagation is by the means of phase field methods, in which the discrete fracture is replaced by a continuous phase-field variable. The phase-field variable then indicates the material state. Together with the material behavior, it can be obtained by the numerical solution of partial differential equations. This replacement however introduces additional...

553. Continuation methods for higher-order topology optimization

Michael Winkler

08/04/2025, 16:30

We aim to solve a topology optimization problem where the distribution of material in the design domain is represented by a scalar valued level set function. The assignment of material at a point in the design domain is then determined by the sign of the level set function which is approximated by a smoothed Heaviside function.

To obtain candidates for local minima, we want to solve the...

540. Goal-oriented optimal sensor placement for PDE-constrained inverse problems

Marco Mattuschka

08/04/2025, 16:50

This talk presents an application of a linear transient PDE-constrained inverse problem to advection-diffusion problems in a real-world environment Ω ⊂ ℝⁿ for n ∈ {2,3}. We consider a forward operator F(m) = ℬu, where u ∈ V is the solution of the linear partial differential equation r(m, u) = f, and ℬ : V → ℝ^q is the observation operator. The objective is to utilize discrete measurements,...

541. Material Law Identification in Boundary Value Problems for Fiber Spinning

Lukas Kannengießer

08/04/2025, 17:10

Industrial manufacturing processes are often described by systems of differential equations with respective material laws. Since these material laws depend on material-specific parameters, which are frequently unknown or can only be obtained through a range of expensive experiments, there is a need for numerical parameter estimation strategies. In this context, we consider a fiber spinning...

542. New results on optimal control problems with total variaton penalty

Nico Haaf

08/04/2025, 17:30

CANCELLED
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We consider elliptic optimal control problems with total variation penalty of the control. Using the predual of the BV space, we provide a new form of optimality conditions. We discuss algorithmic opportunities to solve the control problems and present numerical results.

543. Conditional gradient methods for total variation regularization with PDE constraints

José A. Iglesias

08/04/2025, 17:50

A common ansatz in PDE-constrained problems for distributed parameters is that the sought solutions are piecewise constant, modelling situations like localized inclusions of different material properties within an otherwise homogeneous medium. In this situation, variational regularization with a total variation penalty balances being compatible with piecewise constant minimizers with retaining...

544. Optimal control of a Fokker-Planck/transport equation with BV-drift using renormalized solutions

Christian Lange

08/04/2025, 18:10

This talk focuses on the bilinear optimal control of a Fokker-Planck or a transport equation. We consider a drift field with very low regularity (specifically, BV-regularity in space), which necessitates the use of renormalized solutions, a technique developed by Ronald DiPerna and Pierre-Louis Lions. This foundational theory has been significantly extended by Luigi Ambrosio to encompass cases...

545. Sampling, optimization, SDEs and gradient flows

Mateusz Majka

09/04/2025, 08:30

We discuss connections between the problem of approximate sampling from a given probability measure and the problem of minimizing functions defined on the space of probability measures, motivated by machine learning applications. For both these problems, one can construct continuous-time stochastic processes that converge to the target measure as time goes to infinity. Among many ways of...

546. Spatial decay of perturbations in optimal control

Manuel Schaller

09/04/2025, 09:10

We analyze the robustness of optimal control problems with respect to spatially localized perturbations. We prove that, if the involved dynamic or stationary constraint satisfies a domain-uniform stabilizability and detectability assumption, then these localized perturbations only have a local effect on the optimal solution, even if the (uncontrolled) state equation gives rise to global...

547. A novel distributed method for PDE-constrained GNEPs

Felix Sauer

09/04/2025, 09:30

Solving PDE-constrained generalized Nash equilibrium problems (GNEPs) poses significant mathematical and computational challenges. Inspired by applications in gas markets, we study a GNEP where multiple agents seek to maximise their individual profit on a commodity transported through the network. The dynamics of the shared state variable, representing the commodity flow, are modeled using a...

548. Strategies for robust optimal control of chromatographic separation processes

Dominik H. Cebulla

09/04/2025, 09:50

Column liquid chromatography plays a vital role in the so-called downstream processing of biopharmaceuticals, where the goal is to extract and purify certain proteins from a mixture. Our goal is to employ a model-based approach for process optimization to improve the yield and purity of the product, while also achieving further economical and ecological benefits, e.g., due to an economical use...

549. Optimal control of an ill-posed bloodflow model: Navier-Stokes with do-nothing boundary controls

Jakob Wagner

09/04/2025, 16:30

In this talk, we will present a well posed optimal control problem subject to the Navier-Stokes equations with inhomogeneous do-nothing boundary conditions, as they are frequently used in hemodynamic computations. The simulation of cardiovascular blood flow is an area of research rapidly gaining in significance due to the increased availability of computational resources, and the widespread...

550. Output-based receding horizon stabilizing control for linear parabolic equations

Sergio S. Rodrigues

09/04/2025, 16:50

A receding horizon control framework is coupled with a Luenberger observer to construct an output-based control input stabilizing parabolic equations. The actuators and sensors are indicator functions of small subdomains, representing localized actuation and localized measurements. It is shown that, for a class of explicitly given sets of actuators and sensors, we can guarantee the stabilizing...

551. A machine learning based approximation of semi-concave functions with applications to optimal control

Donato Vasquez-Varas

09/04/2025, 17:10

The synthesis of optimal feedback-laws for optimal control problems is a challenging topic. Classical approaches consist in constructing an optimal feedback-law by approximating the value function of the problem. However, these types of methods suffer from the curse of dimensionality, namely, its computational cost increases exponentially with the dimension of the underlaying control problem....

552. The minimum energy estimator for a cubic wave equation

Jesper Schröder

09/04/2025, 17:30

The minimum energy estimator - also called Mortensen observer - was originally designed for the reconstruction of the state of nonlinear finite-dimensional dynamical systems subject to deterministic disturbances based on partial and flawed measurements. In this presentation we propose a generalization to systems governed by PDEs. Using the example of a nonlinear defocusing wave equation we...

539. Convergence of variational and iterative regularization methods under a range invariance condition

Barbara Kaltenbacher

09/04/2025, 17:50

Range invariance is a property that - like the tangential cone condition - enables a proof of convergence of iterative methods for inverse problems. In contrast to the tangential cone condition it can also be verified for some parameter identification problems in partial differential equations PDEs from boundary measurements, as relevant, e.g., in tomographic applications. The goal of this...

554. Numerical Methods and Optimality Conditions for PDE Constrained Optimal Control Problems with Control Variables Appearing Linearly

Georg Vossen

10/04/2025, 08:30

Optimal control problems with the control variable appearing linearly occur in many applications. However, this setting provides several challenges, from the analytical as well as from the numerical point of view. The optimal control can be of bang-bang or singular type, or even a combination of both is possible.

In the case of ordinary differential equations (ODE), the situation is...

555. Newton's method for nonlinear mappings into vector bundles

Laura Weigl

10/04/2025, 08:50

In this talk, we consider Newton’s method for finding zeros of nonlinear mappings from a manifold X into a vector bundle E. This setting requires a connection on E to render the Newton equation well defined, and a retraction on X is needed to compute a Newton update. As applications we will discuss variational problems involving mappings between manifolds, and, in particular, the numerical...

556. Decomposition methods for mixed-integer optimal control using Pontryagin's principle

Falk Hante

10/04/2025, 09:10

We explore ideas drawn from discrete optimization and domain decomposition to solve necessary optimality conditions given by Pontryagin's Principle, specifically when applied to non-convex optimal control problems. By leveraging these concepts, we aim to identify potential advantages compared to classical solution approaches and demonstrate practical applications for mixed-integer optimal...

557. Finite Element Error Analysis of the Beckmann Problem of Optimal Transport

Niklas Eidecker

10/04/2025, 09:30

The talk is concerned with the numerical analysis of the Beckmann problem of optimal transport. We apply a quadratic regularization of the Beckmann problem and discretize the regularized problem by means of Raviart Thomas finite elements. Moreover, we provide a priori estimates of regularization and discretization error for the minimal objective value. Together with H²-regularity estimates for...

558. Optimal control of rate-independent systems with non-convex energy

Merlin Andreia

10/04/2025, 09:50

This talk deals with an optimal control problem, where the state variable is given as a parametrized balanced viscosity solution of a rate-independent system. Under certain assumptions on the data one can prove the existence of globally optimal solutions for external loads in H¹(0,T). Moreover, we investigate the approximability of optimal solutions by viscous regularized problems. However,...

536. Low-regret shape optimization in the presence of missing data

John Sebastian Simon