arXiv subjects

Max Winkler

Publications and source records attributed to Max Winkler.

14 recordsLinked to original sources

arXiv2025

A stochastic Galerkin method for optimal Dirichlet boundary control problems with uncertain data

The paper deals with a stochastic Galerkin approximation of elliptic Dirichlet boundary control problems with random input data. The expectation of a tracking cost functional with the deterministic constrained control is minimized. Error estimates are derived for the control variable in $L^2(\partial \mathcal D)$-norm and state variable in $L^2(\Omega\times\mathcal D)$-norm. To solve large linear systems, appropriate preconditioners are proposed for both unconstrained and constrained scenarios. To illustrate the validity and efficiency of the proposed approaches, some numerical experiments are performed.

math.OC↗

arXiv2025

Physics-Informed DeepONets for drift-diffusion on metric graphs: simulation and parameter identification

We develop a novel physics informed deep learning approach for solving nonlinear drift-diffusion equations on metric graphs. These models represent an important model class with a large number of applications in areas ranging from transport in biological cells to the motion of human crowds. While traditional numerical schemes require a large amount of tailoring, especially in the case of model design or parameter identification problems, physics informed deep operator networks (DeepONet) have emerged as a versatile tool for the solution of partial differential equations with the particular advantage that they easily incorporate parameter identification questions. We here present an approach where we first learn three DeepONet models for representative inflow, inner and outflow edges, resp., and then subsequently couple these models for the solution of the drift-diffusion metric graph problem by relying on an edge-based domain decomposition approach. We illustrate that our framework is applicable for the accurate evaluation of graph-coupled physics models and is well suited for solving optimization or inverse problems on these coupled networks.

cs.LG↗

arXiv2023

A free boundary model for transport induced neurite growth

We introduce a free boundary model to example the effect of vesicle transport onto neurite growth. It consists of systems of drift-diffusion equations describing the evolution of the density of antero- and retrograde vesicles in each neurite coupled to reservoirs located at the soma and the growth cones of the neurites, respectively. The model allows for a change of neurite length depending on the vesicle concentration in the growth cones. After establishing existence and uniqueness for the time-dependent problem, we briefly comment on possible types of stationary solutions. Finally, we provide numerical studies on biologically relevant scales using a finite volume scheme. We illustrate the capability of the model to reproduce cycles of extension and retraction.

math.AP↗

arXiv2023

Numerical investigation of agent controlled pedestrian dynamics using a structure preserving finite volume scheme

We provide a numerical realisation of an optimal control problem for pedestrian motion with agents that was analysed in Herzog, Pietschmann, Winkler: "Optimal Control of Hughes' Model for Pedestrian Flow via Local Attraction.", arXiv 2011.03580, 2020. The model consists of a regularized variant of Hughes' model for pedestrian dynamics coupled to ordinary differential equations that describe the motion of agents which are able to influence the crowd via attractive forces. We devise a finite volume scheme that preserves the box constraints that are inherent in the model and discuss some of its properties. We apply our scheme to an objective functional tailored to the case of an evacuation scenario. Finally, numerical simulations for several practically relevant geometries are performed.

math.OC↗

arXiv2022

A comparison of PINN approaches for drift-diffusion equations on metric graphs

In this paper we focus on comparing machine learning approaches for quantum graphs, which are metric graphs, i.e., graphs with dedicated edge lengths, and an associated differential operator. In our case the differential equation is a drift-diffusion model. Computational methods for quantum graphs require a careful discretization of the differential operator that also incorporates the node conditions, in our case Kirchhoff-Neumann conditions. Traditional numerical schemes are rather mature but have to be tailored manually when the differential equation becomes the constraint in an optimization problem. Recently, physics informed neural networks (PINNs) have emerged as a versatile tool for the solution of partial differential equations from a range of applications. They offer flexibility to solve parameter identification or optimization problems by only slightly changing the problem formulation used for the forward simulation. We compare several PINN approaches for solving the drift-diffusion on the metric graph.

stat.ML↗

arXiv2020

UV resistance of nucleosides -- an experimental approach

The emergence of life on Earth is a highly discussed but still unsolved question. Current research underlines the importance of environments within close proximity to the Earth's surface as they can solve long-standing problems such as polymerization of nucleotides and phosphorylation of nucleosides. However, near-surface settings, for example, ponds or ice shields, are prone to UV irradiation. We investigated the photosensitivity of uracil, uridine, adenosine, cytidine, and guanosine by using Raman microscopy. The samples were irradiated by a UV source with 150~mW/cm\textsuperscript{2} for 10 min. Uracil and uridine showed the highest photosensitivity, while adenosine, cytidine, and guanosine remained stable. The change of spectral features and ab initio quantum calculations indicate the formation of uracil's trans-syn cyclobutane dimer during UV irradiation.

physics.chem-ph↗

arXiv2020

Optimal Control of Hughes' Model for Pedestrian Flow via Local Attraction

We discuss the control of a human crowd whose dynamics is governed by a regularized version of Hughes' model, cf. Hughes: A continuum theory for the flow of pedestrians. Transportation research part B: methodological, 36 (2002). We assume that a finite number of agents act on the crowd and try to optimize their paths in a given time interval. The objective functional can be general and it can correspond, for instance, to the desire for fast evacuation or to maintain a single group of individuals. We provide an existence result for the forward model, study differentiability properties of the control-to-state map, establish the existence of a globally optimal control and formulate optimality conditions.

math.OC↗

arXiv2020

Error estimation for second-order PDEs in non-variational form

Second-order partial differential equations in non-divergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton-Jacobi-Bellman equations in the context of stochastic optimal control, or as the linearization of fully nonlinear second-order PDEs. The non-divergence form in these problems is natural. If the coefficients of the diffusion matrix are not differentiable, the problem can not be transformed into the more convenient variational form. We investigate tailored non-conforming finite element approximations of second-order PDEs in non-divergence form, utilizing finite element Hessian recovery strategies to approximate second derivatives in the equation. We study both approximations with continuous and discontinuous trial functions. Of particular interest are a priori and a posteriori error estimates as well as adaptive finite element methods. In numerical experiments our method is compared with other approaches known from the literature.

math.NA↗

arXiv2020

Habitability of the early Earth: Liquid water under a faint young Sun facilitated by strong tidal heating due to a closer Moon

Geological evidence suggests liquid water near the Earth's surface as early as 4.4 billion years ago when the faint young Sun only radiated about 70% of its modern power output. At this point, the Earth should have been a global snowball if it possessed atmospheric properties similar to those of the modern Earth. An extreme atmospheric greenhouse effect, an initially more massive Sun, release of heat acquired during the accretion process of protoplanetary material, and radioactivity of the early Earth material have been proposed as reservoirs or traps for heat. For now, the faint-young-sun paradox persists as an important problem in our understanding of the origin of life on Earth. Here we use the constant-phase-lag tidal theory to explore the possibility that the new-born Moon, which formed about 69 million years after the ignition of the Sun, generated extreme tidal friction - and therefore heat - in the Hadean and possibly the Archean Earth. We show that the Earth-Moon system has lost about $3~ \times ~10^{31}$ J (99% of its initial mechanical energy budget) as tidal heat. Tidal heating of roughly 10 W/m$^{-2}$ through the surface on a time scale of 100 million years could have accounted for a temperature increase of up to 5 degrees Celsius on the early Earth. This heating effect alone does not solve the faint-young-sun paradox but it could have played a key role in combination with other effects. Future studies of the interplay of tidal heating, the evolution of the solar power output, and the atmospheric (greenhouse) effects on the early Earth could help in solving the faint-young-sun paradox.

astro-ph.EP↗

arXiv2019

Optimization of a partial differential equation on a complex network

Differential equations on metric graphs can describe many phenomena in the physical world but also the spread of information on social media. To efficiently compute the solution is a hard task in numerical analysis. Solving a design problem, where the optimal setup for a desired state is given, is even more challenging. In this work, we focus on the task of solving an optimization problem subject to a differential equation on a metric graph with the control defined on a small set of Dirichlet nodes. We discuss the discretization by finite elements and provide rigorous error bounds as well as an efficient preconditioning strategy to deal with the large-scale case. We show in various examples that the method performs very robustly.

math.OC↗

arXiv2019

Error estimates for the finite element approximation of bilinear boundary control problems

In this article a special class of nonlinear optimal control problems involving a bilinear term in the boundary condition is studied. These kind of problems arise for instance in the identification of an unknown space-dependent Robin coefficient from a given measurement of the state, or when the Robin coefficient can be controlled in order to reach a desired state. To this end, necessary and sufficient optimality conditions are derived and several discretization approaches for the numerical solution the optimal control problem are investigated. Considered are both a full discretization and the postprocessing approach meaning that we compute an improved control by a pointwise evaluation of the first-order optimality condition. For both approaches finite element error estimates are shown and the validity of these results is confirmed by numerical experiments.

math.NA↗

arXiv2018

Error estimates for variational normal derivatives and Dirichlet control problems with energy regularization

This article deals with error estimates for the finite element approximation of variational normal derivatives and, as a consequence, error estimates for the finite element approximation of Dirichlet boundary control problems with energy regularization. The regularity of the solution is carefully carved out exploiting weighted Sobolev and H\"older spaces. This allows to derive a sharp relation between the convergence rates for the approximation and the structure of the geometry, more precisely, the largest opening angle at the vertices of polygonal domains. Numerical experiments confirm that the derived convergence rates are sharp.

math.NA↗

arXiv2018

$L^{\infty}$-error estimates for Neumann boundary value problems on graded meshes

This paper deals with a priori pointwise error estimates for the finite element solution of boundary value problems with Neumann boundary conditions in polygonal domains. Due to the corners of the domain, the convergence rate of the numerical solutions can be lower than in case of smooth domains. As a remedy the use of local mesh refinement near the corners is investigated. In order to prove quasi-optimal a priori error estimates regularity results in weighted Sobolev spaces are exploited. This is the first work on the Neumann boundary value problem where both the regularity of the data is exactly specified and the sharp convergence order $h^2 \lvert \ln h \rvert $ in the case of piecewise linear finite element approximations is obtained. As an extension we show the same rate for the approximate solution of a semilinear boundary value problem. The proof relies in this case on the supercloseness between the Ritz projection to the continuous solution and the finite element solution.

math.NA↗

arXiv2013

Anisotropic mesh refinement in polyhedral domains: error estimates with data in L^2(\Omega)

The paper is concerned with the finite element solution of the Poisson equation with homogeneous Dirichlet boundary condition in a three-dimensional domain. Anisotropic, graded meshes from a former paper are reused for dealing with the singular behaviour of the solution in the vicinity of the non-smooth parts of the boundary. The discretization error is analyzed for the piecewise linear approximation in the H^1(\Omega)- and L^2(\Omega)-norms by using a new quasi-interpolation operator. This new interpolant is introduced in order to prove the estimates for L^2(\Omega)-data in the differential equation which is not possible for the standard nodal interpolant. These new estimates allow for the extension of certain error estimates for optimal control problems with elliptic partial differential equation and for a simpler proof of the discrete compactness property for edge elements of any order on this kind of finite element meshes.

math.NA↗