Lehrstuhl für Inverse Probleme
Refine
Year of publication
Document Type
- Article (57)
- Book (6)
- Preprint (3)
- Part of a Book (2)
- Conference Proceeding (1)
- Doctoral Thesis (1)
Keywords
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.
This thesis deals with the analysis and numerical simulation of different partial differential equation models arising in socioeconomic sciences. It is divided into two parts: The first part deals with a mean-field price formation model introduced by Lasry and Lions in 2007. This model describes the dynamic behaviour of the price of a good being traded between a group of buyers and a group of vendors. Existence (locally in time) of smooth solutions is established, and obstructions to proving a global existence result are examined. Also, properties of a regularised version of the model are explored and numerical examples are shown. Furthermore, the possibility of reconstructing the initial datum given a number of observations, regarding the price and the transaction rate, is considered. Using a variational approach, the problem can be expressed as a non-linear constrained minimization problem. We show that the initial datum is uniquely determined by the price (identifiability). Furthermore, a numerical scheme is implemented and a variety of examples are presented. The second part of this thesis treats two different models describing the motion of (large) human crowds. For the first model, introduced by R.L. Hughes in 2002, several regularised versions are considered. Existence and uniqueness of entropy solutions are proven using the technique of vanishing viscosity. In one space dimension, the dynamic behaviour of solutions of the original model is explored for some special cases. These results are compared to numerical simulations. Moreover, we consider a discrete cellular automaton model introduced by A. Kirchner and A. Schadschneider in 2002. By (formally) passing to the continuum limit, we obtain a system of partial differential equations. Some analytical properties, such as linear stability of stationary states, are examined and extensive numerical simulations show capabilities and limitations of the model in both the discrete and continuous setting.
Data-driven gradient flows
(2022)
We present a framework enabling variational data assimilation for gradient flows in general metric spaces, based on the minimizing movement (or Jordan–Kinderlehrer–Otto) approximation scheme. After discussing stability properties in the most general case, we specialize to the space of probability measures endowed with the Wasserstein distance. This setting covers many non-linear partial differential equations (PDEs), such as the porous-medium equation or general drift–diffusion–aggregation equations, which can be treated by our methods independently of their respective properties (such as finite speed of propagation or blow-up). We then focus on the numerical implementation using a primal–dual algorithm. The strength of our approach lies in the fact that, by simply changing the driving functional, a wide range of PDEs can be treated without the need to adopt the numerical scheme. We conclude by presenting several numerical examples.
Active crowds
(2022)
This chapter focuses on the mathematical modelling of active particles (or agents) in crowded environments. We discuss several microscopic models found in the literature and the derivation of the respective macroscopic partial differential equations for the particle density. The macroscopic models share common features, such as cross-diffusion or degenerate mobilities. We then take the diversity of macroscopic models to a uniform structure and work out potential similarities and differences. Moreover, we discuss boundary effects and possible applications in life and social sciences. This is complemented by numerical simulations that highlight the effects of different boundary conditions.
Rectifying nanopores feature ion currents that are higher for voltages of one polarity compared to the currents recorded for corresponding voltages of the opposite polarity. Rectification is important for many physiological functions e.g., transducing information along axons, and functioning of the heart. Synthetic pores with asymmetric current-voltage curves can function as switches for ions and charged molecules in solutions and could become building blocks for ionic circuits used in logic and sensory systems. Rectification of nanopores has been found to depend on the pore opening diameter, distribution of surface charges of the pore walls as well as pore geometry. Very little is however known on the dependence of ionic rectification on the type of transported ions. We performed experiments with single conically shaped nanopores in a polymer film, and recorded current-voltage curves in three electrolytes: LiCl, NaCl and KCl. Rectification degree of the pores, quantified as the ratio of currents recorded for voltages of opposite polarities, was the highest for KCl, and the lowest for LiCl. The currents recorded in LiCl as the bulk electrolyte were also several times lower than predicted from relative bulk conductivities of KCl and LiCl. The experimental observations could not be explained by a continuum modeling based on the Poisson-Nernst-Planck equations. All-atom molecular dynamics simulations revealed differential binding between Li+, Na+, and K+ ions and carboxylate residues on the pore wall, resulting in changes to both the effective surface charge of the nanopore and cation mobility within the pore. Differences in ionic current rectification follows from these factors.
The aim of this paper is to further develop mathematical models for bleb formation in cells, including cell membrane interactions with linker proteins. This leads to nonlinear reaction–diffusion equations on a surface coupled to fluid dynamics in the bulk. We provide a detailed mathematical analysis and investigate some singular limits of the model, connecting it to previous literature. Moreover, we provide numerical simulations in different scenarios, confirming that the model can reproduce experimental results on bleb initiation.
One of the key challenges for the future development of efficient and stable metal-halide perovskite solar cells is related to the migration of ions in these materials. Mobile ions have been linked to the observation of hysteresis in the current-voltage characteristics, shown to reduce device stability against degradation and act as recombination centers within the band gap of the active layer. In the literature, one finds a broad spread of reported ionic defect parameters (e.g., activation energies) for seemingly similar perovskite materials, rendering the identification of the nature of these species difficult. In this work, we perform temperature-dependent deep-level transient spectroscopy (DLTS) measurements on methylammonium-lead-iodide perovskite solar cells and develop a extended regularization algorithm for inverting the Laplace transform. Our results indicate that mobile ions form a distribution of emission rates (i.e., a distribution of diffusion constants) for each observed ionic species, which may be responsible for the differences in the previously reported defect parameters. Importantly, different DLTS modes such as optical and current DLTS yield the same defect distributions. Finally, the comparison of our results with conventional boxcar DLTS and impedance spectroscopy verifies our evaluation algorithm.
Motivated by modeling transport processes in the growth of neurons, we present results on (nonlinear) Fokker-Planck equations where the total mass is not conserved. This is either due to in- and outflow boundary conditions or to spatially distributed reaction terms. We are able to prove exponential decay towards equilibrium using entropy methods in several situations. As there is no conservation of mass it is difficult to exploit the gradient flow structure of the differential operator which renders the analysis more challenging. In particular, classical logarithmic Sobolev inequalities are not applicable any more. Our analytic results are illustrated by extensive numerical studies.
In this paper, we derive and analyse mean-field models for the dynamics of groups of individuals undergoing a random walk. The random motion of individuals is only influenced by the perceived densities of the different groups present as well as the available space. All individuals have the tendency to stay within their own group and avoid the others. These interactions lead to the formation of aggregates in case of a single species and to segregation in the case of multiple species. We derive two different mean-field models, which are based on these interactions and weigh local and non-local effects differently. We discuss existence and stability properties of solutions for both models and illustrate the rich dynamics with numerical simulations.
In this paper, we extend the results of [8] by proving exponential asymptotic H1-convergence of solutions to a one-dimensional singular heat equation with L2-source term that describe evolution of viscous thin liquid sheets while considered in the Lagrange coordinates. Furthermore, we extend this asymptotic convergence result to the case of a time inhomogeneous source. This study has also independent interest for the porous medium equation theory.
We study evolution equations of drift-diffusion type when various parameters are random. Motivated by applications in pedestrian dynamics, we focus on the case when the total mass is, due to boundary or reaction terms, not conserved. After providing existence and stability for the deterministic problem, we consider uncertainty in the data. Instead of a sensitivity analysis we propose to measure functionals of the solution, so-called quantities of interest (QoI), by involving scalarizing statistics. For these summarizing statistics we provide probabilistic continuity results.
In this paper we study double phase problems with nonlinear boundary condition and gradient dependence. Under quite general assumptions we prove existence results for such problems where the perturbations satisfy a suitable behavior in the origin and at infinity. Our proofs make use of variational tools, truncation techniques and comparison methods. The obtained solutions depend on the first eigenvalues of the Robin and Steklov eigenvalue problems for the p-Laplacian.
Boundary value problems for second-order elliptic equations in divergence form, whose nonlinearity is governed by a convex function of non-necessarily power type, are considered. The global boundedness of their solutions is established under boundary conditions of Dirichlet, or Neumann, or Robin type. A decisive role in the results is played by optimal forms of Orlicz-Sobolev embeddings and boundary trace embeddings, which allow for critical growths of the coefficients.
