arXiv subjects

Jan-Frederik Pietschmann

Publications and source records attributed to Jan-Frederik Pietschmann.

At least 37 records · Page 2Linked to original sources

arXiv2020

Data assimilation in price formation

We consider the problem of estimating the density of buyers and vendors in a nonlinear parabolic price formation model using measurements of the price and the transaction rate. Our approach is based on a work by Puel et al., see \cite{Puel2002}, and results in a optimal control problem. We analyse this problems and provide stability estimates for the controls as well as the unknown density in the presence of measurement errors. Our analytic findings are supported with numerical experiments.

math.OC↗

arXiv2020

Improved evaluation of deep-level transient spectroscopy on perovskite solar cells reveals ionic defect distribution

One of the key challenges for 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 performed temperature dependent deep-level transient spectroscopy (DLTS) measurements on methylammonium lead iodide perovskite solar cells and developed 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 (IS) verifies our evaluation algorithm.

physics.app-ph↗

arXiv2019

A PDE model for bleb formation and interaction with linker proteins

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 initation.

math.AP↗

arXiv2019

On the Role of Vesicle Transport in Neurite Growth: Modelling and Experiments

The processes that determine the establishment of the complex morphology of neurons during development are still poorly understood. We present experiments that use live imaging to examine the role of vesicle transport and propose a lattice-based model that shows symmetry breaking features similar to a neuron during its polarization. In a otherwise symmetric situation our model predicts that a difference in neurite length increases the growth potential of the longer neurite indicating that vesicle transport can be regarded as a major factor in neurite growth.

q-bio.SC↗

arXiv2019

Strong solutions and weak-strong stability in a system of cross-diffusion equations

Proving the uniqueness of solutions to multi-species cross-diffusion systems is a difficult task in the general case, and there exist very few results in this direction. In this work, we study a particular system with zero-flux boundary conditions for which the existence of a weak solution has been proven in [Ehrlacher2017]. Under additional assumptions on the value of the cross-diffusion coefficients, we are able to show the existence and uniqueness of strong solutions. The proof relies on the use of an appropriate approximation and a fixed-point argument. In addition, a weak-strong stability result is obtained for this system in dimension one.

math.AP↗

arXiv2019

Mean field models for segregation dynamics

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.

math.AP↗

arXiv2018

On Fokker-Planck Equations with In- and Outflow of Mass

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.

math.AP↗

arXiv2018

Parameter identification in a structured population model

We study parameter identification problems in a structured population model without mutations. Given measurements of the total population size or critical points of the population, we aim to recover its growth rate, death rate or initial distribution. We present uniqueness results under suitable assumptions and present counterexamples when these assumptions are violated. Our results a supplemented by numerical studies, either based on Tikhonov regularization or the use of explicit reconstruction formulas.

math.AP↗

arXiv2017

On a Reaction-Cross-Diffusion System Modelling the Growth of Glioblastoma

We investigate a recently proposed cross-diffusion system modelling the growth of gliobastoma taking into account size exclusion both in the migration and proliferation process. In addition to degenerate nonlinear cross-diffusion the model includes reaction terms as in the Fisher-Kolmogorov equation and linear ones modelling transition between states of proliferation and migration. We discuss the mathematical structure of the system and provide a complete existence analysis in spatial dimension one. The proof is based on exploiting partial entropy dissipation techniques and fully implicit time discretisations. In order to prove existence of the latter appropriate variational and fixed-point techniques are used, together with suitable a-priori estimates. Moreover, we review the existence of travelling waves and their relation to potential growth or decay of glioblastoma. Finally we provide extensive numerical studies in one and two spatial dimensions, including the effect of anisotropic diffusions as found in neural tissues.

math.AP↗

arXiv2017

On the uniqueness of nonlinear diffusion coefficients in the presence of lower order terms

We consider the identification of nonlinear diffusion coefficients of the form $a(t,u)$ or $a(u)$ in quasi-linear parabolic and elliptic equations. Uniqueness for this inverse problem is established under very general assumptions using partial knowledge of the Dirichlet-to-Neumann map. The proof of our main result relies on the construction of a series of appropriate Dirichlet data and test functions with a particular singular behavior at the boundary. This allows us to localize the analysis and to separate the principal part of the equation from the remaining terms. We therefore do not require specific knowledge of lower order terms or initial data which allows to apply our results to a variety of applications. This is illustrated by discussing some typical examples in detail.

math.AP↗

arXiv2017

On a cross-diffusion model for multiple species with nonlocal interaction and size exclusion

The aim of this paper is to study a PDE model for two diffusing species interacting by local size exclusion and global attraction. This leads to a nonlinear degenerate cross-diffusion system, for which we provide a global existence result. The analysis is motivated by the formulation of the system as a formal gradient flow for an appropriate energy functional consisting of entropic terms as well as quadratic nonlocal terms. Key ingredients are entropy dissipation methods as well as the recently developed boundedness by entropy principle. Moreover, we investigate phase separation effects inherent in the cross-diffusion model by an analytical and numerical study of minimizers of the energy functional and their asymptotics to a previously studied case as the diffusivity tends to zero. Finally we briefly discuss coarsening dynamics in the system, which can be observed in numerical results and is motivated by rewriting the PDEs as a system of nonlocal Cahn-Hilliard equations.

math.AP↗

arXiv2017

Multiscale modeling of a rectifying bipolar nanopore: Comparing Poisson-Nernst-Planck to Monte Carlo

In the framework of a multiscale modeling approach, we present a systematic study of a bipolar rectifying nanopore using a continuum and a particle simulation method. The common ground in the two methods is the application of the Nernst-Planck (NP) equation to compute ion transport in the framework of the implicit-water electrolyte model. The difference is that the Poisson-Boltzmann theory is used in the Poisson-Nernst-Planck (PNP) approach, while the Local Equilibrium Monte Carlo (LEMC) method is used in the particle simulation approach (NP+LEMC) to relate the concentration profile to the electrochemical potential profile. Since we consider a bipolar pore which is short and narrow, we perform simulations using two-dimensional PNP. In addition, results of a non-linear version of PNP that takes crowding of ions into account are shown. We observe that the mean field approximation applied in PNP is appropriate to reproduce the basic behavior of the bipolar nanopore (e.g., rectification) for varying parameters of the system (voltage, surface charge, electrolyte concentration, and pore radius). We present current data that characterize the nanopore's behavior as a device, as well as concentration, electrical potential, and electrochemical potential profiles.

cond-mat.soft↗

arXiv2015

Analysis and numerical solution of coupled volume-surface reaction-diffusion systems with application to cell biology

We consider the numerical solution of coupled volume-surface reaction-diffusion systems having a detailed balance equilibrium. Based on the conservation of mass, an appropriate quadratic entropy functional is identified and an entropy-entropy dissipation inequality is proven. This allows us to show exponential convergence to equilibrium by the entropy method. We then investigate the discretization of the system by a finite element method and an implicit time stepping scheme including the domain approximation by polyhedral meshes. Mass conservation and exponential convergence to equilibrium are established on the discrete level by arguments similar to those on the continuous level and we obtain estimates of optimal order for the discretization error which hold uniformly in time. Some numerical tests are presented to illustrate these theoretical results. The analysis and the numerical approximation are discussed in detail for a simple model problem. The basic arguments however apply also in a more general context. This is demonstrated by investigation of a particular volume-surface reaction-diffusion system arising as a mathematical model for asymmetric stem cell division.

math.NA↗

arXiv2015

Flow Characteristics in a Crowded Transport Model

The aim of this paper is to discuss the appropriate modelling of in- and outflow boundary conditions for nonlinear drift-diffusion models for the transport of particles including size exclusion and their effect on the behaviour of solutions. We use a derivation from a microscopic asymmetric exclusion process and its extension to particles entering or leaving on the boundaries. This leads to specific Robin-type boundary conditions for inflow and outflow, respectively. For the stationary equation we prove the existence of solutions in a suitable setup. Moreover, we investigate the flow characteristics for small diffusion, which yields the occurence of a maximal current phase in addition to well-known one-sided boundary layer effects for linear drift-diffusion problems. In a one-dimensional setup we provide rigorous estimates in terms of $ε$, which confirm three different phases. Finally, we derive a numerical approach to solve the problem also in multiple dimensions. This provides further insight and allows for the investigation of more complicated geometric setups.

math.AP↗

arXiv2015

Identification of Chemotaxis Models with Volume Filling

Chemotaxis refers to the directed movement of cells in response to a chemical signal called chemoattractant. A crucial point in the mathematical modeling of chemotactic processes is the correct description of the chemotactic sensitivity and of the production rate of the chemoattractant. In this paper, we investigate the identification of these non-linear parameter functions in a chemotaxis model with volume filling. We also discuss the numerical realization of Tikhonov regularization for the stable solution of the inverse problem. Our theoretical findings are supported by numerical tests.

math.AP↗

arXiv2014

Identification of nonlinear heat conduction laws

We consider the identification of nonlinear heat conduction laws in stationary and instationary heat transfer problems. Only a single additional measurement of the temperature on a curve on the boundary is required to determine the unknown parameter function on the range of observed temperatures. We first present a new proof of Cannon's uniqueness result for the stationary case, then derive a corresponding stability estimate, and finally extend our argument to instationary problems.

math.AP↗

arXiv2013

Simultaneous identification of diffusion and absorption coefficients in a quasilinear elliptic problem

In this work we consider the identifiability of two coefficients $a(u)$ and $c(x)$ in a quasilinear elliptic partial differential equation from observation of the Dirichlet-to-Neumann map. We use a linearization procedure due to Isakov [On uniqueness in inverse problems for semilinear parabolic equations. Archive for Rational Mechanics and Analysis, 1993] and special singular solutions to first determine $a(0)$ and $c(x)$ for $x \in Ω$. Based on this partial result, we are then able to determine $a(u)$ for $u \in \mathbb{R}$ by an adjoint approach.

math.AP↗

arXiv2013

Numerical identification of a nonlinear diffusion law via regularization in Hilbert scales

We consider the reconstruction of a diffusion coefficient in a quasilinear elliptic problem from a single measurement of overspecified Neumann and Dirichlet data. The uniqueness for this parameter identification problem has been established by Cannon and we therefore focus on the stable solution in the presence of data noise. For this, we utilize a reformulation of the inverse problem as a linear ill-posed operator equation with perturbed data and operators. We are able to explicitly characterize the mapping properties of the corresponding operators which allow us to apply regularization in Hilbert scales. We can then prove convergence and convergence rates of the regularized reconstructions under very mild assumptions on the exact parameter. These are, in fact, already needed for the analysis of the forward problem and no additional source conditions are required. Numerical tests are presented to illustrate the theoretical statements.

math.NA↗