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- ItemA Bayesian approach to parameter identification in gas networks(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Hajian, Soheil; Hintermüller, Michael; Schillings, Claudia; Strogies, NikolaiThe inverse problem of identifying the friction coefficient in an isothermal semilinear Euler system is considered. Adopting a Bayesian approach, the goal is to identify the distribution of the quantity of interest based on a finite number of noisy measurements of the pressure at the boundaries of the domain. First well-posedness of the underlying non-linear PDE system is shown using semigroup theory, and then Lipschitz continuity of the solution operator with respect to the friction coefficient is established. Based on the Lipschitz property, well-posedness of the resulting Bayesian inverse problem for the identification of the friction coefficient is inferred. Numerical tests for scalar and distributed parameters are performed to validate the theoretical results.
- ItemAuditory cortex modelled as a dynamical network of oscillators: Understanding event-related fields and their adaptation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Hajizadeh, Aida; Matysiak, Artur; Wolfrum, Matthias; May, Patrick J. C.Adaptation, the reduction of neuronal responses by repetitive stimulation, is a ubiquitous feature of auditory cortex (AC). It is not clear what causes adaptation, but short-term synaptic depression (STSD) is a potential candidate for the underlying mechanism. We examined this hypothesis via a computational model based on AC anatomy, which includes serially connected core, belt, and parabelt areas. The model replicates the event-related field (ERF) of the magnetoencephalogram as well as ERF adaptation. The model dynamics are described by excitatory and inhibitory state variables of cell populations, with the excitatory connections modulated by STSD. We analysed the system dynamics by linearizing the firing rates and solving the STSD equation using time-scale separation. This allows for characterization of AC dynamics as a superposition of damped harmonic oscillators, so-called normal modes. We show that repetition suppression of the N1m is due to a mixture of causes, with stimulus repetition modifying both the amplitudes and the frequencies of the normal modes. In this view, adaptation results from a complete reorganization of AC dynamics rather than a reduction of activity in discrete sources. Further, both the network structure and the balance between excitation and inhibition contribute significantly to the rate with which AC recovers from adaptation. This lifetime of adaptation is longer in the belt and parabelt than in the core area, despite the time constants of STSD being spatially constant. Finally, we critically evaluate the use of a single exponential function to describe recovery from adaptation.
- ItemA Redlich-Kister type free energy model for Li-intercalation compounds with variable lattice occupation numbers(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Landstorfer, ManuelOne of the central quantities of a lithium ion intercalation compound is the open circuit potential, the voltage a battery material delivers in thermodynamic equilibrium. This voltage is related to the chemical potential of lithium in the insertion material and in general a non-linear function of the mole fraction of intercalated lithium. Experimental data shows further that it is specific for various materials. The open circuit voltage is a central ingredient for mathematical models of whole battery cells, which are used to investigate and simulate the charge and discharge behavior and to interpret experimental data on non-equilibrium processes. However, since no overall predictive theoretical method presently exists for the open circuit voltage, it is commonly fitted to experimental data. Simple polynomial fitting approaches are widely used, but they lack any thermodynamic interpretation. More recently systematically and thermodynamically motivated approaches are used to model the open circuit potential. We provide here an explicit free energy density which accounts for variable occupation numbers of Li on the intercalation lattice as well as RedlichKister-type enthalpy contributions. The derived chemical potential is validated by experimental data of Liy(Ni1/3Mn1/3Co1/3)O2 and we show that only two parameters are sufficient to obtain an overall agreement of the non-linear open circuit potential within the experimental error.
- ItemFrom nonlinear to linear elasticity in a coupled rate-dependent/independent system for brittle delamination(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Rossi, Riccarda; Thomas, MaritaWe revisit the weak, energetic-type existence results obtained in [RT15] for a system for rateindependent, brittle delamination between two visco-elastic, physically nonlinear bulk materials and explain how to rigorously extend such results to the case of visco-elastic, linearly elastic bulk materials. Our approximation result is essentially based on deducing the MOSCO-convergence of the functionals involved in the energetic formulation of the system. We apply this approximation result in two different situations at small strains: Firstly, to pass from a nonlinearly elastic to a linearly elastic, brittle model on the time-continuous level, and secondly, to pass from a time-discrete to a time-continuous model using an adhesive contact approximation of the brittle model, in combination with a vanishing, super-quadratic regularization of the bulk energy. The latter approach is beneficial if the model also accounts for the evolution of temperature.
- ItemWeak-strong uniqueness for energy-reaction-diffusion systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Hopf, KatharinaWe establish weak-strong uniqueness and stability properties of renormalised solutions to a class of energy-reaction-diffusion systems, which genuinely feature cross-diffusion effects. The systems considered are motivated by thermodynamically consistent models, and their formal entropy structure allows us to use as a key tool a suitably adjusted relative entropy method. Weak-strong uniqueness is obtained for general entropy-dissipating reactions without growth restrictions, and certain models with a non-integrable diffusive flux. The results also apply to a class of (isoenergetic) reaction-cross-diffusion systems.
- ItemNumerical analysis for nematic electrolytes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Baňas, L'ubomír; Lasarzik, Robert; Prohl, AndreasWe consider a system of nonlinear PDEs modeling nematic electrolytes, and construct a dissipative solution with the help of its implementable, structure-inheriting space-time discretization. Computational studies are performed to study the mutual effects of electric, elastic, and viscous effects onto the molecules in a nematic electrolyte.
- ItemWeak solutions and weak-strong uniqueness for a thermodynamically consistent phase-field model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Lasarzik, Robert; Rocca, Elisabetta; Schimperna, GiulioIn this paper we prove the existence of weak solutions for a thermodynamically consistent phase-field model introduced in [26] in two and three dimensions of space. We use a notion of solution inspired by [18], where the pointwise internal energy balance is replaced by the total energy inequality complemented with a weak form of the entropy inequality. Moreover, we prove existence of local-in-time strong solutions and, finally, we show weak-strong uniqueness of solutions, meaning that every weak solution coincides with a local strong solution emanating from the same initial data, as long as the latter exists.
- ItemFast reaction limits via $Gamma$-convergence of the flux rate functional(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Peletier, Mark A.; Renger, D. R. MichielWe study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; `slow' rates are constant, and `fast' rates are scaled as 1/∈, and we prove the convergence in the fast-reaction limit ∈ → 0. We establish a Γ-convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterizes both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the Γ-convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.
- ItemHigh order discretization methods for spatial-dependent epidemic models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Takács, Bálint; Hadjimichael, YiannisIn this paper, an SIR model with spatial dependence is studied and results regarding its stability and numerical approximation are presented. We consider a generalization of the original Kermack and McKendrick model in which the size of the populations differs in space. The use of local spatial dependence yields a system of integro-differential equations. The uniqueness and qualitative properties of the continuous model are analyzed. Furthermore, different choices of spatial and temporal discretizations are employed, and step-size restrictions for population conservation, positivity, and monotonicity preservation of the discrete model are investigated. We provide sufficient conditions under which high order numerical schemes preserve the discrete properties of the model. Computational experiments verify the convergence and accuracy of the numerical methods.
- ItemPeriodic Lp estimates by R-boundedness: Applications to the Navier--Stokes equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Eiter, Thomas; Kyed, Mads; Shibata, YoshihiroGeneral evolution equations in Banach spaces are investigated. Based on an operator-valued version of de Leeuw's transference principle, time-periodic Lp estimates of maximal regularity type are established from R-bounds of the family of solution operators (R-solvers) to the corresponding resolvent problems. With this method, existence of time-periodic solutions to the Navier--Stokes equations is shown for two configurations: in a periodically moving bounded domain and in an exterior domain, subject to prescribed time-periodic forcing and boundary data.