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- ItemUtilizing anatomical information for signal detection in functional magnetic resonance imaging(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Neumann, André; Peitek, Norman; Brechmann, André; Tabelow, Karsten; Dickhaus, ThorstenWe are considering the statistical analysis of functional magnetic resonance imaging (fMRI) data. As demonstrated in previous work, grouping voxels into regions (of interest) and carrying out a multiple test for signal detection on the basis of these regions typically leads to a higher sensitivity when compared with voxel-wise multiple testing approaches. In the case of a multi-subject study, we propose to define the regions for each subject separately based on their individual brain anatomy, represented, e.g., by so-called Aparc labels. The aggregation of the subject-specific evidence for the presence of signals in the different regions is then performed by means of a combination function for p-values. We apply the proposed methodology to real fMRI data and demonstrate that our approach can perform comparably to a two-stage approach for which two independent experiments are needed, one for defining the regions and one for actual signal detection.
- ItemStability of the solution set of quasi-variational inequalities and optimal control(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Alphonse, Amal; Hintermüller, Michael; Rautenberg, Carlos N.For a class of quasivariational inequalities (QVIs) of obstacle-type the stability of its solution set and associated optimal control problems are considered. These optimal control problems are non-standard in the sense that they involve an objective with set-valued arguments. The approach to study the solution stability is based on perturbations of minimal and maximal elements to the solution set of the QVI with respect to monotonic perturbations of the forcing term. It is shown that different assumptions are required for studying decreasing and increasing perturbations and that the optimization problem of interest is well-posed.
- ItemA mathematical model for Alzheimer's disease: An approach via stochastic homogenization of the Smoluchowski equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Franchi, Bruno; Heida, Martin; Lorenzani, SilviaIn this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of the solution of a set of Smoluchowski's coagulation-diffusion equations with non-homogeneous Neumann boundary conditions. This system is meant to model the aggregation and diffusion of β-amyloid peptide (Aβ) in the cerebral tissue, a process associated with the development of Alzheimer's disease. In contrast to the approach used in our previous works, in the present paper we account for the non-periodicity of the cellular structure of the brain by assuming a stochastic model for the spatial distribution of neurons. Further, we consider non-periodic random diffusion coefficients for the amyloid aggregates and a random production of Aβ in the monomeric form at the level of neuronal membranes.
- ItemThe geometry of controlled rough paths(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Varzaneh, Mazyar Ghani; Riedel, Sebastian; Schmeding, Alexander; Tapia, NikolasWe prove that the spaces of controlled (branched) rough paths of arbitrary order form a continuous field of Banach spaces. This structure has many similarities to an (infinite-dimensional) vector bundle and allows to define a topology on the total space, the collection of all controlled path spaces, which turns out to be Polish in the geometric case. The construction is intrinsic and based on a new approximation result for controlled rough paths. This framework turns well-known maps such as the rough integration map and the Itô-Lyons map into continuous (structure preserving) mappings. Moreover, it is compatible with previous constructions of interest in the stability theory for rough integration.
- ItemOn compatibility of the natural configuration framework with general equation for non-equilibrium reversible-irreversible coupling (GENERIC): Derivation of anisotropic rate-type models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Pelech, Petr; Tůma, Karel; Pavelka, Michal; Šípka, Martin; Sýkora, MartinWithin the framework of natural configurations developed by Rajagopal and Srinivasa, evolution within continuum thermodynamics is formulated as evolution of a natural configuration linked with the current configuration. On the other hand, withing the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) framework, the evolution is split into Hamiltonian mechanics and (generalized) gradient dynamics. These seemingly radically different approaches have actually a lot in common and we show their compatibility on a wide range of models. Both frameworks are illustrated on isotropic and anisotropic rate-type fluid models. We propose an interpretation of the natural configurations within GENERIC and vice versa (when possible).
- ItemA kinetic model of a polyelectrolyte gel undergoing phase separation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Celora, Giulia L.; Hennessy, Matthew G.; Münch, Andreas; Wagner, Barbara; Waters, Sarah L.In this study we use non-equilibrium thermodynamics to systematically derive a phase-field model of a polyelectrolyte gel coupled to a thermodynamically consistent model for the salt solution surrounding the gel. The governing equations for the gel account for the free energy of the internal interfaces which form upon phase separation, as well as finite elasticity and multi-component transport. The fully time-dependent model describes the evolution of small changes in the mobile ion concentrations and follows their impact on the large-scale solvent flux and the emergence of long-time pattern formation in the gel. We observe a strong acceleration of the evolution of the free surface when the volume phase transition sets in, as well as the triggering of spinodal decomposition that leads to strong inhomogeneities in the lateral stresses, potentially leading to experimentally visible patterns.
- ItemA continuum model for yttria-stabilised zirconia incorporating triple phase boundary, lattice structure and immobile oxide ions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Vágner, Petr; Guhlke, Clemens; Miloš, Vojtěch; Müller, Rüdiger; Fuhrmann, JürgenA continuum model for yttria-stabilised zirconia (YSZ) in the framework of non-equilibrium thermodynamics is developed. Particular attention is given to i) modeling of the YSZ-metal-gas triple phase boundary, ii) incorporation of the lattice structure and immobile oxide ions within the free energy model and iii) surface reactions. A finite volume discretization method based on modified Scharfetter-Gummel fluxes is derived in order to perform numerical simulations. The model is used to study the impact of yttria and immobile oxide ions on the structure of the charged boundary layer and the double layer capacitance. Cyclic voltammograms of an air-half cell are simulated to study the effect of parameter variations on surface reactions, adsorption and anion diffusion.
- ItemNoise-induced dynamical regimes in a system of globally coupled excitable units(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Klinshov, Vladimir V.; Kirillov, Sergey Yu.; Nekorkin, Vladimir I.; Wolfrum, MatthiasWe study the interplay of global attractive coupling and individual noise in a system of identical active rotators in the excitable regime. Performing a numerical bifurcation analysis of the nonlocal nonlinear Fokker-Planck equation for the thermodynamic limit, we identify a complex bifurcation scenario with regions of different dynamical regimes, including collective oscillations and coexistence of states with different levels of activity. In systems of finite size this leads to additional dynamical features, such as collective excitability of different types, noise-induced switching and bursting. Moreover, we show how characteristic quantities such as macroscopic and microscopic variability of inter spike intervals can depend in a non-monotonous way on the noise level.
- ItemBumps, chimera states, and Turing patterns in systems of coupled active rotators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Franović, Igor; Omel'chenko, Oleh E.; Wolfrum, MatthiasSelf-organized coherence-incoherence patterns, called chimera states, have first been reported in systems of Kuramoto oscillators. For coupled excitable units, similar patterns where coherent units are at rest, are called bump states. Here, we study bumps in an array of active rotators coupled by non-local attraction and global repulsion. We demonstrate how they can emerge in a supercritical scenario from completely coherent Turing patterns: a single incoherent unit appears in a homoclinic bifurcation, undergoing subsequent transitions to quasiperiodic and chaotic behavior, which eventually transforms into extensive chaos with many incoherent units. We present different types of transitions and explain the formation of coherence-incoherence patterns according to the classical paradigm of short-range activation and long-range inhibition.
- ItemFrom optimal martingales to randomized dual optimal stopping(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Belomestny, Denis; Schoenmakers, John G. M.In this article we study and classify optimal martingales in the dual formulation of optimal stopping problems. In this respect we distinguish between weakly optimal and surely optimal martingales. It is shown that the family of weakly optimal and surely optimal martingales may be quite large. On the other hand it is shown that the Doob-martingale, that is, the martingale part of the Snell envelope, is in a certain sense the most robust surely optimal martingale under random perturbations. This new insight leads to a novel randomized dual martingale minimization algorithm that does`nt require nested simulation. As a main feature, in a possibly large family of optimal martingales the algorithm efficiently selects a martingale that is as close as possible to the Doob martingale. As a result, one obtains the dual upper bound for the optimal stopping problem with low variance.