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  • Forschungsbericht  (17)
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  • Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e.V.  (17)
  • 1
    Online Resource
    Online Resource
    Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e.V.
    Keywords: Forschungsbericht
    Description / Table of Contents: We study the class of continuous polynomial Volterra processes, which we define as solutions to stochastic Volterra equations driven by a continuous semimartingale with affine drift and quadratic diffusion matrix in the state of the Volterra pro- cess. To demonstrate the versatility of possible state spaces within our framework, we construct polynomial Volterra processes on the unit ball. This construction is based on a stochastic invariance principle for stochastic Volterra equations with possibly singular kernels. Similarly to classical polynomial processes, polynomial Volterra processes allow for tractable expressions of the moments in terms of the unique solution to a system of deterministic integral equations, which reduce to a system of ODEs in the classical case. By applying this observation to the moments of the finite-dimensional distributions we derive a uniqueness result for polynomial Volterra processes. Moreover, we prove that the moments are polynomials with respect to the initial condition, another crucial property shared by classical polyno- mial processes. The corresponding coefficients can be interpreted as a deterministic dual process and solve integral equations dual to those verified by the moments themselves. Additionally, we obtain a representation of the moments in terms of a pure jump process with killing, which corresponds to another non-deterministic dual process.
    Type of Medium: Online Resource
    Pages: 1 Online-Ressource (34 Seiten, 405,07 KB) , Illustrationen, Diagramme
    Series Statement: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik no. 3098
    Language: English
    Note: Literaturverzeichnis: Seite 30-32
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  • 2
    Online Resource
    Online Resource
    Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e.V.
    Keywords: Forschungsbericht
    Type of Medium: Online Resource
    Pages: 1 Online-Ressource (17 Seiten, 431,44 KB) , Diagramme
    Series Statement: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik no. 3100
    Language: English
    Note: Literaturverzeichnis: Seite 14-15
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  • 3
    Online Resource
    Online Resource
    Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e.V.
    Keywords: Forschungsbericht
    Description / Table of Contents: This work focuses on the numerical solution of the dynamics of a poroelastic material in the frequency domain. We provide a detailed stability analysis based on the application of the Fredholm alternative in the continuous case, considering a total pressure formulation of the Biot’s equations. In the discrete setting, we propose a stabilized equal order finite element method complemented by an additional pressure stabilization to enhance the robustness of the numerical scheme with respect to the fluid permeability. Utilizing the Fredholm alternative, we extend the well-posedness results to the discrete setting, obtaining theoretical optimal convergence for the case of linear finite elements. We present different numerical experiments to validate the proposed method. First, we consider model problems with known analytic solutions in two and three dimensions. As next, we show that the method is robust for a wide range of permeabilities, including the case of discontinuous coefficients. Lastly, we show the application for the simulation of brain elastography on a realistic brain geometry obtained from medical imaging.
    Type of Medium: Online Resource
    Pages: 1 Online-Ressource (31 Seiten, 7,22 MB) , Illustrationen, Diagramme
    Series Statement: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik no. 3101
    Language: English
    Note: Literaturverzeichnis: Seite 26-29
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  • 4
    Online Resource
    Online Resource
    Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e.V.
    Keywords: Forschungsbericht
    Description / Table of Contents: In the field of quantitative imaging, the image information at a pixel or voxel in an underlying domain entails crucial information about the imaged matter. This is particularly important in medical imaging applications, such as quantitative Magnetic Resonance Imaging (qMRI), where quantitative maps of biophysical parameters can characterize the imaged tissue and thus lead to more accurate diagnoses. Such quantitative values can also be useful in subsequent, automatized classification tasks in order to discriminate normal from abnormal tissue, for instance. The accurate reconstruction of these quantitative maps is typically achieved by solving two coupled inverse problems which involve a (forward) measurement operator, typically ill-posed, and a physical process that links the wanted quantitative parameters to the reconstructed qualitative image, given some underlying measurement data. In this review, by considering qMRI as a prototypical application, we provide a mathematically-oriented overview on how data-driven approaches can be employed in these inverse problems eventually improving the reconstruction of the associated quantitative maps.
    Type of Medium: Online Resource
    Pages: 1 Online-Ressource (39 Seiten, 7,27 MB) , Illustrationen, Diagramme
    Series Statement: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik no. 3105
    Language: English
    Note: Literaturverzeichnis: Seite 30-37
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  • 5
    Online Resource
    Online Resource
    Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e.V.
    Keywords: Forschungsbericht
    Description / Table of Contents: We consider the soft Boolean model, a model that interpolates between the Boolean model and long-range percolation, where vertices are given via a stationary Poisson point process. Each vertex carries an independent Pareto-distributed radius and each pair of vertices is assigned another independent Pareto weight with a potentially different tail exponent. Two vertices are now connected if they are within distance of the larger radius multiplied by the edge weight. We determine the tail behaviour of the Euclidean diameter and the number of points of a typical maximally connected component in a subcritical percolation phase. For this, we present a sharp criterion in terms of the tail exponents of the edge-weight and radius distributions that distinguish a regime where the tail behaviour is controlled only by the edge exponent from a regime in which both exponents are relevant. Our proofs rely on fine path-counting arguments identifying the precise order of decay of the probability that far-away vertices are connected.
    Type of Medium: Online Resource
    Pages: 1 Online-Ressource (45 Seiten, 455,48 KB) , Diagramme
    Series Statement: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik no. 3106
    Language: English
    Note: Literaturverzeichnis: Seite 40-43
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  • 6
    Online Resource
    Online Resource
    Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e.V.
    Keywords: Forschungsbericht
    Description / Table of Contents: A thermodynamically consistent visco-elastodynamical model at finite strains is derived that also allows for inelasticity (like plasticity or creep), thermal coupling, and poroelasticity with diffusion. The theory is developed in the Eulerian framework and is shown to be consistent with the thermodynamic framework given by General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC). For the latter we use that the transport terms are given in terms of Lie derivatives. Application is illustrated by two examples, namely volumetric phase transitions with dehydration in rocks and martensitic phase transitions in shape-memory alloys. A strategy towards a rigorous mathematical analysis is only very briefly outlined.
    Type of Medium: Online Resource
    Pages: 1 Online-Ressource (32 Seiten, 499,20 KB) , Diagramme
    Series Statement: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik no. 3107
    Language: English
    Note: Literaturverzeichnis: Seite 27-30
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  • 7
    Online Resource
    Online Resource
    Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e.V.
    Keywords: Forschungsbericht
    Type of Medium: Online Resource
    Pages: 1 Online-Ressource (72 Seiten, 2,15 MB) , Diagramme, Illustrationen
    Series Statement: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik no. 3111
    Language: English
    Note: Literaturverzeichnis: Seite 66-68
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  • 8
    Online Resource
    Online Resource
    Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e.V.
    Keywords: Forschungsbericht
    Description / Table of Contents: We exhibit a percolating ergodic and isotropic lattice model in all but at least two dimensions that has zero effective conductivity in all spatial directions and for all non-trivial choices of the connectivity parameter. The model is based on the so-called randomly stretched lattice where we additionally elongate layers containing few open edges.
    Type of Medium: Online Resource
    Pages: 1 Online-Ressource (16 Seiten, 387,47 KB) , Diagramme
    Series Statement: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik no. 3095
    Language: English
    Note: Literaturverzeichnis: Seite 13-14
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  • 9
    Online Resource
    Online Resource
    Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e.V.
    Keywords: Forschungsbericht
    Description / Table of Contents: We derive large- and moderate-deviation results in random networks given as planar directed navigations on homogeneous Poisson point processes. In this non-Markovian routing scheme, starting from the origin, at each consecutive step a Poisson point is joined by an edge to its nearest Poisson point to the right within a cone. We establish precise exponential rates of decay for the probability that the vertical displacement of the random path is unexpectedly large. The proofs rest on controlling the dependencies of the individual steps and the randomness in the horizonal displacement as well as renewal-process arguments.
    Type of Medium: Online Resource
    Pages: 1 Online-Ressource (28 Seiten, 333,53 KB) , Diagramme
    Series Statement: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik no. 3096
    Language: English
    Note: Literaturverzeichnis: Seite 25-26
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  • 10
    Online Resource
    Online Resource
    Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e.V.
    Keywords: Forschungsbericht
    Description / Table of Contents: We consider the edge-triangle model (or Strauss model), and focus on the asymptotic behavior of the triangle density when the size of the graph increases to infinity. In the analyticity region of the free energy, we prove a law of large numbers for the triangle density. Along the critical curve, where analyticity breaks down, we show that the triangle density concentrates with high probability in a neighborhood of its typical value. A predominant part of our work is devoted to the study of a mean-field approximation of the edge-triangle model, where explicit computations are possible. In this setting we can go further, and additionally prove a standard and non-standard central limit theorem at the critical point, together with many concentration results obtained via large deviations and statistical mechanics techniques. Despite a rigorous comparison between these two models is still lacking, we believe that they are asymptotically equivalent in many respects, therefore we formulate conjectures on the edge-triangle model, partially supported by simulations, based on the mean-field investigation.
    Type of Medium: Online Resource
    Pages: 1 Online-Ressource (37 Seiten, 754,05 KB) , Diagramme
    Series Statement: Preprint / Weierstraß-Institut für Angewandte Analysis und Stochastik no. 3102
    Language: English
    Note: Literaturverzeichnis: Seite 33-35
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