Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics
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Limiting shape for first-passage percolation models on random geometric graphs
No full textLet a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape and we show that the shape is an Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. For the Richardson model we further show that it converges weakly to a branching process in the joint limit of large intensities and slow passing times
A model of gravitational differentiation of compressible self-gravitating planets
Get PDFWe present a dynamic model for inhomogeneous viscoelastic media at finite strains. The model features a Kelvin--Voigt rheology, and includes a self-generated gravitational field in the actual evolving configuration. In particular, a fully Eulerian approach is adopted. We specialize the model to viscoelastic (barotropic) fluids and prove existence and a certain regularity of global weak solutions by a Faedo--Galerkin semi-discretization technique. Then, an extension to multi-component chemically reacting viscoelastic fluids based on a phenomenological approach by Eckart and Prigogine, is advanced and studied. The model is inspired by planetary geophysics. In particular, it describes gravitational differentiation of inhomogeneous planets and moons, possibly undergoing volumetric phase transitions
Sharp deviation bounds and concentration phenomenon for the squared norm of a sub-gaussian vector
Get PDFLet \Xv be a Gaussian zero mean vector with \Var(\Xv) = \BBH . Then \| \Xv \|^{2} well concentrates around its expectation \dimA = \E \| \Xv \|^{2} = \tr \BBH provided that the latter is sufficiently large. Namely, ¶\bigl( \| \Xv \|^{2} - \tr \BBH > 2 \sqrt{\xx \tr(\BBH^{2})} + 2 \| \BBH \| \xx \bigr) \leq \ex^{-\xx} and ¶\bigl( \| \Xv \|^{2} - \tr \BBH < - 2 \sqrt{\xx \tr(\BBH^{2})} \bigr) \leq \ex^{-\xx} ; see \cite{laurentmassart2000}. This note provides an extension of these bounds to the case of a sub-gaussian vector \Xv . The results are based on the recent advances in Laplace approximation from \cite{SpLaplace2022}
Mixed Laplace approximation for marginal posterior and Bayesian inference in error-in-operator model
Get PDFLaplace approximation is a very useful tool in Bayesian inference and it claims a nearly Gaussian behavior of the posterior. \cite{SpLaplace2022} established some rather accurate finite sample results about the quality of Laplace approximation in terms of the so called effective dimension p under the critical dimension constraint p3≪n. However, this condition can be too restrictive for many applications like error-in-operator problem or Deep Neuronal Networks. This paper addresses the question whether the dimensionality condition can be relaxed and the accuracy of approximation can be improved if the target of estimation is low dimensional while the nuisance parameter is high or infinite dimensional. Under mild conditions, the marginal posterior can be approximated by a Gaussian mixture and the accuracy of the approximation only depends on the target dimension. Under the condition p2≪n or in some special situation like semi-orthogonality, the Gaussian mixture can be replaced by one Gaussian distribution leading to a classical Laplace result. The second result greatly benefits from the recent advances in Gaussian comparison from \cite{GNSUl2017}. The results are illustrated and specified for the case of error-in-operator model
Existence of subcritical percolation phases for generalised weight-dependent random connection models
Get PDFWe derive a sufficient condition for the existence of a subcritical percolation phase for a wide range of continuum percolation models where each vertex is embedded into Euclidean space and carries an independent weight. In contrast to many established models, the presence of an edge is not only allowed to depend on the distance and weights of its end vertices but can also depend on the surrounding vertex set. Our result can be applied in particular to models combining heavy-tailed degree distributions and long-range effects, which are typically well connected. Moreover, we establish bounds on the tail-distribution of the number of points and the diameter of the subcritical component of a typical point. The proofs rest on a multi-scale argument
Second-order sufficient conditions in the sparse optimal control of a phase field tumor growth model with logarithmic potential
Get PDFThis paper treats a distributed optimal control problem for a tumor growth model of viscous Cahn--Hilliard type. The evolution of the tumor fraction is governed by a thermodynamic force induced by a double-well potential of logarithmic type. The cost functional contains a nondifferentiable term in order to enhance the occurrence of sparsity effects in the optimal controls, i.e., of subdomains of the space-time cylinder where the controls vanish. In the context of cancer therapies, sparsity is very important in order that the patient is not exposed to unnecessary intensive medical treatment. In this work, we focus on the derivation of second-order sufficient optimality conditions for the optimal control problem. While in previous works on the system under investigation such conditions have been established for the case without sparsity, the case with sparsity has not been treated before. The results obtained in this paper also improve the known results on this phase field model for the case without sparsity
Reconstructing Volatility: Pricing of Index Options under Rough Volatility
No full textAvellaneda et al. (2002, 2003) pioneered the pricing and hedging of index options – products highly sensitive to implied volatility and correlation assumptions – with large deviations methods, assuming local volatility dynamics for all components of the index. We present an extension applicable to non-Markovian dynamics and in particular the case of rough volatility dynamics
Off-diagonal long-range order for the free Bose gas via the Feynman--Kac formula
Get PDFWe consider the path-integral representation of the ideal Bose gas under various boundary conditions. We show that Bose--Einstein condensation occurs at the famous critical density threshold, by proving that its -particle-reduced density matrix exhibits off-diagonal long-range order above that threshold, but not below. Our proofs are based on the well-known Feynman--Kac formula and a representation in terms of a crucial Poisson point process. Furthermore, in the condensation regime, we derive a law of large numbers with strong concentration for the number of particles in short loops. In contrast to the situation for free boundary conditions, where the entire condensate sits in just one loop, for all other boundary conditions we obtain the limiting Poisson--Dirichlet distribution for the collection of the lengths of all long loops. Our proofs are new and purely probabilistic (a part from a standard eigenvalue expansion), using elementary tools like Markov's inequality, Poisson point processes, combinatorial formulas for cardinalities of particular partition sets and asymptotics for random walks with Pareto-distributed steps
Stochastic homogenization on perforated domains III -- General estimates for stationary ergodic random connected Lipschitz domains
No full textThis is Part III of a series on the existence of uniformly bounded extension operators on randomly perforated domains in the context of homogenization theory. Recalling that randomly perforated domains are typically not John and hence extension is possible only from W 1,p to W 1,r, r < p, we will show that the existence of such extension operators can be guarantied if the weighted expectations of four geometric characterizing parameters are bounded: The local Lipschitz constant M, the local Lipschitz radius Δ , the mesoscopic Voronoi diameter ∂ and the local connectivity radius R
Modeling cellular self-organization in strain-stiffening hydrogels
Get PDFWe develop a three-dimensional mathematical model framework for the collective evolution of cell populations by an agent-based model (ABM) that mechanically interacts with the surrounding extracellular matrix (ECM) modeled as a hydrogel. We derive effective two dimensional models for the geometrical set-up of a thin hydrogel sheet to study cell-cell and cell-hydrogel mechanical interactions for a range of external conditions and intrinsic material properties. We show that without any stretching of the hydrogel sheets, cells show the well-known tendency to form long chains with varying orientations. Our results further show that external stretching of the sheet produces the expected nonlinear strain-softening or stiffening response, with, however, little qualitative variation of the overall cell dynamics for all the materials considered. The behavior is remarkably different when solvent is entering or leaving from strain softening or stiffening hydrogels, respectively