Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics
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On a sufficient condition for explosion in CMJ branching processes and applications to recursive trees
Get PDFWe provide sufficient criteria for explosion in Crump-Mode-Jagers branching process, via the process producing an infinite path in finite time. As an application, we deduce a curious phase-transition in the infinite tree associated with a class of recursive tree models with fitness, showing that in one regime every node in the tree has infinite degree, whilst in another, the tree is locally finite, with a unique infinite path. The latter class encompasses many models studied in the literature, including the weighted random recursive tree, the preferential attachment tree with additive fitness, and the Bianconi-Barabási model, or preferential attachment tree with multiplicative fitness
Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability
No full textWe study a particular group law on formal power series in non-commuting variables induced by their interpretation as linear forms on a suitable graded connected word Hopf algebra. This group law is left-linear and is therefore associated to a pre-Lie structure on formal power series. We study these structures and show how they can be used to recast in a group theoretic form various identities and transformations on formal power series that have been central in the context of non-commutative probability theory, in particular in Voiculescu's theory of free probability
Well-Posedness of Stochastic Heat Equation with Distributional Drift and Skew Stochastic Heat Equation
No full textWe study stochastic reaction--diffusion equation ∂tut(x)=12∂2xxut(x)+b(ut(x))+W˙t(x),t>0,x∈D where b is a generalized function in the Besov space Bβq,∞(R), D⊂R and W˙ is a space-time white noise on R+×D. We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever β−1/q≥−1, β>−1 and q∈[1,∞]. This class includes equations with b being measures, in particular, b=δ0 which corresponds to the skewed stochastic heat equation. For β−1/q>−3/2, we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalizes the results of Gyöngy and Pardoux (1993) to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in Lê~(2020)
Displacement and Pressure Reconstruction from Magnetic Resonance Elastography Images: Application to an In Silico Brain Model
No full textMagnetic resonance elastography is a motion-sensitive image modality that allows to measure in vivo tissue displacement fields in response to mechanical excitations. This paper investigates a data assimilation approach for reconstructing tissue displacement and pressure fields in an in silico brain model from partial elastography data. The data assimilation is based on a parametrized-background data-weak methodology, in which the state of the physical system—tissue displacements and pressure fields—is reconstructed from the available data assuming an underlying poroelastic biomechanics model. For this purpose, a physics-informed manifold is built by sampling the space of parameters describing the tissue model close to their physiological ranges to simulate the corresponding poroelastic problem and computing a reduced basis via proper orthogonal decomposition. Displacements and pressure reconstruction are sought in a reduced space after solving a minimization problem that encompasses both the structure of the reduced-order model and the available measurements. The proposed pipeline is validated using synthetic data obtained after simulating the poroelastic mechanics of a physiological brain. The numerical experiments demonstrate that the framework can exhibit accurate joint reconstructions of both displacement and pressure fields. The methodology can be formulated for an arbitrary resolution of available displacement data from pertinent images. It can also inherently handle uncertainty on the physical parameters of the mechanical model by enlarging the physics-informed manifold accordingly. Moreover, the framework can be used to characterize, in silico, biomarkers for pathological conditions by appropriately training the reduced-order model. A first application for the noninvasive estimation of ventricular pressure as an indicator of abnormal intracranial pressure is shown in this contribution
A Micro-Macro Variational Formula for the Free Energy of a Many-Body System with Unbounded Marks
No full textThe interacting quantum Bose gas is a random ensemble of many Brownian bridges (cycles) of various lengths with interactions between any pair of legs of the cycles. It is one of the standard mathematical models in which a proof for the famous Bose–Einstein condensation phase transition is sought for. A qualitative understanding of the free energy would be helpful, but this is currently far out of reach. In this paper, we demonstrate a path towards gaining such an understanding for a simplified version of the model with deterministic boxes instead of Brownian cycles. This model is a marked Poisson point process with unbounded marks containing particles and bounded-reach interactions between the particles. Even though it is not a quantum model, it is close to that in spirit. We derive an explicit and interpretable variational formula in the thermodynamic limit for the limiting free energy of the canonical ensemble for any value of the particle density. This formula features all relevant physical quantities of the model, like the microscopic and the macroscopic particle densities, together with their mutual and self-energies and their entropies. The proof method comprises a two-step meso-macro large-deviation approach for marked Poisson point processes and an explicit distinction into small and large marks; an application of well-known level-three principles á la Georgii/Zessin is not possible because of the appearance of macro marks. The characteristic variational formula enables us to prove a number of properties of the limiting free energy as a function of the particle density, like differentiability and explicit upper and lower bounds, and a qualitative picture below and above the critical threshold (if it is finite). This proves a modified saturation nature of the phase transition. However, we have not yet succeeded in proving the existence of this phase transition
Inexact Tensor Methods and their Application to Stochastic Convex Optimization
No full textWe propose general non-accelerated [The results for non-accelerated methods first appeared in December 2020 in the preprint (A. Agafonov, D. Kamzolov, P. Dvurechensky, and A. Gasnikov, Inexact tensor methods and their application to stochastic convex optimization, preprint 2020. arXiv:2012.15636)] and accelerated tensor methods under inexact information on the derivatives of the objective, analyse their convergence rate. Further, we provide conditions for the inexactness in each derivative that is sufficient for each algorithm to achieve a desired accuracy. As a corollary, we propose stochastic tensor methods for convex optimization and obtain sufficient mini-batch sizes for each derivative
Accelerated Gradient Methods with Absolute and Relative Noise in the Gradient
No full textIn this paper, we investigate accelerated first-order methods for smooth convex optimization problems under inexact information on the gradient of the objective. The noise in the gradient is considered to be additive with two possibilities: absolute noise bounded by a constant, and relative noise proportional to the norm of the gradient. We investigate the accumulation of the errors in the convex and strongly convex settings with the main difference with most of the previous works being that the feasible set can be unbounded. The key to the latter is to prove a bound on the trajectory of the algorithm. We also give a stopping criterion for the algorithm and consider extensions to the cases of stochastic optimization and composite nonsmooth problems
Guaranteed quasi-error reduction of adaptive Galerkin FEM for parametric PDEs with lognormal coefficients
Get PDFSolving high-dimensional random parametric PDEs poses a challenging computational problem. It is well-known that numerical methods can greatly benefit from adaptive refinement algorithms, in particular when functional approximations in polynomials are computed as in stochastic Galerkin and stochastic collocations methods. This work investigates a residual based adaptive algorithm used to approximate the solution of the stationary diffusion equation with lognormal coefficients. It is known that the refinement procedure is reliable, but the theoretical convergence of the scheme for this class of unbounded coefficients remains a challenging open question. This paper advances the theoretical results by providing a quasi-error reduction results for the adaptive solution of the lognormal stationary diffusion problem. A computational example supports the theoretical statement
Energy level alignment of confined hole states in InAs1-x-ySbxPy double quantum dots
Get PDFWe present a combined experimental and theoretical study of uncapped In(As,Sb,P) double quantum dots (DQD), suited for application in novel resonant tunneling nanodiods or single-photon nanooptical up- and down-converters in the mid-infrared spectral range. We provide details on the growth process using liquid-phase epitaxy (LPE), as well as on the characterization using atomic-force microscopy (AFM) and scanning electron microscopy (SEM). We find that most DQDs exhibit an asymmetry such that the two QDs of each pair have different dimensions, giving rise to correspondingly different quantum confinement of hole states localized in each QD. Based on these data, we have performed systematic simulations based on an eight-band model to identify the relationship between QD dimensions and the energy difference between corresponding confined hole states in the two QDs. Finally, we have determined the strength of an applied electric field required to energetically align the hole ground states of two QDs of different dimensions in order to facilitate hole tunneling
Energy-variational solutions for viscoelastic fluid models
Get PDFIn this article, we introduce the concept of energy-variational solutions for a large class of systems of nonlinear evolutionary partial differential equations. Under certain convexity assumptions, the existence of such solutions can be shown constructively by an adapted minimizing movement scheme. Weak-strong uniqueness follows by a suitable relative energy inequality. Our main motivation is to apply the general framework to viscoelastic fluid models. Therefore, we give a short overview on different versions of such models and their derivation. The abstract result is applied to two of these viscoelastic fluid models in full detail. In the conclusion, we comment on further applications of the general theory and its possible impact