Kaiserslautern - Fachbereich Mathematik
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First essential m-dissipativity of an infinite-dimensional Ornstein-Uhlenbeck operator N, perturbed by the gradient of a potential, on a domain FC
∞
b
of finitely based, smooth and bounded functions, is shown. Our considerations allow unbounded diffusion operators as coefficients. We derive corresponding second order regularity estimates for solutions f of the Kolmogorov equation ◂−▸αf−Nf=g, ◂+▸α∈(0,∞), generalizing some results of Da Prato and Lunardi. Second, we prove essential m-dissipativity for generators (◂,▸LΦ,FC
∞
b
) of infinite-dimensional degenerate diffusion processes. We emphasize that the essential m-dissipativity of (◂,▸LΦ,FC
∞
b
) is useful to apply general resolvent methods developed by Beznea, Boboc and Röckner, in order to construct martingale/weak solutions to infinite-dimensional non-linear degenerate stochastic differential equations. Furthermore, the essential m-dissipativity of (◂,▸LΦ,FC
∞
b
) and (◂,▸N,FC
∞
b
), as well as the regularity estimates are essential to apply the general abstract Hilbert space hypocoercivity method from Dolbeault, Mouhot, Schmeiser and Grothaus, Stilgenbauer, respectively, to the corresponding diffusions.
We provide a complete elaboration of the L2-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behavior of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker–Planck framework and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to weakly differentiable diffusion coefficient matrices, introducing multiplicative noise for the corresponding stochastic differential equation. The rate of convergence is explicitly computed depending on the choice of these coefficients and the potential giving the outer force. In order to obtain a solution to the abstract Cauchy problem, we first prove essential self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert spaces, using prior elliptic regularity results and techniques from Bogachev, Krylov and Röckner. We apply operator perturbation theory to obtain essential m-dissipativity of the Kolmogorov operator, extending the m-dissipativity results from Conrad and Grothaus. We emphasize that the chosen Kolmogorov approach is natural, as the theory of generalized Dirichlet forms implies a stochastic representation of the Langevin semigroup as the transition kernel of a diffusion process which provides a martingale solution to the Langevin equation with multiplicative noise. Moreover, we show that even a weak solution is obtained this way.
This article presents a methodology whereby adjoint solutions for partitioned multiphysics problems can be computed efficiently, in a way that is completely independent of the underlying physical sub-problems, the associated numerical solution methods, and the number and type of couplings between them. By applying the reverse mode of algorithmic differentiation to each discipline, and by using a specialized recording strategy, diagonal and cross terms can be evaluated individually, thereby allowing different solution methods for the generic coupled problem (for example block-Jacobi or block-Gauss-Seidel). Based on an implementation in the open-source multiphysics simulation and design software SU2, we demonstrate how the same algorithm can be applied for shape sensitivity analysis on a heat exchanger (conjugate heat transfer), a deforming wing (fluid–structure interaction), and a cooled turbine blade where both effects are simultaneously taken into account.
In this note, we define one more way of quantization of classical systems. The quantization we consider is an analogue of classical Jordan–Schwinger map which has been known and used for a long time by physicists. The difference, compared to Jordan–Schwinger map, is that we use generators of Cuntz algebra O∞ (i.e. countable family of mutually orthogonal partial isometries of separable Hilbert space) as a “building blocks” instead of creation–annihilation operators. The resulting scheme satisfies properties similar to Van Hove prequantization, i.e. exact conservation of Lie brackets and linearity.
In a widely-studied class of multi-parametric optimization problems, the objective value of each solution is an affine function of real-valued parameters. Then, the goal is to provide an optimal solution set, i.e., a set containing an optimal solution for each non-parametric problem obtained by fixing a parameter vector. For many multi-parametric optimization problems, however, an optimal solution set of minimum cardinality can contain super-polynomially many solutions. Consequently, no polynomial-time exact algorithms can exist for these problems even if P=NP. We propose an approximation method that is applicable to a general class of multi-parametric optimization problems and outputs a set of solutions with cardinality polynomial in the instance size and the inverse of the approximation guarantee. This method lifts approximation algorithms for non-parametric optimization problems to their parametric version and provides an approximation guarantee that is arbitrarily close to the approximation guarantee of the approximation algorithm for the non-parametric problem. If the non-parametric problem can be solved exactly in polynomial time or if an FPTAS is available, our algorithm is an FPTAS. Further, we show that, for any given approximation guarantee, the minimum cardinality of an approximation set is, in general, not ℓ-approximable for any natural number ℓ less or equal to the number of parameters, and we discuss applications of our results to classical multi-parametric combinatorial optimizations problems. In particular, we obtain an FPTAS for the multi-parametric minimum s-t-cut problem, an FPTAS for the multi-parametric knapsack problem, as well as an approximation algorithm for the multi-parametric maximization of independence systems problem.
Over the past 2 decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete but there is an infinite series of groups in dimension 4—the symplectically primitive but complex imprimitive groups—and 10 exceptional groups up to dimension 10, for which it is still open. In this paper, we treat the remaining infinite series and prove that for all but possibly 39 cases there is no symplectic resolution. We thereby reduce the classification problem to finitely many open cases. We furthermore prove non-existence of a symplectic resolution for one exceptional group, leaving 39+9=48 open cases in total. We do not expect any of the remaining cases to admit a symplectic resolution.
Consider the primitive equations on ◂+▸R2×(◂,▸z0,z1) with initial data a of the form a=◂+▸a1+a2, where ◂+▸a1∈◂◽.▸BUCσ(◂,▸R2;L1(◂,▸z0,z1)) and ◂+▸a2∈L
∞
σ
(◂,▸R2;L1(◂,▸z0,z1)). These spaces are scaling-invariant and represent the anisotropic character of these equations. It is shown that for a1 arbitrary large and a2 sufficiently small, this set of equations admits a unique strong solution which extends to a global one and is thus strongly globally well posed for these data provided a is periodic in the horizontal variables. The approach presented depends crucially on mapping properties of the hydrostatic Stokes semigroup in the L∞(L1)-setting. It can be seen as the counterpart of the classical iteration schemes for the Navier–Stokes equations, now for the primitive equations in the L∞(L1)-setting.
Die Möglichkeit einer Prämienanpassung in der deutschen PKV ist vom Wert des sogenannten auslösenden Faktors abhängig, der mittels einer linearen Extrapolation der Schadenquotienten der vergangenen drei Jahre berechnet wird. Seine frühzeitige, verlässliche Vorhersage ist aus Sicht des Risikomanagements von großer Bedeutung. Wir untersuchen deshalb vielfältige Vorhersageansätze, die von klassischen Zeitreihenansätzen und Regression über neuronale Netze bis hin zu hybriden Modellen reichen. Während bei den klassischen Methoden Regression mit ARIMA-Fehlern am besten abschneidet, zeigt ein neuronales Netz, das mit Zeitreihenvorhersage kombiniert oder auf desaisonalisierten und trendbereinigten Daten trainiert wurde, das insgesamt beste Verhalten.
Gliomas are primary brain tumors with a high invasive potential and infiltrative spread. Among them, glioblastoma multiforme (GBM) exhibits microvascular hyperplasia and pronounced necrosis triggered by hypoxia. Histological samples showing garland-like hypercellular structures (so-called pseudopalisades) centered around the occlusion site of a capillary are typical for GBM and hint on poor prognosis of patient survival. We propose a multiscale modeling approach in the kinetic theory of active particles framework and deduce by an upscaling process a reaction-diffusion model with repellent pH-taxis. We prove existence of a unique global bounded classical solution for a version of the obtained macroscopic system and investigate the asymptotic behavior of the solution. Moreover, we study two different types of scaling and compare the behavior of the obtained macroscopic PDEs by way of simulations. These show that patterns (not necessarily of Turing type), including pseudopalisades, can be formed for some parameter ranges, in accordance with the tumor grade. This is true when the PDEs are obtained via parabolic scaling (undirected tissue), while no such patterns are observed for the PDEs arising by a hyperbolic limit (directed tissue). This suggests that brain tissue might be undirected - at least as far as glioma migration is concerned. We also investigate two different ways of including cell level descriptions of response to hypoxia and the way they are related .
Understanding human crowd behaviour has been an intriguing topic of interdisciplinary research in recent decades. Modelling of crowd dynamics using differential equations is an indispensable approach to unraveling the various complex dynamics involved in such interacting particle systems. Numerical simulation of pedestrian crowd via these mathematical models allows us to study different realistic scenarios beyond the limitations of studies via controlled experiments.
In this thesis, the main objective is to understand and analyse the dynamics in a domain shared by both pedestrians and moving obstacles. We model pedestrian motion by combining the social force concept with the idea of optimal path computation. This leads to a system of ordinary differential equations governing the dynamics of individual pedestrians via the interaction forces (social forces) between them. Additionally, a non-local force term involving the optimal path and desired velocity governs the pedestrian trajectory. The optimal path computation involves solving a time-independent Eikonal equation, which is coupled to the system of ODEs. A hydrodynamic model is developed from this microscopic model via the mean-field limit.
To consider the interaction with moving obstacles in the domain, we model a set of kinematic equations for the obstacle motion. Two kinds of obstacles are considered - "passive", which move in their predefined trajectories and have only a one-way interaction with pedestrians, and "dynamic", which have a feedback interaction with pedestrians and have their trajectories changing dynamically. The coupled model of pedestrians and obstacles is used to discern pedestrian collision avoidance behaviour in different computational scenarios in a long rectangular domain. We observe that pedestrians avoid collisions through route choice strategies that involve changes in speed and path. We extend this model to consider the interaction between pedestrians and vehicular traffic. We appropriately model the interactions of vehicles, following lane traffic, based on the car-following approach. We observe how the deceleration and braking mechanism of vehicles is executed at pedestrian crossings depending on the right of way on the roads.
As a second objective, we study the disease contagion in moving crowds. We consider the influence of the crowd motion in a complex dynamical environment on the course of infection of pedestrians. A hydrodynamic model for multi-group pedestrian flow is derived from the kinetic equations based on a social force model. It is coupled along with an Eikonal equation to a non-local SEIS contagion model for disease spread. Here, apart from the description of local contacts, the influence of contact times has also been modelled. We observe that the nature of the flow and the geometry of the domain lead to changes in density which affect the contact time and, consequently, the rate of spread of infection.
Finally, the social force model is compared to a variable speed based rational behaviour pedestrian model. We derive a hierarchy of the heuristics-based model from microscopic to macroscopic scales and numerically investigate these models in different density scenarios. Various numerical test cases are considered, including uni- and bi-directional flows and scenarios with and without obstacles. We observe that in low-density scenarios, collision avoidance forces arising from the behavioural heuristics give valid results. Whereas in high-density scenarios, repulsive force terms are essential.
The numerical simulations of all the models are carried out using a mesh-free particle method based on least square approximations. The meshfree numerical framework provides an efficient and elegant way to handle complex geometric situations involving boundaries and stationary or moving obstacles.