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Optimization with learning-informed differential equation constraints and its applications
No full textInspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis is provided, and particular properties resulting from artificial neural network based approximations are addressed. Moreover, for each of the two inspiring applications analytical details are presented and numerical results are provided
Macroscopic loops in the Bose gas, Spin O(N) and related models
Get PDFWe consider a general system of interacting random loops which includes several models of interest, such as the Spin O(N) model, random lattice permutations, a version of the interacting Bose gas in discrete space and of the loop O(N) model. We consider the system in ℤd, d ≥ 3, and prove the occurrence of macroscopic loops whose length is proportional to the volume of the system. More precisely, we approximate ℤd by finite boxes and, given any two vertices whose distance is proportional to the diameter of the box, we prove that the probability of observing a loop visiting both is uniformly positive. Our results hold under general assumptions on the interaction potential, which may have bounded or unbounded support or introduce hard-core constraints
Connectivity in mobile device-to-device networks in urban environments
Get PDFIn this article we setup a dynamic device-to-device communication system where devices, given as a Poisson point process, move in an environment, given by a street system of random planar-tessellation type, via a random-waypoint model. Every device independently picks a target location on the street system using a general waypoint kernel, and travels to the target along the shortest path on the streets with an individual velocity. Then, any pair of devices becomes connected whenever they are on the same street in sufficiently close proximity, for a sufficiently long time. After presenting some general properties of the multi-parameter system, we focus on an analysis of the clustering behavior of the random connectivity graph. In our main results we isolate regimes for the almost-sure absence of percolation if, for example, the device intensity is too small, or the connectivity time is too large. On the other hand, we exhibit parameter regimes of sufficiently large intensities of devices, under favorable choices of the other parameters, such that percolation is possible with positive probability. Most interestingly, we also show an in-and-out of percolation as the velocity increases. The rigorous analysis of the system mainly rests on comparison arguments with simplified models via spatial coarse graining and thinning approaches. Here we also make contact to geostatistical percolation models with infinite-range dependencies
Incompressible limit for a fluid mixture
Get PDFIn this paper we discuss the incompressible limit for multicomponent fluids in the isothermal ideal case. Both a direct limit-passage in the equation of state and the low Mach-number limit in rescaled PDEs are investigated. Using the relative energy inequality, we obtain convergence results for the densities and the velocity-field under the condition that the incompressible model possesses a sufficiently smooth solution, which is granted at least for a short time. Moreover, in comparison to single-component flows, uniform estimates and the convergence of the pressure are needed in the multicomponent case because the incompressible velocity field is not divergence-free. We show that certain constellations of the mobility tensor allow to control gradients of the entropic variables and yield the convergence of the pressure in L1
On a technique for reducing spurious oscillations in DG solutions of convection-diffusion equations
Get PDFThis note studies a generalization of a post-processing technique and a novel method inspired by the same technique which significantly reduce spurious oscillations in discontinuous Galerkin solutions of convection-diffusion equations in the convection-dominated regime
Decoherence and turbulence sources in a long laser
Get PDFWe investigate the turn-on process in a laser cavity where the roundtrip time is several orders of magnitude greater than the active medium timescales. In this long delay limit the electromagnetic field build-up can be mapped experimentally roundtrip after roundtrip. We show how coherence settles down starting from a stochastic initial condition. In the early stages of the turn-on, we show that power drop-outs emerge, persist for several round-trips and seed dark solitons. These latter structures exhibit a chaotic dynamics and emit radiation that can lead to an overall turbulent dynamics depending on the cavity dispersion
Modeling of ultrashort optical pulses in nonlinear fibers
Get PDFThis work deals with theoretical aspects of pulse propagation. The core focus is on extreme, few-cycle pulses in optical fibers, pulses that are strongly affected by both dispersion and nonlinearity. Using Hamil- tonian methods, we discuss how the meaning of pulse envelope changes, as pulses become shorter and shorter, and why an envelope equation can still be used. We also discuss how the standard set of dispersion coefficients yields useful rational approximations for the chromatic dispersion in optical fibers. Three more specific problems are addressed thereafter. First, we present an alternative framework for ultra- short pulses in which non-envelope propagation models are used. The approach yields the limiting, shortest solitons and reveals their universal features. Second, we describe how one can manipulate an ultrashort pulse, i.e., to change its amplitude and duration in a predictable manner. Quantitative theory of the manipu- lation is presented based on perturbation theory for solitons and analogy between classical fiber optics and quantum mechanics. Last but not least, we consider a recently found alternative to the standard split-step approach for numerical solutions of the pulse propagation equations
Pressure-Robustness in the Context of Optimal Control
No full textThis paper studies the benefits of pressure-robust discretizations in the scope of optimal control of incompressible flows. Gradient forces that may appear in the data can have a negative impact on the accuracy of state and control and can only be correctly balanced if their L2-orthogonality onto discretely divergence-free test functions is restored. Perfectly orthogonal divergence-free discretizations or divergence-free reconstructions of these test functions do the trick and lead to much better analytic a priori estimates that are also validated in numerical examples
Gibbsianness of Locally Thinned Random Fields
No full textWe consider the locally thinned Bernoulli field on \zd, which is the lattice version of the Type-I Mat\'ern hardcore process in Euclidean space. It is given as the lattice field of occupation variables, obtained as image of an i.i.d.~Bernoulli lattice field with occupation probability , under the map which removes all particles with neighbors, while keeping the isolated particles. We prove that the thinned measure has a Gibbsian representation and provide control on its quasilocal dependence, both in the regime of small , but also in the regime of large , where the thinning transformation changes the Bernoulli measure drastically. Our methods rely on Dobrushin uniqueness criteria, disagreement percolation arguments~\cite{BeMa94}, and cluster expansion
