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Global well-posedness for the thermodynamically refined passively transported nonlinear moisture dynamics with phase changes
No full textIn this work we study the global solvability of moisture dynamics with phase changes for warm clouds. We thereby in comparison to previous studies (Hittmeir et al. in Nonlinearity 30:3676–3718, 2017) take into account the different gas constants for dry air and water vapor as well as the different heat capacities for dry air, water vapor and liquid water, which leads to a much stronger coupling of the moisture balances and the thermodynamic equation. This refined thermodynamic setting has been demonstrated to be essential, e.g. in the case of deep convective cloud columns in Hittmeir and Klein (Theoret Comput Fluid Dyn 32(2):137–164, 2017). The more complicated structure requires careful derivations of sufficient a priori estimates for proving global existence and uniqueness of solutions
Non-equilibrium steady states as saddle points and EDP-convergence for slow-fast gradient systems
No full textThe theory of slow-fast gradient systems leads in a natural way to non-equilibrium steady states, because on the slow time scale the fast subsystem stays in steady states that are controlled by the interaction with the slow system. Using the theory of convergence of gradient systems depending on a small parameter ɛ (here the ratio between the slow and the fast time scale) in the sense of the energy-dissipation principle shows that there is a natural characterization of these non-equilibrium steady states as saddle points of a so-called B-function where the slow variables are fixed. We give applications to slow-fast reaction-diffusion systems based on the so-called cosh-type gradient structure for reactions. It is shown that two binary reactions give rise to a ternary reaction with a state-dependent reaction coefficient. Moreover, we show that a reaction-diffusion equation with a thin membrane-like layer convergences to a transmission condition, where the formerly quadratic dissipation potential for diffusion convergences to a cosh-type dissipation potential for the transmission in the membrane limit
Finite-size effects and thermodynamic accuracy in many-particle systems
No full textFinite-size effects arise when a sample of particles is not sufficient to provide a statistically satisfactory description of the bulk environment of a physical system. As a consequence, a reliable estimate of finite-size effects in many-particle systems is key to judge the validity of a theoretical model or the accuracy of a numerical simulation. In this context, we propose the use of a theorem on the free-energy cost for separating a system into smaller independent subsystems [J. Stat. Mech.: Theory Exp. (2017) 083201; Lett. Math. Phys. 112, 97 (2022)] to estimate the relevance of finite-size effects in thermodynamic quantities from computer simulations. The key aspect of this study is that for two-body potentials, as mostly occurring in physics, the method requires only two-body distribution functions and the particle number density. The calculation of the involved physical quantities can be done numerically on a three-dimensional grid. In some cases even analytical estimates are possible and as an example the uniform interacting electron gas in the ground state is considered; we derive an approximating scaling law for the finite-size effects
3D surface reconstruction of cellular cryo-soft X-ray microscopy tomograms using semisupervised deep learning
No full textCryo-soft X-ray tomography (cryo-SXT) is a powerful method to investigate the ultrastructure of cells, offering resolution in the tens of nanometer range and strong contrast for membranous structures without requiring labeling or chemical fixation. The short acquisition time and the relatively large field of view leads to fast acquisition of large amounts of tomographic image data. Segmentation of these data into accessible features is a necessary step in gaining biologically relevant information from cryo-soft X-ray tomograms. However, manual image segmentation still requires several orders of magnitude more time than data acquisition. To address this challenge, we have here developed an end-to-end automated 3D segmentation pipeline based on semisupervised deep learning. Our approach is suitable for high-throughput analysis of large amounts of tomographic data, while being robust when faced with limited manual annotations and variations in the tomographic conditions. We validate our approach by extracting three-dimensional information on cellular ultrastructure and by quantifying nanoscopic morphological parameters of filopodia in mammalian cells
Improved sampling via learned diffusions
No full textRecently, a series of papers proposed deep learning-based approaches to sample from unnormalized target densities using controlled diffusion processes. In this work, we identify these approaches as special cases of the Schrödinger bridge problem, seeking the most likely stochastic evolution between a given prior distribution and the specified target. We further generalize this framework by introducing a variational formulation based on divergences between path space measures of time-reversed diffusion processes. This abstract perspective leads to practical losses that can be optimized by gradient-based algorithms and includes previous objectives as special cases. At the same time, it allows us to consider divergences other than the reverse Kullback-Leibler divergence that is known to suffer from mode collapse. In particular, we propose the so-called log-variance loss, which exhibits favorable numerical properties and leads to significantly improved performance across all considered approaches
Scale-interactions between the meso- and synoptic scales and the impact of diabatic heating
No full textFor both the meso- and synoptic scales, reduced mathematical models give insight into their dynamical behaviour. For the mesoscale, the weak temperature gradient approximation is one of several approaches, while for the synoptic scale the quasigeostrophic theory is well established. However, the way these two scales interact with each other is usually not included in such reduced models, thereby limiting our current perception of flow-dependent predictability and upscale error growth. Here, we address the scale interactions explicitly by developing a two-scale asymptotic model for the meso- and synoptic scales with two coupled sets of equations for the meso- and synoptic scales respectively. The mesoscale equations follow a weak temperature gradient balance and the synoptic-scale equations align with quasigeostrophic theory. Importantly, the equation sets are coupled via scale-interaction terms: eddy correlations of mesoscale variables impact the synoptic potential vorticity tendency and synoptic variables force the mesoscale vorticity (for instance due to tilting of synoptic-scale wind shear). Furthermore, different diabatic heating rates—representing the effect of precipitation—define different flow characteristics. With weak mesoscale heating relatable to precipitation rates of
, the mesoscale dynamics resembles two-dimensional incompressible vorticity dynamics and the upscale impact of the mesoscale on the synoptic scale is only of a dynamical nature. With a strong mesosocale heating relatable to precipitation rates of
, divergent motions and three-dimensional effects become relevant for the mesoscale dynamics and the upscale impact also includes thermodynamical effects
On the concentration of subgaussian vectors and positive quadratic forms in Hilbert spaces
No full textIn these notes, we investigate the tail behaviour of the norm of subgaussian vectors in a Hilbert space. The subgaussian variance proxy is given as a trace class operator, allowing for a precise control of the moments along each dimension of the space. This leads to useful extensions and analogues of known Hoeffding-type inequalities and deviation bounds for positive random quadratic forms. We give a straightforward application in terms of a variance bound for the regularisation of statistical inverse problems
Mathematik entdecken (Berlin)
No full textAn der Freien Universität Berlin wurde zum Wintersemester 2017/2018 der Einstieg ins Mathematikstudium der angehenden Sekundarstufenlehrkräfte reformiert. Neuere Ansätze und Diskurse zur Qualitätssicherung in Studium und Lehre aufnehmend, setzt dieses Projekt mit innovativen Lehr- und Lernformaten an der Schnittstelle von Schule und Hochschule an (siehe Beutelspacher et al., 2012). Insbesondere wurden die Fachvorlesungen des ersten Semesters um hochschulmathematikdidaktische Anteile ergänzt, um den Anforderungen einer zeitgemäßen und bedarfsgerechten Mathematikausbildung gerecht zu werden. Der Fokus liegt zu Studienbeginn auf dem Kennenlernen elaborierter mathematischer Denkweisen und Problemlösestrategien und der mathematischen Enkulturation angehender Lehrkräfte, ohne dass dabei fachliche Inhalte vernachlässigt werden. Die Konzeption und Zielsetzungen des Projekts werden in diesem Kapitel vorgestellt
Function spaces, time derivatives and compactness for evolving families of Banach spaces with applications to PDEs
No full textWe develop a functional framework suitable for the treatment of partial differential equations and variational problems on evolving families of Banach spaces. We propose a definition for the weak time derivative that does not rely on the availability of a Hilbertian structure and explore conditions under which spaces of weakly differentiable functions (with values in an evolving Banach space) relate to classical Sobolev–Bochner spaces. An Aubin–Lions compactness result is proved. We analyse concrete examples of function spaces over time-evolving spatial domains and hypersurfaces for which we explicitly provide the definition of the time derivative and verify isomorphism properties with the aforementioned Sobolev–Bochner spaces. We conclude with the proof of well posedness for a class of nonlinear monotone problems on an abstract evolving space (generalising the evolutionary p-Laplace equation on a moving domain or surface) and identify some additional problems that can be formulated with the setting developed in this work
Flow-Matching: Efficient Coarse-Graining of Molecular Dynamics without Forces
No full textCoarse-grained (CG) molecular simulations have become a standard tool to study molecular processes on time and length scales inaccessible to all-atom simulations. Parametrizing CG force fields to match all-atom simulations has mainly relied on force-matching or relative entropy minimization, which require many samples from costly simulations with all-atom or CG resolutions, respectively. Here we present flow-matching, a new training method for CG force fields that combines the advantages of both methods by leveraging normalizing flows, a generative deep learning method. Flow-matching first trains a normalizing flow to represent the CG probability density, which is equivalent to minimizing the relative entropy without requiring iterative CG simulations. Subsequently, the flow generates samples and forces according to the learned distribution in order to train the desired CG free energy model via force-matching. Even without requiring forces from the all-atom simulations, flow-matching outperforms classical force-matching by an order of magnitude in terms of data efficiency and produces CG models that can capture the folding and unfolding transitions of small proteins