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Truncated Nonsmooth Newton Multigrid for phase-field brittle-fracture problems
Get PDFWe propose the Truncated Nonsmooth Newton Multigrid Method (TNNMG) as a solver for the spatial problems of the small-strain brittle-fracture phase-field equations. TNNMG is a nonsmooth multigrid method that can solve biconvex, block-separably nonsmooth minimization problems in roughly the time of solving one linear system of equations. It exploits the variational structure inherent in the problem, and handles the pointwise irreversibility constraint on the damage variable directly, without penalization or the introduction of a local history field. Memory consumption is significantly lower compared to approaches based on direct solvers. In the paper we introduce the method and show how it can be applied to several established models of phase-field brittle fracture. We then prove convergence of the solver to a solution of the nonsmooth Euler-Lagrange equations of the spatial problem for any load and initial iterate. Numerical comparisons to an operator-splitting algorithm show a speed increase of more than one order of magnitude, without loss of robustness
A deeper look into natural sciences with physics-based and data-driven measures
No full textWith the development of machine learning in recent years, it is possible to glean much more information from an experimental data set to study matter. In this perspective, we discuss some state-of-the-art data-driven tools to analyze latent effects in data and explain their applicability in natural science, focusing on two recently introduced, physics-motivated computationally cheap tools—latent entropy and latent dimension. We exemplify their capabilities by applying them on several examples in the natural sciences and show that they reveal so far unobserved features such as, for example, a gradient in a magnetic measurement and a latent network of glymphatic channels from the mouse brain microscopy data. What sets these techniques apart is the relaxation of restrictive assumptions typical of many machine learning models and instead incorporating aspects that best fit the dynamical systems at hand
Dynamics of tilted atmospheric vortices under asymmetric diabatic heating
Get PDFPäschke et al. (J Fluid Mech, 2012) studied the nonlinear dynamics of strongly tilted vortices subject to asymmetric diabatic heating by asymptotic methods. They found, inter alia, that an azimuthal Fourier mode 1 heating pattern can intensify or attenuate such a vortex depending on the relative orientation of the tilt and the heating asymmetries. The theory originally addressed the gradient wind regime which, asymptotically speaking, corresponds to vortex Rossby numbers of order unity in the limit. Formally, this restricts the applicability of the theory to rather weak vortices. It is shown below that said theory is, in contrast, uniformly valid for vanishing Coriolis parameter and thus applicable to vortices up to low hurricane strengths. An extended discussion of the asymptotics as regards their physical interpretation and their implications for the overall vortex dynamics is also provided in this context. The paper’s second contribution is a series of three-dimensional numerical simulations examining the effect of different orientations of dipolar diabatic heating on idealized tropical cyclones. Comparisons with numerical solutions of the asymptotic equations yield evidence that supports the original theoretical predictions of Päschke et al. In addition, the influence of asymmetric diabatic heating on the time evolution of the vortex centerline is further analyzed, and a steering mechanism that depends on the orientation of the heating dipole is revealed. Finally, the steering mechanism is traced back to the correlation of dipolar perturbations of potential temperature, induced by the vortex tilt, and vertical velocity, for which diabatic heating not necessarily needs to be responsible, but which may have other origins
Generalized master equation for first-passage problems in partitioned spaces
Get PDFAbstract
Motivated by a range of biological applications related to the transport of molecules
in cells, we present a modular framework to treat first-passage problems for diffusion
in partitioned spaces. The spatial domains can differ with respect to their diffusivity,
geometry, and dimensionality, but can also refer to transport modes alternating
between diffusive, driven, or anomalous motion. The approach relies on a coarsegraining
of the motion by dissecting the trajectories on domain boundaries or when
the mode of transport changes, yielding a small set of states. The time evolution of
the reduced model follows a generalized master equation, which takes the form of a
set of linear integro-differential equations in the occupation probabilities of the states
and the corresponding probability fluxes. Further building blocks of the model are
partial first-passage time (FPT) densities, which encode the transport behavior in
each domain or state. The approach is exemplified and validated for a target search
problem with two domains in one- and three-dimensional space, first by exactly reproducing
known results for an artificially divided, homogeneous space, and second by
considering the situation of domains with distinct diffusivities. Analytical solutions
for the FPT densities are given in Laplace domain and are complemented by numerical
backtransform yielding FPT densities over many decades in time, confirming that
the geometry and heterogeneity of the space can introduce additional characteristic
timescales
eIF5A hypusination, boosted by dietary spermidine, protects from premature brain aging and mitochondrial dysfunction
No full textSummary
Mitochondrial function declines during brain aging and is suspected to play a key role in age-induced cognitive decline and neurodegeneration. Supplementing levels of spermidine, a body-endogenous metabolite, has been shown to promote mitochondrial respiration and delay aspects of brain aging. Spermidine serves as the amino-butyl group donor for the synthesis of hypusine (Nε-[4-amino-2-hydroxybutyl]-lysine) at a specific lysine residue of the eukaryotic translation initiation factor 5A (eIF5A). Here, we show that in the Drosophila brain, hypusinated eIF5A levels decline with age but can be boosted by dietary spermidine. Several genetic regimes of attenuating eIF5A hypusination all similarly affect brain mitochondrial respiration resembling age-typical mitochondrial decay and also provoke a premature aging of locomotion and memory formation in adult Drosophilae. eIF5A hypusination, conserved through all eukaryotes as an obviously critical effector of spermidine, might thus be an important diagnostic and therapeutic avenue in aspects of brain aging provoked by mitochondrial decline
SeqAn3 – Sequence Analysis and Modern C++
No full textThis thesis introduces SeqAn3, a new software library built with Modern C++ to solve problems from the domain of sequence analysis in bioinformatics. It discusses previous versions of the library in detail and explains the importance of highly performing programming languages like C++. Complexity in the design of the library and of the programming language itself are identified as the major obstacles to user satisfaction, widespread adoption and long-term viability of the project. Therefore, based on very fundamental changes in the C++ programming language, a new library design is formulated and implemented. Its impact is showcased by porting the local aligner called Lambda from SeqAn2 to SeqAn3. Both, the library and the application are highly relevant in practice and prove that simpler and more compact solutions are possible. This thesis documents the process of creating said software, contributing vital information to the fields of research software engineering, library design and to a certain degree also applied programming language research. As one of the first larger projects to be designed fully around C++20 features, it has instructive value beyond bioinformatics
Waves in the gas centrifuge: Asymptotic theory and similarities with the atmosphere
No full textWe study the stratified gas in a rapidly rotating centrifuge as a model for the Earth's atmosphere. Based on methods of perturbation theory, it is shown that in certain regimes, internal waves in the gas centrifuge have the same dispersion relation to leading order as their atmospheric siblings. Assuming an air filled centrifuge with a radius of around 50 cm, the optimal rotational frequency for realistic atmosphere-like waves is around 10 000 revolutions per minute. Using gases of lower heat capacities at constant pressure, such as xenon, the rotational frequencies can be even halved to obtain the same results. Similar to the atmosphere, it is feasible in the gas centrifuge to generate a clear scale separation of wave frequencies and therefore phase speeds between acoustic waves and internal waves. In addition to the centrifugal force, the Coriolis force acts in the same plane. However, its influence on axially homogeneous internal waves appears only as a higher-order correction. We conclude that the gas centrifuge provides an unprecedented opportunity to investigate atmospheric internal waves experimentally with a compressible working fluid
The linear conditional expectation in Hilbert space
No full textThe linear conditional expectation (LCE) provides a best linear (or rather, affine) estimate of the conditional expectation and hence plays an important rôle in approximate Bayesian inference, especially the Bayes linear approach. This article establishes the analytical properties of the LCE in an infinite-dimensional Hilbert space context. In addition, working in the space of affine Hilbert–Schmidt operators, we establish a regularisation procedure for this LCE. As an important application, we obtain a simple alternative derivation and intuitive justification of the conditional mean embedding formula, a concept widely used in machine learning to perform the conditioning of random variables by embedding them into reproducing kernel Hilbert spaces
On differentiability of the membrane-mediated mechanical interaction energy of discrete-continuum membrane-particle models
Get PDFWe consider a discrete-continuum model of a biomembrane with embedded particles. While the membrane is represented by a continuous surface, embedded particles are described by rigid discrete objects which are free to move and rotate in lateral direction. For the membrane we consider a linearized Canham-Helfrich energy functional and height and slope boundary conditions imposed on the particle boundaries resulting in a coupled minimization problem for the membrane shape and particle positions.
When considering the energetically optimal membrane shape for each particle position we obtain a reduced energy functional that models the implicitly given interaction potential for the membrane-mediated mechanical particle-particle interactions. We show that this interaction potential is differentiable with respect to the particle positions and orientations. Furthermore we derive a fully practical representation of the derivative only in terms of well defined derivatives of the membrane. This opens the door for the application of minimization algorithms for the computation of minimizers of the coupled system and for further investigation of the interaction potential of membrane-mediated mechanical particle--particle interaction.
The results are illustrated with numerical examples comparing the explicit derivative formula with difference quotient approximations. We furthermore demonstrate the application of the derived formula to implement a gradient flow for the approximation of optimal particle configurations
Coupling Particle-Based Reaction-Diffusion Simulations with Reservoirs Mediated by Reaction-Diffusion PDEs
No full textOpen biochemical systems of interacting molecules are ubiquitous in life-related processes. However, established computational methodologies, like molecular dynamics, are still mostly constrained to closed systems and timescales too small to be relevant for life processes. Alternatively, particle-based reaction-diffusion models are currently the most accurate and computationally feasible approach at these scales. Their efficiency lies in modeling entire molecules as particles that can diffuse and interact with each other. In this work, we develop modeling and numerical schemes for particle-based reaction-diffusion in an open setting, where the reservoirs are mediated by reaction-diffusion PDEs. We derive two important theoretical results. The first one is the mean-field for open systems of diffusing particles; the second one is the mean-field for a particle-based reaction-diffusion system with second-order reactions. We employ these two results to develop a numerical scheme that consistently couples particle-based reaction-diffusion processes with reaction-diffusion PDEs. This allows modeling open biochemical systems in contact with reservoirs that are time-dependent and spatially inhomogeneous, as in many relevant real-world applications