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Skorokhod topologies: What they are and why we should care
No full textThis paper presents a gentle and informal introduction to the Skorokhod topologies. Focus is on motivating examples and concepts
Improving control based importance sampling strategies for metastable diffusions via adapted metadynamics
No full textSampling rare events in metastable dynamical systems is often a computationally expensive task and one needs to resort to enhanced sampling methods such as importance sampling. Since we can formulate the problem of finding optimal importance sampling controls as a stochastic optimization problem, this then brings additional numerical challenges and the convergence of corresponding algorithms might as well suffer from metastabilty. In this article, we address this issue by combining systematic control approaches with the heuristic adaptive metadynamics method. Crucially, we approximate the importance sampling control by a neural network, which makes the algorithm in principle feasible for high-dimensional applications. We can numerically demonstrate in relevant metastable problems that our algorithm is more effective than previous attempts and that only the combination of the two approaches leads to a satisfying convergence and therefore to an efficient sampling in certain metastable settings
On posterior consistency of data assimilation with Gaussian process priors: the 2D Navier-Stokes equations
Get PDFWe consider a non-linear Bayesian data assimilation model for the periodic
two-dimensional Navier-Stokes equations with initial condition modelled by a
Gaussian process prior. We show that if the system is updated with sufficiently
many discrete noisy measurements of the velocity field, then the posterior distri-
bution eventually concentrates near the ground truth solution of the time evolu-
tion equation, and in particular that the initial condition is recovered consistently
by the posterior mean vector field. We further show that the convergence rate
can in general not be faster than inverse logarithmic in sample size, but describe
specific conditions on the initial conditions when faster rates are possible. In the
proofs we provide an explicit quantitative estimate for backward uniqueness of
solutions of the two-dimensional Navier-Stokes equations
On the motion of hairpin filaments in the atmospheric boundary layer
No full textA recent work of Harikrishnan et al. [“Geometry and organization of coherent structures in stably stratified atmospheric boundary layers,” arXiv:2110.02253 (2021)] has revealed an abundance of hairpin-like vortex structures, oriented in a similar direction, in the turbulent patches of a stably stratified Ekman flow. In this study, hairpin-like structures are investigated by treating them as slender vortex filaments, i.e., a vortex filament whose diameter d is small when compared to its radius of curvature R. The corrected thin-tube model of Klein and Knio [“Asymptotic vorticity structure and numerical simulation of slender vortex filaments,” J. Fluid Mech. 284, 275 (1995)] is used to compute the motion of these filaments with the atmospheric boundary layer as a background flow. Our results suggest that the orientation of the hairpin filament in the spanwise direction is linked to its initial starting height under stable stratification, whereas no such dependency can be observed with the neutrally stratified background flow. An improved feature tracking scheme based on spatial overlap for tracking Q-criterion vortex structures on the direct numerical simulation data is also developed. It overcomes the limitation of using a constant threshold in time by dynamically adjusting the thresholds to accommodate the growth or deterioration of a feature. A comparison between the feature tracking and the filament simulation reveals qualitatively similar temporal developments. Finally, an extension of the asymptotic analysis of Callegari and Ting [“Motion of a curved vortex filament with decaying vortical core and axial velocity,” J. Appl. Math. 35, 148–175 (1978)] is carried out to include the effect of gravity. The results show that, in the regime considered here, a contribution from the gravity term occurs only when the tail of an infinitely long filament is tilted at an angle relative to the wall
Investigation of DNA methylation heterogeneity in cancer
No full textWithin the body, every cell contains the same genetic blueprint, the DNA, which is wrapped around histones and densely packed in the nucleus. Given the same genome, the identity of each cell is in part defined by modifications to the histones but also the genomic sequence itself, such as DNA methylation, that define active and inactive parts of the DNA. In somatic cells, DNA methylation levels are largely bimodal, with a high genome-wide methylation average that predominantly excludes CpG islands (CGIs), features often found near gene promoters that remain free of methylation. These patterns change across the majority of human cancer types, which exhibit global loss of methylation accompanied by a gain of methylation at select CGIs. To date, bisulfite sequencing represents the gold-standard method to profile DNA methylation at single-base resolution and has been widely used to characterize and understand DNA methylation landscapes in healthy and tumor cells. This thesis presents advancements in the computational analysis of bisulfite sequencing data sets, as well as applications to large-scale studies of DNA methylation in cancer. It showcases the adaptation of a local alignment tool to enable homology search for bisulfite-converted sequences, which outperforms established semi-global alignment tools when applied to the search of metagenomic data sets. Additionally, this thesis describes the development of a new application that provides fast and simplified extraction of DNA methylation heterogeneity metrics from single reads of bisulfite sequencing data. The importance of such metrics is demonstrated in the context of two studies that focus on DNA methylation changes within primary tumors and cancer cell lines. Single-read metrics and single-cell methylome profiling show that primary tumors are mainly characterized by heterogeneous, intermediate global and CGI DNA methylation that is intrinsic to the underlying single tumor cells. In contrast, cancer cell lines mostly assume one of two different states, where global DNA methylation levels are either drastically decreased or comparable to healthy tissues, while CGIs become almost fully methylated in both scenarios. Although rarely seen in solid tumors, extremely high genome-wide methylation levels can also be observed in an exceptional primary tumor type, acute lymphoblastic leukemia, where this landscape is influenced by specific epigenetic regulators. Together, the findings of this thesis advance our ability to analyze bisulfite sequencing data sets as well as to apply these more nuanced measurements to understand DNA methylation changes during tumorigenesis and in culture
Optimal Reaction Coordinates: Variational Characterization and Sparse Computation
Get PDFReaction coordinates (RCs) are indicators of hidden, low-dimensional mechanisms that govern the long-term behavior of high-dimensional stochastic processes. We present a novel and general variational characterization of optimal RCs and provide conditions for their existence. Optimal RCs are minimizers of a certain loss function, and reduced models based on them guarantee a good approximation of the statistical long-term properties of the original high-dimensional process. We show that for slow-fast systems, metastable systems, and other systems with known good RCs, the novel theory reproduces previous insight. Remarkably, for reversible systems, the numerical effort required to evaluate the loss function scales only with the variability of the underlying, low-dimensional mechanism, and not with that of the full system. The theory provided lays the foundation for an efficient and data-sparse computation of RCs via modern machine learning techniques
Numerical Simulations Reproduce Field Observations Showing Transient Weakening During Shear Zone Formation by Diffusional Hydrogen Influx and H2O Inflow
No full textExposures on Holsnøy island (Bergen Arcs, Norway) indicate fluid infiltration through fractures into a dry, metastable granulite, which triggered a kinetically delayed eclogitization, a transient weakening during fluid-rock interaction, and formation of shear zones that widened during shearing. It remains unclear whether the effects of grain boundary-assisted aqueous fluid inflow on the duration of granulite hydration were influenced by a diffusional hydrogen influx accompanying the fluid inflow. To better estimate the fluid infiltration efficiencies and the parameter interdependencies, a 1D numerical model of a viscous shear zone is utilized and validated using measured mineral phase abundance distributions and H2O-contents in nominally anhydrous minerals of the original granulite assemblage to constrain the hydration by aqueous fluid inflow and diffusional hydrogen influx, respectively. Both hydrations are described with a diffusion equation and affect the effective viscosity. Shear zone kinematics are constrained by the observed shear strain and thickness. The model fits the phase abundance and H2O-content profiles if the effective hydrogen diffusivity is approximately one order of magnitude higher than the diffusivity for aqueous fluid inflow. The observed shear zone thickness is reproduced if the viscosity ratio between dry granulite and deforming, reequilibrating eclogite is ∼104 and that between dry granulite and hydrated granulite is ∼102. The results suggest shear velocities <10−2 cm/a, hydrogen diffusivities of ∼10−13±1 m2/s, and a shearing duration of <10 years. This study successfully links and validates field data to a shear zone model and highlights the importance of hydrogen diffusion for shear zone widening and eclogitization
Scaling approaches to quasi-geostrophic theory for moist, precipitating air
No full textQuasi-geostrophic (QG) theory is of fundamental importance in the study of large-scale atmospheric flows. In recent years, there has been growing interest in extending the classical QG plus Ekman friction layer model (QG-Ekman) to systematically include additional physical processes known to significantly contribute to real-life weather phenomena. This paper lays the foundation for combining two of these developments, namely Smith and Stechmann's family of \emph{Precipitating Quasi-Geostrophic} (PQG) models (J.\ Atmos.\ Sci, {\bfseries 74}, 3285--3303, 2017) on the one hand, and the extension of QG-Ekman for dry air by a strongly \emph{Diabatic Layer} (DL) of intermediate height (QG-DL-Ekman) in (J.\ Atmos.\ Sci, {\bfseries 79}, 887--905, 2022) on the other hand. To this end, Smith and Stechmann's PQG equations for sound-proof motions are first corroborated within a general asymptotic modeling framework starting from a full compressible flow model. The derivations show that the PQG model family is naturally embedded in the asymptotic hierarchy of scale-dependent atmospheric flow models introduced by one of the present authors in (Ann.\ Rev.\ Fluid Mech., {\bfseries 42}, 249--274). Particular emphasis is then placed on an asymptotic scaling regime for PQG that accounts for a generic Kessler-type bulk microphysics closure and is compatible with QG-DL-Ekman theory. The detailed derivation of a moist QG-DL-Ekman model is deferred to a future publication
Quantum Dynamics of Coupled Excitons and Phonons in Chain-Like Systems: Tensor Train Approaches and Higher-Order Propagators
No full textWe investigate the use of tensor-train approaches to the solution of the time-dependent Schrödinger equation for chain-like quantum systems with on-site and nearest-neighbor interactions only. Using efficient low-rank tensor train representations, we aim at reducing the memory consumption as well as the computation costs. As an example, coupled excitons and phonons modeled in terms of Fröhlich-Holstein type Hamiltonians are studied here. By comparing our tensor-train based results with semi-analytical results, we demonstrate the key role of the ranks of the quantum state vectors. Typically, an excellent quality of the solutions is found only when the maximum number of ranks exceed a certain value. One class of propagation schemes builds on splitting the Hamiltonian into two groups of interleaved nearest-neighbor interactions which commutate within each of the groups. In particular, the 4-th order Yoshida-Neri and the 8-th order Kahan-Li symplectic compositions are demonstrated to yield very accurate results, close to machine precision. However, due to the computational costs, currently their use is restricted to rather short chains. That also applies to propagations based on the time-dependent variational principle, typically used in the context of matrix product states. Yet another class of propagators involves explicit, time-symmetrized Euler integrators. Especially the 4-th order variant is recommended for quantum simulations of longer chains, even though the high precision of the splitting schemes cannot be reached. Moreover, the scaling of the computational effort with the dimensions of the local Hilbert spaces is much more favorable for the differencing than for the splitting or variational schemes
Machine Learning Coarse-Grained Potentials of Protein Thermodynamics
No full textA generalized understanding of protein dynamics is an unsolved scientific problem, the solution of which is critical to the interpretation of the structure-function relationships that govern essential biological processes. Here, we approach this problem by constructing coarse-grained molecular potentials based on artificial neural networks and grounded in statistical mechanics. For training, we build a unique dataset of unbiased all-atom molecular dynamics simulations of approximately 9 ms for twelve different proteins with multiple secondary structure arrangements. The coarse-grained models are capable of accelerating the dynamics by more than three orders of magnitude while preserving the thermodynamics of the systems. Coarse-grained simulations identify relevant structural states in the ensemble with comparable energetics to the all-atom systems. Furthermore, we show that a single coarse-grained potential can integrate all twelve proteins and can capture experimental structural features of mutated proteins. These results indicate that machine learning coarse-grained potentials could provide a feasible approach to simulate and understand protein dynamics