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Linear Parabolic Problems in Random Moving Domains
No full textWe consider linear parabolic equations on a random non-cylindrical domain. Utilizing the domain mapping method, we write the problem as a partial differential equation with random coefficients on a cylindrical deterministic domain. Exploiting the deterministic results concerning equations on non-cylindrical domains, we state the necessary assumptions about the velocity filed and in addition, about the flow transformation that this field generates. In this paper we consider both cases, the uniformly bounded with respect to the sample and log-normal type transformation. In addition, we give an explicit example of a log-normal type transformation and prove that it does not satisfy the uniformly bounded condition. We define a general framework for considering linear parabolic problems on random non-cylindrical domains. As the first example, we consider the heat equation on a random tube domain and prove its well-posedness. Moreover, as the other example we consider the parabolic Stokes equation which illustrates the case when it is not enough just to study the plain-back transformation of the function, but instead to consider for example the Piola type transformation, in order to keep the divergence free property
Orthogonality of fluxes in general nonlinear reaction networks
Get PDFWe consider the chemical reaction networks and study currents in these systems. Reviewing recent decomposition of rate functionals from large deviation theory for Markov processes, we adapt these results for reaction networks. In particular, we state a suitable generalisation of orthogonality of forces in these systems, and derive an inequality that bounds the free energy loss and Fisher information by the rate functional
Genome Analysis Methods using Long Read Nanopore Sequencing
No full textThird-generation long-read technologies denote the latest progression in high throughput DNA and RNA sequence analysis. Complementing the widespread second-generation short-read platforms, long-read sequencing adds unique application opportunities by generating previously unattainable read lengths. Despite the remaining higher error rate compared to short reads, single-molecule real-time sequencing (SMRT) and nanopore sequencing advanced to be state-of-the-art for \textit{de-novo} genome assemblies and identification of structural variants. Continuous throughput and accuracy improvements lead to development of novel methods and applications at a fast pace. We identify major application fields and key bioinformatic software for long-read sequencing data analysis by employing a data driven literature research. The integration of citations and keywords into a literature graph provides a scaling approach to analyze an exponentially growing number of third-generation sequencing related publications. Even though sparking the development of countless bioinformatics software, the streamlined nanopore data processing into standardized formats is still lacking. As an enabling step for its successful application, we developed \textit{Nanopype}, a modular and scalable pipeline. Our approach facilitates the basic steps of basecalling, alignment, methylation- and structural variant detection with exchangeable tools in each module. Optimized for the usage on high performance compute clusters, we propose a raw data management, capable of handling multiple sequencing devices placed locally and remotely. Strict version control of integrated tools and deployment as containerized software, ensure reproducibility across projects and laboratories. Finally, we analyze disease associated repeat regions utilizing targeted nanopore sequencing and the \textit{Nanopype} processing infrastructure. The expansion of unstable genomic short tandem repeats (STRs) is of particular interest as it causes more than 30 Mendelian human disorders. Long stretches of repetitive sequence render these regions inaccessible for short-read sequencing by synthesis. Furthermore, finding current nanopore basecalling algorithms insufficient to resolve the repeat length, we developed \textit{STRique}, a raw nanopore signal based repeat detection and quantification software. We demonstrate the precise analysis of repeat lengths on patient-derived samples containing C9orf72 and FMR1 repeat expansions. The additional integration of repeat- and nearby promoter-methylation levels reveal a repeat length depending gain, suggesting an epigenetic response to the expansion. Taken together, this work contributes to further increase the usability and provides novel insights based on third-generation nanopore sequencing
A large‐deviations principle for all the cluster sizes of a sparse Erdős–Rényi graph
No full textAbstract
Let urn:x-wiley:rsa:media:rsa21007:rsa21007-math-0001 be the Erdős–Rényi graph with connection probability urn:x-wiley:rsa:media:rsa21007:rsa21007-math-0002 as N → ∞ for a fixed t ∈ (0, ∞). We derive a large‐deviations principle for the empirical measure of the sizes of all the connected components of urn:x-wiley:rsa:media:rsa21007:rsa21007-math-0003, registered according to microscopic sizes (i.e., of finite order), macroscopic ones (i.e., of order N), and mesoscopic ones (everything in between). The rate function explicitly describes the microscopic and macroscopic components and the fraction of vertices in components of mesoscopic sizes. Moreover, it clearly captures the well known phase transition at t = 1 as part of a comprehensive picture. The proofs rely on elementary combinatorics and on known estimates and asymptotics for the probability that subgraphs are connected. We also draw conclusions for the strongly related model of the multiplicative coalescent, the Marcus–Lushnikov coagulation model with monodisperse initial condition, and its gelation phase transition
Agent-based modeling: Population limits and large timescales
No full textModeling, simulation, and analysis of interacting agent systems is a broad field of research, with existing approaches reaching from informal descriptions of interaction dynamics to more formal, mathematical models. In this paper, we study agent-based models (ABMs) given as continuous-time stochastic processes and their pathwise approximation by ordinary and stochastic differential equations (SDEs) for medium to large populations. By means of an appropriately adapted transfer operator approach, we study the behavior of the ABM process on long time scales. We show that, under certain conditions, the transfer operator approach allows us to bridge the gap between the pathwise results for large populations on finite timescales, i.e., the SDE limit model, and approaches built to study dynamical behavior on long time scales like large deviation theory. The latter provides a rigorous analysis of rare events including the associated asymptotic rates on timescales that scale exponentially with the population size. We demonstrate that it is possible to reveal metastable structures and timescales of rare events of the ABM process by finite-length trajectories of the SDE process for large enough populations. This approach has the potential to drastically reduce computational effort for the analysis of ABMs
STOCHASTIC HOMOGENIZATION OF L-CONVEX GRADIENT FLOWS
Get PDFAbstract. In this paper we present a stochastic homogenization result for
a class of Hilbert space evolutionary gradient systems driven by a quadratic
dissipation potential and a �-convex energy functional featuring random and
rapidly oscillating coe�cients. Speci�c examples included in the result are
Allen-Cahn type equations and evolutionary equations driven by the p-Laplace
operator with p 2 (1;1). The homogenization procedure we apply is based
on a stochastic two-scale convergence approach. In particular, we de�ne a stochastic
unfolding operator which can be considered as a random counterpart of
the well-established notion of periodic unfolding. The stochastic unfolding procedure
grants a very convenient method for homogenization problems de�ned
in terms of (�-)convex functionals
The Collaborative Research Center FONDA
No full textToday’s scientific data analysis very often requires complex Data Analysis Workflows (DAWs) executed over distributed computational infrastructures, e.g., clusters. Much research effort is devoted to the tuning and performance optimization of specific workflows for specific clusters. However, an arguably even more important problem for accelerating research is the reduction of development, adaptation, and maintenance times of DAWs. We describe the design and setup of the Collaborative Research Center (CRC) 1404 “FONDA -– Foundations of Workflows for Large-Scale Scientific Data Analysis”, in which roughly 50 researchers jointly investigate new technologies, algorithms, and models to increase the portability, adaptability, and dependability of DAWs executed over distributed infrastructures. We describe the motivation behind our project, explain its underlying core concepts, introduce FONDA’s internal structure, and sketch our vision for the future of workflow-based scientific data analysis. We also describe some lessons learned during the “making of” a CRC in Computer Science with strong interdisciplinary components, with the aim to foster similar endeavors
Estimation of the Koopman generator by Newton’s extrapolation
No full textThis article addresses the problem of estimating the Koopman generator of a Markov process. The direct computation of the infinitesimal generator is not easy because of the discretization of the state space, in particular because of the trade-off inherent in the choice of the best lag time to study the process. Short lag times implies a strong discretization of the state space and a consequent loss of Markovianity. Large lag times bypass events on fast timescales. We propose a method to approximate the generator with the computation of the Newton polynomial extrapolation. This technique is a multistep approach which uses as its input Koopman transfer operators evaluated for a series of lag times. Thus, the estimated infinitesimal generator combines information from different time resolutions and does not bias only fast- or slow-decaying dynamics. We show that the multi-scale Newton method can improve the estimation of the generator in comparison to the computation using finite difference or matrix logarithm methods.
Read More: https://epubs.siam.org/doi/abs/10.1137/20M1333006?mobileUi=
The augmented jump chain – a sparse representationof time-dependent Markov jump processes
No full textAbstract
Modern methods of simulating molecular systems are based on the mathematical theory of Markov operators with a focus on autonomous equilibrated systems. However, non-autonomous physical systems or non-autonomous simulation processes are becoming more and more important. A representation of non-autonomous Markov jump processes is presented as autonomous Markov chains on space-time. Augmenting the spatial information of the embedded Markov chain by the temporal information of the associated jump times, the so-called augmented jump chain is derived. The augmented jump chain inherits the sparseness of the infinitesimal generator of the original process and therefore provides a useful tool for studying time-dependent dynamics even in high dimensions. Furthermore, possible generalizations and applications to the computation of committor functions and coherent sets in the non-autonomous setting are discussed. After deriving the theoretical foundations, the concepts with a proof-of-concept Galerkin discretization of the transfer operator of the augmented jump chain applied to simple examples are illustrated
Estimation of the Koopman Generator by Newton's Extrapolation
No full textThis article addresses the problem of estimating the Koopman generator of a Markov process. The direct computation of the infinitesimal generator is not easy because of the discretization of the state space, in particular because of the trade-off inherent in the choice of the best lag time to study the process. Short lag times implies a strong discretization of the state space and a consequent loss of Markovianity. Large lag times bypass events on fast timescales. We propose a method to approximate the generator with the computation of the Newton polynomial extrapolation. This technique is a multistep approach which uses as its input Koopman transfer operators evaluated for a series of lag times. Thus, the estimated infinitesimal generator combines information from different time resolutions and does not bias only fast- or slow-decaying dynamics. We show that the multi-scale Newton method can improve the estimation of the generator in comparison to the computation using finite difference or matrix logarithm methods.
Read More: https://epubs.siam.org/doi/abs/10.1137/20M1333006?mobileUi=