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Hoelder regularity of the pressure for weak solutions of the 3D Euler equations in bounded domains
Get PDFWe consider the three-dimensional incompressible Euler equations on a bounded
domain
with C3 boundary. We prove that if the velocity field u 2 C0;�(
) with � > 0
(where we are omitting the time dependence), it follows that the pressure p 2 C0;�(
).
In order to prove this result we use a local parametrisation of the boundary and a very
weak formulation of the boundary condition for the pressure, as was introduced in [C.
Bardos and E.S. Titi, Philos. Trans. Royal Soc. A, 380 (2022), 20210073]. Moreover,
we provide an example illustrating the necessity of this new very weak formulation of
the boundary condition for the pressure. This result is of importance for the proof of
the �rst half of the Onsager Conjecture, the su�cient conditions for energy conservation
of weak solutions to the three-dimensional incompressible Euler equations in bounded
domains
Mistle: bringing spectral library predictions to metaproteomics with an efficient search index
No full textMotivation
Deep learning has moved to the forefront of tandem mass spectrometry-driven proteomics and authentic prediction for peptide fragmentation is more feasible than ever. Still, at this point spectral prediction is mainly used to validate database search results or for confined search spaces. Fully predicted spectral libraries have not yet been efficiently adapted to large search space problems that often occur in metaproteomics or proteogenomics.
Results
In this study, we showcase a workflow that uses Prosit for spectral library predictions on two common metaproteomes and implement an indexing and search algorithm, Mistle, to efficiently identify experimental mass spectra within the library. Hence, the workflow emulates a classic protein sequence database search with protein digestion but builds a searchable index from spectral predictions as an in-between step. We compare Mistle to popular search engines, both on a spectral and database search level, and provide evidence that this approach is more accurate than a database search using MSFragger. Mistle outperforms other spectral library search engines in terms of run time and proves to be extremely memory efficient with a 4- to 22-fold decrease in RAM usage. This makes Mistle universally applicable to large search spaces, e.g. covering comprehensive sequence databases of diverse microbiomes.
Availability and implementation
Mistle is freely available on GitHub at https://github.com/BAMeScience/Mistle
Transporting Higher-Order Quadrature Rules: Quasi-Monte Carlo Points and Sparse Grids for Mixture Distributions
No full textIntegration against, and hence sampling from, high-dimensional probability distributions is of essential importance in many application areas and has been an active research area for decades. One approach that has drawn increasing attention in recent years has been the generation of samples from a target distribution Ptar using transport maps: if Ptar=T#Pref is the pushforward of an easily-sampled probability distribution Pref under the transport map T, then the application of T to Pref-distributed samples yields Ptar-distributed samples. This paper proposes the application of transport maps not just to random samples, but also to quasi-Monte Carlo points, higher-order nets, and sparse grids in order for the transformed samples to inherit the original convergence rates that are often better than N−1/2, N being the number of samples/quadrature nodes. Our main result is the derivation of an explicit transport map for the case that Ptar is a mixture of simple distributions, e.g.\ a Gaussian mixture, in which case application of the transport map T requires the solution of an \emph{explicit} ODE with \emph{closed-form} right-hand side. Mixture distributions are of particular applicability and interest since many methods proceed by first approximating Ptar by a mixture and then sampling from that mixture (often using importance reweighting). Hence, this paper allows for the sampling step to provide a better convergence rate than N−1/2 for all such methods
Algorithms for finding RNA sequence-structure motifs
No full textThe function of non-coding RNA sequences is largely determined by their spatial conformation. This is the secondary structure of the molecule, which is formed by Watson–Crick interactions between nucleotides. Hence, modern RNA alignment algorithms routinely take structural information into account. Essential tasks for discovering yet unknown RNA families and inferring their possible functions are the structural alignment of RNAs and the subsequent search of the derived structural motifs. These tasks demand a lot of computational resources, especially for aligning many long sequences, and it therefore requires efficient algorithms that utilize modern hardware when available. A subset of the secondary structures contains pseudoknots, which are overlapping interactions that add additional complexity to the analysis and are often ignored in available software.
In this thesis, I present LaRA 2 and MaRs, two SeqAn-based software tools that implement algorithms for finding sequence-structure motifs in genomic sequences. In contrast to other programs, my tools can handle arbitrary pseudoknots. They use multithreading for parallel execution and are implemented in modern C++ code for maximal longevity and performance.
LaRA2 is significantly faster than comparable software for accurate pairwise and multiple alignments of structured RNA sequences. It uses a new heuristic for computing a lower boundary to the solution and employs vectorization techniques for speeding up the time-critical parts of the algorithm.
MaRs can be applied in a workflow right after LaRA2 and derives sequence-structure motifs from the structural alignments. The motifs are descriptors of the RNA sequences’ properties and drive the search for homologs in genomic sequences. MaRs employs a bidirectional index on the genomic sequences and an optimized multithreaded search strategy for finding the matches really fast. The use of a thread pool, effective pruning strategies, and a low memory footprint ensure that MaRs is capable of processing extremely large data sets
Performance-Driven Algorithm Engineering
No full textA number of technological advancements in high-throughput genome sequencing have led to the generation of exabyte-scale sequencing data worldwide in recent years. These developments have facilitated large-scale resequencing projects like the 1000 Genomes Project, which aim to catalog the genetic diversity of organisms and specific populations. In the context of medical research, incorporating population data, is of particular interest. However, existing applications are often optimised for analysing a few sequences, and the methods used cannot be easily transferred to these vast datasets without exceeding system resources or producing results within a suitable time frame. Simultaneously, the execution model of processors has evolved from sequential to highly parallel process execution, thanks to the addition of processor cores, vector processing units, and advances in superscalar processor designs. This ongoing development in high-performance computing requires applications and algorithms to scale with the growing levels of parallelism. However, highly optimised algorithms are often embedded within applications, making them practically inaccessible to the scientific community. In this dissertation, we first investigate a generic approach to parallelise and vectorise pairwise sequence alignments using a dynamically scalable concurrency model. We explore various techniques and code-level optimisations to effectively utilise the available parallelisms on modern high-performance CPUs. The results demonstrate that the dynamically accelerated pairwise sequence alignment scales proportionally with the number of cores and provides speed-ups of up to a factor of 2500 compared to the sequential reference implementation on modern hardware. Second, we propose a general solution for data-compressed acceleration of pattern matching algorithms by compressing a large collection of related DNA sequences and providing a set of composable algorithms to refine and optimise the applicable operations. Our research on data-compressed acceleration shows hundred-fold speed-ups for online searching over a pangenome comprising over 5000 reference sequences compared to the naive approach of individually searching all sequences. These speed-ups are achieved while utilising only a fraction of the main memory. Moreover, we implemented these features in dedicated modules of the open-source software library SeqAn (https://www.seqan.de/), making them accessible and adoptable by the entire research community. While doing so, we strived for a user-friendly API design so that these methods can be easily customised and extended, making them applicable to a wide range of applications in the domain of computational sequence analysis
Bravo MaRDI: A Wikibase Knowledge Graph on Mathematics
No full textMathematical world knowledge is a fundamental component of Wikidata. However, to date, no expertly curated knowledge graph has focused specifically on contemporary mathematics. Addressing this gap, the Mathematical Research Data Initiative (MaRDI) has developed a comprehensive knowledge graph that links multimodal research data in mathematics. This encompasses traditional research data items like datasets, software, and publications and includes semantically advanced objects such as mathematical formulae and hypotheses. This paper details the abilities of the MaRDI knowledge graph, which is based on Wikibase, leading up to its inaugural public release, codenamed Bravo, available on https://portal.mardi4nfdi.de
Efficient global sensitivity analysis of kinetic Monte Carlo simulations using Cramérs–von Mises distance
No full textTypically, the parameters entering a physical simulation model carry some kind of uncertainty, e.g., due to the intrinsic approximations in a higher fidelity theory from which they have been obtained. Global sensitivity analysis (GSA) targets quantifying which parameter uncertainties impact the accuracy of the simulation results, e.g., to identify which parameters need to be determined more accurately. We present a GSA approach based on the Cramérs–von Mises distance. Unlike prevalent approaches, it combines the following properties: (i) it is equally suited for deterministic as well as stochastic model outputs, (ii) it does not require gradients, and (iii) it can be estimated from numerical quadrature without further numerical approximations. Using quasi-Monte Carlo for numerical integration and a first-principles kinetic Monte Carlo model for the CO oxidation on RuO2(110), we examine the performance of the approach. We find that the results agree very well with what is known in the literature about the sensitivity of this model and that the approach converges in a modest number of quadrature points. Furthermore, it appears to be robust against even extreme relative noise. All these properties make the method particularly suited for expensive (kinetic) Monte Carlo models because we can reduce the number of simulations as well as the target variance of each of these
Chemical diffusion master equation: formulations of reaction–diffusion processes on the molecular level
No full textThe chemical diffusion master equation (CDME) describes the probabilistic dynamics of reaction–diffusion systems at the molecular level [del Razo et al., Lett. Math. Phys. 112, 49 (2022)]; it can be considered as the master equation for reaction–diffusion processes. The CDME consists of an infinite ordered family of Fokker–Planck equations, where each level of the ordered family corresponds to a certain number of particles and each particle represents a molecule. The equations at each level describe the spatial diffusion of the corresponding set of particles, and they are coupled to each other via reaction operators—linear operators representing chemical reactions. These operators change the number of particles in the system and, thus, transport probability between different levels in the family. In this work, we present three approaches to formulate the CDME and show the relations between them. We further deduce the non-trivial combinatorial factors contained in the reaction operators, and we elucidate the relation to the original formulation of the CDME, which is based on creation and annihilation operators acting on many-particle probability density functions. Finally, we discuss applications to multiscale simulations of biochemical systems among other future prospects
Derivation of a generalized quasi-geostrophic approximation for inviscid ows in a channel domain: The fast waves correction
Get PDFThis paper is devoted to investigating the rotating Boussinesq equations
of inviscid, incompressible
ows with both fast Rossby waves and
fast internal gravity waves. The main objective is to establish a rigorous
derivation and justi�cation of a new generalized quasi-geostrophic
approximation in a channel domain with no normal
ow at the upper
and lower solid boundaries, taking into account the resonance terms
due to the fast and slow waves interactions. Under these circumstances,
We are able to obtain uniform estimates and compactness without the
requirement of either well-prepared initial data (as in [10]) or domain
with no boundary (as in [17]). In particular, the nonlinear resonances
and the new limit system, which takes into account the fast waves
correction to the slow waves dynamics, are also identi�ed without introducing
Fourier series expansion. The key ingredient includes the
introduction of (full) generalized potential vorticity
Statistically Optimal Force Aggregation for Coarse-Graining Molecular Dynamics
No full textMachine-learned coarse-grained (CG) models have the potential for simulating large molecular complexes beyond what is possible with atomistic molecular dynamics. However, training accurate CG models remains a challenge. A widely used methodology for learning CG force-fields maps forces from all-atom molecular dynamics to the CG representation and matches them with a CG force-field on average. We show that there is flexibility in how to map all-atom forces to the CG representation, and that the most commonly used mapping methods are statistically inefficient and potentially even incorrect in the presence of constraints in the all-atom simulation. We define an optimization statement for force mappings and demonstrate that substantially improved CG force-fields can be learned from the same simulation data when using optimized force maps. The method is demonstrated on the miniproteins Chignolin and Tryptophan Cage and published as open-source code