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Institute
We establish weighted Lp-Fourier extension estimates for O(N−k)×O(k)-invariant functions defined on the unit sphere SN−1, allowing for exponents p below the Stein–Tomas critical exponent 2(N+1)/N−1. Moreover, in the more general setting of an arbitrary closed subgroup G⊂O(N) and G-invariant functions, we study the implications of weighted Fourier extension estimates with regard to boundedness and nonvanishing properties of the corresponding weighted Helmholtz resolvent operator. Finally, we use these properties to derive new existence results for G-invariant solutions to the nonlinear Helmholtz equation −Δu−u = Q(x)|u|p−2u,u∈W2,p(RN), where Q is a nonnegative bounded and G-invariant weight function.
This thesis concerns three specific constraint satisfaction problems: the k-SAT problem, random linear equations and the Potts model. We investigated a phenomenon called replica symmetry, its consequences and its limitation. For the $k$-SAT problem, we were able to show that replica symmetry holds up to a threshold $d^{*}$. However, after another critical threshold $d^{**}$, we discovered that replica symmetry could not hold anymore, which enabled us to establish the existence of a replica symmetry breaking region. For the random linear problem, a peculiar phenomenon occurs. We observed that a more robust version of replica symmetry (strong replica symmetry) holds up to a threshold $d=e$ and ceases to hold after. This phenomenon is linked to the fact that before the threshold $d=e$, the fraction of frozen variables, i.e. variable forced to take the same value in all solutions, is concentrated around a deterministic value but vacillates between two values with equal probability for $d>e$. Lastly, for the Potts model, we show that a phenomenon called metastability occurs. The latter phenomenon can be understood as a consequence of trivial replica symmetry breaking scheme. This metastability phenomenon further produces slow mixing results for two famous Markov chains, the Glauber and the Swendsen-Wang dynamics.
In this survey paper, we present a multiscale post-processing method in exploration. Based on a physically relevant mollifier technique involving the elasto-oscillatory Cauchy–Navier equation, we mathematically describe the extractable information within 3D geological models obtained by migration as is commonly used for geophysical exploration purposes. More explicitly, the developed multiscale approach extracts and visualizes structural features inherently available in signature bands of certain geological formations such as aquifers, salt domes etc. by specifying suitable wavelet bands.
The relevant field of interest in High Energy Physics experiments is shifting to searching and studying extremely rare particles and phenomena. The search for rare probes requires an increase in the number of available statistics by increasing the particle interaction rate. The structure of the events also becomes more complicated, the multiplicity of particles in each event increases, and a pileup appears. Due to technical limitations, such data flow becomes impossible to store fully on available storage devices. The solution to the problem is the correct triggering of events and real-time data processing.
In this work, the issue of accelerating and improving the algorithms for reconstruction of the charged particles' trajectories based on the Cellular Automaton in the STAR experiment is considered to implement them for track reconstruction in real-time within the High-Level Trigger. This is an important step in the preparation of the CBM experiment as part of the FAIR Phase-0 program. The study of online data processing methods in real conditions at similar interaction energies allows us to study this process and determine the possible weaknesses of the approach.
Two versions of the Cellular Automaton based track reconstruction are discussed, which are used, depending on the detecting systems' features. HFT~CA Track Finder, similar to the tracking algorithm of the CBM experiment, has been accelerated by several hundred times, using both algorithm optimization and data-level parallelism. TPC~CA Track Finder has been upgraded to improve the reconstruction quality while maintaining high calculation speed. The algorithm was tuned to work with the new iTPC geometry and provided an additional module for very low momentum track reconstruction.
The improved track reconstruction algorithm for the TPC detector in the STAR experiment was included in the HLT reconstruction chain and successfully tested in the express production for the online real data analysis. This made it possible to obtain important physical results during the experiment runtime without the full offline data processing. The tracker is also being prepared for integration into a standard offline data processing chain, after which it will become the basic track search algorithm in the STAR experiment.
Monte Carlo methods : barrier option pricing with stable Greeks and multilevel Monte Carlo learning
(2021)
For discretely observed barrier options, there exists no closed solution under the Black-Scholes model. Thus, it is often helpful to use Monte Carlo simulations, which are easily adapted to these models. However, as presented above, the discontinuous payoff may lead to instability in option's sensitivities for Monte Carlo algorithms.
This thesis presents a new Monte Carlo algorithm that can calculate the pathwise sensitivities for discretely monitored barrier options. The idea is based on Glasserman and Staum's one-step survival strategy and the results of Alm et al., with which we can stably determine the option's sensitivities such as Delta and Vega by finite-differences. The basic idea of Glasserman and Staum is to use a truncated normal distribution, which excludes the values above the barrier (e.g.\ for knock-up-out options), instead of sampling from the full normal distribution. This approach avoids the discontinuity generated by any Monte Carlo path crossing the barrier and yields a Lipschitz-continuous payoff function.
The new part will be to develop an extended algorithm that estimates the sensitivities directly, without simulation at multiple parameter values as in finite-difference.
Consider the local volatility model, which is a generalisation of the Black-Scholes model. Although standard Monte Carlo algorithms work well for the pricing of continuously monitored barrier options within this model, they often do not behave stably with respect to numerical differentiation.
To bypass this problem, one would generally either resort to regularised differentiation schemes or derive an algorithm for precise differentiation. Unfortunately, while the widespread solution of using a Brownian bridge approach leads to accurate first derivatives, they are not Lipschitz-continuous. This leads to instability with respect to numerical differentiation for second-order Greeks.
To alleviate this problem - i.e. produce Lipschitz-continuous first-order derivatives - and reduce variance, we generalise the idea of one-step survival to general scalar stochastic differential equations. This approach leads to the new one-step survival Brownian bridge approximation, which allows for stable second-order Greeks calculations.
To show the new approach's numerical efficiency, we present a new respective Monte Carlo pathwise sensitivity estimator for the first-order Greeks and study different methods to compute second-order Greeks stably. Finally, we develop a one-step survival Brownian bridge multilevel Monte Carlo algorithm to reduce the computational cost in practice.
This thesis proves unbiasedness and variance reduction of our new, one-step survival version with respect to the classical, Brownian bridge approach. Furthermore, we will present a new convergence result for the Brownian bridge approach using the Milstein scheme under certain conditions. Overall, these properties imply convergence of the new one-step survival Brownian bridge approach.
In recent years, deep learning has become pervasive in various fields. As a family of machine learning methods it is used in a broad set of applications, such as image processing, voice recognition, email filtering, computer vision. Most modern deep learning algorithms are based on artificial neural networks inspired by the biological neural networks constituting animal brains. Also in computational finance deep learning may be of use: Consider there is no closed-solution available for an option price, Monte Carlo simulations are substantially for estimation. Instead of persistently contributing new price computations arising from an updated volatility term, one could replace these by evaluating a neural network.
If an according neural network is available, the evaluation could lead to substantial savings and be highly efficient. I.e., once trained, a neural network could save further expensive estimations. However, in practice, the challenge is the training process of the neural network.
We study and compare two generic neural network training algorithms' computational complexity. Then, we introduce a new multilevel training algorithm that combines a deep learning algorithm with the idea of multilevel Monte Carlo path simulation. The idea is to train several neural networks with training data computed from the so-called level estimators of the multilevel Monte Carlo approach introduced by Giles. We show that the new method can reduce computational complexity by formulating a complexity theorem.
We show how nonlocal boundary conditions of Robin type can be encoded in the pointwise expression of the fractional operator. Notably, the fractional Laplacian of functions satisfying homogeneous nonlocal Neumann conditions can be expressed as a regional operator with a kernel having logarithmic behaviour at the boundary.
This article deals with the solution of linear ill-posed equations in Hilbert spaces. Often, one only has a corrupted measurement of the right hand side at hand and the Bakushinskii veto tells us, that we are not able to solve the equation if we do not know the noise level. But in applications it is ad hoc unrealistic to know the error of a measurement. In practice, the error of a measurement may often be estimated through averaging of multiple measurements. We integrated that in our anlaysis and obtained convergence to the true solution, with the only assumption that the measurements are unbiased, independent and identically distributed according to an unknown distribution.
We prove new existence results for a nonlinear Helmholtz equation with sign-changing nonlinearity of the form − delta u−k2u=Q(x)/u/p−2u, uEW2, p(RN) – delta u − k2u=Q(x)/u/p−2u, uEW2, p(RN) with k>0, k>0, N≥3N≥3, pE[2(N+1)N − 1, 2NN − 2)pE[2(N+1)N − 1, 2NN−2) and QEL ∞ (RN)QEL ∞ (RN). Due to the sign-changes of Q, our solutions have infinite Morse-Index in the corresponding dual variational formulation.
Objectives: To analyze the performance of radiological assessment categories and quantitative computational analysis of apparent diffusion coefficient (ADC) maps using variant machine learning algorithms to differentiate clinically significant versus insignificant prostate cancer (PCa). Methods: Retrospectively, 73 patients were included in the study. The patients (mean age, 66.3 ± 7.6 years) were examined with multiparametric MRI (mpMRI) prior to radical prostatectomy (n = 33) or targeted biopsy (n = 40). The index lesion was annotated in MRI ADC and the equivalent histologic slides according to the highest Gleason Grade Group (GrG). Volumes of interest (VOIs) were determined for each lesion and normal-appearing peripheral zone. VOIs were processed by radiomic analysis. For the classification of lesions according to their clinical significance (GrG ≥ 3), principal component (PC) analysis, univariate analysis (UA) with consecutive support vector machines, neural networks, and random forest analysis were performed. Results: PC analysis discriminated between benign and malignant prostate tissue. PC evaluation yielded no stratification of PCa lesions according to their clinical significance, but UA revealed differences in clinical assessment categories and radiomic features. We trained three classification models with fifteen feature subsets. We identified a subset of shape features which improved the diagnostic accuracy of the clinical assessment categories (maximum increase in diagnostic accuracy ΔAUC = + 0.05, p < 0.001) while also identifying combinations of features and models which reduced overall accuracy. Conclusions: The impact of radiomic features to differentiate PCa lesions according to their clinical significance remains controversial. It depends on feature selection and the employed machine learning algorithms. It can result in improvement or reduction of diagnostic performance.
The recently introduced Lipschitz–Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz–Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.
The thesis is composed of four Chapters.
In the first Chapter, the boundary expression of the one-sided shape derivative of nonlocal Sobolev best constants is derived. As a simple consequence, we obtain the fractional version of the so-called Hadamard formula for the torsional rigidity and the first Dirichlet eigenvalue. An application to the optimal obstacle placement problem for the torsional rigidity and the first eigenvalue of the fractional Laplacian is given.
In the second Chapter, we introduce and prove a new maximum principle for doubly antisymmetric functions. The latter can be seen as the first step towards studying the optimal obstacle placement problem for the second fractional eigenvalue. Using the new maximum principle we derive new symmetry results for odd solutions to semilinear Dirichlet boundary value problems with Lipschitz nonlinearity.
In the third Chapter, we derive new integration by parts formula for the fractional Laplace operator with a general globally Lipschitz vector field and in particular, we obtain a new Pohozaev type identity generalizing the one obtained by X. Ros-Oton and J. Serra. As an application we obtain nonexistence results for semilinear Dirichlet boundary problems in bounded domains that are not necessarly starshaped.
In the last Chapter, we study symmetry properties of second eigenfunctions of annuli. Using results from the first Chapter and the maximum principle in Chpater 2, we extend the result on the optimal obstacle placement problem from the first eigenvalue to the second eigenvalue.
Reproducible annotations
(2022)
This bachelor thesis presents a software solution which implements reproducible annotations in the context of the UIMA framework. This is achieved by creating an automated containerization of arbitrary analysis engines and annotating every analysis engine configuration in the processed CAS document. Any CAS document created by this solution is self sufficient and able to reproduce the exact environment under which it was created.
A review of the state-of-the art software in the field of UIMA reveals that there are many implementations trying to increase reproducibility for a given application relying on UIMA, but no publication trying to increase the reproducibility of UIMA itself. This thesis improves upon that technological gap and provides a throughout analysis at the end which shows a negligible overhead in memory consumption, but a significant performance regression depending on the complexity of the analysis engine which was examined.
Ein aktuelles Forschungsthema ist die automatische Generierung von 3D-Szenen ausgehend von Beschreibungen in natürlicher Sprache. S.g. Text2Scene-Anwendungen sollen Objekte und räumliche Relationen in einer Texteingabe identifizieren und mit 3D-Modellen eine visuelle Repräsentation der Beschreibung konstruieren. Bisherige Ansätze kombinieren eine
stichwortbasierte Erkennung von explizit gemachten Angaben mit vorher gelerntem Allgemeinwissen über die sinnvolle Anordnung von Objekten. Den Anwendungen fehlt jedoch ein tiefergehendes Verständnis von räumlicher Sprache.
Mit dem Annotationsschema ISOSpace können Texte mit detaillierten räumlichen Informationen angereichert und so für NLP-Anwendungen verständlicher gemacht werden. Bereits in einer früheren Arbeit wurde der SemAF-Annotator zum Erstellen von ISOSpaceAnnotationen als Modul für den TextAnnotator entwickelt. In dieser Arbeit wurde der SemAF-Annotator zusätzlich um eine Funktionalität zur Szenenerstellung erweitert: Benutzer können einzelnen Wörtern in der Weboberfläche des TextAnnotators Objekte aus dem ShapeNet Datensatz zuordnen und diese in einer zweidimensionalen Darstellung einer Szene räumlich anordnen. Trotz einiger Einschränkungen durch die fehlende dritte Dimension lassen sich in vielen Fällen gute Ergebnisse erzielen. Die auf diese Weise erzeugten Szenen sollen später in Kombination mit den ISOSpace-Annotionen verwendet werden, um Text2SceneAnwendungen zu entwickeln, die ein umfassenderes räumliches Verständnis aufweisen.
Kleinere Nebenaufgaben dieser Arbeit waren die Erweiterung des SemAF-Annotators um zusätzliche Annotationstypen sowie diverse Nachbesserungen der bereits bestehenden Funktionalität zur ISOSpace Annotation.
The recognition of pharmacological substances, compounds and proteins is an essential preliminary work for the recognition of relations between chemicals and other biomedically relevant units. In this paper, we describe an approach to Task 1 of the PharmaCoNER Challenge, which involves the recognition of mentions of chemicals and drugs in Spanish medical texts. We train a state-of-the-art BiLSTM-CRF sequence tagger with stacked Pooled Contextualized Embeddings, word and sub-word embeddings using the open-source framework FLAIR. We present a new corpus composed of articles and papers from Spanish health science journals, termed the Spanish Health Corpus, and use it to train domain-specific embeddings which we incorporate in our model training. We achieve a result of 89.76% F1-score using pre-trained embeddings and are able to improve these results to 90.52% F1-score using specialized embeddings.
Despite the great importance of the Latin language in the past, there are relatively few resources available today to develop modern NLP tools for this language. Therefore, the EvaLatin Shared Task for Lemmatization and Part-of-Speech (POS) tagging was published in the LT4HALA workshop. In our work, we dealt with the second EvaLatin task, that is, POS tagging. Since most of the available Latin word embeddings were trained on either few or inaccurate data, we trained several embeddings on better data in the first step. Based on these embeddings, we trained several state-of-the-art taggers and used them as input for an ensemble classifier called LSTMVoter. We were able to achieve the best results for both the cross-genre and the cross-time task (90.64% and 87.00%) without using additional annotated data (closed modality). In the meantime, we further improved the system and achieved even better results (96.91% on classical, 90.87% on cross-genre and 87.35% on cross-time).
We present new results on nonlocal Dirichlet problems established by means of suitable spectral theoretic and variational methods, taking care of the nonlocal feature of the operators. We mainly address: First, we estimate the Morse index of radially symmetric sign changing bounded weak solutions to a semilinear Dirichlet problem involving the fractional Laplacian. In particular, we derive a conjecture due to Bañuelos and Kulczycki on the geometric structure of the second Dirichlet eigenfunctions. Secondly, we study a small order asymptotics with respect to the parameter s of the Dirichlet eigenvalues problem for the fractional Laplacian. Thirdly, we deal with the logarithmic Schrödinger operator. In particular, we provide an alternative to derive the singular integral representation corresponding to the associated Fourier symbol and introduce tools and functional analytic framework for variational studies. Finaly, we study nonlocal operators of order strictly below one. In particular, we investigate interior regularity properties of weak solutions to the associated Poisson problem depending on the regularity of the right-hand side.
Biodiversity information is contained in countless digitized and unprocessed scholarly texts. Although automated extraction of these data has been gaining momentum for years, there are still innumerable text sources that are poorly accessible and require a more advanced range of methods to extract relevant information. To improve the access to semantic biodiversity information, we have launched the BIOfid project (www.biofid.de) and have developed a portal to access the semantics of German language biodiversity texts, mainly from the 19th and 20th century. However, to make such a portal work, a couple of methods had to be developed or adapted first. In particular, text-technological information extraction methods were needed, which extract the required information from the texts. Such methods draw on machine learning techniques, which in turn are trained by learning data. To this end, among others, we gathered the BIOfid text corpus, which is a cooperatively built resource, developed by biologists, text technologists, and linguists. A special feature of BIOfid is its multiple annotation approach, which takes into account both general and biology-specific classifications, and by this means goes beyond previous, typically taxon- or ontology-driven proper name detection. We describe the design decisions and the genuine Annotation Hub Framework underlying the BIOfid annotations and present agreement results. The tools used to create the annotations are introduced, and the use of the data in the semantic portal is described. Finally, some general lessons, in particular with multiple annotation projects, are drawn.
Are nearby places (e.g., cities) described by related words? In this article, we transfer this research question in the field of lexical encoding of geographic information onto the level of intertextuality. To this end, we explore Volunteered Geographic Information (VGI) to model texts addressing places at the level of cities or regions with the help of so-called topic networks. This is done to examine how language encodes and networks geographic information on the aboutness level of texts. Our hypothesis is that the networked thematizations of places are similar, regardless of their distances and the underlying communities of authors. To investigate this, we introduce Multiplex Topic Networks (MTN), which we automatically derive from Linguistic Multilayer Networks (LMN) as a novel model, especially of thematic networking in text corpora. Our study shows a Zipfian organization of the thematic universe in which geographical places (especially cities) are located in online communication. We interpret this finding in the context of cognitive maps, a notion which we extend by so-called thematic maps. According to our interpretation of this finding, the organization of thematic maps as part of cognitive maps results from a tendency of authors to generate shareable content that ensures the continued existence of the underlying media. We test our hypothesis by example of special wikis and extracts of Wikipedia. In this way, we come to the conclusion that geographical places, whether close to each other or not, are located in neighboring semantic places that span similar subnetworks in the topic universe.
The annotation of texts and other material in the field of digital humanities and Natural Language Processing (NLP) is a common task of research projects. At the same time, the annotation of corpora is certainly the most time- and cost-intensive component in research projects and often requires a high level of expertise according to the research interest. However, for the annotation of texts, a wide range of tools is available, both for automatic and manual annotation. Since the automatic pre-processing methods are not error-free and there is an increasing demand for the generation of training data, also with regard to machine learning, suitable annotation tools are required. This paper defines criteria of flexibility and efficiency of complex annotations for the assessment of existing annotation tools. To extend this list of tools, the paper describes TextAnnotator, a browser-based, multi-annotation system, which has been developed to perform platform-independent multimodal annotations and annotate complex textual structures. The paper illustrates the current state of development of TextAnnotator and demonstrates its ability to evaluate annotation quality (inter-annotator agreement) at runtime. In addition, it will be shown how annotations of different users can be performed simultaneously and collaboratively on the same document from different platforms using UIMA as the basis for annotation.