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- Zustandsschätzung (1)
- Zählprozesse (1)
- a posteriori stopping rule (1)
- absorbing boundary (1)
- absorbing set (1)
- absorption (1)
- accelerated life time model (1)
- accelerated small (1)
- accuracy (1)
- acute severe (1)
- adaptive (1)
- adaptivity (1)
- adipose tissue (1)
- adrenal insufficiency (1)
- aerosol (1)
- aerosol distribution (1)
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- aerosols (1)
- affine (1)
- affine invariance (1)
- algebra (1)
- algebra of rank n (1)
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- algebras (1)
- alignment (1)
- alternating direction implicit (1)
- alternative variety (1)
- amoeboid motion (1)
- amöboide Bewegung (1)
- analytic functional (1)
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- asymptotic (1)
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- asymptotic expansions (1)
- asymptotic methods (1)
- asymptotic properties of eigenfunctions (1)
- asymptotic stable (1)
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- asymptotics (1)
- asymptotics of solutions (1)
- asymptotische Entwicklung (1)
- asymptotische Normalverteilung (1)
- attenuated Radon transform (1)
- aurora (1)
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- b-value (1)
- backtrajectories; (1)
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- balanced dynamics (1)
- bar with variable cross-section (1)
- basic ideas ('Grundvorstellungen') (1)
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- bedingter Erwartungswert (1)
- behavior (1)
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- compressible Euler equations (1)
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- conditioned (1)
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- conditions (1)
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- congenital adrenal hyperplasia (1)
- congruence (1)
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- convergence assessment (1)
- convergence rate (1)
- corner parametrices (1)
- corona virus (1)
- correlated noise (1)
- cortisol (1)
- coupled solution (1)
- covering (1)
- critical and subcritical Dawson-Watanabe process (1)
- criticality theorem (1)
- criticality theory (1)
- critically ill (1)
- crohn's disease (1)
- crossed product (1)
- curvature varifold (1)
- cusp (1)
- cusped bar (1)
- das Cauchyproblem (1)
- das Goursatproblem (1)
- das charakteristische Cauchyproblem (1)
- data-driven (1)
- dbar-Neumann problem (1)
- de Sitter model ; Fundamental solutions ; Decay estimates (1)
- decay of eigenfunctions (1)
- decomposition (1)
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- degenerate elliptic equations (1)
- degenerate elliptic systems (1)
- delaney-dress tiling theory (1)
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- disagreement percolation (1)
- discontinuous Robin condition (1)
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- discrete Schrodinger (1)
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- diskreter Witten-Laplace-Operator (1)
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- distribution (1)
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- distributions with one-sided support (1)
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- doppelsemigroup (1)
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- edge Sobolev spaces (1)
- edge algebra (1)
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- edge quantizations (1)
- edge spaces (1)
- edge symbol (1)
- edge- and corner-degenerate symbols (1)
- eigenfunction (1)
- eigenvalue asymptotics (1)
- eigenvalue decay (1)
- eigenvalues (1)
- elastic bar (1)
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- elliptic boundary (1)
- elliptic boundary conditions (1)
- elliptic complex (1)
- elliptic differential operators of firstorder (1)
- elliptic equation (1)
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- elliptic morphism (1)
- elliptic operators in subspaces (1)
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- elliptic problem (1)
- elliptic problems (1)
- elliptic quasicomplexes (1)
- elliptic systems (1)
- ellipticity in the edge calculus (1)
- ellipticity of cone operators (1)
- ellipticity of corners operators (1)
- ellipticity with interface conditions (1)
- ellipticity with parameter (1)
- ellipticity with respect to interior and edge symbols (1)
- elliptische Gleichungen (1)
- elliptische Quasi-Komplexe (1)
- embedded Markov chain (1)
- embeddings (1)
- empirical Wasserstein distance (1)
- endomorphism semigroup (1)
- energetic space (1)
- enlargement of filtration (1)
- ensembles (1)
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- equatorial ionosphere (1)
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- equivalence (1)
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- error diagram (1)
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- essential position in terms (1)
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- expansion (1)
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- exponential function (1)
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- exterior tensor product (1)
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- fat-free mass (1)
- feedback (1)
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- fibration (1)
- fibre coordinates (1)
- fibroblasts (1)
- field-aligned currents (1)
- filtering (1)
- finite transformation semigroup (1)
- finiteness theorem (1)
- finsler distance (1)
- first exit location (1)
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- illposed problem (1)
- indecomposable varifold (1)
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- inegral formulas (1)
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- invariance (1)
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- manifold (1)
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- minor planets, asteroids: individual: (162173) Ryugu (1)
- mit Anwendungen in der Laufzeittomographie, Seismischer Quellinversion und Magnetfeldmodellierung (1)
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Institute
- Institut für Mathematik (2160) (remove)
The canonical trace and the Wodzicki residue on classical pseudo-differential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral zeta-invariants using lifted defect formulae which express discrepancies of zeta-regularised traces in terms of Wodzicki residues. We derive Atiyah's L-2-index theorem as an instance of the Z(2)-graded generalisation of the canonical lift of spectral zeta-invariants and we show that certain lifted spectral zeta-invariants for geometric operators are integrals of Pontryagin and Chern forms.
We investigate if kernel regularization methods can achieve minimax convergence rates over a source condition regularity assumption for the target function. These questions have been considered in past literature, but only under specific assumptions about the decay, typically polynomial, of the spectrum of the the kernel mapping covariance operator. In the perspective of distribution-free results, we investigate this issue under much weaker assumption on the eigenvalue decay, allowing for more complex behavior that can reflect different structure of the data at different scales.
LetH be a Schrodinger operator defined on a noncompact Riemannianmanifold Omega, and let W is an element of L-infinity (Omega; R). Suppose that the operator H + W is critical in Omega, and let phi be the corresponding Agmon ground state. We prove that if u is a generalized eigenfunction ofH satisfying vertical bar u vertical bar <= C-phi in Omega for some constant C > 0, then the corresponding eigenvalue is in the spectrum of H. The conclusion also holds true if for some K is an element of Omega the operator H admits a positive solution in (Omega) over bar = Omega \ K, and vertical bar u vertical bar <= C psi in (Omega) over bar for some constant C > 0, where psi is a positive solution of minimal growth in a neighborhood of infinity in Omega. Under natural assumptions, this result holds also in the context of infinite graphs, and Dirichlet forms.
This paper further improves the Lie group method with Magnus expansion proposed in a previous paper by the authors, to solve some types of direct singular Sturm-Liouville problems. Next, a concrete implementation to the inverse Sturm-Liouville problem algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm-Liouville problems of higher order (for n=2,4) are verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied. In conclusion, this work provides a method that can be adapted successfully for solving a direct (regular/singular) or inverse Sturm-Liouville problem (SLP) of an arbitrary order with arbitrary boundary conditions.
The estimation of a log-concave density on R is a canonical problem in the area of shape-constrained nonparametric inference. We present a Bayesian nonparametric approach to this problem based on an exponentiated Dirichlet process mixture prior and show that the posterior distribution converges to the log-concave truth at the (near-) minimax rate in Hellinger distance. Our proof proceeds by establishing a general contraction result based on the log-concave maximum likelihood estimator that prevents the need for further metric entropy calculations. We further present computationally more feasible approximations and both an empirical and hierarchical Bayes approach. All priors are illustrated numerically via simulations.
Im Jahre 1960 behauptete Yamabe folgende Aussage bewiesen zu haben: Auf jeder kompakten Riemannschen Mannigfaltigkeit (M,g) der Dimension n ≥ 3 existiert eine zu g konform äquivalente Metrik mit konstanter Skalarkrümmung. Diese Aussage ist äquivalent zur Existenz einer Lösung einer bestimmten semilinearen elliptischen Differentialgleichung, der Yamabe-Gleichung. 1968 fand Trudinger einen Fehler in seinem Beweis und infolgedessen beschäftigten sich viele Mathematiker mit diesem nach Yamabe benannten Yamabe-Problem. In den 80er Jahren konnte durch die Arbeiten von Trudinger, Aubin und Schoen gezeigt werden, dass diese Aussage tatsächlich zutrifft. Dadurch ergeben sich viele Vorteile, z.B. kann beim Analysieren von konform invarianten partiellen Differentialgleichungen auf kompakten Riemannschen Mannigfaltigkeiten die Skalarkrümmung als konstant vorausgesetzt werden.
Es stellt sich nun die Frage, ob die entsprechende Aussage auch auf Lorentz-Mannigfaltigkeiten gilt. Das Lorentz'sche Yamabe Problem lautet somit: Existiert zu einer gegebenen räumlich kompakten global-hyperbolischen Lorentz-Mannigfaltigkeit (M,g) eine zu g konform äquivalente Metrik mit konstanter Skalarkrümmung? Das Ziel dieser Arbeit ist es, dieses Problem zu untersuchen.
Bei der sich aus dieser Fragestellung ergebenden Yamabe-Gleichung handelt es sich um eine semilineare Wellengleichung, deren Lösung eine positive glatte Funktion ist und aus der sich der konforme Faktor ergibt. Um die für die Behandlung des Yamabe-Problems benötigten Grundlagen so allgemein wie möglich zu halten, wird im ersten Teil dieser Arbeit die lokale Existenztheorie für beliebige semilineare Wellengleichungen für Schnitte auf Vektorbündeln im Rahmen eines Cauchy-Problems entwickelt. Hierzu wird der Umkehrsatz für Banachräume angewendet, um mithilfe von bereits existierenden Existenzergebnissen zu linearen Wellengleichungen, Existenzaussagen zu semilinearen Wellengleichungen machen zu können. Es wird bewiesen, dass, falls die Nichtlinearität bestimmte Bedingungen erfüllt, eine fast zeitglobale Lösung des Cauchy-Problems für kleine Anfangsdaten sowie eine zeitlokale Lösung für beliebige Anfangsdaten existiert.
Der zweite Teil der Arbeit befasst sich mit der Yamabe-Gleichung auf global-hyperbolischen Lorentz-Mannigfaltigkeiten. Zuerst wird gezeigt, dass die Nichtlinearität der Yamabe-Gleichung die geforderten Bedingungen aus dem ersten Teil erfüllt, so dass, falls die Skalarkrümmung der gegebenen Metrik nahe an einer Konstanten liegt, kleine Anfangsdaten existieren, so dass die Yamabe-Gleichung eine fast zeitglobale Lösung besitzt. Mithilfe von Energieabschätzungen wird anschließend für 4-dimensionale global-hyperbolische Lorentz-Mannigfaltigkeiten gezeigt, dass unter der Annahme, dass die konstante Skalarkrümmung der konform äquivalenten Metrik nichtpositiv ist, eine zeitglobale Lösung der Yamabe-Gleichung existiert, die allerdings nicht notwendigerweise positiv ist. Außerdem wird gezeigt, dass, falls die H2-Norm der Skalarkrümmung bezüglich der gegebenen Metrik auf einem kompakten Zeitintervall auf eine bestimmte Weise beschränkt ist, die Lösung positiv auf diesem Zeitintervall ist. Hierbei wird ebenfalls angenommen, dass die konstante Skalarkrümmung der konform äquivalenten Metrik nichtpositiv ist. Falls zusätzlich hierzu gilt, dass die Skalarkrümmung bezüglich der gegebenen Metrik negativ ist und die Metrik gewisse Bedingungen erfüllt, dann ist die Lösung für alle Zeiten in einem kompakten Zeitintervall positiv, auf dem der Gradient der Skalarkrümmung auf eine bestimmte Weise beschränkt ist. In beiden Fällen folgt unter den angeführten Bedingungen die Existenz einer zeitglobalen positiven Lösung, falls M = I x Σ für ein beschränktes offenes Intervall I ist. Zum Schluss wird für M = R x Σ ein Beispiel für die Nichtexistenz einer globalen positiven Lösung angeführt.
Flood loss modeling is a central component of flood risk analysis. Conventionally, this involves univariable and deterministic stage-damage functions. Recent advancements in the field promote the use of multivariable and probabilistic loss models, which consider variables beyond inundation depth and account for prediction uncertainty. Although companies contribute significantly to total loss figures, novel modeling approaches for companies are lacking. Scarce data and the heterogeneity among companies impede the development of company flood loss models. We present three multivariable flood loss models for companies from the manufacturing, commercial, financial, and service sector that intrinsically quantify prediction uncertainty. Based on object-level loss data (n = 1,306), we comparatively evaluate the predictive capacity of Bayesian networks, Bayesian regression, and random forest in relation to deterministic and probabilistic stage-damage functions, serving as benchmarks. The company loss data stem from four postevent surveys in Germany between 2002 and 2013 and include information on flood intensity, company characteristics, emergency response, private precaution, and resulting loss to building, equipment, and goods and stock. We find that the multivariable probabilistic models successfully identify and reproduce essential relationships of flood damage processes in the data. The assessment of model skill focuses on the precision of the probabilistic predictions and reveals that the candidate models outperform the stage-damage functions, while differences among the proposed models are negligible. Although the combination of multivariable and probabilistic loss estimation improves predictive accuracy over the entire data set, wide predictive distributions stress the necessity for the quantification of uncertainty.
Die Erweiterung des natürlichen Zahlbereichs um die positiven Bruchzahlen und die negativen ganzen Zahlen geht für Schülerinnen und Schüler mit großen gedanklichen Hürden und einem Umbruch bis dahin aufgebauter Grundvorstellungen einher. Diese Masterarbeit trägt wesentliche Veränderungen auf der Vorstellungs- und Darstellungsebene für beide Zahlbereiche zusammen und setzt sich mit den kognitiven Herausforderungen für Lernende auseinander. Auf der Grundlage einer Diskussion traditioneller sowie alternativer Lehrgänge der Zahlbereichserweiterung wird eine Unterrichtskonzeption für den Mathematikunterricht entwickelt, die eine parallele Einführung der Bruchzahlen und der negativen Zahlen vorschlägt. Die Empfehlungen der Unterrichtkonzeption erstrecken sich über den Zeitraum von der ersten bis zur siebten Klassenstufe, was der behutsamen Weiterentwicklung und Modifikation des Zahlbegriffs viel Zeit einräumt, und enthalten auch didaktische Überlegungen sowie konkrete Hinweise zu möglichen Aufgabenformaten.
This thesis aims at presenting in an organized fashion the required basics to understand the Glauber dynamics as a way of simulating configurations according to the Gibbs distribution of the Curie-Weiss Potts model. Therefore, essential aspects of discrete-time Markov chains on a finite state space are examined, especially their convergence behavior and related mixing times. Furthermore, special emphasis is placed on a consistent and comprehensive presentation of the Curie-Weiss Potts model and its analysis. Finally, the Glauber dynamics is studied in general and applied afterwards in an exemplary way to the Curie-Weiss model as well as the Curie-Weiss Potts model. The associated considerations are supplemented with two computer simulations aiming to show the cutoff phenomenon and the temperature dependence of the convergence behavior.
The Willmore functional is a function that maps an immersed Riemannian manifold to its total mean curvature. Finding closed surfaces that minimizes the Willmore energy, or more generally finding critical surfaces, is a classic problem of differential geometry.
In this thesis we will develop the concept of generalized Willmore functionals for surfaces in Riemannian manifolds. We are guided by models in mathematical physics, such as the Hawking energy of general relativity and the bending energies for thin membranes.
We prove the existence of minimizers under area constraint for these generalized Willmore functionals in a suitable class of generalized surfaces. In particular, we construct minimizers of the bending energy mentioned above for prescribed area and enclosed volume.
Furthermore, we prove that critical surfaces of generalized Willmore functionals with prescribed area are smooth, away from finitely many points. These results and the following are based on the existing theory for the Willmore functional.
This general discussion is succeeded by a detailed analysis of the Hawking energy. In the context of general relativity the surrounding manifold describes the space at a given time, hence we strive to understand the interplay between the Hawking energy and the ambient space. We characterize points in the surrounding manifold for which there are small critical spheres with prescribed area in any neighborhood. These points are interpreted as concentration points of the Hawking energy.
Additionally, we calculate an expansion of the Hawking energy on small, round spheres. This allows us to identify a kind of energy density of the Hawking energy.
It needs to be mentioned that our results stand in contrast to previous expansions of the Hawking energy. However, these expansions are obtained on spheres along the light cone at a given point. At this point it is not clear how to explain the discrepancy.
Finally, we consider asymptotically Schwarzschild manifolds. They are a special case of asymptotically flat manifolds, which serf as models for isolated systems. The Schwarzschild spacetime itself is a classical solution to the Einstein equations and yields a simple description of a black hole.
In these asymptotically Schwarzschild manifolds we construct a foliation of the exterior region by critical spheres of the Hawking energy with prescribed large area. This foliation can be seen as a generalized notion of the center of mass of the isolated system. Additionally, the Hawking energy of grows along the foliation as the area of the surfaces grows.
We prove a Feynman path integral formula for the unitary group exp(-itL(nu,theta)), t >= 0, associated with a discrete magnetic Schrodinger operator L-nu,L-theta on a large class of weighted infinite graphs. As a consequence, we get a new Kato-Simon estimate
vertical bar exp(- itL(nu,theta))(x,y)vertical bar <= exp( -tL(-deg,0))(x,y),
which controls the unitary group uniformly in the potentials in terms of a Schrodinger semigroup, where the potential deg is the weighted degree function of the graph.
Let M be a compact manifold of dimension n. In this paper, we introduce the Mass Function a >= 0 bar right arrow X-+(M)(a) (resp. a >= 0 bar right arrow X--(M)(a)) which is defined as the supremum (resp. infimum) of the masses of all metrics on M whose Yamabe constant is larger than a and which are flat on a ball of radius 1 and centered at a point p is an element of M. Here, the mass of a metric flat around p is the constant term in the expansion of the Green function of the conformal Laplacian at p. We show that these functions are well defined and have many properties which allow to obtain applications to the Yamabe invariant (i.e. the supremum of Yamabe constants over the set of all metrics on M).
For the time stationary global geomagnetic field, a new modelling concept is presented. A Bayesian non-parametric approach provides realistic location dependent uncertainty estimates. Modelling related variabilities are dealt with systematically by making little subjective apriori assumptions. Rather than parametrizing the model by Gauss coefficients, a functional analytic approach is applied. The geomagnetic potential is assumed a Gaussian process to describe a distribution over functions. Apriori correlations are given by an explicit kernel function with non-informative dipole contribution. A refined modelling strategy is proposed that accommodates non-linearities of archeomagnetic observables: First, a rough field estimate is obtained considering only sites that provide full field vector records. Subsequently, this estimate supports the linearization that incorporates the remaining incomplete records. The comparison of results for the archeomagnetic field over the past 1000 yr is in general agreement with previous models while improved model uncertainty estimates are provided.
This work provides a necessary and sufficient condition for a symbolic dynamical system to admit a sequence of periodic approximations in the Hausdorff topology. The key result proved and applied here uses graphs that are called De Bruijn graphs, Rauzy graphs, or Anderson-Putnam complex, depending on the community. Combining this with a previous result, the present work justifies rigorously the accuracy and reliability of algorithmic methods used to compute numerically the spectra of a large class of self-adjoint operators. The so-called Hamiltonians describe the effective dynamic of a quantum particle in aperiodic media. No restrictions on the structure of these operators other than general regularity assumptions are imposed. In particular, nearest-neighbor correlation is not necessary. Examples for the Fibonacci and the Golay-Rudin-Shapiro sequences are explicitly provided illustrating this discussion. While the first sequence has been thoroughly studied by physicists and mathematicians alike, a shroud of mystery still surrounds the latter when it comes to spectral properties. In light of this, the present paper gives a new result here that might help uncovering a solution.
In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.
We describe a new, original approach to the modelling of the Earth's magnetic field. The overall objective of this study is to reliably render fast variations of the core field and its secular variation. This method combines a sequential modelling approach, a Kalman filter, and a correlation-based modelling step. Sources that most significantly contribute to the field measured at the surface of the Earth are modelled. Their separation is based on strong prior information on their spatial and temporal behaviours. We obtain a time series of model distributions which display behaviours similar to those of recent models based on more classic approaches, particularly at large temporal and spatial scales. Interesting new features and periodicities are visible in our models at smaller time and spatial scales. An important aspect of our method is to yield reliable error bars for all model parameters. These errors, however, are only as reliable as the description of the different sources and the prior information used are realistic. Finally, we used a slightly different version of our method to produce candidate models for the thirteenth edition of the International Geomagnetic Reference Field.
In this article, we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existing fully explicit space-time discrete approximation scheme, in particular the fact that it satisfies suitable a priori estimates. We also obtain almost sure and strong convergence of the approximation scheme to the mild solutions of the considered SPDEs. We conclude by applying the main result of the article to the stochastic Burgers equations with additive space-time white noise.
We show how to deduce Rellich inequalities from Hardy inequalities on infinite graphs. Specifically, the obtained Rellich inequality gives an upper bound on a function by the Laplacian of the function in terms of weighted norms. These weights involve the Hardy weight and a function which satisfies an eikonal inequality. The results are proven first for Laplacians and are extended to Schrodinger operators afterwards.
The purpose of this paper is to build an algebraic framework suited to regularize branched structures emanating from rooted forests and which encodes the locality principle. This is achieved by means of the universal properties in the locality framework of properly decorated rooted forests. These universal properties are then applied to derive the multivariate regularization of integrals indexed by rooted forests. We study their renormalization, along the lines of Kreimer's toy model for Feynman integrals.
We present a new model of the geomagnetic field spanning the last 20 years and called Kalmag. Deriving from the assimilation of CHAMP and Swarm vector field measurements, it separates the different contributions to the observable field through parameterized prior covariance matrices. To make the inverse problem numerically feasible, it has been sequentialized in time through the combination of a Kalman filter and a smoothing algorithm. The model provides reliable estimates of past, present and future mean fields and associated uncertainties. The version presented here is an update of our IGRF candidates; the amount of assimilated data has been doubled and the considered time window has been extended from [2000.5, 2019.74] to [2000.5, 2020.33].
Global numerical weather prediction (NWP) models have begun to resolve the mesoscale k(-5/3) range of the energy spectrum, which is known to impose an inherently finite range of deterministic predictability per se as errors develop more rapidly on these scales than on the larger scales. However, the dynamics of these errors under the influence of the synoptic-scale k(-3) range is little studied. Within a perfect-model context, the present work examines the error growth behavior under such a hybrid spectrum in Lorenz's original model of 1969, and in a series of identical-twin perturbation experiments using an idealized two-dimensional barotropic turbulence model at a range of resolutions. With the typical resolution of today's global NWP ensembles, error growth remains largely uniform across scales. The theoretically expected fast error growth characteristic of a k(-5/3) spectrum is seen to be largely suppressed in the first decade of the mesoscale range by the synoptic-scale k(-3) range. However, it emerges once models become fully able to resolve features on something like a 20-km scale, which corresponds to a grid resolution on the order of a few kilometers.
Inferring causal relations from observational time series data is a key problem across science and engineering whenever experimental interventions are infeasible or unethical. Increasing data availability over the past few decades has spurred the development of a plethora of causal discovery methods, each addressing particular challenges of this difficult task. In this paper, we focus on an important challenge that is at the core of time series causal discovery: regime-dependent causal relations. Often dynamical systems feature transitions depending on some, often persistent, unobserved background regime, and different regimes may exhibit different causal relations. Here, we assume a persistent and discrete regime variable leading to a finite number of regimes within which we may assume stationary causal relations. To detect regime-dependent causal relations, we combine the conditional independence-based PCMCI method [based on a condition-selection step (PC) followed by the momentary conditional independence (MCI) test] with a regime learning optimization approach. PCMCI allows for causal discovery from high-dimensional and highly correlated time series. Our method, Regime-PCMCI, is evaluated on a number of numerical experiments demonstrating that it can distinguish regimes with different causal directions, time lags, and sign of causal links, as well as changes in the variables' autocorrelation. Furthermore, Regime-PCMCI is employed to observations of El Nino Southern Oscillation and Indian rainfall, demonstrating skill also in real-world datasets.
Concurrent observation technologies have made high-precision real-time data available in large quantities. Data assimilation (DA) is concerned with how to combine this data with physical models to produce accurate predictions. For spatial-temporal models, the ensemble Kalman filter with proper localisation techniques is considered to be a state-of-the-art DA methodology. This article proposes and investigates a localised ensemble Kalman Bucy filter for nonlinear models with short-range interactions. We derive dimension-independent and component-wise error bounds and show the long time path-wise error only has logarithmic dependence on the time range. The theoretical results are verified through some simple numerical tests.
Process-oriented theories of cognition must be evaluated against time-ordered observations. Here we present a representative example for data assimilation of the SWIFT model, a dynamical model of the control of fixation positions and fixation durations during natural reading of single sentences. First, we develop and test an approximate likelihood function of the model, which is a combination of a spatial, pseudo-marginal likelihood and a temporal likelihood obtained by probability density approximation Second, we implement a Bayesian approach to parameter inference using an adaptive Markov chain Monte Carlo procedure. Our results indicate that model parameters can be estimated reliably for individual subjects. We conclude that approximative Bayesian inference represents a considerable step forward for computational models of eye-movement control, where modeling of individual data on the basis of process-based dynamic models has not been possible so far.
Concurrent observation technologies have made high-precision real-time data available in large quantities. Data assimilation (DA) is concerned with how to combine this data with physical models to produce accurate predictions. For spatial-temporal models, the ensemble Kalman filter with proper localisation techniques is considered to be a state-of-the-art DA methodology. This article proposes and investigates a localised ensemble Kalman Bucy filter for nonlinear models with short-range interactions. We derive dimension-independent and component-wise error bounds and show the long time path-wise error only has logarithmic dependence on the time range. The theoretical results are verified through some simple numerical tests.
Synthetic Aperture Radar (SAR) amplitude measurements from spaceborne sensors are sensitive to surface roughness conditions near their radar wavelength. These backscatter signals are often exploited to assess the roughness of plowed agricultural fields and water surfaces, and less so to complex, heterogeneous geological surfaces. The bedload of mixed sand- and gravel-bed rivers can be considered a mixture of smooth (compacted sand) and rough (gravel) surfaces. Here, we assess backscatter gradients over a large high-mountain alluvial river in the eastern Central Andes with aerially exposed sand and gravel bedload using X-band TerraSAR-X/TanDEM-X, C-band Sentinel-1, and L-band ALOS-2 PALSAR-2 radar scenes. In a first step, we present theory and hypotheses regarding radar response to an alluvial channel bed. We test our hypotheses by comparing backscatter responses over vegetation-free endmember surfaces from inside and outside of the active channel-bed area. We then develop methods to extract smoothed backscatter gradients downstream along the channel using kernel density estimates. In a final step, the local variability of sand-dominated patches is analyzed using Fourier frequency analysis, by fitting stretched-exponential and power-law regression models to the 2-D power spectrum of backscatter amplitude. We find a large range in backscatter depending on the heterogeneity of contiguous smooth- and rough-patches of bedload material. The SAR amplitude signal responds primarily to the fraction of smooth-sand bedload, but is further modified by gravel elements. The sensitivity to gravel is more apparent in longer wavelength L-band radar, whereas C- and X-band is sensitive only to sand variability. Because the spatial extent of smooth sand patches in our study area is typically< 50 m, only higher resolution sensors (e.g., TerraSAR-X/TanDEM-X) are useful for power spectrum analysis. Our results show the potential for mapping sand-gravel transitions and local geomorphic complexity in alluvial rivers with aerially exposed bedload using SAR amplitude.
Author summary <br /> Switching between local and global attention is a general strategy in human information processing. We investigate whether this strategy is a viable approach to model sequences of fixations generated by a human observer in a free viewing task with natural scenes. Variants of the basic model are used to predict the experimental data based on Bayesian inference. Results indicate a high predictive power for both aggregated data and individual differences across observers. The combination of a novel model with state-of-the-art Bayesian methods lends support to our two-state model using local and global internal attention states for controlling eye movements. <br /> Understanding the decision process underlying gaze control is an important question in cognitive neuroscience with applications in diverse fields ranging from psychology to computer vision. The decision for choosing an upcoming saccade target can be framed as a selection process between two states: Should the observer further inspect the information near the current gaze position (local attention) or continue with exploration of other patches of the given scene (global attention)? Here we propose and investigate a mathematical model motivated by switching between these two attentional states during scene viewing. The model is derived from a minimal set of assumptions that generates realistic eye movement behavior. We implemented a Bayesian approach for model parameter inference based on the model's likelihood function. In order to simplify the inference, we applied data augmentation methods that allowed the use of conjugate priors and the construction of an efficient Gibbs sampler. This approach turned out to be numerically efficient and permitted fitting interindividual differences in saccade statistics. Thus, the main contribution of our modeling approach is two-fold; first, we propose a new model for saccade generation in scene viewing. Second, we demonstrate the use of novel methods from Bayesian inference in the field of scan path modeling.
In this paper, we present the convergence rate analysis of the modified Landweber method under logarithmic source condition for nonlinear ill-posed problems. The regularization parameter is chosen according to the discrepancy principle. The reconstructions of the shape of an unknown domain for an inverse potential problem by using the modified Landweber method are exhibited.
Several numerical tools designed to overcome the challenges of smoothing in a non-linear and non-Gaussian setting are investigated for a class of particle smoothers. The considered family of smoothers is induced by the class of linear ensemble transform filters which contains classical filters such as the stochastic ensemble Kalman filter, the ensemble square root filter, and the recently introduced nonlinear ensemble transform filter. Further the ensemble transform particle smoother is introduced and particularly highlighted as it is consistent in the particle limit and does not require assumptions with respect to the family of the posterior distribution. The linear update pattern of the considered class of linear ensemble transform smoothers allows one to implement important supplementary techniques such as adaptive spread corrections, hybrid formulations, and localization in order to facilitate their application to complex estimation problems. These additional features are derived and numerically investigated for a sequence of increasingly challenging test problems.
In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly nonorientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and unambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces.
Purpose The anatomy of the circle of Willis (CoW), the brain's main arterial blood supply system, strongly differs between individuals, resulting in highly variable flow fields and intracranial vascularization patterns. To predict subject-specific hemodynamics with high certainty, we propose a data assimilation (DA) approach that merges fully 4D phase-contrast magnetic resonance imaging (PC-MRI) data with a numerical model in the form of computational fluid dynamics (CFD) simulations. Methods To the best of our knowledge, this study is the first to provide a transient state estimate for the three-dimensional velocity field in a subject-specific CoW geometry using DA. High-resolution velocity state estimates are obtained using the local ensemble transform Kalman filter (LETKF). Results Quantitative evaluation shows a considerable reduction (up to 90%) in the uncertainty of the velocity field state estimate after the data assimilation step. Velocity values in vessel areas that are below the resolution of the PC-MRI data (e.g., in posterior communicating arteries) are provided. Furthermore, the uncertainty of the analysis-based wall shear stress distribution is reduced by a factor of 2 for the data assimilation approach when compared to the CFD model alone. Conclusion This study demonstrates the potential of data assimilation to provide detailed information on vascular flow, and to reduce the uncertainty in such estimates by combining various sources of data in a statistically appropriate fashion.
The XI international conference Stochastic and Analytic Methods in Mathematical Physics was held in Yerevan 2 – 7 September 2019 and was dedicated to the memory of the great mathematician Robert Adol’fovich Minlos, who passed away in January 2018.
The present volume collects a large majority of the contributions presented at the conference on the following domains of contemporary interest: classical and quantum statistical physics, mathematical methods in quantum mechanics, stochastic analysis, applications of point processes in statistical mechanics. The authors are specialists from Armenia, Czech Republic, Denmark, France, Germany, Italy, Japan, Lithuania, Russia, UK and Uzbekistan.
A particular aim of this volume is to offer young scientists basic material in order to inspire their future research in the wide fields presented here.
In this article, we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities. The proof of this result exploits the properties of an existing fully explicit space-time discrete approximation scheme, in particular the fact that it satisfies suitable a priori estimates. We also obtain almost sure and strong convergence of the approximation scheme to the mild solutions of the considered SPDEs. We conclude by applying the main result of the article to the stochastic Burgers equations with additive space-time white noise.
Relationship between large-scale ionospheric field-aligned currents and electron/ion precipitations
(2020)
In this study, we have derived field-aligned currents (FACs) from magnetometers onboard the Defense Meteorological Satellite Project (DMSP) satellites. The magnetic latitude versus local time distribution of FACs from DMSP shows comparable dependences with previous findings on the intensity and orientation of interplanetary magnetic field (IMF)B(y)andB(z)components, which confirms the reliability of DMSP FAC data set. With simultaneous measurements of precipitating particles from DMSP, we further investigate the relation between large-scale FACs and precipitating particles. Our result shows that precipitation electron and ion fluxes both increase in magnitude and extend to lower latitude for enhanced southward IMFBz, which is similar to the behavior of FACs. Under weak northward and southwardB(z)conditions, the locations of the R2 current maxima, at both dusk and dawn sides and in both hemispheres, are found to be close to the maxima of the particle energy fluxes; while for the same IMF conditions, R1 currents are displaced further to the respective particle flux peaks. Largest displacement (about 3.5 degrees) is found between the downward R1 current and ion flux peak at the dawn side. Our results suggest that there exists systematic differences in locations of electron/ion precipitation and large-scale upward/downward FACs. As outlined by the statistical mean of these two parameters, the FAC peaks enclose the particle energy flux peaks in an auroral band at both dusk and dawn sides. Our comparisons also found that particle precipitation at dawn and dusk and in both hemispheres maximizes near the mean R2 current peaks. The particle precipitation flux maxima closer to the R1 current peaks are lower in magnitude. This is opposite to the known feature that R1 currents are on average stronger than R2 currents.
We show how to deduce Rellich inequalities from Hardy inequalities on infinite graphs. Specifically, the obtained Rellich inequality gives an upper bound on a function by the Laplacian of the function in terms of weighted norms. These weights involve the Hardy weight and a function which satisfies an eikonal inequality. The results are proven first for Laplacians and are extended to Schrodinger operators afterwards.
The rational Krylov subspace method (RKSM) and the low-rank alternating directions implicit (LR-ADI) iteration are established numerical tools for computing low-rank solution factors of large-scale Lyapunov equations. In order to generate the basis vectors for the RKSM, or extend the low-rank factors within the LR-ADI method, the repeated solution to a shifted linear system of equations is necessary. For very large systems this solve is usually implemented using iterative methods, leading to inexact solves within this inner iteration (and therefore to "inexact methods"). We will show that one can terminate this inner iteration before full precision has been reached and still obtain very good accuracy in the final solution to the Lyapunov equation. In particular, for both the RKSM and the LR-ADI method we derive theory for a relaxation strategy (e.g. increasing the solve tolerance of the inner iteration, as the outer iteration proceeds) within the iterative methods for solving the large linear systems. These theoretical choices involve unknown quantities, therefore practical criteria for relaxing the solution tolerance within the inner linear system are then provided. The theory is supported by several numerical examples, which show that the total amount of work for solving Lyapunov equations can be reduced significantly.
Europa Universalis IV
(2020)
We extend our approach of asymptotic parametrix construction for Hamiltonian operators from conical to edge-type singularities which is applicable to coalescence points of two particles of the helium atom and related two electron systems including the hydrogen molecule. Up to second-order, we have calculated the symbols of an asymptotic parametrix of the nonrelativistic Hamiltonian of the helium atom within the Born-Oppenheimer approximation and provide explicit formulas for the corresponding Green operators which encode the asymptotic behavior of the eigenfunctions near an edge.
Based on an analysis of continuous monitoring of farm animal behavior in the region of the 2016 M6.6 Norcia earthquake in Italy, Wikelski et al., 2020; (Seismol Res Lett, 89, 2020, 1238) conclude that animal activity can be anticipated with subsequent seismic activity and that this finding might help to design a "short-term earthquake forecasting method." We show that this result is based on an incomplete analysis and misleading interpretations. Applying state-of-the-art methods of statistics, we demonstrate that the proposed anticipatory patterns cannot be distinguished from random patterns, and consequently, the observed anomalies in animal activity do not have any forecasting power.
We construct marked Gibbs point processes in R-d under quite general assumptions. Firstly, we allow for interaction functionals that may be unbounded and whose range is not assumed to be uniformly bounded. Indeed, our typical interaction admits an a.s. finite but random range. Secondly, the random marks-attached to the locations in R-d-belong to a general normed space G. They are not bounded, but their law should admit a super-exponential moment. The approach used here relies on the so-called entropy method and large-deviation tools in order to prove tightness of a family of finite-volume Gibbs point processes. An application to infinite-dimensional interacting diffusions is also presented.
Purpose This review provides an overview of the current challenges in oral targeted antineoplastic drug (OAD) dosing and outlines the unexploited value of therapeutic drug monitoring (TDM). Factors influencing the pharmacokinetic exposure in OAD therapy are depicted together with an overview of different TDM approaches. Finally, current evidence for TDM for all approved OADs is reviewed. Methods A comprehensive literature search (covering literature published until April 2020), including primary and secondary scientific literature on pharmacokinetics and dose individualisation strategies for OADs, together with US FDA Clinical Pharmacology and Biopharmaceutics Reviews and the Committee for Medicinal Products for Human Use European Public Assessment Reports was conducted. Results OADs are highly potent drugs, which have substantially changed treatment options for cancer patients. Nevertheless, high pharmacokinetic variability and low treatment adherence are risk factors for treatment failure. TDM is a powerful tool to individualise drug dosing, ensure drug concentrations within the therapeutic window and increase treatment success rates. After reviewing the literature for 71 approved OADs, we show that exposure-response and/or exposure-toxicity relationships have been established for the majority. Moreover, TDM has been proven to be feasible for individualised dosing of abiraterone, everolimus, imatinib, pazopanib, sunitinib and tamoxifen in prospective studies. There is a lack of experience in how to best implement TDM as part of clinical routine in OAD cancer therapy. Conclusion Sub-therapeutic concentrations and severe adverse events are current challenges in OAD treatment, which can both be addressed by the application of TDM-guided dosing, ensuring concentrations within the therapeutic window.
Aim Quantitative and kinetic insights into the drug exposure-disease response relationship might enhance our knowledge on loss of response and support more effective monitoring of inflammatory activity by biomarkers in patients with inflammatory bowel disease (IBD) treated with infliximab (IFX). This study aimed to derive recommendations for dose adjustment and treatment optimisation based on mechanistic characterisation of the relationship between IFX serum concentration and C-reactive protein (CRP) concentration. <br /> Methods Data from an investigator-initiated trial included 121 patients with IBD during IFX maintenance treatment. Serum concentrations of IFX, antidrug antibodies (ADA), CRP, and disease-related covariates were determined at the mid-term and end of a dosing interval. Data were analysed using a pharmacometric nonlinear mixed-effects modelling approach. An IFX exposure-CRP model was generated and applied to evaluate dosing regimens to achieve CRP remission. <br /> Results The generated quantitative model showed that IFX has the potential to inhibit up to 72% (9% relative standard error [RSE]) of CRP synthesis in a patient. IFX concentration leading to 90% of the maximum CRP synthesis inhibition was 18.4 mu g/mL (43% RSE). Presence of ADA was the most influential factor on IFX exposure. With standard dosing strategy, >= 55% of ADA+ patients experienced CRP nonremission. Shortening the dosing interval and co-therapy with immunomodulators were found to be the most beneficial strategies to maintain CRP remission. <br /> Conclusions With the generated model we could for the first time establish a robust relationship between IFX exposure and CRP synthesis inhibition, which could be utilised for treatment optimisation in IBD patients.
Background:
Anti-TNFα monoclonal antibodies (mAbs) are a well-established treatment for patients with Crohn’s disease (CD). However, subtherapeutic concentrations of mAbs have been related to a loss of response during the first year of therapy1. Therefore, an appropriate dosing strategy is crucial to prevent the underexposure of mAbs for those patients. The aim of our study was to assess the impact of different dosing strategies (fixed dose or body size descriptor adapted) on drug exposure and the target concentration attainment for two different anti-TNFα mAbs: infliximab (IFX, body weight (BW)-based dosing) and certolizumab pegol (CZP, fixed dosing). For this purpose, a comprehensive pharmacokinetic (PK) simulation study was performed.
Methods:
A virtual population of 1000 clinically representative CD patients was generated based on the distribution of CD patient characteristics from an in-house clinical database (n = 116). Seven dosing regimens were investigated: fixed dose and per BW, lean BW (LBW), body surface area, height, body mass index and fat-free mass. The individual body size-adjusted doses were calculated from patient generated body size descriptor values. Then, using published PK models for IFX and CZP in CD patients2,3, for each patient, 1000 concentration–time profiles were simulated to consider the typical profile of a specific patient as well as the range of possible individual profiles due to unexplained PK variability across patients. For each dosing strategy, the variability in maximum and minimum mAb concentrations (Cmax and Cmin, respectively), area under the concentration-time curve (AUC) and the per cent of patients reaching target concentration were assessed during maintenance therapy.
Results:
For IFX and CZP, Cmin showed the highest variability between patients (CV ≈110% and CV ≈80%, respectively) with a similar extent across all dosing strategies. For IFX, the per cent of patients reaching the target (Cmin = 5 µg/ml) was similar across all dosing strategies (~15%). For CZP, the per cent of patients reaching the target average concentration of 17 µg/ml ranged substantially (52–71%), being the highest for LBW-adjusted dosing.
Conclusion:
By using a PK simulation approach, different dosing regimen of IFX and CZP revealed the highest variability for Cmin, the most commonly used PK parameter guiding treatment decisions, independent upon dosing regimen. Our results demonstrate similar target attainment with fixed dosing of IFX compared with currently recommended BW-based dosing. For CZP, the current fixed dosing strategy leads to comparable percentage of patients reaching target as the best performing body size-adjusted dosing (66% vs. 71%, respectively).
The Coulomb failure stress (CFS) criterion is the most commonly used method for predicting spatial distributions of aftershocks following large earthquakes. However, large uncertainties are always associated with the calculation of Coulomb stress change. The uncertainties mainly arise due to nonunique slip inversions and unknown receiver faults; especially for the latter, results are highly dependent on the choice of the assumed receiver mechanism. Based on binary tests (aftershocks yes/no), recent studies suggest that alternative stress quantities, a distance-slip probabilistic model as well as deep neural network (DNN) approaches, all are superior to CFS with predefined receiver mechanism. To challenge this conclusion, which might have large implications, we use 289 slip inversions from SRCMOD database to calculate more realistic CFS values for a layered half-space and variable receiver mechanisms. We also analyze the effect of the magnitude cutoff, grid size variation, and aftershock duration to verify the use of receiver operating characteristic (ROC) analysis for the ranking of stress metrics. The observations suggest that introducing a layered half-space does not improve the stress maps and ROC curves. However, results significantly improve for larger aftershocks and shorter time periods but without changing the ranking. We also go beyond binary testing and apply alternative statistics to test the ability to estimate aftershock numbers, which confirm that simple stress metrics perform better than the classic Coulomb failure stress calculations and are also better than the distance-slip probabilistic model.
We study superharmonic functions for Schrodinger operators on general weighted graphs. Specifically, we prove two decompositions which both go under the name Riesz decomposition in the literature. The first one decomposes a superharmonic function into a harmonic and a potential part. The second one decomposes a superharmonic function into a sum of superharmonic functions with certain upper bounds given by prescribed superharmonic functions. As application we show a Brelot type theorem.
We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of computational cost O(n(beta+1)) where nis the size of the matrix and O(n(beta)) is the cost of multiplying n x n-matrices, beta is an element of [2, 2.37286). We compare its performance to that of other algorithms and show how it can be used to compute the Euler form of a Riemannian manifold using computer algebra.