TY - JOUR
A1 - Junek, Heinz
T1 - Zyklizität in Raum, zeit und geist : über Pflasterungen, Rollkurven, Dezimalbrüche, Schwingungen, Wellen, Iteration und Neuronale Netze
JF - Zyklizität & Rhythmik: eine multidisziplinäre Vorlesungsreihe
Y1 - 2020
SN - 978-3-86464-169-5
SP - 85
EP - 103
PB - trafo
CY - Berlin
ER -
TY - JOUR
A1 - Denecke, Klaus-Dieter
T1 - Partial clones
JF - Asian-European journal of mathematics : AEJM
N2 - A set C of operations defined on a nonempty set A is said to be a clone if C is closed under composition of operations and contains all projection mappings. The concept of a clone belongs to the algebraic main concepts and has important applications in Computer Science. A clone can also be regarded as a many-sorted algebra where the sorts are the n-ary operations defined on set A for all natural numbers n >= 1 and the operations are the so-called superposition operations S-m(n) for natural numbers m, n >= 1 and the projection operations as nullary operations. Clones generalize monoids of transformations defined on set A and satisfy three clone axioms. The most important axiom is the superassociative law, a generalization of the associative law. If the superposition operations are partial, i.e. not everywhere defined, instead of the many-sorted clone algebra, one obtains partial many-sorted algebras, the partial clones. Linear terms, linear tree languages or linear formulas form partial clones. In this paper, we give a survey on partial clones and their properties.
KW - Operation
KW - term
KW - formula
KW - superposition of operations
KW - terms and
KW - formulas
KW - linear term
KW - linear formula
KW - linear tree language
KW - clone
KW - partial clone
KW - linear hypersubstitution
KW - dht-symmetric category
KW - partial
KW - theory
Y1 - 2020
U6 - https://doi.org/10.1142/S1793557120501612
SN - 1793-5571
SN - 1793-7183
VL - 13
IS - 8
PB - World Scientific
CY - Singapore
ER -
TY - JOUR
A1 - Somogyvári, Márk
A1 - Reich, Sebastian
T1 - Convergence tests for transdimensional Markov chains in geoscience imaging
JF - Mathematical geosciences : the official journal of the International Association for Mathematical Geosciences
N2 - Classic inversion methods adjust a model with a predefined number of parameters to the observed data. With transdimensional inversion algorithms such as the reversible-jump Markov chain Monte Carlo (rjMCMC), it is possible to vary this number during the inversion and to interpret the observations in a more flexible way. Geoscience imaging applications use this behaviour to automatically adjust model resolution to the inhomogeneities of the investigated system, while keeping the model parameters on an optimal level. The rjMCMC algorithm produces an ensemble as result, a set of model realizations, which together represent the posterior probability distribution of the investigated problem. The realizations are evolved via sequential updates from a randomly chosen initial solution and converge toward the target posterior distribution of the inverse problem. Up to a point in the chain, the realizations may be strongly biased by the initial model, and must be discarded from the final ensemble. With convergence assessment techniques, this point in the chain can be identified. Transdimensional MCMC methods produce ensembles that are not suitable for classic convergence assessment techniques because of the changes in parameter numbers. To overcome this hurdle, three solutions are introduced to convert model realizations to a common dimensionality while maintaining the statistical characteristics of the ensemble. A scalar, a vector and a matrix representation for models is presented, inferred from tomographic subsurface investigations, and three classic convergence assessment techniques are applied on them. It is shown that appropriately chosen scalar conversions of the models could retain similar statistical ensemble properties as geologic projections created by rasterization.
KW - transdimensional inversion
KW - MCMC modelling
KW - convergence assessment
Y1 - 2019
U6 - https://doi.org/10.1007/s11004-019-09811-x
SN - 1874-8961
SN - 1874-8953
VL - 52
IS - 5
SP - 651
EP - 668
PB - Springer
CY - Heidelberg
ER -
TY - JOUR
A1 - Ly, Ibrahim
T1 - A Cauchy problem for the Cauchy-Riemann operator
JF - Afrika Matematika
N2 - We study the Cauchy problem for a nonlinear elliptic equation with data on a piece S of the boundary surface partial derivative X. By the Cauchy problem is meant any boundary value problem for an unknown function u in a domain X with the property that the data on S, if combined with the differential equations in X, allows one to determine all derivatives of u on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy-Kovalevskaya theorem. We discuss a variational setting of the Cauchy problem which always possesses a generalized solution.
KW - nonlinear PDI
KW - Cauchy problem
KW - Zaremba problem
Y1 - 2020
U6 - https://doi.org/10.1007/s13370-020-00810-4
SN - 1012-9405
SN - 2190-7668
VL - 32
IS - 1-2
SP - 69
EP - 76
PB - Springer
CY - Heidelberg
ER -
TY - JOUR
A1 - Malass, Ihsane
A1 - Tarkhanov, Nikolaj Nikolaevič
T1 - A perturbation of the de Rham complex
T1 - Возмущение комплекса де Рама
JF - Journal of Siberian Federal University : Mathematics & Physics
JF - Žurnal Sibirskogo Federalʹnogo Universiteta : Matematika i fizika
N2 - We consider a perturbation of the de Rham complex on a compact manifold with boundary. This perturbation goes beyond the framework of complexes, and so cohomology does not apply to it. On the other hand, its curvature is "small", hence there is a natural way to introduce an Euler characteristic and develop a Lefschetz theory for the perturbation. This work is intended as an attempt to develop a cohomology theory for arbitrary sequences of linear mappings.
N2 - Рассмотрим возмущение комплекса де Рама на компактном многообразии с краем. Это возмущение выходит за рамки комплексов, и поэтому когомологии к нему не относятся. С другой стороны, его кривизна "мала", поэтому существует естественный способ ввести характеристику Эйлера и разработать теорию Лефшеца для возмущения. Данная работа предназначена для попытки разработать теорию когомологий для произвольных последовательностей линейных отображений.
KW - de Rham complex
KW - cohomology
KW - Hodge theory
KW - Neumann problem
KW - комплекс де Рама
KW - когомологии
KW - теория Ходжа
KW - проблема Неймана
Y1 - 2020
U6 - https://doi.org/10.17516/1997-1397-2020-13-5-519-532
SN - 1997-1397
SN - 2313-6022
VL - 13
IS - 5
SP - 519
EP - 532
PB - Siberian Federal University
CY - Krasnojarsk
ER -
TY - JOUR
A1 - Al-Saedy, Ammar Jaffar Muhesin
A1 - Tarchanov, Nikolaj Nikolaevič
T1 - A degree theory for Lagrangian boundary value problems
JF - Žurnal Sibirskogo Federalʹnogo Universiteta = Journal of Siberian Federal University; mathematics & physics
N2 - We study those nonlinear partial differential equations which appear as Euler-Lagrange equations of variational problems. On defining weak boundary values of solutions to such equations we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to Lagrangian problems.
N2 - Мы изучаем те нелинейные уравнения с частными производными, которые возникают как уравнения Эйлера-Лагранжа вариационных задач. Определяя слабые граничные значения решений таких уравнений, мы инициируем теорию лагранжевых краевых задач в функциональных пространствах подходящей гладкости. Мы также анализируем, применяется ли современная концепция степени отображения к лагранжевым проблемам.
KW - nonlinear equations
KW - Lagrangian system
KW - weak boundary values
KW - quasilinear Fredholm operators
KW - mapping degree
Y1 - 2020
U6 - https://doi.org/10.17516/1997-1397-2020-13-1-5-25
SN - 1997-1397
SN - 2313-6022
VL - 13
IS - 1
SP - 5
EP - 25
PB - Sibirskij Federalʹnyj Universitet
CY - Krasnojarsk
ER -
TY - CHAP
A1 - Clavier, Pierre J.
A1 - Guo, Li
A1 - Paycha, Sylvie
A1 - Zhang, Bin
T1 - Renormalisation and locality
BT - branched zeta values
T2 - Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA)
Volume 2
Y1 - 2020
SN - 978-3-03719-205-4 print
SN - 978-3-03719-705-9 online
U6 - https://doi.org/10.4171/205
SP - 85
EP - 132
PB - European Mathematical Society Publishing House
CY - Zürich
ER -
TY - JOUR
A1 - Chelkh, W.
A1 - Ly, Ibrahim
A1 - Tarkhanov, Nikolai
T1 - A remark on the Laplace transform
JF - Siberian Mathematical Journal
N2 - The study of the Cauchy problem for solutions of the heat equation in a cylindrical domain with data on the lateral surface by the Fourier method raises the problem of calculating the inverse Laplace transform of the entire function cos root z. This problem has no solution in the standard theory of the Laplace transform. We give an explicit formula for the inverse Laplace transform of cos root z using the theory of analytic functionals. This solution suits well to efficiently develop the regularization of solutions to Cauchy problems for parabolic equations with data on noncharacteristic surfaces.
KW - Fourier-Laplace transform
KW - distributions with one-sided support
KW - holomorphic function
KW - analytic functional
Y1 - 2020
U6 - https://doi.org/10.1134/S0037446620040151
SN - 0037-4466
SN - 1573-9260
VL - 61
IS - 4
SP - 755
EP - 762
PB - Consultants Bureau, Springer
CY - New York
ER -
TY - JOUR
A1 - Keller, Matthias
A1 - Schwarz, Michael
T1 - Courant’s nodal domain theorem for positivity preserving forms
JF - Journal of spectral theory
N2 - We introduce a notion of nodal domains for positivity preserving forms. This notion generalizes the classical ones for Laplacians on domains and on graphs. We prove the Courant nodal domain theorem in this generalized setting using purely analytical methods.
KW - Nodal domain
KW - eigenfunction
KW - Dirichlet form
KW - compact resolvent
Y1 - 2020
U6 - https://doi.org/10.4171/JST/292
SN - 1664-039X
SN - 1664-0403
VL - 10
IS - 1
SP - 271
EP - 309
PB - EMS Publishing House
CY - Zürich
ER -
TY - JOUR
A1 - Ly, Ibrahim
A1 - Tarkhanov, Nikolaj Nikolaevič
T1 - Asymptotic expansions at nonsymmetric cuspidal points
JF - Mathematical notes
N2 - We study the asymptotics of solutions to the Dirichlet problem in a domain X subset of R3 whose boundary contains a singular point O. In a small neighborhood of this point, the domain has the form {z > root x(2) + y(4)}, i.e., the origin is a nonsymmetric conical point at the boundary. So far, the behavior of solutions to elliptic boundary-value problems has not been studied sufficiently in the case of nonsymmetric singular points. This problem was posed by V.A. Kondrat'ev in 2000. We establish a complete asymptotic expansion of solutions near the singular point.
KW - Dirichlet problem
KW - singular points
KW - asymptotic expansions
Y1 - 2020
U6 - https://doi.org/10.1134/S0001434620070238
SN - 0001-4346
SN - 1573-8876
VL - 108
IS - 1-2
SP - 219
EP - 228
PB - Springer Science
CY - New York
ER -
TY - JOUR
A1 - Clavier, Pierre J.
T1 - Double shuffle relations for arborified zeta values
JF - Journal of algebra
N2 - Arborified zeta values are defined as iterated series and integrals using the universal properties of rooted trees. This approach allows to study their convergence domain and to relate them to multiple zeta values. Generalisations to rooted trees of the stuffle and shuffle products are defined and studied. It is further shown that arborified zeta values are algebra morphisms for these new products on trees.
KW - Rooted trees
KW - Multiple zeta values
KW - Shuffle products
KW - Rota-Baxter
KW - algebras
Y1 - 2020
U6 - https://doi.org/10.1016/j.jalgebra.2019.10.015
SN - 0021-8693
SN - 1090-266X
VL - 543
SP - 111
EP - 155
PB - Elsevier
CY - San Diego
ER -
TY - JOUR
A1 - Hamm, Maximilian
A1 - Pelivan, Ivanka
A1 - Grott, Matthias
A1 - de Wiljes, Jana
T1 - Thermophysical modelling and parameter estimation of small solar system bodies via data assimilation
JF - Monthly notices of the Royal Astronomical Society
N2 - Deriving thermophysical properties such as thermal inertia from thermal infrared observations provides useful insights into the structure of the surface material on planetary bodies. The estimation of these properties is usually done by fitting temperature variations calculated by thermophysical models to infrared observations. For multiple free model parameters, traditional methods such as least-squares fitting or Markov chain Monte Carlo methods become computationally too expensive. Consequently, the simultaneous estimation of several thermophysical parameters, together with their corresponding uncertainties and correlations, is often not computationally feasible and the analysis is usually reduced to fitting one or two parameters. Data assimilation (DA) methods have been shown to be robust while sufficiently accurate and computationally affordable even for a large number of parameters. This paper will introduce a standard sequential DA method, the ensemble square root filter, for thermophysical modelling of asteroid surfaces. This method is used to re-analyse infrared observations of the MARA instrument, which measured the diurnal temperature variation of a single boulder on the surface of near-Earth asteroid (162173) Ryugu. The thermal inertia is estimated to be 295 +/- 18 Jm(-2) K-1 s(-1/2), while all five free parameters of the initial analysis are varied and estimated simultaneously. Based on this thermal inertia estimate the thermal conductivity of the boulder is estimated to be between 0.07 and 0.12,Wm(-1) K-1 and the porosity to be between 0.30 and 0.52. For the first time in thermophysical parameter derivation, correlations and uncertainties of all free model parameters are incorporated in the estimation procedure that is more than 5000 times more efficient than a comparable parameter sweep.
KW - radiation mechanisms: thermal
KW - methods: data analysis
KW - methods
KW - statistical
KW - minor planets, asteroids: individual: (162173) Ryugu
Y1 - 2020
U6 - https://doi.org/10.1093/mnras/staa1755
SN - 0035-8711
SN - 1365-2966
VL - 496
IS - 3
SP - 2776
EP - 2785
PB - Oxford Univ. Press
CY - Oxford
ER -
TY - JOUR
A1 - Ludewig, Matthias
A1 - Roos, Saskia
T1 - The chiral anomaly of the free fermion in functorial field theory
JF - Annales Henri Poincaré : a journal of theoretical and mathematical physics
N2 - When trying to cast the free fermion in the framework of functorial field theory, its chiral anomaly manifests in the fact that it assigns the determinant of the Dirac operator to a top-dimensional closed spin manifold, which is not a number as expected, but an element of a complex line. In functorial field theory language, this means that the theory is twisted, which gives rise to an anomaly theory. In this paper, we give a detailed construction of this anomaly theory, as a functor that sends manifolds to infinite-dimensional Clifford algebras and bordisms to bimodules.
Y1 - 2020
U6 - https://doi.org/10.1007/s00023-020-00893-6
SN - 1424-0637
SN - 1424-0661
VL - 21
IS - 4
SP - 1191
EP - 1233
PB - Springer International Publishing AG
CY - Cham (ZG)
ER -
TY - JOUR
A1 - Hartung, Niklas
A1 - Borghardt, Jens Markus
T1 - A mechanistic framework for a priori pharmacokinetic predictions of orally inhaled drugs
JF - PLoS Computational Biology : a new community journal
N2 - Author summary
The use of orally inhaled drugs for treating lung diseases is appealing since they have the potential for lung selectivity, i.e. high exposure at the site of action -the lung- without excessive side effects. However, the degree of lung selectivity depends on a large number of factors, including physiochemical properties of drug molecules, patient disease state, and inhalation devices. To predict the impact of these factors on drug exposure and thereby to understand the characteristics of an optimal drug for inhalation, we develop a predictive mathematical framework (a "pharmacokinetic model"). In contrast to previous approaches, our model allows combining knowledge from different sources appropriately and its predictions were able to adequately predict different sets of clinical data. Finally, we compare the impact of different factors and find that the most important factors are the size of the inhaled particles, the affinity of the drug to the lung tissue, as well as the rate of drug dissolution in the lung. In contrast to the common belief, the solubility of a drug in the lining fluids is not found to be relevant. These findings are important to understand how inhaled drugs should be designed to achieve best treatment results in patients.
The fate of orally inhaled drugs is determined by pulmonary pharmacokinetic processes such as particle deposition, pulmonary drug dissolution, and mucociliary clearance. Even though each single process has been systematically investigated, a quantitative understanding on the interaction of processes remains limited and therefore identifying optimal drug and formulation characteristics for orally inhaled drugs is still challenging. To investigate this complex interplay, the pulmonary processes can be integrated into mathematical models. However, existing modeling attempts considerably simplify these processes or are not systematically evaluated against (clinical) data. In this work, we developed a mathematical framework based on physiologically-structured population equations to integrate all relevant pulmonary processes mechanistically. A tailored numerical resolution strategy was chosen and the mechanistic model was evaluated systematically against data from different clinical studies. Without adapting the mechanistic model or estimating kinetic parameters based on individual study data, the developed model was able to predict simultaneously (i) lung retention profiles of inhaled insoluble particles, (ii) particle size-dependent pharmacokinetics of inhaled monodisperse particles, (iii) pharmacokinetic differences between inhaled fluticasone propionate and budesonide, as well as (iv) pharmacokinetic differences between healthy volunteers and asthmatic patients. Finally, to identify the most impactful optimization criteria for orally inhaled drugs, the developed mechanistic model was applied to investigate the impact of input parameters on both the pulmonary and systemic exposure. Interestingly, the solubility of the inhaled drug did not have any relevant impact on the local and systemic pharmacokinetics. Instead, the pulmonary dissolution rate, the particle size, the tissue affinity, and the systemic clearance were the most impactful potential optimization parameters. In the future, the developed prediction framework should be considered a powerful tool for identifying optimal drug and formulation characteristics.
Y1 - 2020
U6 - https://doi.org/10.1371/journal.pcbi.1008466
SN - 1553-734X
SN - 1553-7358
VL - 16
IS - 12
PB - PLoS
CY - San Fransisco
ER -
TY - THES
A1 - Reinhardt, Maria
T1 - Hybrid filters and multi-scale models
N2 - This thesis is concerned with Data Assimilation, the process of combining model predictions with observations. So called filters are of special interest. One is inter- ested in computing the probability distribution of the state of a physical process in the future, given (possibly) imperfect measurements. This is done using Bayes’ rule. The first part focuses on hybrid filters, that bridge between the two main groups of filters: ensemble Kalman filters (EnKF) and particle filters. The first are a group of very stable and computationally cheap algorithms, but they request certain strong assumptions. Particle filters on the other hand are more generally applicable, but computationally expensive and as such not always suitable for high dimensional systems. Therefore it exists a need to combine both groups to benefit from the advantages of each. This can be achieved by splitting the likelihood function, when assimilating a new observation and treating one part of it with an EnKF and the other part with a particle filter.
The second part of this thesis deals with the application of Data Assimilation to multi-scale models and the problems that arise from that. One of the main areas of application for Data Assimilation techniques is predicting the development of oceans and the atmosphere. These processes involve several scales and often balance rela- tions between the state variables. The use of Data Assimilation procedures most often violates relations of that kind, which leads to unrealistic and non-physical pre- dictions of the future development of the process eventually. This work discusses the inclusion of a post-processing step after each assimilation step, in which a minimi- sation problem is solved, which penalises the imbalance. This method is tested on four different models, two Hamiltonian systems and two spatially extended models, which adds even more difficulties.
N2 - Diese Dissertation beschäftigt sich mit Daten Assimilation - die Kombination von Modellvorhersagen mit Beobachtungen. Sogenannte Filter sind dabei von beson- derem Interesse. Diese Algorithmen berechnen die Wahrscheinlichkeitsverteilung des Zustandes eines physikalischen Prozesses in der Zukunft unter der Bedingung, dass wir (meist) fehlerbehaftete Messungen vorliegen haben. Der erste Teil bezieht sich auf Hybridfilter, welche eine Brücke zwischen den beiden Hauptgruppen von Filtern schlagen: Ensemble-Kalman-Filter (EnKF) und Teilchenfilter. Die erst- genannten sind sehr stabil und rechnerisch unaufwändig, aber basieren auf recht starken Voraussetzungen. Teilchenfilter hingegen sind allgemeiner aber recheninten- siv und daher nicht immer geeignet für höherdimensionale Systeme. Daher besteht die Notwen-
digkeit beide Gruppen zu kombinieren um von den Vorteilen beider Filter zu prof- itieren. Dies kann erreicht werden, indem man, wenn eine Beobachtung assimiliert werden soll, die Likelihood-Funktion in zwei Teile spaltet und auf den einen Teil einen EnKF und auf den anderen einen Teilchenfilter anwendet.
Der zweite Teil dieser Arbeit befasst sich mit der Anwendung von Daten Assimilation auf mehrskalige Modelle und die Probleme die daraus entstehen. Eines der Haup- tanwendungsgebiete für Daten Assimilation ist die Vorhersage der Entwicklung von Ozeanen und der Atmosphäre. Diese Prozesse finden auf mehreren Skalen statt und häufig bestehen Balancerelationen zwischen den Zustandsvariablen. Die Nutzung von Daten Assimilationstechniken zerstört diese Beziehungen häufig, was schließlich zu unrealistischen und unphysikalischen Vorhersagen führt. In dieser Dissertation wird vorgeschlagen, nach jedem Assimilationsschritt ein Minimierungsproblem zu lösen, welches die Imbalance als Strafterm beinhaltet. Diese Methode wird an vier verschiedenen Modellen getestet, zwei Hamiltonische Systeme und zwei Modelle mit räumlicher Ausdehnung, was zusätzliche Schwierigkeiten schafft.
T2 - Hybridfilter und Multiskalen-Modelle
KW - Data Assimilation
KW - Daten Assimilation
KW - Bayesian Inference
KW - bayessche Inferenz
KW - Uncertainty Quantification
KW - Quantifizierung von Unsicherheit
KW - Data-Driven Methods
KW - Datengetriebene Methoden
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-474356
ER -
TY - GEN
A1 - Pornsawad, Pornsarp
A1 - Sungcharoen, Parada
A1 - Böckmann, Christine
T1 - Convergence rate of the modified Landweber method for solving inverse potential problems
T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2 - 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.
T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1034
KW - nonlinear operator
KW - regularization
KW - modified Landweber method
KW - discrepancy principle
KW - logarithmic source condition
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-471942
SN - 1866-8372
IS - 1034
ER -
TY - JOUR
A1 - Zagrebnov, Valentin
T1 - Trotter product formula on Hilbert and Banach spaces for operator-norm convergence
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-471971
SN - 978-3-86956-485-2
SN - 2199-4951
SN - 2199-496X
SP - 23
EP - 34
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Zass, Alexander
T1 - A Gibbs point process of diffusions: Existence and uniqueness
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-471951
SN - 978-3-86956-485-2
SN - 2199-4951
SN - 2199-496X
IS - 6
SP - 13
EP - 22
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Sukiasyan, Hayk
A1 - Melkonyan, Tatev
T1 - Semi-recursive algorithm of piecewise linear approximation of two-dimensional function by the method of worst segment dividing
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-471982
SN - 978-3-86956-485-2
SN - 2199-4951
SN - 2199-496X
IS - 6
SP - 35
EP - 44
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Boldrighini, Carlo
A1 - Frigio, Sandro
A1 - Maponi, Pierluigi
A1 - Pellegrinotti, Alessandro
A1 - Sinai, Yakov G.
T1 - 3-D incompressible Navier-Stokes equations: Complex blow-up and related real flows
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-472201
SN - 978-3-86956-485-2
SN - 2199-4951
SN - 2199-496X
IS - 6
SP - 185
EP - 194
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Figari, Rodolfo
A1 - Teta, Alessandro
T1 - Zero-range hamiltonians for three quantum particles
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-472189
SN - 978-3-86956-485-2
SN - 2199-4951
SN - 2199-496X
IS - 6
SP - 175
EP - 184
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Houdebert, Pierre
T1 - Numerical study for the phase transition of the area-interaction model
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-472177
SN - 978-3-86956-485-2
SN - 2199-4951
SN - 2199-496X
IS - 6
SP - 165
EP - 174
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Jansen, Sabine
A1 - Kuna, Tobias
A1 - Tsagkarogiannis, Dimitrios
T1 - Virial inversion for inhomogeneous systems
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-472111
SN - 978-3-86956-485-2
SN - 2199-4951
SN - 2199-496X
IS - 6
SP - 135
EP - 144
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Hryniv, Ostap
A1 - Wallace, Clare
T1 - Phase separation and sharp large deviations
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-472168
SN - 978-3-86956-485-2
SN - 2199-4951
SN - 2199-496X
IS - 6
SP - 155
EP - 164
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Jansen, Sabine
A1 - Kolesnikov, Leonid
T1 - Activity expansions for Gibbs correlation functions
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-472121
SN - 978-3-86956-485-2
SN - 2199-4951
SN - 2199-496X
IS - 6
SP - 145
EP - 154
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Jansen, Sabine
A1 - Tsagkarogiannis, Dimitrios
T1 - Mayer expansion for the Asakura-Oosawa model of colloid theory
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-472109
SN - 978-3-86956-485-2
SN - 2199-4951
SN - 2199-496X
IS - 6
SP - 127
EP - 134
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Piatnitski, Andrey
A1 - Zhizhina, Elena
T1 - Non-local convolution type parabolic equations with fractional and regular time derivative
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-472024
SN - 978-3-86956-485-2
SN - 2199-4951
SN - 2199-496X
IS - 6
SP - 65
EP - 67
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Jursenas, Rytis
T1 - The peak model for finite rank supersingular perturbations
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-472090
IS - 6
SP - 117
EP - 126
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Mazzonetto, Sara
T1 - On an approximation of 2-D stochastic Navier-Stokes equations
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-472053
SN - 978-3-86956-485-2
SN - 2199-4951
SN - 2199-496X
IS - 6
SP - 87
EP - 96
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Pechersky, Eugeny
A1 - Pirogov, Sergei
A1 - Yambartsev, Anatoly
T1 - Large emissions
BT - Hawking-Penrose black hole model
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-472049
SN - 978-3-86956-485-2
SN - 2199-4951
SN - 2199-496X
IS - 6
SP - 77
EP - 86
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Poghosyan, Suren
A1 - Zessin, Hans
T1 - Construction of limiting Gibbs processes and the uniqueness of Gibbs processes
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-472015
SN - 978-3-86956-485-2
SN - 2199-4951
SN - 2199-496X
IS - 6
SP - 55
EP - 64
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Lykov, Alexander
A1 - Malyshev, Vadim
T1 - When bounded chaos becomes unbounded
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-472060
SN - 978-3-86956-485-2
SN - 2199-4951
SN - 2199-496X
IS - 6
SP - 97
EP - 106
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Khachatryan, Linda
A1 - Nahapetian, Boris
T1 - On direct and inverse problems in the description of lattice random fields
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-472083
SN - 978-3-86956-485-2
SN - 2199-4951
SN - 2199-496X
IS - 6
SP - 107
EP - 116
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Rafler, Mathias
T1 - Pinned Gibbs processes
JF - Lectures in pure and applied mathematics
KW - random point processes
KW - statistical mechanics
KW - stochastic analysis
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-472007
SN - 978-3-86956-485-2
SN - 2199-4951
SN - 2199-496X
IS - 6
SP - 45
EP - 53
PB - Universitätsverlag Potsdam
CY - Potsdam
ER -
TY - JOUR
A1 - Rœlly, Sylvie
A1 - Zass, Alexander
T1 - Marked Gibbs point processes with unbounded interaction
BT - An existence result
JF - Journal of statistical physics
N2 - 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.
KW - Marked Gibbs process
KW - Infinite-dimensional interacting diffusion
KW - Specific entropy
KW - DLR equation
Y1 - 2020
U6 - https://doi.org/10.1007/s10955-020-02559-3
SN - 0022-4715
SN - 1572-9613
VL - 179
IS - 4
SP - 972
EP - 996
PB - Springer
CY - New York
ER -
TY - JOUR
A1 - Sanchez, Sabrina
A1 - Wicht, Johannes
A1 - Bärenzung, Julien
T1 - Predictions of the geomagnetic secular variation based on the ensemble sequential assimilation of geomagnetic field models by dynamo simulations
JF - Earth, planets and space
N2 - The IGRF offers an important incentive for testing algorithms predicting the Earth's magnetic field changes, known as secular variation (SV), in a 5-year range. Here, we present a SV candidate model for the 13th IGRF that stems from a sequential ensemble data assimilation approach (EnKF). The ensemble consists of a number of parallel-running 3D-dynamo simulations. The assimilated data are geomagnetic field snapshots covering the years 1840 to 2000 from the COV-OBS.x1 model and for 2001 to 2020 from the Kalmag model. A spectral covariance localization method, considering the couplings between spherical harmonics of the same equatorial symmetry and same azimuthal wave number, allows decreasing the ensemble size to about a 100 while maintaining the stability of the assimilation. The quality of 5-year predictions is tested for the past two decades. These tests show that the assimilation scheme is able to reconstruct the overall SV evolution. They also suggest that a better 5-year forecast is obtained keeping the SV constant compared to the dynamically evolving SV. However, the quality of the dynamical forecast steadily improves over the full assimilation window (180 years). We therefore propose the instantaneous SV estimate for 2020 from our assimilation as a candidate model for the IGRF-13. The ensemble approach provides uncertainty estimates, which closely match the residual differences with respect to the IGRF-13. Longer term predictions for the evolution of the main magnetic field features over a 50-year range are also presented. We observe the further decrease of the axial dipole at a mean rate of 8 nT/year as well as a deepening and broadening of the South Atlantic Anomaly. The magnetic dip poles are seen to approach an eccentric dipole configuration.
KW - Earth's magnetic field
KW - Geomagnetic secular variation
KW - Dynamo
KW - simulations
KW - Data assimilation
Y1 - 2020
U6 - https://doi.org/10.1186/s40623-020-01279-y
SN - 1880-5981
VL - 72
IS - 1
PB - Springer
CY - New York
ER -
TY - JOUR
A1 - Garbuno-Inigo, Alfredo
A1 - Nüsken, Nikolas
A1 - Reich, Sebastian
T1 - Affine invariant interacting Langevin dynamics for Bayesian inference
JF - SIAM journal on applied dynamical systems
N2 - We propose a computational method (with acronym ALDI) for sampling from a given target distribution based on first-order (overdamped) Langevin dynamics which satisfies the property of affine invariance. The central idea of ALDI is to run an ensemble of particles with their empirical covariance serving as a preconditioner for their underlying Langevin dynamics. ALDI does not require taking the inverse or square root of the empirical covariance matrix, which enables application to high-dimensional sampling problems. The theoretical properties of ALDI are studied in terms of nondegeneracy and ergodicity. Furthermore, we study its connections to diffusion on Riemannian manifolds and Wasserstein gradient flows. Bayesian inference serves as a main application area for ALDI. In case of a forward problem with additive Gaussian measurement errors, ALDI allows for a gradient-free approximation in the spirit of the ensemble Kalman filter. A computational comparison between gradient-free and gradient-based ALDI is provided for a PDE constrained Bayesian inverse problem.
KW - Langevin dynamics
KW - interacting particle systems
KW - Bayesian inference
KW - gradient flow
KW - multiplicative noise
KW - affine invariance
KW - gradient-free
Y1 - 2020
U6 - https://doi.org/10.1137/19M1304891
SN - 1536-0040
VL - 19
IS - 3
SP - 1633
EP - 1658
PB - Society for Industrial and Applied Mathematics
CY - Philadelphia
ER -
TY - JOUR
A1 - Froyland, Gary
A1 - Koltai, Peter
A1 - Stahn, Martin
T1 - Computation and optimal perturbation of finite-time coherent sets for aperiodic flows without trajectory integration
JF - SIAM journal on applied dynamical systems
N2 - Understanding the macroscopic behavior of dynamical systems is an important tool to unravel transport mechanisms in complex flows. A decomposition of the state space into coherent sets is a popular way to reveal this essential macroscopic evolution. To compute coherent sets from an aperiodic time-dependent dynamical system we consider the relevant transfer operators and their infinitesimal generators on an augmented space-time manifold. This space-time generator approach avoids trajectory integration and creates a convenient linearization of the aperiodic evolution. This linearization can be further exploited to create a simple and effective spectral optimization methodology for diminishing or enhancing coherence. We obtain explicit solutions for these optimization problems using Lagrange multipliers and illustrate this technique by increasing and decreasing mixing of spatial regions through small velocity field perturbations.
KW - coherent set
KW - mixing
KW - transfer operator
KW - infinitesimal generator
KW - unsteady flow
KW - mixing optimization
Y1 - 2020
U6 - https://doi.org/10.1137/19M1261791
SN - 1536-0040
VL - 19
IS - 3
SP - 1659
EP - 1700
PB - Society for Industrial and Applied Mathematics
CY - Philadelphia
ER -
TY - JOUR
A1 - Rastogi, Abhishake
T1 - Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems
JF - Communications on Pure and Applied Analysis
N2 - In this paper, we consider the nonlinear ill-posed inverse problem with noisy data in the statistical learning setting. The Tikhonov regularization scheme in Hilbert scales is considered to reconstruct the estimator from the random noisy data. In this statistical learning setting, we derive the rates of convergence for the regularized solution under certain assumptions on the nonlinear forward operator and the prior assumptions. We discuss estimates of the reconstruction error using the approach of reproducing kernel Hilbert spaces.
KW - Statistical inverse problem
KW - Tikhonov regularization
KW - Hilbert Scales
KW - reproducing kernel Hilbert space
KW - minimax convergence rates
Y1 - 2020
U6 - https://doi.org/10.3934/cpaa.2020183
SN - 1534-0392
SN - 1553-5258
VL - 19
IS - 8
SP - 4111
EP - 4126
PB - American Institute of Mathematical Sciences
CY - Springfield
ER -
TY - JOUR
A1 - Gräter, Joachim
T1 - Free division rings of fractions of crossed products of groups with Conradian left-orders
JF - Forum mathematicum
N2 - Let D be a division ring of fractions of a crossed product F[G, eta, alpha], where F is a skew field and G is a group with Conradian left-order <=. For D we introduce the notion of freeness with respect to <= and show that D is free in this sense if and only if D can canonically be embedded into the endomorphism ring of the right F-vector space F((G)) of all formal power series in G over F with respect to <=. From this we obtain that all division rings of fractions of F[G, eta, alpha] which are free with respect to at least one Conradian left-order of G are isomorphic and that they are free with respect to any Conradian left-order of G. Moreover, F[G, eta, alpha] possesses a division ring of fraction which is free in this sense if and only if the rational closure of F[G, eta, alpha] in the endomorphism ring of the corresponding right F-vector space F((G)) is a skew field.
KW - crossed product
KW - group ring
KW - ordered group
KW - Conradian left-order
KW - locally indicable group
KW - division ring of fractions
KW - Hughes-free
KW - formal
KW - power series
Y1 - 2020
U6 - https://doi.org/10.1515/forum-2019-0264
SN - 0933-7741
SN - 1435-5337
VL - 32
IS - 3
SP - 739
EP - 772
PB - De Gruyter
CY - Berlin
ER -
TY - JOUR
A1 - Ludewig, Matthias
A1 - Rosenberger, Elke
T1 - Asymptotic eigenfunctions for Schrödinger operators on a vector bundle
JF - Reviews in mathematical physics
N2 - In the limit (h) over bar -> 0, we analyze a class of Schrödinger operators H-(h) over bar = (h) over bar L-2 + (h) over barW + V .id(epsilon) acting on sections of a vector bundle epsilon over a Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has a non-degenerate minimum at some point p is an element of M. We construct quasimodes of WKB-type near p for eigenfunctions associated with the low-lying eigenvalues of H-(h) over bar. These are obtained from eigenfunctions of the associated harmonic oscillator H-p,H-(h) over bar at p, acting on smooth functions on the tangent space.
KW - Semi-classical analysis
KW - WKB approximation
KW - Schrödinger operators
KW - semi-classical limit
Y1 - 2020
U6 - https://doi.org/10.1142/S0129055X20500208
SN - 0129-055X
SN - 1793-6659
VL - 32
IS - 7
PB - World Scientific
CY - Singapore
ER -
TY - JOUR
A1 - Maoutsa, Dimitra
A1 - Reich, Sebastian
A1 - Opper, Manfred
T1 - Interacting particle solutions of Fokker–Planck equations through gradient–log–density estimation
JF - Entropy
N2 - Fokker-Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often it is inevitable to resort to numerical solutions. Here, we develop a computational approach for simulating the time evolution of Fokker-Planck solutions in terms of a mean field limit of an interacting particle system. The interactions between particles are determined by the gradient of the logarithm of the particle density, approximated here by a novel statistical estimator. The performance of our method shows promising results, with more accurate and less fluctuating statistics compared to direct stochastic simulations of comparable particle number. Taken together, our framework allows for effortless and reliable particle-based simulations of Fokker-Planck equations in low and moderate dimensions. The proposed gradient-log-density estimator is also of independent interest, for example, in the context of optimal control.
KW - stochastic systems
KW - Fokker-Planck equation
KW - interacting particles
KW - multiplicative noise
KW - gradient flow
KW - stochastic differential equations
Y1 - 2020
U6 - https://doi.org/10.3390/e22080802
SN - 1099-4300
VL - 22
IS - 8
PB - MDPI
CY - Basel
ER -
TY - JOUR
A1 - Bandara, Lashi
A1 - Bryan, Paul
T1 - Heat kernels and regularity for rough metrics on smooth manifolds
JF - Mathematische Nachrichten
N2 - We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are Holder continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with bounded measurable coefficients in weighted Sobolev spaces.
KW - heat kernel
KW - parabolic Harnack estimate
KW - rough metrics
Y1 - 2020
U6 - https://doi.org/10.1002/mana.201800459
SN - 0025-584X
SN - 1522-2616
VL - 293
IS - 12
SP - 2255
EP - 2270
PB - Wiley-VCH
CY - Weinheim
ER -
TY - JOUR
A1 - Azzali, Sara
A1 - Paycha, Sylvie
T1 - Spectral zeta-invariants lifted to coverings
JF - Transactions of the American Mathematical Society
N2 - 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.
Y1 - 2020
U6 - https://doi.org/10.1090/tran/8067
SN - 0002-9947
SN - 1088-6850
VL - 373
IS - 9
SP - 6185
EP - 6226
PB - American Mathematical Society
CY - Providence, RI
ER -
TY - JOUR
A1 - Blanchard, Gilles
A1 - Mücke, Nicole
T1 - Kernel regression, minimax rates and effective dimensionality
BT - beyond the regular case
JF - Analysis and applications
N2 - 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.
KW - Kernel regression
KW - minimax optimality
KW - eigenvalue decay
Y1 - 2020
U6 - https://doi.org/10.1142/S0219530519500258
SN - 0219-5305
SN - 1793-6861
VL - 18
IS - 4
SP - 683
EP - 696
PB - World Scientific
CY - New Jersey
ER -
TY - JOUR
A1 - Beckus, Siegfried
A1 - Pinchover, Yehuda
T1 - Shnol-type theorem for the Agmon ground state
JF - Journal of spectral theory
N2 - 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.
KW - Shnol theorem
KW - Caccioppoli inequality
KW - Schrodinger operators
KW - generalized eigenfunction
KW - ground state
KW - positive solutions
KW - weighted
KW - graphs
Y1 - 2020
U6 - https://doi.org/10.4171/JST/296
SN - 1664-039X
SN - 1664-0403
VL - 10
IS - 2
SP - 355
EP - 377
PB - EMS Publishing House
CY - Zürich
ER -
TY - JOUR
A1 - Perera, Upeksha
A1 - Böckmann, Christine
T1 - Solutions of Sturm-Liouville problems
JF - Mathematics
N2 - 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.
KW - Sturm-Liouville problems of higher order
KW - singular Sturm-Liouville
KW - problems
KW - inverse Sturm-Liouville problems
Y1 - 2020
U6 - https://doi.org/10.3390/math8112074
SN - 2227-7390
VL - 8
IS - 11
PB - MDPI
CY - Basel
ER -
TY - JOUR
A1 - Mariucci, Ester
A1 - Ray, Kolyan
A1 - Szabo, Botond
T1 - A Bayesian nonparametric approach to log-concave density estimation
JF - Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability
N2 - 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.
KW - convergence rate
KW - density estimation
KW - Dirichlet mixture
KW - log-concavity
KW - nonparametric hypothesis testing
KW - posterior distribution
Y1 - 2020
U6 - https://doi.org/10.3150/19-BEJ1139
SN - 1350-7265
SN - 1573-9759
VL - 26
IS - 2
SP - 1070
EP - 1097
PB - International Statistical Institute
CY - The Hague
ER -
TY - THES
A1 - Rothe, Viktoria
T1 - Das Yamabe-Problem auf global-hyperbolischen Lorentz-Mannigfaltigkeiten
N2 - 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.
N2 - Yamabe claimed in 1960 that he had proven the following theorem: Any Riemannian metric g on a compact smooth manifold M of dimension n ≥ 3 is conformal to a metric with constant scalar curvature. An equivalent formulation of this theorem is the existence of a solution to a certain semilinear elliptic differential equation, the so-called Yamabe equation. In 1968 Trudinger found a mistake in Yamabe's paper and consequently many mathematicians dealt with this so-called Yamabe problem. In the 80s Trudinger, Aubin and Shoen were able to fix the mistake and prove that Yamabe's theorem was indeed true. This has many advantages, for example when analyzing a conformally invariant partial differential equation on compact Riemannian manifolds one can assume that the scalar curvature is constant.
The question now arises whether the analogous statement on Lorentzian manifolds also applies. The Lorentzian Yamabe Problem can be stated as follows: Given a spatially compact globally hyperbolic Lorentzian manifold (M, g), does there exist a metric conformal to g with constant scalar curvature? The goal of this dissertation is to examine this problem.
The Yamabe equation which arises from this question is a semilinear wave equation which must have a positive smooth solution. In the first part of this dissertation the local theory of existence of general semilinear wave equations for sections on vector bundles was developed. For this the inverse function theorem and already existing statements about the existence of solutions to linear wave equation on Lorentzian manifolds were used. It will be proven that there exists an almost global solution to the corresponding Cauchy problem for small initial data as well as a time local solution for arbitrary initial data if the nonlinearity fulfills certain conditions.
The second part of the dissertation deals with the Yamabe equation on globally hyperbolic Lorentzian manifolds. First by using the results of the first part it will be proven that there exist initial data such that the Yamabe equation has an almost time global solution if the scalar curvature of the given metric is sufficiently close to a constant. Afterwards by using energy estimates it will be shown in the case of 4-dimensional Lorentzian manifolds that under the assumption that the constant scalar curvature of the conformal metric is non-positive there exists a global smooth solution to the Yamabe equation which is not necessarily positive. But it will be proven that the solution is positive on a compact time interval if the H2-Norm of the scalar curvature of the given metric is bounded on this time interval in a certain way or if the scalar curvature is negative and the gradient of the scalar curvature is bounded in a specific way. In both cases the existence of a global positive smooth solution follows, if the Lorentzian manifold has the form M = I x Σ where I is an open bounded time interval and Σ is a Riemannian manifold. At the end an example for the nonexistence of a global positive solution in the case of M= R x Σ will be presented.
T2 - The Yamabe problem on globally hyperbolic Lorentzian manifolds
KW - Yamabe-Problem
KW - Yamabe problem
KW - wave equation
KW - globally hyperbolic
KW - global-hyperbolisch
KW - Wellengleichung
Y1 - 2020
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-486012
ER -
TY - JOUR
A1 - Schoppa, Lukas
A1 - Sieg, Tobias
A1 - Vogel, Kristin
A1 - Zöller, Gert
A1 - Kreibich, Heidi
T1 - Probabilistic flood loss models for companies
JF - Water resources research
N2 - 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.
KW - flood loss estimation
KW - probabilistic modeling
KW - companies
KW - multivariable
KW - models
Y1 - 2020
U6 - https://doi.org/10.1029/2020WR027649
SN - 0043-1397
SN - 1944-7973
VL - 56
IS - 9
PB - American Geophysical Union
CY - Washington
ER -