TY - JOUR A1 - Elin, Mark A1 - Shoikhet, David A1 - Tarkhanov, Nikolai Nikolaevich T1 - Analytic Semigroups of Holomorphic Mappings and Composition Operators JF - Computational Methods and Function Theory N2 - In this manuscript we provide a review on the classical and resent results related to the problem of analytic extension in parameter for a semigroup of holomorphic self-mappings of the unit ball in a complex Banach space and its relation to the linear continuous semigroup of composition operators. KW - Non-linear semigroups KW - Composition operators KW - Analytic extension KW - Holomorphic mappings Y1 - 2017 U6 - https://doi.org/10.1007/s40315-017-0227-x SN - 1617-9447 SN - 2195-3724 VL - 18 IS - 2 SP - 269 EP - 294 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Carpentier, Alexandra A1 - Klopp, Olga A1 - Löffler, Matthias A1 - Nickl, Richard T1 - Adaptive confidence sets for matrix completion JF - Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability N2 - In the present paper, we study the problem of existence of honest and adaptive confidence sets for matrix completion. We consider two statistical models: the trace regression model and the Bernoulli model. In the trace regression model, we show that honest confidence sets that adapt to the unknown rank of the matrix exist even when the error variance is unknown. Contrary to this, we prove that in the Bernoulli model, honest and adaptive confidence sets exist only when the error variance is known a priori. In the course of our proofs, we obtain bounds for the minimax rates of certain composite hypothesis testing problems arising in low rank inference. KW - adaptivity KW - confidence sets KW - low rank recovery KW - matrix completion KW - minimax hypothesis testing KW - unknown variance Y1 - 2018 U6 - https://doi.org/10.3150/17-BEJ933 SN - 1350-7265 SN - 1573-9759 VL - 24 IS - 4A SP - 2429 EP - 2460 PB - International Statistical Institute CY - Voorburg ER - TY - JOUR A1 - Koltai, Peter A1 - Lie, Han Cheng A1 - Plonka, Martin T1 - Frechet differentiable drift dependence of Perron-Frobenius and Koopman operators for non-deterministic dynamics JF - Nonlinearity N2 - We prove the Fréchet differentiability with respect to the drift of Perron–Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations. This result relies on a similar differentiability result for pathwise expectations of path functionals of the solution of the stochastic differential equation, which we establish using Girsanov's formula. We demonstrate the significance of our result in the context of dynamical systems and operator theory, by proving continuously differentiable drift dependence of the simple eigen- and singular values and the corresponding eigen- and singular functions of the stochastic Perron–Frobenius and Koopman operators. KW - stochastic differential equations KW - transfer operator KW - Koopman operator KW - Perron-Frobenius operator KW - smooth drift dependence KW - linear response KW - pathwise expectations Y1 - 2019 U6 - https://doi.org/10.1088/1361-6544/ab1f2a SN - 0951-7715 SN - 1361-6544 VL - 32 IS - 11 SP - 4232 EP - 4257 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Kollosche, David T1 - Entdeckendes Lernen BT - eine Problematisierung JF - Journal für Mathematik-Didaktik N2 - Trotz der nachweislichen Popularität des Entdeckenden Lernens in der deutschsprachigen Mathematikdidaktik finden sich aktuell keine kritischen Beiträge, die dazu beitragen könnten, dieses grundlegende Unterrichtskonzept zu hinterfragen und auszuschärfen. In diesem Diskussionsbeitrag werden zunächst die Theorie und einige Umsetzungsbeispiele des Entdeckenden Lernens herausgearbeitet, um aufzuzeigen, dass das Entdeckende Lernen einem vagen Sammelbegriff gleicht, unter dem oft fragwürdige Unterrichtsumgebungen legitimiert werden. Anschließend werden an Hand erkenntnistheoretischer, lerntheoretischer, didaktischer und soziokultureller Betrachtungen Probleme des Entdeckenden Lernens im Mathematikunterricht und Möglichkeiten ihrer Überwindung thematisiert. Dabei zeigt sich, dass die Konzeption des Entdeckenden Lernens hinter dem aktuellen mathematikdidaktischen Erkenntnisstand zurückfällt und Lehrer sowie Schüler mit unmöglichen Forderungen konfrontiert, dass lerntheoretische Vorteile des Entdeckenden Lernens oft nicht nachweisbar sind, dass die Idee des Entdeckens auf einem problematischen platonistischen Verständnis von Erkenntnis beruht und dass Entdeckendes Lernen bildungsferne Schüler zu benachteiligen droht. Abschließend werden Forschungsdesiderata abgeleitet, deren Bearbeitung dazu beitragen könnte, die aufgezeigten Problemfelder zu überwinden. N2 - Despite the attested popularity of discovery learning in germanophone research on mathematics education, we contemporarily lack critical contributions which may contribute to questioning and sharpening this fundamental teaching concept. Here, in this discussion paper the theory and some implementations of discovery learning are being discussed in order to show that discovery learning constitutes a vague umbrella term, with which often questionable learning environments are being legitimised. Afterwards, it is outlined which objections from the perspectives of epistemology, learning theory, didactics and social theory may be brought forward against discovery learning in mathematics education, and how these objections may be overcome. Thereby it becomes obvious that the conception of discovery learning falls behind the current level of knowledge in mathematics education research and confronts teachers as well as students with impossible demands; that learning benefits of discovery learning often cannot be substantiated; that the idea of discovery rests on a problematic platonic understanding of insight; and that discovery learning threatens to disadvantage educationally alienated students. As a last point, research desiderata, whose study may contribute to the negotiation of the described obstacles, are deduced. T2 - Discovery learning: A problematization KW - Discovery learning KW - Inquiry-based learning KW - Epistemology KW - Learning theory KW - Sociolinguistics KW - Entdeckendes Lernen KW - Epistemologie KW - Lerntheorie KW - Soziolinguistik Y1 - 2017 U6 - https://doi.org/10.1007/s13138-017-0116-x SN - 0173-5322 SN - 1869-2699 VL - 38 SP - 209 EP - 237 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Bandara, Menaka Lashitha A1 - Rosen, Andreas T1 - Riesz continuity of the Atiyah-Singer Dirac operator under perturbations of local boundary conditions JF - Communications in partial differential equations N2 - On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator in depends Riesz continuously on perturbations of local boundary conditions The Lipschitz bound for the map depends on Lipschitz smoothness and ellipticity of and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius away from a compact neighbourhood of the boundary. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions. KW - Boundary value problems KW - Dirac operator KW - functional calculus KW - real-variable harmonic analysis KW - Riesz continuity KW - spectral flow Y1 - 2019 U6 - https://doi.org/10.1080/03605302.2019.1611847 SN - 0360-5302 SN - 1532-4133 VL - 44 IS - 12 SP - 1253 EP - 1284 PB - Taylor & Francis Group CY - Philadelphia ER - TY - JOUR A1 - Ringel, Lisa Maria A1 - Somogyvári, Márk A1 - Jalali, Mohammadreza A1 - Bayer, Peter T1 - Comparison of hydraulic and tracer tomography for discrete fracture network inversion JF - Geosciences N2 - Fractures serve as highly conductive preferential flow paths for fluids in rocks, which are difficult to exactly reconstruct in numerical models. Especially, in low-conductive rocks, fractures are often the only pathways for advection of solutes and heat. The presented study compares the results from hydraulic and tracer tomography applied to invert a theoretical discrete fracture network (DFN) that is based on data from synthetic cross-well testing. For hydraulic tomography, pressure pulses in various injection intervals are induced and the pressure responses in the monitoring intervals of a nearby observation well are recorded. For tracer tomography, a conservative tracer is injected in different well levels and the depth-dependent breakthrough of the tracer is monitored. A recently introduced transdimensional Bayesian inversion procedure is applied for both tomographical methods, which adjusts the fracture positions, orientations, and numbers based on given geometrical fracture statistics. The used Metropolis-Hastings-Green algorithm is refined by the simultaneous estimation of the measurement error’s variance, that is, the measurement noise. Based on the presented application to invert the two-dimensional cross-section between source and the receiver well, the hydraulic tomography reveals itself to be more suitable for reconstructing the original DFN. This is based on a probabilistic representation of the inverted results by means of fracture probabilities. KW - hydraulic tomography KW - tracer tomography KW - DFN KW - Bayesian inversion KW - heterogeneity KW - fracture KW - hydrogeophysics Y1 - 2019 U6 - https://doi.org/10.3390/geosciences9060274 SN - 2076-3263 VL - 9 IS - 6 PB - MDPI CY - Basel ER - TY - JOUR A1 - Beckus, Siegfried A1 - Bellissard, Jean A1 - Cornean, Horia T1 - Holder Continuity of the Spectra for Aperiodic Hamiltonians JF - Annales de l'Institut Henri Poincaré N2 - We study the spectral location of a strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are "close to each other" if, up to a translation, they "almost coincide" on a large fixed ball. The larger this ball, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Holder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the spectral distance of two Hamiltonians is estimated by the distance of the corresponding dynamical systems. Y1 - 2019 U6 - https://doi.org/10.1007/s00023-019-00848-6 SN - 1424-0637 SN - 1424-0661 VL - 20 IS - 11 SP - 3603 EP - 3631 PB - Springer CY - Cham ER - TY - JOUR A1 - Blanchard, Gilles A1 - Zadorozhnyi, Oleksandr T1 - Concentration of weakly dependent Banach-valued sums and applications to statistical learning methods JF - Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability N2 - We obtain a Bernstein-type inequality for sums of Banach-valued random variables satisfying a weak dependence assumption of general type and under certain smoothness assumptions of the underlying Banach norm. We use this inequality in order to investigate in the asymptotical regime the error upper bounds for the broad family of spectral regularization methods for reproducing kernel decision rules, when trained on a sample coming from a tau-mixing process. KW - Banach-valued process KW - Bernstein inequality KW - concentration KW - spectral regularization KW - weak dependence Y1 - 2019 U6 - https://doi.org/10.3150/18-BEJ1095 SN - 1350-7265 SN - 1573-9759 VL - 25 IS - 4B SP - 3421 EP - 3458 PB - International Statistical Institute CY - Voorburg ER - TY - JOUR A1 - Hofer-Temmel, Christoph A1 - Houdebert, Pierre T1 - Disagreement percolation for Gibbs ball models JF - Stochastic processes and their application N2 - We generalise disagreement percolation to Gibbs point processes of balls with varying radii. This allows to establish the uniqueness of the Gibbs measure and exponential decay of pair correlations in the low activity regime by comparison with a sub-critical Boolean model. Applications to the Continuum Random Cluster model and the Quermass-interaction model are presented. At the core of our proof lies an explicit dependent thinning from a Poisson point process to a dominated Gibbs point process. (C) 2018 Elsevier B.V. All rights reserved. KW - Continuum random cluster model KW - Disagreement percolation KW - Dependent thinning KW - Boolean model KW - Stochastic domination KW - Phase transition KW - Unique Gibbs state KW - Exponential decay of pair correlation Y1 - 2019 U6 - https://doi.org/10.1016/j.spa.2018.11.003 SN - 0304-4149 SN - 1879-209X VL - 129 IS - 10 SP - 3922 EP - 3940 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Mera, Azal A1 - Stepanenko, Vitaly A. A1 - Tarkhanov, Nikolai Nikolaevich T1 - Successive approximation for the inhomogeneous burgers equation JF - Journal of Siberian Federal University : Mathematics & Physics N2 - The inhomogeneous Burgers equation is a simple form of the Navier-Stokes equations. From the analytical point of view, the inhomogeneous form is poorly studied, the complete analytical solution depending closely on the form of the nonhomogeneous term. KW - Navier-Stokes equations KW - classical solution Y1 - 2018 U6 - https://doi.org/10.17516/1997-1397-2018-11-4-519-531 SN - 1997-1397 SN - 2313-6022 VL - 11 IS - 4 SP - 519 EP - 531 PB - Siberian Federal University CY - Krasnoyarsk ER - TY - BOOK A1 - Van Leeuwen, Peter Jan A1 - Cheng, Yuan A1 - Reich, Sebastian T1 - Nonlinear data assimilation T3 - Frontiers in applied dynamical systems: reviews and tutorials ; 2 N2 - This book contains two review articles on nonlinear data assimilation that deal with closely related topics but were written and can be read independently. Both contributions focus on so-called particle filters. The first contribution by Jan van Leeuwen focuses on the potential of proposal densities. It discusses the issues with present-day particle filters and explorers new ideas for proposal densities to solve them, converging to particle filters that work well in systems of any dimension, closing the contribution with a high-dimensional example. The second contribution by Cheng and Reich discusses a unified framework for ensemble-transform particle filters. This allows one to bridge successful ensemble Kalman filters with fully nonlinear particle filters, and allows a proper introduction of localization in particle filters, which has been lacking up to now. Y1 - 2015 SN - 978-3-319-18346-6 SN - 978-3-319-18347-3 U6 - https://doi.org/10.1007/978-3-319-18347-3 PB - Springer CY - Cham ER - TY - GEN A1 - Blanchard, Gilles A1 - Scott, Clayton T1 - Corrigendum to: Classification with asymmetric label noise BT - Consistency and maximal denoising T2 - Electronic journal of statistics N2 - We point out a flaw in Lemma 15 of [1]. We also indicate how the main results of that section are still valid using a modified argument. Y1 - 2018 U6 - https://doi.org/10.1214/18-EJS1422 SN - 1935-7524 VL - 12 IS - 1 SP - 1779 EP - 1781 PB - Institute of Mathematical Statistics CY - Cleveland ER - TY - JOUR A1 - Shlapunov, Alexander A1 - Tarkhanov, Nikolai Nikolaevich T1 - Golusin-Krylov formulas in complex analysis JF - Complex variables and elliptic equations N2 - This is a brief survey of a constructive technique of analytic continuation related to an explicit integral formula of Golusin and Krylov (1933). It goes far beyond complex analysis and applies to the Cauchy problem for elliptic partial differential equations as well. As started in the classical papers, the technique is elaborated in generalised Hardy spaces also called Hardy-Smirnov spaces. KW - Analytic continuation KW - inegral formulas KW - Cauchy problem Y1 - 2017 U6 - https://doi.org/10.1080/17476933.2017.1395872 SN - 1747-6933 SN - 1747-6941 VL - 63 IS - 7-8 SP - 1142 EP - 1167 PB - Routledge CY - Abingdon ER - TY - JOUR A1 - Mücke, Nicole A1 - Blanchard, Gilles T1 - Parallelizing spectrally regularized kernel algorithms JF - Journal of machine learning research N2 - We consider a distributed learning approach in supervised learning for a large class of spectral regularization methods in an reproducing kernel Hilbert space (RKHS) framework. The data set of size n is partitioned into m = O (n(alpha)), alpha < 1/2, disjoint subsamples. On each subsample, some spectral regularization method (belonging to a large class, including in particular Kernel Ridge Regression, L-2-boosting and spectral cut-off) is applied. The regression function f is then estimated via simple averaging, leading to a substantial reduction in computation time. We show that minimax optimal rates of convergence are preserved if m grows sufficiently slowly (corresponding to an upper bound for alpha) as n -> infinity, depending on the smoothness assumptions on f and the intrinsic dimensionality. In spirit, the analysis relies on a classical bias/stochastic error analysis. KW - Distributed Learning KW - Spectral Regularization KW - Minimax Optimality Y1 - 2018 SN - 1532-4435 VL - 19 PB - Microtome Publishing CY - Cambridge, Mass. ER - TY - JOUR A1 - Blanchard, Gilles A1 - Hoffmann, Marc A1 - Reiss, Markus T1 - Optimal adaptation for early stopping in statistical inverse problems JF - SIAM/ASA Journal on Uncertainty Quantification N2 - For linear inverse problems Y = A mu + zeta, it is classical to recover the unknown signal mu by iterative regularization methods ((mu) over cap,(m) = 0,1, . . .) and halt at a data-dependent iteration tau using some stopping rule, typically based on a discrepancy principle, so that the weak (or prediction) squared-error parallel to A((mu) over cap (()(tau)) - mu)parallel to(2) is controlled. In the context of statistical estimation with stochastic noise zeta, we study oracle adaptation (that is, compared to the best possible stopping iteration) in strong squared- error E[parallel to((mu) over cap (()(tau)) - mu)parallel to(2)]. For a residual-based stopping rule oracle adaptation bounds are established for general spectral regularization methods. The proofs use bias and variance transfer techniques from weak prediction error to strong L-2-error, as well as convexity arguments and concentration bounds for the stochastic part. Adaptive early stopping for the Landweber method is studied in further detail and illustrated numerically. KW - linear inverse problems KW - early stopping KW - discrepancy principle KW - adaptive estimation KW - oracle inequality KW - Landweber iteration Y1 - 2018 U6 - https://doi.org/10.1137/17M1154096 SN - 2166-2525 VL - 6 IS - 3 SP - 1043 EP - 1075 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Shaki, Samuel A1 - Pinhas, Michal A1 - Fischer, Martin H. T1 - Heuristics and biases in mental arithmetic BT - revisiting and reversing operational momentum JF - Thinking & Reasoning N2 - Mental arithmetic is characterised by a tendency to overestimate addition and to underestimate subtraction results: the operational momentum (OM) effect. Here, motivated by contentious explanations of this effect, we developed and tested an arithmetic heuristics and biases model that predicts reverse OM due to cognitive anchoring effects. Participants produced bi-directional lines with lengths corresponding to the results of arithmetic problems. In two experiments, we found regular OM with zero problems (e.g., 3+0, 3-0) but reverse OM with non-zero problems (e.g., 2+1, 4-1). In a third experiment, we tested the prediction of our model. Our results suggest the presence of at least three competing biases in mental arithmetic: a more-or-less heuristic, a sign-space association and an anchoring bias. We conclude that mental arithmetic exhibits shortcuts for decision-making similar to traditional domains of reasoning and problem-solving. KW - Heuristics KW - mental arithmetic KW - mental number line KW - operational momentum KW - problem-solving Y1 - 2017 U6 - https://doi.org/10.1080/13546783.2017.1348987 SN - 1354-6783 SN - 1464-0708 VL - 24 IS - 2 SP - 138 EP - 156 PB - Routledge, Taylor & Francis Group CY - Abingdon ER - TY - JOUR A1 - Blanchard, Gilles A1 - Hoffmann, Marc A1 - Reiss, Markus T1 - Early stopping for statistical inverse problems via truncated SVD estimation JF - Electronic journal of statistics N2 - We consider truncated SVD (or spectral cut-off, projection) estimators for a prototypical statistical inverse problem in dimension D. Since calculating the singular value decomposition (SVD) only for the largest singular values is much less costly than the full SVD, our aim is to select a data-driven truncation level (m) over cap is an element of {1, . . . , D} only based on the knowledge of the first (m) over cap singular values and vectors. We analyse in detail whether sequential early stopping rules of this type can preserve statistical optimality. Information-constrained lower bounds and matching upper bounds for a residual based stopping rule are provided, which give a clear picture in which situation optimal sequential adaptation is feasible. Finally, a hybrid two-step approach is proposed which allows for classical oracle inequalities while considerably reducing numerical complexity. KW - Linear inverse problems KW - truncated SVD KW - spectral cut-off KW - early stopping KW - discrepancy principle KW - adaptive estimation KW - oracle inequalities Y1 - 2018 U6 - https://doi.org/10.1214/18-EJS1482 SN - 1935-7524 VL - 12 IS - 2 SP - 3204 EP - 3231 PB - Institute of Mathematical Statistics CY - Cleveland ER - TY - JOUR A1 - Blanchard, Gilles A1 - Carpentier, Alexandra A1 - Gutzeit, Maurilio T1 - Minimax Euclidean separation rates for testing convex hypotheses in R-d JF - Electronic journal of statistics N2 - We consider composite-composite testing problems for the expectation in the Gaussian sequence model where the null hypothesis corresponds to a closed convex subset C of R-d. We adopt a minimax point of view and our primary objective is to describe the smallest Euclidean distance between the null and alternative hypotheses such that there is a test with small total error probability. In particular, we focus on the dependence of this distance on the dimension d and variance 1/n giving rise to the minimax separation rate. In this paper we discuss lower and upper bounds on this rate for different smooth and non-smooth choices for C. KW - Minimax hypothesis testing KW - Gaussian sequence model KW - nonasymptotic minimax separation rate Y1 - 2018 U6 - https://doi.org/10.1214/18-EJS1472 SN - 1935-7524 VL - 12 IS - 2 SP - 3713 EP - 3735 PB - Institute of Mathematical Statistics CY - Cleveland ER - TY - JOUR A1 - Friedrich, Tobias A1 - Katzmann, Maximilian A1 - Krohmer, Anton T1 - Unbounded Discrepancy of Deterministic Random Walks on Grids JF - SIAM journal on discrete mathematics N2 - Random walks are frequently used in randomized algorithms. We study a derandomized variant of a random walk on graphs called the rotor-router model. In this model, instead of distributing tokens randomly, each vertex serves its neighbors in a fixed deterministic order. For most setups, both processes behave in a remarkably similar way: Starting with the same initial configuration, the number of tokens in the rotor-router model deviates only slightly from the expected number of tokens on the corresponding vertex in the random walk model. The maximal difference over all vertices and all times is called single vertex discrepancy. Cooper and Spencer [Combin. Probab. Comput., 15 (2006), pp. 815-822] showed that on Z(d), the single vertex discrepancy is only a constant c(d). Other authors also determined the precise value of c(d) for d = 1, 2. All of these results, however, assume that initially all tokens are only placed on one partition of the bipartite graph Z(d). We show that this assumption is crucial by proving that, otherwise, the single vertex discrepancy can become arbitrarily large. For all dimensions d >= 1 and arbitrary discrepancies l >= 0, we construct configurations that reach a discrepancy of at least l. KW - deterministic random walk KW - rotor-router model KW - single vertex discrepancy Y1 - 2018 U6 - https://doi.org/10.1137/17M1131088 SN - 0895-4801 SN - 1095-7146 VL - 32 IS - 4 SP - 2441 EP - 2452 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Bourne, D. P. A1 - Cushing, D. A1 - Liu, S. A1 - Münch, Florentin A1 - Peyerimhoff, Norbert T1 - Ollivier-Ricci idleness functions of graphs JF - SIAM Journal on Discrete Mathematics N2 - We study the Ollivier-Ricci curvature of graphs as a function of the chosen idleness. We show that this idleness function is concave and piecewise linear with at most three linear parts, and at most two linear parts in the case of a regular graph. We then apply our result to show that the idleness function of the Cartesian product of two regular graphs is completely determined by the idleness functions of the factors. KW - Ollivier-Ricci KW - idleness KW - optimal transport Y1 - 2018 U6 - https://doi.org/10.1137/17M1134469 SN - 0895-4801 SN - 1095-7146 VL - 32 IS - 2 SP - 1408 EP - 1424 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Gomez, Christophe A1 - Hartung, Niklas T1 - Stochastic and deterministic models for the metastatic emission process BT - Formalisms and Crosslinks JF - Cancer Systems Biology N2 - Although the detection of metastases radically changes prognosis of and treatment decisions for a cancer patient, clinically undetectable micrometastases hamper a consistent classification into localized or metastatic disease. This chapter discusses mathematical modeling efforts that could help to estimate the metastatic risk in such a situation. We focus on two approaches: (1) a stochastic framework describing metastatic emission events at random times, formalized via Poisson processes, and (2) a deterministic framework describing the micrometastatic state through a size-structured density function in a partial differential equation model. Three aspects are addressed in this chapter. First, a motivation for the Poisson process framework is presented and modeling hypotheses and mechanisms are introduced. Second, we extend the Poisson model to account for secondary metastatic emission. Third, we highlight an inherent crosslink between the stochastic and deterministic frameworks and discuss its implications. For increased accessibility the chapter is split into an informal presentation of the results using a minimum of mathematical formalism and a rigorous mathematical treatment for more theoretically interested readers. KW - Poisson process KW - Structured population equation KW - Metastasis KW - Mathematical modeling Y1 - 2018 SN - 978-1-4939-7493-1 SN - 978-1-4939-7492-4 U6 - https://doi.org/10.1007/978-1-4939-7493-1_10 SN - 1064-3745 SN - 1940-6029 VL - 1711 SP - 193 EP - 224 PB - Humana Press Inc. CY - Totowa ER - TY - JOUR A1 - Bär, Christian T1 - The curl operator on odd-dimensional manifolds JF - Journal of mathematical physics N2 - We study the spectral properties of curl, a linear differential operator of first order acting on differential forms of appropriate degree on an odd-dimensional closed oriented Riemannian manifold. In three dimensions, its eigenvalues are the electromagnetic oscillation frequencies in vacuum without external sources. In general, the spectrum consists of the eigenvalue 0 with infinite multiplicity and further real discrete eigenvalues of finite multiplicity. We compute the Weyl asymptotics and study the zeta-function. We give a sharp lower eigenvalue bound for positively curved manifolds and analyze the equality case. Finally, we compute the spectrum for flat tori, round spheres, and 3-dimensional spherical space forms. Published under license by AIP Publishing. Y1 - 2019 U6 - https://doi.org/10.1063/1.5082528 SN - 0022-2488 SN - 1089-7658 VL - 60 IS - 3 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Becker, Christian A1 - Benini, Marco A1 - Schenkel, Alexander A1 - Szabo, Richard J. T1 - Cheeger-Simons differential characters with compact support and Pontryagin duality JF - Communications in analysis and geometry N2 - By adapting the Cheeger-Simons approach to differential cohomology, we establish a notion of differential cohomology with compact support. We show that it is functorial with respect to open embeddings and that it fits into a natural diagram of exact sequences which compare it to compactly supported singular cohomology and differential forms with compact support, in full analogy to ordinary differential cohomology. We prove an excision theorem for differential cohomology using a suitable relative version. Furthermore, we use our model to give an independent proof of Pontryagin duality for differential cohomology recovering a result of [Harvey, Lawson, Zweck - Amer. J. Math. 125 (2003), 791]: On any oriented manifold, ordinary differential cohomology is isomorphic to the smooth Pontryagin dual of compactly supported differential cohomology. For manifolds of finite-type, a similar result is obtained interchanging ordinary with compactly supported differential cohomology. Y1 - 2019 U6 - https://doi.org/10.4310/CAG.2019.v27.n7.a2 SN - 1019-8385 SN - 1944-9992 VL - 27 IS - 7 SP - 1473 EP - 1522 PB - International Press of Boston CY - Somerville ER - TY - JOUR A1 - Reich, Sebastian T1 - Data assimilation BT - the Schrödinger perspective JF - Acta numerica N2 - Data assimilation addresses the general problem of how to combine model-based predictions with partial and noisy observations of the process in an optimal manner. This survey focuses on sequential data assimilation techniques using probabilistic particle-based algorithms. In addition to surveying recent developments for discrete- and continuous-time data assimilation, both in terms of mathematical foundations and algorithmic implementations, we also provide a unifying framework from the perspective of coupling of measures, and Schrödinger’s boundary value problem for stochastic processes in particular. Y1 - 2019 U6 - https://doi.org/10.1017/S0962492919000011 SN - 0962-4929 SN - 1474-0508 VL - 28 SP - 635 EP - 711 PB - Cambridge Univ. Press CY - New York ER - TY - JOUR A1 - Mera, Azal A1 - Shlapunov, Alexander A. A1 - Tarkhanov, Nikolai Nikolaevich T1 - Navier-Stokes Equations for Elliptic Complexes JF - Journal of Siberian Federal University. Mathematics & Physics N2 - We continue our study of invariant forms of the classical equations of mathematical physics, such as the Maxwell equations or the Lam´e system, on manifold with boundary. To this end we interpret them in terms of the de Rham complex at a certain step. On using the structure of the complex we get an insight to predict a degeneracy deeply encoded in the equations. In the present paper we develop an invariant approach to the classical Navier-Stokes equations. KW - Navier-Stokes equations KW - classical solution Y1 - 2019 U6 - https://doi.org/10.17516/1997-1397-2019-12-1-3-27 SN - 1997-1397 SN - 2313-6022 VL - 12 IS - 1 SP - 3 EP - 27 PB - Sibirskij Federalʹnyj Universitet CY - Krasnojarsk ER - TY - JOUR A1 - Nüsken, Nikolas A1 - Pavhotis, Grigorios A. T1 - Constructing Sampling Schemes via Coupling BT - Markov Semigroups and Optimal Transport JF - SIAM ASA journal on uncertainty quantification / Society for Industrial and Applied Mathematics ; American Statistical Association N2 - In this paper we develop a general framework for constructing and analyzing coupled Markov chain Monte Carlo samplers, allowing for both (possibly degenerate) diffusion and piecewise deterministic Markov processes. For many performance criteria of interest, including the asymptotic variance, the task of finding efficient couplings can be phrased in terms of problems related to optimal transport theory. We investigate general structural properties, proving a singularity theorem that has both geometric and probabilistic interpretations. Moreover, we show that those problems can often be solved approximately and support our findings with numerical experiments. For the particular objective of estimating the variance of a Bayesian posterior, our analysis suggests using novel techniques in the spirit of antithetic variates. Addressing the convergence to equilibrium of coupled processes we furthermore derive a modified Poincare inequality. KW - sampling KW - optimal transport KW - particle methods KW - Markov semigroups KW - MCMC Y1 - 2019 U6 - https://doi.org/10.1137/18M119896X SN - 2166-2525 VL - 7 IS - 1 SP - 324 EP - 382 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Katz-Samuels, Julian A1 - Blanchard, Gilles A1 - Scott, Clayton T1 - Decontamination of Mutual Contamination Models JF - Journal of machine learning research N2 - Many machine learning problems can be characterized by mutual contamination models. In these problems, one observes several random samples from different convex combinations of a set of unknown base distributions and the goal is to infer these base distributions. This paper considers the general setting where the base distributions are defined on arbitrary probability spaces. We examine three popular machine learning problems that arise in this general setting: multiclass classification with label noise, demixing of mixed membership models, and classification with partial labels. In each case, we give sufficient conditions for identifiability and present algorithms for the infinite and finite sample settings, with associated performance guarantees. KW - multiclass classification with label noise KW - classification with partial labels KW - mixed membership models KW - topic modeling KW - mutual contamination models Y1 - 2019 UR - http://arxiv.org/abs/1710.01167 SN - 1532-4435 VL - 20 PB - Microtome Publishing CY - Cambridge, Mass. ER - TY - JOUR A1 - Malass, Ihsane A1 - Tarkhanov, Nikolai Nikolaevich T1 - The de Rham Cohomology through Hilbert Space Methods JF - Journal of Siberian Federal University. Mathematics & physics N2 - We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler-Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer. KW - De Rham complex KW - cohomology KW - Hodge theory KW - Neumann problem Y1 - 2019 U6 - https://doi.org/10.17516/1997-1397-2019-12-4-455-465 SN - 1997-1397 SN - 2313-6022 VL - 12 IS - 4 SP - 455 EP - 465 PB - Sibirskij Federalʹnyj Universitet CY - Krasnoyarsk ER - TY - JOUR A1 - Dombrowsky, Charlotte A1 - Und, Myriam Fradon A1 - Roelly, Sylvie T1 - Packungen aus Kreisscheiben BT - Wie eine wahrscheinlichkeitstheoretische Sichtweise eine geometrische Analyse vervollständigen kann JF - Elemente der Mathematik N2 - Der englische Seefahrer Sir Walter Raleigh fragte sich einst, wie er in seinem Schiffsladeraum moeglichst viele Kanonenkugeln stapeln koennte. Johannes Kepler entwickelte daraufhin 1611 eine Vermutung ueber die optimale Anordnung der Kugeln. Diese Vermutung sollte sich als eine der haertesten mathematischen Nuesse der Geschichte erweisen. Selbst in der Ebene sind dichteste Packungen kongruenter Kreise eine Herausforderung. 1892 und 1910 veroeffentlichte Axel Thue (kritisierte) Beweise, dass die hexagonale Kreispackung optimal sei. Erst 1940 lieferte Laszlo Fejes Toth schliesslich einen wasserdichten Beweis fuer diese Tatsache. Eine Variante des Problems verlangt, Packungen mit endlich vielen kongruenten Kugeln zu finden, die eine gewisse quadratische Energie minimieren: Diese spannende geometrische Aufgabe wurde 1967 von Toth gestellt. Sie ist auch heute noch nicht vollstaendig gelaest. In diesem Beitrag schlagen die Autorinnen eine originelle wahrscheinlichkeitstheoretische Methode vor, um in der Ebene Näherungen der Lösung zu konstruieren. Y1 - 2019 U6 - https://doi.org/10.4171/EM/381 SN - 0013-6018 SN - 1420-8962 VL - 74 IS - 2 SP - 45 EP - 62 PB - EMS Publ. CY - Zürich ER - TY - JOUR A1 - Klein, Markus A1 - Rosenberger, Elke T1 - Tunneling for a class of difference operators BT - Complete asymptotics JF - Annales Henri Poincaré : a journal of theoretical and mathematical physics N2 - We analyze a general class of difference operators Hε=Tε+Vε on ℓ2((εZ)d), where Vε is a multi-well potential and ε is a small parameter. We derive full asymptotic expansions of the prefactor of the exponentially small eigenvalue splitting due to interactions between two “wells” (minima) of the potential energy, i.e., for the discrete tunneling effect. We treat both the case where there is a single minimal geodesic (with respect to the natural Finsler metric induced by the leading symbol h0(x,ξ) of Hε) connecting the two minima and the case where the minimal geodesics form an ℓ+1 dimensional manifold, ℓ≥1. These results on the tunneling problem are as sharp as the classical results for the Schrödinger operator in Helffer and Sjöstrand (Commun PDE 9:337–408, 1984). Technically, our approach is pseudo-differential and we adapt techniques from Helffer and Sjöstrand [Analyse semi-classique pour l’équation de Harper (avec application à l’équation de Schrödinger avec champ magnétique), Mémoires de la S.M.F., 2 series, tome 34, pp 1–113, 1988)] and Helffer and Parisse (Ann Inst Henri Poincaré 60(2):147–187, 1994) to our discrete setting. Y1 - 2018 U6 - https://doi.org/10.1007/s00023-018-0732-0 SN - 1424-0637 SN - 1424-0661 VL - 19 IS - 11 SP - 3511 EP - 3559 PB - Springer International Publishing CY - Cham ER - TY - JOUR A1 - Paycha, Sylvie T1 - When the market wins over research and higher education JF - Sustainable Futures for Higher Education : the Making of Knowledge Makers N2 - In this chapter, an overview of systematic eradication of basic science foci in European universities in the last two decades is given. This happens under the slogan of optimisation of the university education to the needs and demands of the society. It is pointed out that reliance on “market demands” brings with it long-term deficiencies in the maintenance of basic and advanced knowledge construction in societies necessary for long-term future technological advances. University policies that claim improvement of higher education towards more immediate efficiency may end up with the opposite effect of affecting its quality and long term expected positive impact on society. Y1 - 2018 SN - 978-3-319-96035-7 SN - 978-3-319-96034-0 U6 - https://doi.org/10.1007/978-3-319-96035-7_2 SN - 2364-6799 VL - 7 SP - 23 EP - 28 PB - Springer CY - Cham ER - TY - JOUR A1 - Bachoc, Francois A1 - Blanchard, Gilles A1 - Neuvial, Pierre T1 - On the post selection inference constant under restricted isometry properties JF - Electronic journal of statistics N2 - Uniformly valid confidence intervals post model selection in regression can be constructed based on Post-Selection Inference (PoSI) constants. PoSI constants are minimal for orthogonal design matrices, and can be upper bounded in function of the sparsity of the set of models under consideration, for generic design matrices. In order to improve on these generic sparse upper bounds, we consider design matrices satisfying a Restricted Isometry Property (RIP) condition. We provide a new upper bound on the PoSI constant in this setting. This upper bound is an explicit function of the RIP constant of the design matrix, thereby giving an interpolation between the orthogonal setting and the generic sparse setting. We show that this upper bound is asymptotically optimal in many settings by constructing a matching lower bound. KW - Inference post model-selection KW - confidence intervals KW - PoSI constants KW - linear regression KW - high-dimensional inference KW - sparsity KW - restricted isometry property Y1 - 2018 U6 - https://doi.org/10.1214/18-EJS1490 SN - 1935-7524 VL - 12 IS - 2 SP - 3736 EP - 3757 PB - Institute of Mathematical Statistics CY - Cleveland ER - TY - JOUR A1 - Conforti, Giovanni A1 - Pra, Paolo Dai A1 - Roelly, Sylvie T1 - Reciprocal Class of Jump Processes JF - Journal of theoretical probability N2 - Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of compound Poisson processes whose jumps belong to a finite set . We propose a characterization of the reciprocal class as the unique set of probability measures on which a family of time and space transformations induces the same density, expressed in terms of the reciprocal invariants. The geometry of plays a crucial role in the design of the transformations, and we use tools from discrete geometry to obtain an optimal characterization. We deduce explicit conditions for two Markov jump processes to belong to the same class. Finally, we provide a natural interpretation of the invariants as short-time asymptotics for the probability that the reference process makes a cycle around its current state. KW - Reciprocal processes KW - Stochastic bridges KW - Jump processes KW - Compound Poisson processes Y1 - 2015 U6 - https://doi.org/10.1007/s10959-015-0655-3 SN - 0894-9840 SN - 1572-9230 VL - 30 SP - 551 EP - 580 PB - Springer CY - New York ER - TY - JOUR A1 - Nüsken, Nikolas A1 - Reich, Sebastian A1 - Rozdeba, Paul J. T1 - State and parameter estimation from observed signal increments JF - Entropy : an international and interdisciplinary journal of entropy and information studies N2 - The success of the ensemble Kalman filter has triggered a strong interest in expanding its scope beyond classical state estimation problems. In this paper, we focus on continuous-time data assimilation where the model and measurement errors are correlated and both states and parameters need to be identified. Such scenarios arise from noisy and partial observations of Lagrangian particles which move under a stochastic velocity field involving unknown parameters. We take an appropriate class of McKean-Vlasov equations as the starting point to derive ensemble Kalman-Bucy filter algorithms for combined state and parameter estimation. We demonstrate their performance through a series of increasingly complex multi-scale model systems. KW - parameter estimation KW - continuous-time data assimilation KW - ensemble Kalman filter KW - correlated noise KW - multi-scale diffusion processes Y1 - 2019 U6 - https://doi.org/10.3390/e21050505 SN - 1099-4300 VL - 21 IS - 5 PB - MDPI CY - Basel ER - TY - JOUR A1 - Mera, Azal Jaafar Musa A1 - Tarchanov, Nikolaj Nikolaevič T1 - The Neumann Problem after Spencer JF - Žurnal Sibirskogo Federalʹnogo Universiteta = Journal of Siberian Federal University : Matematika i fizika = Mathematics & physics N2 - When trying to extend the Hodge theory for elliptic complexes on compact closed manifolds to the case of compact manifolds with boundary one is led to a boundary value problem for the Laplacian of the complex which is usually referred to as Neumann problem. We study the Neumann problem for a larger class of sequences of differential operators on a compact manifold with boundary. These are sequences of small curvature, i.e., bearing the property that the composition of any two neighbouring operators has order less than two. KW - elliptic complexes KW - manifolds with boundary KW - Hodge theory KW - Neumann problem Y1 - 2017 U6 - https://doi.org/10.17516/1997-1397-2017-10-4-474-493 SN - 1997-1397 SN - 2313-6022 VL - 10 SP - 474 EP - 493 PB - Sibirskij Federalʹnyj Universitet CY - Krasnojarsk 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 - Vasiliev, Sergey B. A1 - Tarchanov, Nikolaj Nikolaevič T1 - Construction of series of perfect lattices by layer superposition JF - Journal of Siberian Federal University : Mathematics & physics JF - Žurnal Sibirskogo Federalʹnogo Universiteta = Journal of Siberian Federal University : Serija Matematika i fizika = Mathematics & physics N2 - We construct a new series of perfect lattices in n dimensions by the layer superposition method of Delaunay-Barnes. KW - lattice packing and covering KW - polyhedra and polytopes KW - regular figures KW - division of spaces Y1 - 2017 U6 - https://doi.org/10.17516/1997-1397-2017-10-3-353-361 SN - 1997-1397 SN - 2313-6022 VL - 10 IS - 3 SP - 353 EP - 361 PB - Sibirskij Federalʹnyj Universitet CY - Krasnojarsk ER - TY - GEN A1 - Bandara, Menaka Lashitha A1 - Rosén, Andreas T1 - Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of local boundary conditions T2 - Postprints der Universität Potsdam Mathematisch-Naturwissenschaftliche Reihe N2 - On a smooth complete Riemannian spin manifold with smooth compact boundary, we demonstrate that Atiyah-Singer Dirac operator in depends Riesz continuously on perturbations of local boundary conditions The Lipschitz bound for the map depends on Lipschitz smoothness and ellipticity of and bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius away from a compact neighbourhood of the boundary. More generally, we prove perturbation estimates for functional calculi of elliptic operators on manifolds with local boundary conditions. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 758 KW - boundary value problems KW - Dirac operator KW - functional calculus KW - real-variable harmonic analysis KW - Riesz continuity KW - spectral flow Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-434078 SN - 1866-8372 IS - 758 SP - 1253 EP - 1284 ER - TY - THES A1 - Angwenyi, David T1 - Time-continuous state and parameter estimation with application to hyperbolic SPDEs T1 - Zeitkontinuierliche Zustands- und Parameterschätzung bei Anwendung auf hyperbolische SPDEs N2 - Data assimilation has been an active area of research in recent years, owing to its wide utility. At the core of data assimilation are filtering, prediction, and smoothing procedures. Filtering entails incorporation of measurements' information into the model to gain more insight into a given state governed by a noisy state space model. Most natural laws are governed by time-continuous nonlinear models. For the most part, the knowledge available about a model is incomplete; and hence uncertainties are approximated by means of probabilities. Time-continuous filtering, therefore, holds promise for wider usefulness, for it offers a means of combining noisy measurements with imperfect model to provide more insight on a given state. The solution to time-continuous nonlinear Gaussian filtering problem is provided for by the Kushner-Stratonovich equation. Unfortunately, the Kushner-Stratonovich equation lacks a closed-form solution. Moreover, the numerical approximations based on Taylor expansion above third order are fraught with computational complications. For this reason, numerical methods based on Monte Carlo methods have been resorted to. Chief among these methods are sequential Monte-Carlo methods (or particle filters), for they allow for online assimilation of data. Particle filters are not without challenges: they suffer from particle degeneracy, sample impoverishment, and computational costs arising from resampling. The goal of this thesis is to:— i) Review the derivation of Kushner-Stratonovich equation from first principles and its extant numerical approximation methods, ii) Study the feedback particle filters as a way of avoiding resampling in particle filters, iii) Study joint state and parameter estimation in time-continuous settings, iv) Apply the notions studied to linear hyperbolic stochastic differential equations. The interconnection between Itô integrals and stochastic partial differential equations and those of Stratonovich is introduced in anticipation of feedback particle filters. With these ideas and motivated by the variants of ensemble Kalman-Bucy filters founded on the structure of the innovation process, a feedback particle filter with randomly perturbed innovation is proposed. Moreover, feedback particle filters based on coupling of prediction and analysis measures are proposed. They register a better performance than the bootstrap particle filter at lower ensemble sizes. We study joint state and parameter estimation, both by means of extended state spaces and by use of dual filters. Feedback particle filters seem to perform well in both cases. Finally, we apply joint state and parameter estimation in the advection and wave equation, whose velocity is spatially varying. Two methods are employed: Metropolis Hastings with filter likelihood and a dual filter comprising of Kalman-Bucy filter and ensemble Kalman-Bucy filter. The former performs better than the latter. N2 - Die Datenassimilation war in den letzten Jahren aufgrund ihres breiten Nutzens ein aktives Forschungsgebiet. Im Zentrum der Datenassimilation stehen Filter-, Vorhersage- und Glättungsverfahren. Die Filterung beinhaltet die Einbeziehung von Messinformationen in das Modell, um einen besseren Einblick in einen gegebenen Zustand zu erhalten, der durch ein verrauschtes Zustandsraummodell gesteuert wird. Die meisten Naturgesetze werden von zeitkontinuierlichen nichtlinearen Modellen bestimmt. Das verfügbare Wissen über ein Modell ist größtenteils unvollständig; und daher werden Unsicherheiten mittels Wahrscheinlichkeiten angenähert. Die zeitkontinuierliche Filterung verspricht daher eine größere Nützlichkeit, denn sie bietet die Möglichkeit, verrauschte Messungen mit einem unvollkommenen Modell zu kombinieren, um mehr Einblick in einen bestimmten Zustand zu erhalten. Das Problem der zeitkontinuierlichen nichtlinearen Gaußschen Filterung wird durch die Kushner-Stratonovich-Gleichung gelöst. Leider fehlt der Kushner-Stratonovich-Gleichung eine geschlossene Lösung. Darüber hinaus sind die numerischen Näherungen, die auf der Taylor-Erweiterung über der dritten Ordnung basieren, mit rechnerischen Komplikationen behaftet. Aus diesem Grund wurde auf numerische Methoden zurückgegriffen, die auf Monte-Carlo-Methoden basieren. Die wichtigsten dieser Methoden sind sequentielle Monte-Carlo-Methoden (oder Partikelfilter), da sie die Online-Assimilation von Daten ermöglichen. Partikelfilter sind nicht unproblematisch: Sie leiden unter Partikelentartung, Probenverarmung und Rechenkosten, die sich aus der Neuabtastung ergeben. Das Ziel dieser Arbeit ist es, i) die Ableitung der Kushner-Stratonovich-Gleichung aus den ersten Prinzipien und ihre vorhandenen numerischen Approximationsmethoden zu überprüfen, ii) die Rückkopplungs-Partikelfilter zu untersuchen, um eine Neuabtastung in Partikelfiltern zu vermeiden, iii) Studieren Sie die Zustands- und Parameterschätzung in zeitkontinuierlichen Einstellungen, iv) Wenden Sie die untersuchten Begriffe auf lineare hyperbolische stochastische Differentialgleichungen an. Die Verbindung zwischen Itô Integralen und stochastischen partiellen Differentialgleichungen und denen von Stratonovich wird in Erwartung von Rückkopplungs-Partikelfiltern eingeführt. Mit diesen Ideen und motiviert durch die Varianten von Kalman-Bucy-Filtern, die auf der Struktur des Innovationsprozesses gegründet, wird ein Feedback-Partikelfilter mit zufällig gestörter Innovation vorgeschlagen. Darüber hinaus werden Rückkopplungspartikelfilter basierend auf der Kopplung von Vorhersage- und Analysemaßnahmen vorgeschlagen. Diese Feedback-Partikelfiltern haben eine bessere Leistung als der Bootstrap-Partikelfilter bei niedrigeren Ensemble-Größen. Wir untersuchen gemeinsame Zustands- und Parameterschätzungen, sowohl durch erweiterte Zustandsräume als auch durch Verwendung von Doppelfiltern. Rückkopplungs-Partikelfilter scheinen in beiden Fällen gut zu funktionieren. Schließlich wenden wir eine gemeinsame Zustands- und Parameterschätzung in der Advektions-und Wellengleichung an, deren Geschwindigkeit räumlich variiert. Es werden zwei Verfahren verwendet: Metropolis-Hastings mit Filterwahrscheinlichkeit und ein Doppelfilter bestehend aus Kalman-Bucy-Filter und Ensemble-Kalman-Bucy-Filter. Ersteres schneidet besser ab als letzteres. KW - state estimation KW - filtering KW - parameter estimation KW - Zustandsschätzung KW - Filterung KW - Parameter Schätzung Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-436542 ER - TY - GEN A1 - Molahajloo, Shahla A1 - Pirhayati, Mohammad T1 - Traces of pseudo-differential operators on compact and Hausdorff groups T2 - Postprints der Universität Potsdam : Mathematisch Naturwissenschaftliche Reihe N2 - We give a characterization of and a trace formula for trace class pseudo-differential operators on compact Hausdorff groups. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 893 KW - pseudo-differential operators KW - compact groups Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-436371 SN - 1866-8372 IS - 893 SP - 361 EP - 369 ER - TY - JOUR A1 - Dereudre, David A1 - Mazzonetto, Sara A1 - Roelly, Sylvie T1 - Exact simulation of Brownian diffusions with drift admitting jumps JF - SIAM journal on scientific computing N2 - In this paper, using an algorithm based on the retrospective rejection sampling scheme introduced in [A. Beskos, O. Papaspiliopoulos, and G. O. Roberts,Methodol. Comput. Appl. Probab., 10 (2008), pp. 85-104] and [P. Etore and M. Martinez, ESAIM Probab.Stat., 18 (2014), pp. 686-702], we propose an exact simulation of a Brownian di ff usion whose drift admits several jumps. We treat explicitly and extensively the case of two jumps, providing numerical simulations. Our main contribution is to manage the technical di ffi culty due to the presence of t w o jumps thanks to a new explicit expression of the transition density of the skew Brownian motion with two semipermeable barriers and a constant drift. KW - exact simulation methods KW - skew Brownian motion KW - skew diffusions KW - Brownian motion with discontinuous drift Y1 - 2017 U6 - https://doi.org/10.1137/16M107699X SN - 1064-8275 SN - 1095-7197 VL - 39 IS - 3 SP - A711 EP - A740 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - GEN A1 - Devchand, Chandrashekar A1 - Nuyts, Jean A1 - Weingart, Gregor T1 - Matryoshka of special democratic forms T2 - Postprints der Universität Potsdam : Mathematisch Naturwissenschaftliche Reihe N2 - Special p-forms are forms which have components fµ1…µp equal to +1, -1 or 0 in some orthonormal basis. A p-form ϕ ∈ � pRd is called democratic if the set of nonzero components {ϕμ1...μp} is symmetric under the transitive action of a subgroup of O(d,Z) on the indices {1, . . . , d}. Knowledge of these symmetry groups allows us to define mappings of special democratic p-forms in d dimensions to special democratic P-forms in D dimensions for successively higher P = p and D = d. In particular, we display a remarkable nested structure of special forms including a U(3)-invariant 2-form in six dimensions, a G2-invariant 3-form in seven dimensions, a Spin(7)-invariant 4-form in eight dimensions and a special democratic 6-form O in ten dimensions. The latter has the remarkable property that its contraction with one of five distinct bivectors, yields, in the orthogonal eight dimensions, the Spin(7)-invariant 4-form. We discuss various properties of this ten dimensional form. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 841 KW - commutator subgroup KW - transitive action KW - cycle decomposition KW - democratic form KW - special holonomy Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-429002 SN - 1866-8372 IS - 841 SP - 545 EP - 562 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 - Chauhan, Ankit A1 - Friedrich, Tobias A1 - Rothenberger, Ralf T1 - Greed is good for deterministic scale-free networks JF - Algorithmica : an international journal in computer science N2 - Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. Therefore, Brach et al. (27th symposium on discrete algorithms (SODA), pp 1306-1325, 2016) introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both properties and exploit them to design faster algorithms for a number of classical graph problems. We complement their work by showing that some well-studied random graph models exhibit both of the mentioned PLB properties. PLB-U and PLB-N hold with high probability for Chung-Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability or almost surely for those random graph classes. In the second part we study three classical NP-hard optimization problems on PLB networks. It is known that on general graphs with maximum degree Delta, a greedy algorithm, which chooses nodes in the order of their degree, only achieves a Omega (ln Delta)-approximation forMinimum Vertex Cover and Minimum Dominating Set, and a Omega(Delta)-approximation forMaximum Independent Set. We prove that the PLB-U property with beta>2 suffices for the greedy approach to achieve a constant-factor approximation for all three problems. We also show that these problems are APX-hard even if PLB-U, PLB-N, and an additional power-law lower bound on the degree distribution hold. Hence, a PTAS cannot be expected unless P = NP. Furthermore, we prove that all three problems are in MAX SNP if the PLB-U property holds. KW - random graphs KW - deterministic properties KW - power-law KW - approximation KW - APX-hardness Y1 - 2020 U6 - https://doi.org/10.1007/s00453-020-00729-z SN - 0178-4617 SN - 1432-0541 VL - 82 IS - 11 SP - 3338 EP - 3389 PB - Springer CY - New York ER - TY - JOUR A1 - Klein, Markus A1 - Rosenberger, Elke T1 - The tunneling effect for Schrödinger operators on a vector bundle JF - Analysis and mathematical physics N2 - In the semiclassical limit (h) over bar -> 0, we analyze a class of self-adjoint Schrodinger operators H-(h) over bar = (h) over bar L-2 + (h) over barW + V center dot id(E) acting on sections of a vector bundle E over an oriented Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has non-degenerate minima at a finite number of points m(1),... m(r) is an element of M, called potential wells. Using quasimodes of WKB-type near m(j) for eigenfunctions associated with the low lying eigenvalues of H-(h) over bar, we analyze the tunneling effect, i.e. the splitting between low lying eigenvalues, which e.g. arises in certain symmetric configurations. Technically, we treat the coupling between different potential wells by an interaction matrix and we consider the case of a single minimal geodesic (with respect to the associated Agmon metric) connecting two potential wells and the case of a submanifold of minimal geodesics of dimension l + 1. This dimension l determines the polynomial prefactor for exponentially small eigenvalue splitting. KW - Laplace-type operator KW - Vector bundle KW - WKB-expansion KW - Quasimodes KW - Tunneling KW - Spectral gap KW - Complete asymptotics Y1 - 2021 U6 - https://doi.org/10.1007/s13324-021-00485-5 SN - 1664-2368 SN - 1664-235X VL - 11 IS - 2 PB - Springer International Publishing AG CY - Cham (ZG) ER - TY - JOUR A1 - Schlosser, Rainer T1 - Heuristic mean-variance optimization in Markov decision processes using state-dependent risk aversion JF - IMA journal of management mathematics / Institute of Mathematics and Its Applications N2 - In dynamic decision problems, it is challenging to find the right balance between maximizing expected rewards and minimizing risks. In this paper, we consider NP-hard mean-variance (MV) optimization problems in Markov decision processes with a finite time horizon. We present a heuristic approach to solve MV problems, which is based on state-dependent risk aversion and efficient dynamic programming techniques. Our approach can also be applied to mean-semivariance (MSV) problems, which particularly focus on the downside risk. We demonstrate the applicability and the effectiveness of our heuristic for dynamic pricing applications. Using reproducible examples, we show that our approach outperforms existing state-of-the-art benchmark models for MV and MSV problems while also providing competitive runtimes. Further, compared to models based on constant risk levels, we find that state-dependent risk aversion allows to more effectively intervene in case sales processes deviate from their planned paths. Our concepts are domain independent, easy to implement and of low computational complexity. KW - risk aversion KW - mean-variance optimization KW - Markov decision process; KW - dynamic programming KW - dynamic pricing KW - heuristics Y1 - 2021 U6 - https://doi.org/10.1093/imaman/dpab009 SN - 1471-678X SN - 1471-6798 VL - 33 IS - 2 SP - 181 EP - 199 PB - Oxford Univ. Press CY - Oxford 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 - THES A1 - Busaman, Saofee T1 - Hyperequational theory for partial algebras T1 - Hyperequationale Theorie für partielle Algebren N2 - Our work goes in two directions. At first we want to transfer definitions, concepts and results of the theory of hyperidentities and solid varieties from the total to the partial case. (1) We prove that the operators chi^A_RNF and chi^E_RNF are only monotone and additive and we show that the sets of all fixed points of these operators are characterized only by three instead of four equivalent conditions for the case of closure operators. (2) We prove that V is n − SF-solid iff clone^SF V is free with respect to itself, freely generated by the independent set {[fi(x_1, . . . , x_n)]Id^SF_n V | i \in I}. (3) We prove that if V is n-fluid and ~V |P(V ) =~V −iso |P(V ) then V is kunsolid for k >= n (where P(V ) is the set of all V -proper hypersubstitutions of type \tau ). (4) We prove that a strong M-hyperquasi-equational theory is characterized by four equivalent conditions. The second direction of our work is to follow ideas which are typical for the partial case. (1) We characterize all minimal partial clones which are strongly solidifyable. (2)We define the operator Chi^A_Ph where Ph is a monoid of regular partial hypersubstitutions.Using this concept, we define the concept of a Phyp_R(\tau )-solid strong regular variety of partial algebras and we prove that a PHyp_R(\tau )-solid strong regular variety satisfies four equivalent conditions. KW - partial algebras KW - hyperequational theory Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-12048 ER -