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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.
Data assimilation
(2019)
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.
For a finite measure space X, we characterize strongly continuous Markov lattice semigroups on Lp(X) by showing that their generator A acts as a derivation on the dense subspace D(A)L(X). We then use this to characterize Koopman semigroups on Lp(X) if X is a standard probability space. In addition, we show that every measurable and measure-preserving flow on a standard probability space is isomorphic to a continuous flow on a compact Borel probability space.
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.
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.
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.
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.
We provide explicit examples of positive and power-bounded operators on c(0) and l(infinity) which are mean ergodic but not weakly almost periodic. As a consequence we prove that a countably order complete Banach lattice on which every positive and power-bounded mean ergodic operator is weakly almost periodic is necessarily a KB-space. This answers several open questions from the literature. Finally, we prove that if T is a positive mean ergodic operator with zero fixed space on an arbitrary Banach lattice, then so is every power of T .
Packungen aus Kreisscheiben
(2019)
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.
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.