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 - 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 - JOUR A1 - Hoegele, Michael A1 - Ruffino, Paulo T1 - Averaging along foliated Levy diffusions JF - Nonlinear analysis : theory, methods & applications ; an international multidisciplinary journal N2 - This article studies the dynamics of the strong solution of a SDE driven by a discontinuous Levy process taking values in a smooth foliated manifold with compact leaves. It is assumed that it is foliated in the sense that its trajectories stay on the leaf of their initial value for all times almost surely. Under a generic ergodicity assumption for each leaf, we determine the effective behaviour of the system subject to a small smooth perturbation of order epsilon > 0, which acts transversal to the leaves. The main result states that, on average, the transversal component of the perturbed SDE converges uniformly to the solution of a deterministic ODE as e tends to zero. This transversal ODE is generated by the average of the perturbing vector field with respect to the invariant measures of the unperturbed system and varies with the transversal height of the leaves. We give upper bounds for the rates of convergence and illustrate these results for the random rotations on the circle. This article complements the results by Gonzales and Ruffino for SDEs of Stratonovich type to general Levy driven SDEs of Marcus type. KW - Averaging principle KW - Levy diffusions on manifolds KW - Foliated spaces KW - Marcus canonical equation KW - Stochastic Hamiltonian KW - Stochastic geometry KW - Perturbation theory Y1 - 2015 U6 - https://doi.org/10.1016/j.na.2014.09.006 SN - 0362-546X SN - 1873-5215 VL - 112 SP - 1 EP - 14 PB - Elsevier CY - Oxford ER - TY - JOUR A1 - Escribano, Bruno A1 - Akhmatskaya, Elena A1 - Reich, Sebastian A1 - Azpiroz, Jon M. T1 - Multiple-time-stepping generalized hybrid Monte Carlo methods JF - Journal of computational physics N2 - Performance of the generalized shadow hybrid Monte Carlo (GSHMC) method [1], which proved to be superior in sampling efficiency over its predecessors [2-4], molecular dynamics and hybrid Monte Carlo, can be further improved by combining it with multi-time-stepping (MTS) and mollification of slow forces. We demonstrate that the comparatively simple modifications of the method not only lead to better performance of GSHMC itself but also allow for beating the best performed methods, which use the similar force splitting schemes. In addition we show that the same ideas can be successfully applied to the conventional generalized hybrid Monte Carlo method (GHMC). The resulting methods, MTS-GHMC and MTS-GSHMC, provide accurate reproduction of thermodynamic and dynamical properties, exact temperature control during simulation and computational robustness and efficiency. MTS-GHMC uses a generalized momentum update to achieve weak stochastic stabilization to the molecular dynamics (MD) integrator. MTS-GSHMC adds the use of a shadow (modified) Hamiltonian to filter the MD trajectories in the HMC scheme. We introduce a new shadow Hamiltonian formulation adapted to force-splitting methods. The use of such Hamiltonians improves the acceptance rate of trajectories and has a strong impact on the sampling efficiency of the method. Both methods were implemented in the open-source MD package ProtoMol and were tested on a water and a protein systems. Results were compared to those obtained using a Langevin Molly (LM) method [5] on the same systems. The test results demonstrate the superiority of the new methods over LM in terms of stability, accuracy and sampling efficiency. This suggests that putting the MTS approach in the framework of hybrid Monte Carlo and using the natural stochasticity offered by the generalized hybrid Monte Carlo lead to improving stability of MTS and allow for achieving larger step sizes in the simulation of complex systems. KW - Force splitting KW - Mollification KW - Generalized hybrid Monte Carlo KW - Molecular dynamics KW - Modified Hamiltonians Y1 - 2015 U6 - https://doi.org/10.1016/j.jcp.2014.08.052 SN - 0021-9991 SN - 1090-2716 VL - 280 SP - 1 EP - 20 PB - Elsevier CY - San Diego ER - TY - JOUR A1 - Graeter, Joachim A1 - Sperner, Robert P. T1 - On embedding left-ordered groups into division rings JF - Forum mathematicum KW - Left-ordered groups KW - division rings KW - embeddings KW - formal power series Y1 - 2015 U6 - https://doi.org/10.1515/forum-2012-0070 SN - 0933-7741 SN - 1435-5337 VL - 27 IS - 1 SP - 485 EP - 518 PB - De Gruyter CY - Berlin ER - TY - JOUR A1 - Kroencke, Klaus T1 - On the stability of Einstein manifolds JF - Annals of global analysis and geometry N2 - Certain curvature conditions for the stability of Einstein manifolds with respect to the Einstein-Hilbert action are given. These conditions are given in terms of quantities involving the Weyl tensor and the Bochner tensor. In dimension six, a stability criterion involving the Euler characteristic is given. KW - Einstein-Hilbert action KW - variational stability KW - Eigenvalue problem KW - Weyl tensor Y1 - 2015 U6 - https://doi.org/10.1007/s10455-014-9436-y SN - 0232-704X SN - 1572-9060 VL - 47 IS - 1 SP - 81 EP - 98 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Aizinger, Vadym A1 - Korn, Peter A1 - Giorgetta, Marco A1 - Reich, Sebastian T1 - Large-scale turbulence modelling via alpha-regularisation for atmospheric simulations JF - Journal of turbulence N2 - We study the possibility of obtaining a computational turbulence model by means of non-dissipative regularisation of the compressible atmospheric equations for climate-type applications. We use an -regularisation (Lagrangian averaging) of the atmospheric equations. For the hydrostatic and compressible atmospheric equations discretised using a finite volume method on unstructured grids, deterministic and non-deterministic numerical experiments are conducted to compare the individual solutions and the statistics of the regularised equations to those of the original model. The impact of the regularisation parameter is investigated. Our results confirm the principal compatibility of -regularisation with atmospheric dynamics and encourage further investigations within atmospheric model including complex physical parametrisations. KW - hydrostatic atmosphere KW - non-dissipative regularisations KW - Lagrangian-averaged equations Y1 - 2015 U6 - https://doi.org/10.1080/14685248.2014.991443 SN - 1468-5248 VL - 16 IS - 4 SP - 367 EP - 391 PB - Routledge, Taylor & Francis Group CY - Abingdon ER - TY - JOUR A1 - Bagderina, Yulia Yu. A1 - Tarkhanov, Nikolai Nikolaevich T1 - Solution of the equivalence problem for the third Painleve equation JF - Journal of mathematical physics N2 - We find necessary conditions for a second order ordinary differential equation to be equivalent to the Painleve III equation under a general point transformation. Their sufficiency is established by reduction to known results for the equations of the form y ' = f (x, y). We consider separately the generic case and the case of reducibility to an autonomous equation. The results are illustrated by the primary resonance equation. Y1 - 2015 U6 - https://doi.org/10.1063/1.4905383 SN - 0022-2488 SN - 1089-7658 VL - 56 IS - 1 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Bär, Christian A1 - Wafo, Roger Tagne T1 - Initial value problems for wave equations on manifolds JF - Mathematical physics, analysis and geometry : an international journal devoted to the theory and applications of analysis and geometry to physics N2 - We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out that certain spaces of finite energy solutions are independent of this choice and hence invariantly defined. We also show existence and uniqueness of solutions for the Goursat problem where one prescribes initial data on a characteristic partial Cauchy hypersurface. This extends classical results due to Hormander. KW - Wave equation KW - Globally hyperbolic Lorentz manifold KW - Cauchy problem KW - Goursat problem KW - Finite energy sections Y1 - 2015 U6 - https://doi.org/10.1007/s11040-015-9176-7 SN - 1385-0172 SN - 1572-9656 VL - 18 IS - 1 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Azzali, Sara A1 - Goette, Sebastian A1 - Schick, Thomas T1 - Large time limit and local L-2-index theorems for families JF - Journal of noncommutative geometry N2 - We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut-Lott type superconnections in the L-2-setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of the heat operator at large time implies refined L-2-index formulas. As applications, we prove a local L-2-index theorem for families of signature operators and an L-2-Bismut-Lott theorem, expressing the Becker-Gottlieb transfer of flat bundles in terms of Kamber-Tondeur classes. With slightly stronger regularity we obtain the respective refined versions: we construct L-2-eta forms and L-2-torsion forms as transgression forms. KW - Local index theory KW - eta forms KW - torsion forms KW - L-2-invariants Y1 - 2015 U6 - https://doi.org/10.4171/JNCG/203 SN - 1661-6952 SN - 1661-6960 VL - 9 IS - 2 SP - 621 EP - 664 PB - EMS Publ. CY - Zürich ER -