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Data Assimilation and Parameter Estimation for a Multiscale Stochastic System with Α-Stable Lévy Noise
Data assimilation and parameter estimation for a multiscale stochastic system with α-stable L´evy noise Yanjie Zhang1, Zhuan Cheng2, Xinyong Zhang3, Xiaoli Chen1, Jinqiao Duan2 and Xiaofan Li2 1Center for Mathematical Sciences & School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China 2Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA 3Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] and [email protected] . ∗This work was partly supported by the NSF grant 1620449, and NSFC grants 11531006, 11371367, and 11271290. Abstract. This work is about low dimensional reduction for a slow-fast data assimilation system with non-Gaussian α stable L´evy noise via stochastic averaging. − When the observations are only available for slow components, we show that the averaged, low dimensional filter approximates the original filter, by examining the corresponding Zakai stochastic partial differential equations. Furthermore, we demonstrate that the low dimensional slow system approximates the slow dynamics of the original system, by examining parameter estimation and most probable paths. Keywords: Multiscale systems, non-Gaussian L´evy noise, averaging principle, Zakai equation, parameter estimation, most probable paths arXiv:1801.02846v1 [math.DS] 9 Jan 2018 1. Introduction Data assimilation is a procedure to extract system state information with the help of observations [18]. The state evolution and the observations are usually under random fluctuations. The general idea is to gain the best estimate for the true system state, in terms of the probability distribution for the system state, given only some noisy observations of the system. -
Limit Theorems for Non-Stationary and Random
LIMIT THEOREMS FOR NON-STATIONARY AND RANDOM DYNAMICAL SYSTEMS by Yaofeng Su A dissertation submitted to the Department of Mathematics, College of Natural Sciences and Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Chair of Committee: Andrew T¨or¨ok Committee Member: Vaughn Climenhaga Committee Member: Gemunu Gunaratne Committee Member: Matthew Nicol University of Houston May 2020 Copyright 2020, Yaofeng Su ACKNOWLEDGMENTS This dissertation would not have been possible without the guidance and the help of several in- dividuals who in one way or another contributed and extended their valuable assistance in the preparation and completion of this study. First, my utmost gratitude to my advisor Dr. Andrew T¨or¨okfor the support and help he provided me throughout my graduate education; my commit- tee members, Dr. Vaughn Climenhaga, Dr. Gemunu Gunaratne, and Dr. Matthew Nicol for their participation and their thoughtful comments; all professors in the dynamical system research group of University of Houston, Dr. Vaughn Climenhaga, Dr. Alan Haynes, Dr. Matthew Nicol, Dr. William Ott, and Dr. Andrew T¨or¨okfor organizing the wonderful \Houston Summer School on Dynamical Systems", where I learned a lot about dynamical systems, ergodic theory, number theory, and quantum mechanics. iii ABSTRACT We study the limit behavior of the non-stationary/random chaotic dynamical systems and prove a strong statistical limit theorem: (vector-valued) almost sure invariance principle for the non- stationary dynamical systems and quenched (vector-valued) almost sure invariance principle for the random dynamical systems. It is a matching of the trajectories of the dynamical system with a Brownian motion in such a way that the error is negligible in comparison with the Birkhoff sum. -
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SYNCHRONIZATION PROPERTIES OF RANDOM PIECEWISE ISOMETRIES VICTOR KLEPTSYN AND ANTON GORODETSKI Abstract. We study the synchronization properties of the random double rotations on tori. We give a criterion that show when synchronization is present in the case of random double rotations on the circle and prove that it is always absent in dimensions two and higher. 1. Introduction The observation of a synchronization effect goes back at least to 17th century, when Huygens [Hu] discovered the synchronization of two linked pendulums. Since then, synchronization phenomena have been observed in numerous systems and settings, see [PRK] for a comprehensive survey of the subject. In the theory of dynamical systems, synchronization usually refers to random dynamical system trajectories of different initial points converging to each other under the applica- tion of a sequence of random transformations. A first such result is the famous Furstenberg’s Theorem [Fur1], stating that under some very mild assumptions, the angle (mod π) between the images of any two vectors under a long product of random matrices (exponentially) tends to zero. Projectivizing the dynamics, it is easy to see that this theorem in fact states that random trajectories of the quotient system on the projective space (exponentially) approach each other. For random dynamical systems on the circle, several results are known. Surely, in random projective dynamics there is a synchronization due to the simplest pos- sible case of Furstenberg’s Theorem. In 1984, this result was generalized to the setting of homeomorphisms (with some very mild and natural assumptions of min- imality of the action and the presence of a North-South map) in the work of a physicist V. -
A Model of Gene Expression Based on Random Dynamical Systems Reveals Modularity Properties of Gene Regulatory Networks†
A Model of Gene Expression Based on Random Dynamical Systems Reveals Modularity Properties of Gene Regulatory Networks† Fernando Antoneli1,4,*, Renata C. Ferreira3, Marcelo R. S. Briones2,4 1 Departmento de Informática em Saúde, Escola Paulista de Medicina (EPM), Universidade Federal de São Paulo (UNIFESP), SP, Brasil 2 Departmento de Microbiologia, Imunologia e Parasitologia, Escola Paulista de Medicina (EPM), Universidade Federal de São Paulo (UNIFESP), SP, Brasil 3 College of Medicine, Pennsylvania State University (Hershey), PA, USA 4 Laboratório de Genômica Evolutiva e Biocomplexidade, EPM, UNIFESP, Ed. Pesquisas II, Rua Pedro de Toledo 669, CEP 04039-032, São Paulo, Brasil Abstract. Here we propose a new approach to modeling gene expression based on the theory of random dynamical systems (RDS) that provides a general coupling prescription between the nodes of any given regulatory network given the dynamics of each node is modeled by a RDS. The main virtues of this approach are the following: (i) it provides a natural way to obtain arbitrarily large networks by coupling together simple basic pieces, thus revealing the modularity of regulatory networks; (ii) the assumptions about the stochastic processes used in the modeling are fairly general, in the sense that the only requirement is stationarity; (iii) there is a well developed mathematical theory, which is a blend of smooth dynamical systems theory, ergodic theory and stochastic analysis that allows one to extract relevant dynamical and statistical information without solving -
«Algebraic and Geometric Methods of Analysis»
International scientific conference «Algebraic and geometric methods of analysis» Book of abstracts May 31 - June 5, 2017 Odessa Ukraine http://imath.kiev.ua/~topology/conf/agma2017/ LIST OF TOPICS • Algebraic methods in geometry • Differential geometry in the large • Geometry and topology of differentiable manifolds • General and algebraic topology • Dynamical systems and their applications • Geometric problems in mathematical analysis • Geometric and topological methods in natural sciences • History and methodology of teaching in mathematics ORGANIZERS • The Ministry of Education and Science of Ukraine • Odesa National Academy of Food Technologies • The Institute of Mathematics of the National Academy of Sciences of Ukraine • Taras Shevchenko National University of Kyiv • The International Geometry Center PROGRAM COMMITTEE Chairman: Prishlyak A. Maksymenko S. Rahula M. (Kyiv, Ukraine) (Kyiv, Ukraine) (Tartu, Estonia) Balan V. Matsumoto K. Sabitov I. (Bucharest, Romania) (Yamagata, Japan) (Moscow, Russia) Banakh T. Mashkov O. Savchenko A. (Lviv, Ukraine) (Kyiv, Ukraine) (Kherson, Ukraine) Fedchenko Yu. Mykytyuk I. Sergeeva А. (Odesa, Ukraine) (Lviv, Ukraine) (Odesa, Ukraine) Fomenko A. Milka A. Strikha M. (Moscow, Russia) (Kharkiv, Ukraine) (Kyiv, Ukraine) Fomenko V. Mikesh J. Shvets V. (Taganrog, Russia) (Olomouc, Czech Republic) (Odesa, Ukraine) Glushkov A. Mormul P. Shelekhov A. (Odesa, Ukraine) (Warsaw, Poland) (Tver, Russia) Haddad М. Moskaliuk S. Shurygin V. (Wadi al-Nasara, Syria) (Wien, Austri) (Kazan, Russia) Herega A. Panzhenskiy V. Vlasenko I. (Odesa, Ukraine) (Penza, Russia) (Kyiv, Ukraine) Khruslov E. Pastur L. Zadorozhnyj V. (Kharkiv, Ukraine) (Kharkiv, Ukraine) (Odesa, Ukraine) Kirichenko V. Plachta L. Zarichnyi M. (Moscow, Russia) (Krakov, Poland) (Lviv, Ukraine) Kirillov V. Pokas S. Zelinskiy Y. (Odesa, Ukraine) (Odesa, Ukraine) (Kyiv, Ukraine) Konovenko N. -
The Mathematics of Critical Shifts in Ecological Networks with Alternative Stable State Theory, a Potential Framework for Early Warning Indicators
The Mathematics of Critical Shifts in Ecological Networks with Alternative Stable State Theory, A Potential Framework for Early Warning Indicators by Michael Kupperman A THESIS submitted to Oregon State University Honors College in partial fulfillment of the requirements for the degree of Honors Bachelors of Science in Mathematics (Honors Scholar) Honors Bachelors of Science in Biochemistry & Molecular Biology (Honors Scholar) Presented June 14, 2019 Commencement June 2020 AN ABSTRACT OF THE THESIS OF Michael Kupperman for the degree of Honors Bachelors of Science in Mathematics and Honors Bachelors of Science in Biochemistry & Molecular Biology presented on June 14, 2019. Title: The Mathematics of Critical Shifts in Ecological Networks with Alternative Stable State Theory, A Potential Framework for Early Warning Indicators Abstract approved: David Koslicki A long running problem in mathematical biology is the prediction of extinction events, a specialized case of the larger global stability problem found in differential equations and dynamical systems theory. A central technical question is how to introduce the randomness observed in real ecological systems not accounted for in deterministic models. This work introduces the SP-system as a new mathematical object in which ecological parameters are treated as sequences of random variables that attain values over intervals of random lengths of time. The SP-system characterization of ecological networks leads to two differ- ent novel approaches for the simulation and extinction prediction. The first approach uses a construct new to modeling literature to describe the probability of a parameter transition event into an extinction event. The second approach utilizes Markov chains for both sim- ulation and extinction prediction. -
CERTAIN FUNDAMENTAL CONGRUENCES on a REGULAR SEMIGROUP! by J
CERTAIN FUNDAMENTAL CONGRUENCES ON A REGULAR SEMIGROUP! by J. M. HOWIE and G. LALLEMENT (Received 21 June, 1965) In recent developments in the algebraic theory of semigroups attention has been focussing increasingly on the study of congruences, in particular on lattice-theoretic properties of the lattice of congruences. In most cases it has been found advantageous to impose some re- striction on the type of semigroup considered, such as regularity, commutativity, or the property of being an inverse semigroup, and one of the principal tools has been the consideration of special congruences. For example, the minimum group congruence on an inverse semigroup has been studied by Vagner [21] and Munn [13], the maximum idempotent-separating con- gruence on a regular or inverse semigroup by the authors separately [9, 10] and by Munn [14], and the minimum semilattice congruence on a general or commutative semigroup by Tamura and Kimura [19], Yamada [22], Clifford [3] and Petrich [15]. In this paper we study regular semigroups and our primary concern is with the minimum group congruence, the minimum band congruence and the minimum semilattice congruence, which we shall con- sistently denote by a, P and t] respectively. In § 1 we establish connections between /? and t\ on the one hand and the equivalence relations of Green [7] (see also Clifford and Preston [4, § 2.1]) on the other. If for any relation H on a semigroup S we denote by K* the congruence on S generated by H, then, in the usual notation, In § 2 we show that the intersection of a with jS is the smallest congruence p on S for which Sip is a UBG-semigroup, that is, a band of groups [4, p. -
EMBEDDING ANY COUNTABLE SEMIGROUP WITHOUT IDEMPOTENTS in a 2-GENERATED SIMPLE SEMIGROUP WITHOUT IDEMPOTENTS by KARL BYLEEN (Received 2 June, 1986)
EMBEDDING ANY COUNTABLE SEMIGROUP WITHOUT IDEMPOTENTS IN A 2-GENERATED SIMPLE SEMIGROUP WITHOUT IDEMPOTENTS by KARL BYLEEN (Received 2 June, 1986) Although the classes of regular simple semigroups and simple semigroups without idempotents are evidently at opposite ends of the spectrum of simple semigroups, their theories involve some interesting connections. Jones [5] has obtained analogues of the bicyclic semigroup for simple semigroups without idempotents. Megyesi and Polldk [7] have classified all combinatorial simple principal ideal semigroups on two generators, showing that all are homomorphic images of one such semigroup Po which has no idempotents. In an earlier paper [1] a construction designed to produce regular simple semigroups was used to show that any countable semigroup can be embedded in a 2-generated bisimple monoid. In this paper a modification of the earlier construction is employed to prove that any countable semigroup without idempotents can be embedded in a 2-generated simple semigroup without idempotents, and to produce certain 2-generated congruence-free semigroups. The reader is referred to the survey paper by Hall [3] and to Chapter 4 of Lyndon and Schupp [6] for discussions of related embedding theorems for semigroups, inverse semigroups, and groups. These references also contain extensive bibliographies. Clifford and Preston [2] or Howie [4] may be consulted for standard definitions and results from the theory of semigroups. The author thanks Peter R. Jones for many helpful discussions. 1. The construction. Let (S, •) be any semigroup and let A and B be nonempty sets which are disjoint from each other and from S. Let a:A x S—*A be a right action of S on A (i.e., a\>(s • t) = (aV>s)\>t for all a eA and s, teS, where a\>s denotes (a, s)a). -
Engel-M-2018-Phd-Thesis.Pdf
Imperial College London, Department of Mathematics Local phenomena in random dynamical systems: bifurcations, synchronisation, and quasi-stationary dynamics A thesis presented for the degree of Doctor of Philosophy of Imperial College London by Maximilian Engel Supervisors: Prof. Jeroen S. W. Lamb, Dr. Martin Rasmussen January 2018 Declaration I declare that this thesis is entirely my own work and that where any material could be construed as the work of others, it is fully cited and referenced, and/or with appropriate acknowledgement given. Maximilian Engel 2 Copyright The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. 3 Acknowledgements I would like to express my gratitude to the following people who have supported me in accomplishing the present work. First of all, I would like to thank my supervisors Jeroen Lamb and Martin Rasmussen for the numerous passionate discussions we have had throughout the past three years, for encouraging me in difficult times and being enthusiastic but still critical in times of apparent breakthroughs. I am extremely grateful for their guidance, support and trust in me as a PhD student. In particular, I have to emphasize their willingness to enable many inspiring research connections and visits to different parts of the world. -
Green's Relations and Dimension in Abstract Semi-Groups
University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 8-1964 Green's Relations and Dimension in Abstract Semi-groups George F. Hampton University of Tennessee - Knoxville Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Mathematics Commons Recommended Citation Hampton, George F., "Green's Relations and Dimension in Abstract Semi-groups. " PhD diss., University of Tennessee, 1964. https://trace.tennessee.edu/utk_graddiss/3235 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by George F. Hampton entitled "Green's Relations and Dimension in Abstract Semi-groups." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Mathematics. Don D. Miller, Major Professor We have read this dissertation and recommend its acceptance: Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) July 13, 1962 To the Graduate Council: I am submitting herewith a dissertation written by George Fo Hampton entitled "Green's Relations and Dimension in Abstract Semi groups.-" I recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosop�y, with a major in Mathematics. -
Stability and Random Attractors for a Reaction-Diffusion Equation with Multiplicative Noise
DISCRETE AND CONTINUOUS Website: http://AIMsciences.org DYNAMICAL SYSTEMS Volume 6, Number 4, October 2000 pp. 875{892 STABILITY AND RANDOM ATTRACTORS FOR A REACTION-DIFFUSION EQUATION WITH MULTIPLICATIVE NOISE Tomas¶ Caraballo & Jose¶ A. Langa Dpto. Ecuaciones Diferenciales y An¶alisisNum¶erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain. James C. Robinson Mathematics Institute, University of Warwick Coventry CV4 7AL, U.K. Abstract. We study the asymptotic behaviour of a reaction-di®usion equation, and prove that the addition of multiplicative white noise (in the sense of It^o)stabilizes the stationary solution x ´ 0. We show in addition that this stochastic equation has a ¯nite-dimensional random attractor, and from our results conjecture a possible bifurcation scenario. 1. Introduction. The study of the asymptotic behaviour of evolution equations is one of the most important problems in the ¯eld of di®erential equations, as the vast literature on the subject shows. Perhaps the ¯rst step in this programme is to investigate the stability of the stationary solutions, and this is at present a well-developed branch of the theory of stochastic ordinary and partial di®erential equations (see, among others, Has'minskii [20], Mao [23], [24], Caraballo and Liu [5]). Such a stability analysis is essentially a local study, as we obtain information on the dynamics only around these points. A more complete analysis of the quali- tative properties of the system as a whole requires information relating not only to the stationary points but also, for example, to the stability or instability of the as- sociated invariant manifolds of these points. -
On the Computation of Invariant Measures in Random Dynamical Systems∗
On the computation of invariant measures in random dynamical systems¤ Peter Imkeller Peter Kloeden Institut f¨urMathematik Fachbereich Mathematik Humboldt-Universit¨atzu Berlin J-W von Goethe-Universit¨at Unter den Linden 6 Postfach 11 19 32 10099 Berlin 60054 Frankfurt am Main Germany Germany May 27, 2003 Abstract Invariant measures of dynamical systems generated e. g. by difference equa- tions can be computed by discretizing the originally continuum state space, and replacing the action of the generator by the transition mechanism of a Markov chain. In fact they are approximated by stationary vectors of these Markov chains. Here we extend this well known approximation result and the underlying algorithm to the setting of random dynamical systems, i.e. dynamical systems on the skew product of a probability space carrying the underlying stationary stochasticity and the state space, a particular non-autonomous framework. The systems are generated by difference equations driven by stationary random pro- cesses modelled on a metric dynamical system. The approximation algorithm involves spatial discretizations and the definition of appropriate random Markov chains with stationary vectors converging to the random invariant measure of the system. 2000 AMS subject classifications: 60 G, 60 H 25, 60 J 10, 65 C 30, 65 Q 05, 37 H 10, 37 M 25. Key words and phrases: dynamical system; random dynamical system; approx- imation; invariant measure; Markov chain; random Markov chain; interval matrix; spatial discretization. ¤This work was partially supported by the DFG research center ‘Mathematics for key technologies’ (FZT 86) in Berlin. 1 1 Introduction Computational methods are a basic tool in the investigation of dynamical systems, both to explore what may happen and to approximate specific dynamical features such as limit cycles or attractors and, more generally, invariant measures; see, e.g., [5, 20, 23].