scholarly journals ‘Intractable and unsolved’: some thoughts on statistical data assimilation with uncertain static parameters

2013 ◽  
Vol 371 (1991) ◽  
pp. 20120297 ◽  
Author(s):  
Jonathan Rougier
Keyword(s):  
Numerical Methods ◽  
Data Assimilation ◽  
State Space ◽  
Stochastic Models ◽  
Environmental Science ◽  
Statistical Data ◽  
Initial Conditions ◽  
The State ◽  
Likelihood Functions ◽  
Sensitive Dependence

If you seem to be able to do data assimilation with uncertain static parameters then you are probably not working in environmental science. In this field, applications are often characterized by sensitive dependence on initial conditions and attracting sets in the state-space, which, taken together, can be a major challenge to numerical methods, leading to very peaky likelihood functions. Inherently stochastic models and uncertain static parameters increase the challenge.

scholarly journals Likelihood functions for state space models with diffuse initial conditions

2010 ◽  
Vol 31 (6) ◽  
pp. 407-414 ◽  
Author(s):  
Marc K. Francke ◽  
Siem Jan Koopman ◽  
Aart F. De Vos
Keyword(s):  
State Space ◽  
Initial Conditions ◽  
State Space Models ◽  
Likelihood Functions

Languages, equicontinuity and attractors in cellular automata

1997 ◽  
Vol 17 (2) ◽  
pp. 417-433 ◽  
Author(s):  
PETR KŮRKA
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
State Space ◽  
Finite Type ◽  
Measure Zero ◽  
Initial Conditions ◽  
The State ◽  
Subshift Of Finite Type ◽  
Topologically Transitive ◽  
The Third

We consider three related classifications of cellular automata: the first is based on the complexity of languages generated by clopen partitions of the state space, i.e. on the complexity of the factor subshifts; the second is based on the concept of equicontinuity and it is a modification of the classification introduced by Gilman [9]. The third one is based on the concept of attractors and it refines the classification introduced by Hurley [16]. We show relations between these classifications and give examples of cellular automata in the intersection classes. In particular, we show that every positively expansive cellular automaton is conjugate to a one-sided subshift of finite type and that every topologically transitive cellular automaton is sensitive to initial conditions. We also construct a cellular automaton with minimal quasi-attractor, whose basin has measure zero, answering a question raised in Hurley [16].

On Dynamic Tooth Load and Stability of a Spur-Gear System Using the State-Space Approach

1985 ◽  
Vol 107 (1) ◽  
pp. 54-60 ◽  
Author(s):  
A. S. Kumar ◽  
T. S. Sankar ◽  
M. O. M. Osman
Keyword(s):  
State Space ◽  
Dynamic Load ◽  
Initial Conditions ◽  
Gear Tooth ◽  
Spur Gear ◽  
The State ◽  
Gear System ◽  
Computational Time ◽  
Steady State Condition ◽  
The Stability

In this study, a new approach using the state-space method is presented for the dynamic load analysis of spur gear systems. This approach gives the dynamic load on gear tooth in mesh as well as information on the stability of the gear system. Also a procedure is given for the selection of proper initial conditions that enable the steady-state condition to be reached faster, conditions that result in considerable savings in computational time. The variations in the dynamic load with respect to changes in contact position, operating speed, backlash, damping, and stiffness are also investigated. In addition, the stability of the gear system is studied, using the Floquet theory and the well-known stability conditions of difference systems.

scholarly journals Inferring the instability of a dynamical system from the skill of data assimilation exercises

2021 ◽  
Vol 28 (4) ◽  
pp. 633-649
Author(s):  
Yumeng Chen ◽  
Alberto Carrassi ◽  
Valerio Lucarini
Keyword(s):  
Dynamical System ◽  
Data Assimilation ◽  
Observational Data ◽  
Initial Conditions ◽  
Model Error ◽  
High Dimensional ◽  
Accurate Knowledge ◽  
Sensitive Dependence ◽  
Abstract Data ◽  
Very High

Abstract. Data assimilation (DA) aims at optimally merging observational data and model outputs to create a coherent statistical and dynamical picture of the system under investigation. Indeed, DA aims at minimizing the effect of observational and model error and at distilling the correct ingredients of its dynamics. DA is of critical importance for the analysis of systems featuring sensitive dependence on the initial conditions, as chaos wins over any finitely accurate knowledge of the state of the system, even in absence of model error. Clearly, the skill of DA is guided by the properties of dynamical system under investigation, as merging optimally observational data and model outputs is harder when strong instabilities are present. In this paper we reverse the usual angle on the problem and show that it is indeed possible to use the skill of DA to infer some basic properties of the tangent space of the system, which may be hard to compute in very high-dimensional systems. Here, we focus our attention on the first Lyapunov exponent and the Kolmogorov–Sinai entropy and perform numerical experiments on the Vissio–Lucarini 2020 model, a recently proposed generalization of the Lorenz 1996 model that is able to describe in a simple yet meaningful way the interplay between dynamical and thermodynamical variables.

ANALYSIS OF COLD-STANDBY SYSTEMS — CUTTING AND CLUSTERING THE STATE SPACE

1995 ◽  
Vol 02 (03) ◽  
pp. 327-340 ◽  
Author(s):  
M.N. GOPALAN ◽  
U. DINESH KUMAR
Keyword(s):  
State Space ◽  
Integral Equations ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
The State ◽  
Convolution Integral ◽  
Cold Standby ◽  
Convolution Integral Equations ◽  
Systems Of Integral Equations ◽  
Repair Facility

In this paper two methods, namely cutting and clustering the state space are discussed to find approximate solutions to n-unit cold-standby system with a single repair facility. It has been assumed that the failure rate is constant and the repair time is arbitrarily distributed. A mathematical model is developed using semi-regenerative phenomena and systems of convolution integral equations satisfied by various state probabilities corresponding to different initial conditions are obtained. Explicit expressions for the availability and the reliability of the system are obtained. An iterative method is used to solve the systems of integral equations obtained and a comparative study has been carried out between exact and approximate solutions.

scholarly journals Inferring the instability of a dynamical system from the skill of data assimilation exercises

2021 ◽  
Author(s):  
Yumeng Chen ◽  
Alberto Carrassi ◽  
Valerio Lucarini
Keyword(s):  
Dynamical System ◽  
Data Assimilation ◽  
Observational Data ◽  
Initial Conditions ◽  
Model Error ◽  
High Dimensional ◽  
Accurate Knowledge ◽  
Sensitive Dependence ◽  
Abstract Data ◽  
Very High

Abstract. Data assimilation (DA) aims at optimally merging observational data and model outputs to create a coherent statistical and dynamical picture of the system under investigation. Indeed, DA aims at minimizing the effect of observational and model error, and at distilling the correct ingredients of its dynamics. DA is of critical importance for the analysis of systems featuring sensitive dependence on the initial conditions, as chaos wins over any finitely accurate knowledge of the state of the system, even in absence of model error. Clearly, the skill of DA is guided by the properties of dynamical system under investigation, as merging optimally observational data and model outputs is harder when strong instabilities are present. In this paper we reverse the usual angle on the problem and show that it is indeed possible to use the skill of DA to infer some basic properties of the tangent space of the system, which may be hard to compute in very high-dimensional systems. Here, we focus our attention on the first Lyapunov exponent and the Kolmogorov-Sinai entropy, and perform numerical experiments on the Vissio-Lucarini 2020 model, a recently proposed generalisation of the Lorenz 1996 model that is able to describe in a simple yet meaningful way the interplay between dynamical and thermodynamical variables.

scholarly journals Converging towards the optimal path to extinction

2011 ◽  
Vol 8 (65) ◽  
pp. 1699-1707 ◽  
Author(s):  
Ira B. Schwartz ◽  
Eric Forgoston ◽  
Simone Bianco ◽  
Leah B. Shaw
Keyword(s):  
Dynamical Systems ◽  
Random Process ◽  
Chemical Reactions ◽  
Stochastic Models ◽  
Population Biology ◽  
Initial Conditions ◽  
Optimal Path ◽  
Rare Event ◽  
Effective Force ◽  
Sensitive Dependence

Extinction appears ubiquitously in many fields, including chemical reactions, population biology, evolution and epidemiology. Even though extinction as a random process is a rare event, its occurrence is observed in large finite populations. Extinction occurs when fluctuations owing to random transitions act as an effective force that drives one or more components or species to vanish. Although there are many random paths to an extinct state, there is an optimal path that maximizes the probability to extinction. In this paper, we show that the optimal path is associated with the dynamical systems idea of having maximum sensitive dependence to initial conditions. Using the equivalence between the sensitive dependence and the path to extinction, we show that the dynamical systems picture of extinction evolves naturally towards the optimal path in several stochastic models of epidemics.

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