scholarly journals Persistence of small noise and random initial conditions

2018 ◽  
Vol 50 (A) ◽  
pp. 67-81 ◽  
Author(s):  
J. Baker ◽  
P. Chigansky ◽  
K. Hamza ◽  
F. C. Klebaner
Keyword(s):  
Initial Conditions ◽  
Scaling Limit ◽  
Unperturbed System ◽  
Time Interval ◽  
Initial Condition ◽  
Small Noise ◽  
Random Initial Conditions ◽  
Unstable Fixed Point ◽  
Smooth Dynamical System ◽  
Positive Half

AbstractThe effect of small noise in a smooth dynamical system is negligible on any finite time interval; in this paper we study situations where the effect persists on intervals increasing to ∞. Such an asymptotic regime occurs when the system starts from an initial condition that is sufficiently close to an unstable fixed point. In this case, under appropriate scaling, the trajectory converges to a solution of the unperturbed system started from a certainrandominitial condition. In this paper we consider the case of one-dimensional diffusions on the positive half-line; this case often arises as a scaling limit in population dynamics.

scholarly journals Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line

2020 ◽  
Vol 8 (3) ◽  
Author(s):  
Alexandre Krajenbrink ◽  
Pierre Le Doussal
Keyword(s):  
Bethe Ansatz ◽  
Initial Conditions ◽  
Fredholm Determinant ◽  
Half Line ◽  
Initial Condition ◽  
Exact Probability ◽  
Exact Probability Distribution ◽  
The Laplace Transform ◽  
Bethe Ansatz Solution ◽  
Positive Half

We consider the Kardar-Parisi-Zhang (KPZ) equation for the stochastic growth of an interface of height h(x,t)h(x,t) on the positive half line with boundary condition \partial_x h(x,t)|_{x=0}=A∂xh(x,t)|x=0=A. It is equivalent to a continuum directed polymer (DP) in a random potential in half-space with a wall at x=0x=0 either repulsive A>0A>0, or attractive A<0A<0. We provide an exact solution, using replica Bethe ansatz methods, to two problems which were recently proved to be equivalent [Parekh, arXiv:1901.09449]: the droplet initial condition for arbitrary A \geqslant -1/2A≥−1/2, and the Brownian initial condition with a drift for A=+\inftyA=+∞ (infinite hard wall). We study the height at x=0x=0 and obtain (i) at all time the Laplace transform of the distribution of its exponential (ii) at infinite time, its exact probability distribution function (PDF). These are expressed in two equivalent forms, either as a Fredholm Pfaffian with a matrix valued kernel, or as a Fredholm determinant with a scalar kernel. For droplet initial conditions and A> - \frac{1}{2}A>−12 the large time PDF is the GSE Tracy-Widom distribution. For A= \frac{1}{2}A=12, the critical point at which the DP binds to the wall, we obtain the GOE Tracy-Widom distribution. In the critical region, A+\frac{1}{2} = \epsilon t^{-1/3} \to 0A+12=ϵt−1/3→0 with fixed \epsilon = \mathcal{O}(1)ϵ=𝒪(1), we obtain a transition kernel continuously depending on \epsilonϵ. Our work extends the results obtained previously for A=+\inftyA=+∞, A=0A=0 and A=- \frac{1}{2}A=−12.

For How Long Should What Data Be Assimilated for the Mesoscale Forecasting of Convection and Why? Part I: On the Propagation of Initial Condition Errors and Their Implications for Data Assimilation

2010 ◽  
Vol 138 (1) ◽  
pp. 242-255 ◽  
Author(s):  
Frédéric Fabry ◽  
Juanzhen Sun
Keyword(s):  
Data Assimilation ◽  
Great Plains ◽  
Incomplete Data ◽  
Weather Forecasting ◽  
Time Window ◽  
Initial Conditions ◽  
Time Interval ◽  
Initial Condition ◽  
Convective Storms ◽  
Assimilation Time

Abstract Data assimilation is used among other things to constrain the initial conditions of weather forecasting models by fitting the model fields to observations made over a certain time interval. In particular, it tries to tie incomplete data with model constraints to detect and correct for initial condition errors. This is possible only if initial condition errors leave their signature on the data assimilated and if the model is capable of faithfully reproducing such signatures. Using simulations of the evolution of convective storms in the Great Plains over an active 6-day period, the propagation of initial condition errors to other variables as well as their effect on the accuracy of the forecasts were investigated. Increasing the assimilation time window boosts the ability of assimilation systems to detect a variety of initial condition errors; however, limits to the predictability of convective events impose a maximum assimilation period that is a function of the type of measurements assimilated as well as of the type of errors one tries to correct for. These findings are then used to suggest changes in assimilation approaches to take into account the different predictability times of the model fields constrained by assimilation.

scholarly journals On the emergence of random initial conditions in fluid limits

2016 ◽  
Vol 53 (4) ◽  
pp. 1193-1205 ◽  
Author(s):  
A. D. Barbour ◽  
P. Chigansky ◽  
F. C. Klebaner
Keyword(s):  
Carrying Capacity ◽  
Classical Result ◽  
Initial Conditions ◽  
Death Process ◽  
Initial Population ◽  
Random Initial Condition ◽  
Initial Condition ◽  
Birth And Death Process ◽  
Fluid Limit ◽  
Random Initial Conditions

Abstract In the paper we present a phenomenon occurring in population processes that start near 0 and have large carrying capacity. By the classical result of Kurtz (1970), such processes, normalized by the carrying capacity, converge on finite intervals to the solutions of ordinary differential equations, also known as the fluid limit. When the initial population is small relative to the carrying capacity, this limit is trivial. Here we show that, viewed at suitably chosen times increasing to ∞, the process converges to the fluid limit, governed by the same dynamics, but with a random initial condition. This random initial condition is related to the martingale limit of an associated linear birth-and-death process.

scholarly journals Advanced Load Effect Model for Probabilistic Structural Design

10.14311/874 ◽  
2006 ◽  
Vol 46 (5) ◽  
Author(s):  
M. Sýkora
Keyword(s):  
Structural Design ◽  
Structural Reliability ◽  
Initial Conditions ◽  
Arbitrary Point ◽  
Future Research ◽  
Time Interval ◽  
Expected Number ◽  
Load Effect ◽  
Effect Model ◽  
Random Initial Conditions

In probabilistic structural design some actions on structures can be well described by renewal processes with intermittencies. The expected number of renewals for a given time interval and the probability of “on“-state at an arbitrary point in time are of a main interest when estimating the structural reliability level related to the observed period. It appears that the expected number of renewals follows the Poisson distribution. The initial probability of “on”-state is derived assuming random initial conditions. Based on a two-state Markov process, the probability of “on”-state at an arbitrary point in time then proves to be a time-invariant quantity under random initial conditions. The results are numerically verified by Monte Carlo simulations. It is anticipated that the proposed load effect model will become a useful tool in probabilistic structural design. The aims of future research are outlined in the conclusions of the paper. 

Heat Conduction in Solids With Random Initial Conditions

1974 ◽  
Vol 96 (4) ◽  
pp. 474-477 ◽  
Author(s):  
G. Ahmadi
Keyword(s):  
Temperature Field ◽  
Heat Conduction ◽  
General Theory ◽  
Unbounded Domain ◽  
Initial Conditions ◽  
Bounded Domains ◽  
Random Initial Condition ◽  
Initial Condition ◽  
Random Initial Conditions ◽  
Random Temperature Field

The problem of heat conduction in solids with random initial condition is studied. A general theory is first discussed and several examples are considered. It is observed that the homogeneity of the random initial condition is sufficient for the homogeneity of the temperature field in an unbounded domain. But in bounded domains the random temperature field becomes nonhomogeneous even though the initial condition is homogeneous.

Is it a Right Assumption That B and Ge are Distributed Randomly After Growing a Strained HBT-Structure?

2002 ◽  
Vol 716 ◽  
Author(s):  
Victor I. Kol'dyaev
Keyword(s):  
Surface Segregation ◽  
Initial Conditions ◽  
Initial Condition ◽  
Basic Assumptions ◽  
Transient Diffusion ◽  
Segregation Process ◽  
Main Observation

AbstractIt is accepted that surface Ge atoms are considered to be responsible for the surface B segregation process. A set of original experiments is carried out. A main observation from the B and Ge profiles grown at different conditions shows that at certain conditions B is taking initiative and determine the Ge surface segregation process. basic assumptions are suggested to self-consistently explain these original experimental features and what is observed in the literature. These results have a strong implication for modeling the B diffusion in Si1-xGex where the initial conditions should be formulated accounting for the correlation in B and Ge distribution. A new assumption for the initial condition to be “all B atoms are captured by Ge” is regarded as a right one implicating that there is no any transient diffusion representing the B capturing kinetics.

scholarly journals Application of Adjoint-Derived Forecast Sensitivities to the 24–25 January 2000 U.S. East Coast Snowstorm

2005 ◽  
Vol 133 (11) ◽  
pp. 3148-3175 ◽  
Author(s):  
Daryl T. Kleist ◽  
Michael C. Morgan
Keyword(s):  
Response Function ◽  
Initial Conditions ◽  
Forecast Error ◽  
Weather Prediction ◽  
Lower Troposphere ◽  
Vertical Motion ◽  
Initial Condition ◽  
Response Functions ◽  
Eastern United States ◽  
Surface Cyclone

Abstract The 24–25 January 2000 eastern United States snowstorm was noteworthy as operational numerical weather prediction (NWP) guidance was poor for lead times as short as 36 h. Despite improvements in the forecast of the surface cyclone position and intensity at 1200 UTC 25 January 2000 with decreasing lead time, NWP guidance placed the westward extent of the midtropospheric, frontogenetically forced precipitation shield too far to the east. To assess the influence of initial condition uncertainties on the forecast of this event, an adjoint model is used to evaluate forecast sensitivities for 36- and 48-h forecasts valid at 1200 UTC 25 January 2000 using as response functions the energy-weighted forecast error, lower-tropospheric circulation about a box surrounding the surface cyclone, 750-hPa frontogenesis, and vertical motion. The sensitivities with respect to the initial conditions for these response functions are in general very similar: geographically isolated, maximized in the middle and lower troposphere, and possessing an upshear vertical tilt. The sensitivities are maximized in a region of enhanced low-level baroclinicity in the vicinity of the surface cyclone’s precursor upper trough. However, differences in the phase and structure of the gradients for the four response functions are evident, which suggests that perturbations could be constructed to alter one response function but not necessarily the others. Gradients of the forecast error response function with respect to the initial conditions are used in an iterative procedure to construct initial condition perturbations that reduce the forecast error. These initial condition perturbations were small in terms of both spatial scale and magnitude. Those initial condition perturbations that were confined primarily to the midtroposphere grew rapidly into much larger amplitude upper-and-lower tropospheric perturbations. The perturbed forecasts were not only characterized by reduced final time forecast error, but also had a synoptic evolution that more closely followed analyses and observations.

Transient elastodynamic analysis with a combination of convolution quadrature method and pseudo-initial condition method

2018 ◽  
Vol 36 (1) ◽  
pp. 334-355
Author(s):  
Yuan Li ◽  
J. Zhang ◽  
Yudong Zhong ◽  
Xiaomin Shu ◽  
Yunqiao Dong
Keyword(s):  
Initial Conditions ◽  
Initial Condition ◽  
Quadrature Method ◽  
Memory Requirement ◽  
Content Type ◽  
Convolution Quadrature ◽  
Force Method ◽  
Cpu Time ◽  
Condition Method ◽  
Convolution Quadrature Method

Purpose The Convolution Quadrature Method (CQM) has been widely applied to solve transient elastodynamic problems because of its stability and generality. However, the CQM suffers from the problems of huge memory requirement in case of direct implementation in time domain or CPU time in case of its reformulation in Laplace domain. The purpose of this paper is to combine the CQM with the pseudo-initial condition method (PICM) to achieve a good balance between memory requirement and CPU time. Design/methodology/approach The combined methods first subdivide the whole analysis into a few sub-analyses, which is dealt with the PICM, namely, the results obtained by previous sub-analysis are used as the initial conditions for the next sub-analysis. In each sub-analysis, the time interval is further discretized into a number of sub-steps and dealt with the CQM. For non-zero initial conditions, the pseudo-force method is used to transform them into equivalent body forces. The boundary face method is employed in the numerical implementation. Three examples are analyzed. Results are compared with analytical solutions or FEM results and the results of reformulated CQM. Findings Results demonstrate that the computation time and the storage requirement can be reduced significantly as compared to the CQM, by using the combined approach. Originality/value The combined methods can be successfully applied to the problems of long-time dynamic response, which requires a large amount of computer memory when CQM is applied, while preserving the CQM stability. If the number of time steps is high, then the accuracy of the proposed approach can be deteriorated because of the pseudo-force method.

scholarly journals Using G-Suite Docs to Improve Students’ Writing Ability at SMA Negeri 1 Batuan, Sumenep

2021 ◽  
Vol 3 (2) ◽  
pp. 1-15
Author(s):  
Norani Yanuar Subandi ◽  
Hablil Warid ◽  
Sulistyaningsih
Keyword(s):  
Action Research ◽  
Data Collection ◽  
Group Work ◽  
Initial Conditions ◽  
Research Data ◽  
Initial Condition ◽  
Writing Ability ◽  
Student Activity ◽  
School Year ◽  
Learning Management

The aim of this research were to describe the implementation of Using G - Suite Docs to Improve Students’ Writing Ability at SMA Negeri 1 Batuan, Sumenep and to find out  the achievement of student’s writing ability in Using G - Suite at SMA Negeri 1 Batuan, Sumenep. The approach used in this research was classroom action research. Data collection tools in this study were observation sheets of learning management through G-Suite Docs media, student activity observation sheets and tests. The objects of this study were 25 students of class XII IPA 1 SMA Negeri 1 Batuan even semester of the 2020/2021 school year. The research showed that the result of implementing of using G-Suite Docs showed that the students were more enthusiastic in writing discussion text and all students could respond to their group work by commenting on the process in writing. Moreover, the students achievement from the initial conditions up to second cycle. The average of initial condition was 66.6 changed to be 79.64 or increased 80% with the result of completeness reaching 92%.   Keywords: G - Suite Docs, Writing ability, Discussion Text

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