On the stability of processes defined by stochastic difference-differential equations
Stability of processes defined by stochastic difference and differential equations
DIFFERENCE EQUATION↗
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Stochastic goal-oriented error estimation with memory
We propose a stochastic dual-weighted error estimator for the viscous shallow-water equation with boundaries. For this purpose, previous work on memory-less stochastic dual-weighted error estimation is extended by incorporating memory effects. The memory is introduced by describing the local truncation error as a sum of time-correlated random variables. The random variables itself represent the temporal fluctuations in local truncation errors and are estimated from high-resolution information at near-initial times. The resulting error estimator is evaluated experimentally in two classical ocean-type experiments, the Munk gyre and the flow around an island. In these experiments, the stochastic process is adapted locally to the respective dynamical flow regime. Our stochastic dual-weighted error estimator is shown to provide meaningful error bounds for a range of physically relevant goals. We prove, as well as show numerically, that our approach can be interpreted as a linearized stochastic-physics ensemble.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
A Scenario Generation Method for Wind Power Ramp Events Forecasting
Wind power ramp events (WPREs) have received increasing attention in recent years due to their significant impact on the reliability of power grid operations. In this paper, a novel WPRE forecasting method is proposed which is able to estimate the probability distributions of three important properties of the WPREs. To do so, a neural network (NN) is first proposed to model the wind power generation (WPG) as a stochastic process so that a number of scenarios of the future WPG can be generated (or predicted). Each possible scenario of the future WPG generated in this manner contains the ramping information, and the distributions of the designated WPRE properties can be stochastically derived based on the possible scenarios. Actual data from a wind power plant in the Bonneville Power Administration (BPA) was selected for testing the proposed ramp forecasting method. Results showed that the proposed method effectively forecasted the probability of ramp events.
17 WIND ENERGY↗
Stochastic simulation for the propagation of high-frequency acoustic waves through a random velocity field
In-service inspection of Sodium-Cooled Fast Reactors (SFR) requires the development of non-destructive techniques adapted to the harsh environment conditions and the examination complexity. From past experiences, ultrasonic techniques are considered as suitable candidates. The ultrasonic telemetry is a technique used to constantly insure the safe functioning of reactor inner components by determining their exact position: it consists in measuring the time of flight of the ultrasonic response obtained after propagation of a pulse emitted by a transducer and its interaction with the targets. While in-service the sodium flow creates turbulences that lead to temperature inhomogeneities, which translates into ultrasonic velocity inhomogeneities. These velocity variations could directly impact the accuracy of the target locating by introducing time of flight variations. A stochastic simulation model has been developed to calculate the propagation of ultrasonic waves in such an inhomogeneous medium. Using this approach, the travel time is randomly generated by a stochastic process whose inputs are the statistical moments of travel times known analytically. The stochastic model predicts beam deviations due to velocity inhomogeneities, which are similar to those provided by a determinist method, such as the ray method.
21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
Analysis of Terminal Deletions using a Generalized Time-Dependent Model of Radiation-Induced Formation of Chromosomal Aberrations
We have developed a model that can simulate different types of radiation induced chromosomal aberrations (CA's) and can provide predictions on the frequency and size of chromosomes with terminal deletions. Chromosomes with terminal deletions lack telomeres and this can elicit sister chromatid unions and the prolonged breakage/fusion/bridge (B/F/B) cycles that have been observed in mammalian tumors. The loss of a single telomere has been shown to cause extensive genomic instability through the B/F/B cycle process. Our model uses a stochastic process of DNA broken end joining, in which a realistic spectrum of CA's is created from improperly joined DNA free ends formed by DNA double strand breaks (DSBs). The distribution of the DNA free ends is given by a mechanistic model that takes into account the chromatin structure and track structure for high-LET radiation. The model allows for DSB clustering from high-LET radiation and simulates the formation of CA's in stages that correspond to the actual time after radiation exposure. The time scale for CA formation is derived from experimental data on DSB repair kinetics. At any given time a nucleus may have intact chromosomes, CA's, and/or unrepaired fragments, some of which are defined as terminal deletions, if they are capped by one telomere. The model produces a spectrum of terminal deletions with their corresponding probabilities and size distributions for different heavy ions exposures for the first division after exposure. This data provides valuable information because there is limited experimental data available in the literature on the on the actual size of terminal deletions. We compare our model output to the available experimental data and make a reasonable extrapolation on the number of chromosomes lacking telomeres in human lymphocytes exposed to heavy ions. This model generates data which may lead to predictions on the rate of genomic instability in cells after exposure to high charge and energy nuclei affecting astronauts during space missions.
Ponomarev, Artem L.↗
Nested polynomial trends for the improvement of Gaussian process-based predictors
The role of simulation keeps increasing for the sensitivity analysis and the uncertainty quantification of complex systems. Such numerical procedures are generally based on the processing of a huge amount of code evaluations. When the computational cost associated with one particular evaluation of the code is high, such direct approaches based on the computer code only, are not affordable. Surrogate models have therefore to be introduced to interpolate the information given by a fixed set of code evaluations to the whole input space. When confronted to deterministic mappings, the Gaussian process regression (GPR), or kriging, presents a good compromise between complexity, efficiency and error control. Such a method considers the quantity of interest of the system as a particular realization of a Gaussian stochastic process, whose mean and covariance functions have to be identified from the available code evaluations. In this context, this work proposes an innovative parametrization of this mean function, which is based on the composition of two polynomials. This approach is particularly relevant for the approximation of strongly non linear quantities of interest from very little information. After presenting the theoretical basis of this method, this work compares its efficiency to alternative approaches on a series of examples.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Almost periodic oscillations for delay impulsive stochastic Nicholson’s blowflies timescale model
In this paper, we employ the new concepts of piecewise almost periodic stochastic process on a new periodic time scale and study a type of delay impulsive stochastic Nicholson’s blowflies timescale model under shifts operators. Some sufficient conditions for the existence and exponential stability of piecewise mean-square almost periodic solutions to the model are established. In addition, we provide numerical examples and computer simulations to demonstrate the feasibility of the obtained results. The methods and definitions introduced in the paper can be applied to study other types of stochastic dynamic models involving q-difference models and more.
97 MATHEMATICS AND COMPUTING↗
First-passage time statistics on surfaces of general shape: Surface PDE solvers using Generalized Moving Least Squares (GMLS)
Here, we develop numerical methods for computing statistics of stochastic processes on surfaces of general shape with drift-diffusion dynamics d X t = a (X t ) dt + b(X t ) d W t . We formulate descriptions of Brownian motion and general drift-diffusion processes on surfaces. We consider statistics of the form u (x) = E x [$∫^{τ}_{0}$ g (X t ) dt ] + E x [ f (X τ )] for a domain Ω and the exit stopping time τ = inf t { t >0 | X i Ω}, where f , g are general smooth functions. For computing these statistics, we develop high-order Generalized Moving Least Squares (GMLS) solvers for associated surface PDE boundary-value problems based on Backward- Kolmogorov equations. We focus particularly on the mean First Passage Times (FPTs) given by the case f = 0, g = 1 where u (x) = E x [τ]. We perform studies for a variety of shapes showing our methods converge with high-order accuracy both in capturing the geometry and the surface PDE solutions. We then perform studies showing how statistics are influenced by the surface geometry, drift dynamics, and spatially dependent diffusivities.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Reconstruction of neuronal input through modeling single-neuron dynamics and computations
Mathematical models provide a mathematical description of neuron activity, which can better understand and quantify neural computations and corresponding biophysical mechanisms evoked by stimulus. In this paper, based on the output spike train evoked by the acupuncture mechanical stimulus, we present two different levels of models to describe the input-output system to achieve the reconstruction of neuronal input. The reconstruction process is divided into two steps: First, considering the neuronal spiking event as a Gamma stochastic process. The scale parameter and the shape parameter of Gamma process are, respectively, defined as two spiking characteristics, which are estimated by a state-space method. Then, leaky integrate-and-fire (LIF) model is used to mimic the response system and the estimated spiking characteristics are transformed into two temporal input parameters of LIF model, through two conversion formulas. We test this reconstruction method by three different groups of simulation data. All three groups of estimates reconstruct input parameters with fairly high accuracy. We then use this reconstruction method to estimate the non-measurable acupuncture input parameters. Results show that under three different frequencies of acupuncture stimulus conditions, estimated input parameters have an obvious difference. The higher the frequency of the acupuncture stimulus is, the higher the accuracy of reconstruction is.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Statistical patterns of theory uncertainties
A comprehensive uncertainty estimation is vital for the precision program of the LHC. While experimental uncertainties are often described by stochastic processes and well-defined nuisance parameters, theoretical uncertainties lack such a description. We study uncertainty estimates for cross-section predictions based on scale variations across a large set of processes. We find patterns similar to a stochastic origin, with accurate uncertainties for processes mediated by the strong force, but a systematic underestimate for electroweak processes. We propose an improved scheme, based on the scale variation of reference processes, which reduces outliers in the mapping from leading order to next-to-leading-order in perturbation theory.
Ghosh, Aishik↗
Robustness analysis of an air heating plant and control law by using polynomial chaos
This paper presents a robustness analysis of an air heating plant with a multivariable closed-loop control law by using the polynomial chaos methodology (MPC). The plant consists of a PVC tube with a fan in the air input (that forces the air through the tube) and a mass flux sensor in the output. A heating resistance warms the air as it flows inside the tube, and a thermo-couple sensor measures the air temperature. The plant has thus two inputs (the fan's rotation intensity and heat generated by the resistance, both measured in percent of the maximum value) and two outputs (air temperature and air mass flux, also in percent of the maximal value). The mathematical model is obtained by System Identification techniques. The mass flux sensor, which is nonlinear, is linearized and the delays in the transfer functions are properly approximated by non-minimum phase transfer functions. The resulting model is transformed to a state-space model, which is used for control design purposes. The multivariable robust control design techniques used is the LQG/LTR, and the controllers are validated in simulation software and in the real plant. Finally, the MPC is applied by considering some of the system's parameters as random variables (one at a time, and the system's stochastic differential equations are solved by expanding the solution (a stochastic process) in an orthogonal basis of polynomial functions of the basic random variables. This method transforms the stochastic equations in a set of deterministic differential equations, which can be solved by traditional numerical methods (That is the MPC). Statistical data for the system (like expected values and variances) are then calculated. The effects of randomness in the parameters are evaluated in the open-loop and closed-loop pole's positions.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Sequential ranking procedures
Sequential ranking procedures, stochastic processes, and statistical analysis
STATISTICAL ANALYSIS↗
Stochastic Community Assembly: Does It Matter in Microbial Ecology?
Understanding the mechanisms controlling community diversity, functions, succession, and biogeography is a central, but poorly understood, topic in ecology, particularly in microbial ecology. Although stochastic processes are believed to play nonnegligible roles in shaping community structure, their importance relative to deterministic processes is hotly debated. The importance of ecological stochasticity in shaping microbial community structure is far less appreciated. Some of the main reasons for such heavy debates are the difficulty in defining stochasticity and the diverse methods used for delineating stochasticity. Here in this paper, we provide a critical review and synthesis of data from the most recent studies on stochastic community assembly in microbial ecology. We then describe both stochastic and deterministic components embedded in various ecological processes, including selection, dispersal, diversification, and drift. We also describe different approaches for inferring stochasticity from observational diversity patterns and highlight experimental approaches for delineating ecological stochasticity in microbial communities. In addition, we highlight research challenges, gaps, and future directions for microbial community assembly research.
59 BASIC BIOLOGICAL SCIENCES↗
Development of reliability methodology for systems engineering. Volume III - Theoretical investigations - An approach to a class of reliability problems Final report
Random quantities from continuous time stochastic process with application to reliability and probability
PROBABILITY↗
Simulating diffusion processes in discontinuous media: A numerical scheme with constant time steps
In this article, we propose new Monte Carlo techniques for moving a diffusive particle in a discontinuous media. In this framework, we characterize the stochastic process that governs the positions of the particle. The key tool is the reduction of the process to a Skew Brownian motion (SBM). In a zone where the coefficients are locally constant on each side of the discontinuity, the new position of the particle after a constant time step is sampled from the exact distribution of the SBM process at the considered time. To do so, we propose two different but equivalent algorithms: a two-steps simulation with a stop at the discontinuity and a one-step direct simulation of the SBM dynamic. Some benchmark tests illustrate their effectiveness.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Sample quadratic variation of sample continuous second order martingales.
Sample continuous second order martingale process, determining existence of limit of stochastic processes using approximation theorems
RANDOM PROCESS↗
Weighted Watson-Crick automata
There are tremendous works in biotechnology especially in area of DNA molecules. The computer society is attempting to develop smaller computing devices through computational models which are based on the operations performed on the DNA molecules. A Watson-Crick automaton, a theoretical model for DNA based computation, has two reading heads, and works on double-stranded sequences of the input related by a complementarity relation similar with the Watson-Crick complementarity of DNA nucleotides. Over the time, several variants of Watson-Crick automata have been introduced and investigated. However, they cannot be used as suitable DNA based computational models for molecular stochastic processes and fuzzy processes that are related to important practical problems such as molecular parsing, gene disease detection, and food authentication. In this paper we define new variants of Watson-Crick automata, called weighted Watson-Crick automata, developing theoretical models for molecular stochastic and fuzzy processes. We define weighted Watson-Crick automata adapting weight restriction mechanisms associated with formal grammars and automata. We also study the generative capacities of weighted Watson-Crick automata, including probabilistic and fuzzy variants. We show that weighted variants of Watson-Crick automata increase their generative power.
60 APPLIED LIFE SCIENCES↗
Data compression - A lower bound for epsilon entropy
Lower bound for asymptotic behavior of epsilon entropy of Gaussian stochastic process on unit interval under square-error fidelity criterion - data compression
ENTROPY↗