Modeling fast diffusion processes in time integration of stiff stochastic differential equations.
Abstract not provided.
Han, Xiaoying↗
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Results for “Stochastic process”
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Stochastic simulation of vertically nonhomogeneous gusts
The small-scale horizontal gust structure of detailed wind profiles along the vertical in the first 20 km of the atmosphere is a vertically nonhomogeneous process. A linear stochastic model was developed based on the process covariance function. This model was formulated through the use of a scaling hypothesis which transforms the nonhomogeneous gust process into a nondimensional gust process which is homogeneous in a nondimensional gust height coordinate. The velocity scaling parameter for the gust process is the gust standard deviation, and the length scale used to nondimensionalize the altitude is the vertical space lag associated with the first zero of the gust covariance function. State space theory was used to derive a digital filter from the model, which can be readily used to simulate gusts for space vehicle design applications.
Fichtl, G. H.↗
On the representation of a continuous stochastic intensity by Poisson shot noise
Poisson shot noise process relationship with continuous stochastic intensity as sample function
Gagliardi, R. M.↗
Stationary conditions for stochastic differential equations
This is a preliminary study of possible necessary and sufficient conditions to insure stationarity in the solution process for a stochastic differential equation. It indirectly sheds some light on ergodicity properties and shows that the spectral density is generally inadequate as a statistical measure of the solution. Further work is proceeding on a more general theory which gives necessary and sufficient conditions in a form useful for applications.
Adomian, G.↗
Validity of path thermodynamic description of reactive systems: Microscopic simulations
Traditional stochastic modeling of reactive systems limits the domain of applicability of the associated path thermodynamics to systems involving a single elementary reaction at the origin of each observed change in composition. An alternative stochastic modeling has recently been proposed to overcome this limitation. These two ways of modeling reactive systems are in principle incompatible. Here, the question thus arises about choosing the appropriate type of modeling to be used in practical situations. In the absence of sufficiently accurate experimental results, one way to address this issue is through the microscopic simulation of reactive fluids, usually based on hard-sphere dynamics in the Boltzmann limit. In this paper, we show that results obtained through such simulations unambiguously confirm the predictions of traditional stochastic modeling, invalidating a recently proposed alternative.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Lecture notes on stochastic control, part I
Stochastic control of diffusion processes, optimal linear filtering and prediction, and optimization of control for Markov chains
CONTROL SYSTEM↗
Bayesian State-Space Modeling Framework for Understanding and Predicting Golden Eagle Movements Using Telemetry Data
Predicting raptor movements through a wind power plant under given atmospheric and topographical conditions is a crucial first step in the overall goal of quantifying the risk of turbine-related collisions and mortalities. Extracting behavioral traits of golden eagles (Aquila chrysaetos) from telemetry data requires the fusion of noisy and sparse movement data (location, heading, velocity) with a stochastic mathematical representation of the eagles' decision-making processes. In this study, we framed this problem in a Bayesian state-space framework where both observations and decision-making are assumed to be stochastic processes connected through hidden states (mode of flight, intent), and the unknown model parameters are assumed to be random variables that are calibrated using the available telemetry data. This framework allowed for rigorous consideration of underlying uncertainties while allowing for both data and prior biological knowledge to contribute to a probabilistic and predictive agent-based movement model. We implemented and applied the Bayesian framework to understand movement behavior of 23 GPS-tagged golden eagles travelling in the western US for years 2019 and 2020. Our preliminary findings show that the Bayesian state-space framework provides a robust inverse modeling apparatus to decode eagle behavioral characteristics from telemetry data. This study was primarily aimed at verifying and validating the framework with selected golden eagle tracks (both long- and short-ranged), with future research aimed at extending the framework to include multi-mode flight, consideration of atmospheric data and uplift mechanisms, eagle-to-eagle interaction, and eagle-to-turbine interaction.
Bayesian modeling↗
Application of Ensemble Detection and Analysis to Modeling Uncertainty in Non Stationary Process
Characterization of non stationary and nonlinear processes is a challenge in many engineering and scientific disciplines. Climate change modeling and projection, retrieving information from Doppler measurements of hydrometeors, and modeling calibration architectures and algorithms in microwave radiometers are example applications that can benefit from improvements in the modeling and analysis of non stationary processes. Analyses of measured signals have traditionally been limited to a single measurement series. Ensemble Detection is a technique whereby mixing calibrated noise produces an ensemble measurement set. The collection of ensemble data sets enables new methods for analyzing random signals and offers powerful new approaches to studying and analyzing non stationary processes. Derived information contained in the dynamic stochastic moments of a process will enable many novel applications.
Racette, Paul↗
Two stochastic models useful in petroleum exploration
A model of the petroleum exploration process that tests empirically the hypothesis that at an early stage in the exploration of a basin, the process behaves like sampling without replacement is proposed along with a model of the spatial distribution of petroleum reserviors that conforms to observed facts. In developing the model of discovery, the following topics are discussed: probabilitistic proportionality, likelihood function, and maximum likelihood estimation. In addition, the spatial model is described, which is defined as a stochastic process generating values of a sequence or random variables in a way that simulates the frequency distribution of areal extent, the geographic location, and shape of oil deposits
Kaufman, G. M.↗
Dynamical Mass Estimates of the β Pictoris Planetary System through Gaussian Process Stellar Activity Modeling
Nearly 15 yr of radial velocity (RV) monitoring and direct imaging enabled the detection of two giant planets orbiting the young, nearby star β Pictoris. The δ Scuti pulsations of the star, which overwhelm planetary signals, need to be carefully suppressed. In this work, we independently revisit the analysis of the RV data following a different approach than available in the literature to model the activity of the star. We show that a Gaussian process (GP) with a stochastically driven damped harmonic oscillator kernel can model the δ Scuti pulsations. It provides similar results to parametric models but with a simpler framework, using only three hyperparameters. It also enables us to model poorly sampled RV data that were excluded from previous analyses, hence extending the RV baseline by nearly five years. Altogether, the orbit and mass of both planets can be constrained from RV only, which was not possible with the parametric modeling. To characterize the system more accurately, we also perform a joint fit of all available relative astrometry and RV data. Our orbital solutions for β Pic b favor a low eccentricity of 0.029$_{−0.024}^{+0.061}$ and a relatively short period of 21.1$_{−0.8}^{+2.0}$ yr. The orbit of β Pic c is eccentric with 0.206$_{−0.063}^{+0.074}$ with a period of 3.36 ± 0.03 yr. We find model-independent masses of 11.7 ± 1.4 and 8.5 ± 0.5 M Jup for β Pic b and c, respectively, assuming coplanarity. The mass of β Pic b is consistent with the hottest start evolutionary models, at an age of 25 ± 3 Myr. A direct detection of β Pic c would provide a second calibration measurement in a coeval system.
79 ASTRONOMY AND ASTROPHYSICS↗
Operator learning for predicting multiscale bubble growth dynamics
We report simulating and predicting multiscale problems that couple multiple physics and dynamics across many orders of spatiotemporal scales is a great challenge that has not been investigated systematically by deep neural networks (DNNs). Herein, we develop a framework based on operator regression, the so-called deep operator network (DeepONet), with the long-term objective to simplify multiscale modeling by avoiding the fragile and time-consuming “hand-shaking” interface algorithms for stitching together heterogeneous descriptions of multiscale phenomena. To this end, as a first step, we investigate if a DeepONet can learn the dynamics of different scale regimes, one at the deterministic macroscale and the other at the stochastic microscale regime with inherent thermal fluctuations. Specifically, we test the effectiveness and accuracy of the DeepONet in predicting multirate bubble growth dynamics, which is described by a Rayleigh–Plesset (R–P) equation at the macroscale and modeled as a stochastic nucleation and cavitation process at the microscale by dissipative particle dynamics (DPD). First, we generate data using the R–P equation for multirate bubble growth dynamics caused by randomly time-varying liquid pressures drawn from Gaussian random fields (GRFs). Our results show that properly trained DeepONets can accurately predict the macroscale bubble growth dynamics and can outperform long short-term memory networks. We also demonstrate that the DeepONet can extrapolate accurately outside the input distribution using only very few new measurements. Subsequently, we train the DeepONet with DPD data corresponding to stochastic bubble growth dynamics. Although the DPD data are noisy and we only collect sparse data points on the trajectories, the trained DeepONet model is able to predict accurately the mean bubble dynamics for time-varying GRF pressures. Taken together, our findings demonstrate that DeepONets can be employed to unify the macroscale and microscale models of the multirate bubble growth problem, hence providing new insight into the role of operator regression via DNNs in tackling realistic multiscale problems and in simplifying modeling with heterogeneous descriptions.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Energy-momentum-conserving stochastic differential equations and algorithms for the nonlinear Landau-Fokker-Planck equation
Coulomb collision is a fundamental diffusion process in plasmas that can be described by the Landau-Fokker-Planck (LFP) equation or the stochastic differential equation (SDE). While energy and momentum are conserved exactly in the LFP equation, they are conserved only on average by the conventional corresponding SDEs, suggesting that the underlying stochastic process may not be well defined by such SDEs. Here, in this study, we derive new SDEs with exact energy-momentum conservation for the Coulomb collision by factorizing the collective effect of field particles into individual particles and enforcing Newton's third law. These SDEs, when interpreted in the Stratonovich sense, have a particularly simple form that represents pure diffusion between particles without drag. To demonstrate that the new SDEs correspond to the LFP equation, we develop numerical algorithms that converge to the SDEs and preserve discrete conservation laws. Simulation results are presented in a benchmark of various relaxation processes.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Computing Nonequilibrium Responses with Score-Shifted Stochastic Differential Equations
Using equilibrium fluctuations to understand the response of a physical system to an externally imposed perturbation is the basis for linear response theory, which is widely used to interpret experiments and shed light on microscopic dynamics. For nonequilibrium systems, perturbations cannot be interpreted simply by monitoring fluctuations in a conjugate observable and general response results rely on path ensemble averaging. Furthermore, these techniques do not apply to perturbations that affect the diffusion tensor in a stochastic system. Here, we introduce an “effective” physical process that represents the diffusion perturbed dynamics and enables accurate calculations of responses to a change in the diffusion. Interestingly, the effective dynamics contain an additional drift involving the instantaneous “score” of the system, and we leverage score matching algorithms to carry out nonequilibrium response calculations on systems for which the exact stationary distribution is unknown.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Sequential Bayesian Parameter Estimation of Stochastic Dynamic Load Models
In this paper we focus on the parameter estimation of dynamic load models with stochastic terms-in particular, load models where protection settings are uncertain, such as in aggregated air conditioning units. We show how the uncertainty in the aggregated protection characteristics can be formulated as a stochastic differential equation with process noise. We cast the parameter inversion within a Bayesian parameter estimation framework, and we present methods to include process noise. We demonstrate the benefits of considering stochasticity in the parameter estimation and the risks of ignoring it.
Bayesian Statistics↗
Gamma-ray emission and electron acceleration in solar flares
Recent observations have extended the spectra of the impulsive phase of flares to the GeV range. Such high-energy photons can be produced either by electron bremsstrahlung or by decay of pions produced by accelerated protons. In this paper we investigate the effects of processes which become important at high energies. We examine the effects of synchrotron losses during the transport of electrons as they travel from the acceleration region in the corona to the gamma-ray emission sites deep in the chromosphere and photosphere, and the effects of scattering and absorption of gamma rays on their way from the photosphere to space instruments. These results are compared with the spectra from so-called electron-dominated flares, observed by GRS on the Solar Maximum Mission, which show negligible or no detectable contribution from accelerated protons. The spectra of these flares show a distinct steepening at energies below 100 keV and a rapid falloff at energies above 50 MeV. Following our earlier results based on lower energy gamma-ray flare emission we have modeled these spectra. We show that neither the radiative transfer effects, which are expected to become important at higher energies, nor the transport effects (Coulomb collisions, synchrotron losses, or magnetic field convergence) can explain such sharp spectral deviations from a simple power law. These spectral deviations from a power law are therefore attributed to the acceleration process. In a stochastic acceleration model the low-energy steepening can be attributed to Coulomb collision and the rapid high-energy steepening can result from synchrotron losses during the acceleration process.
Petrosian, Vahe↗
Fiber Bundle Morphisms as a Framework for Modeling Many-to-Many Maps
While it is not generally reflected in the `nice' datasets used for benchmarking machine learning algorithms, the real-world is full of processes that would be best described as many-to-many. That is, a single input can potentially yield many different outputs (whether due to noise, imperfect measurement, or intrinsic stochasticity in the process) and many different inputs can yield the same output (that is, the map is not injective). For example, imagine a sentiment analysis task where, due to linguistic ambiguity, a single statement can have a range of different sentiment interpretations while at the same time many distinct statements can represent the same sentiment. When modeling such a multivalued function $f: X \rightarrow Y$, it is frequently useful to be able to model the distribution on $f(x)$ for specific input $x$ as well as the distribution on fiber $f^{-1}(y)$ for specific output $y$. Such an analysis helps the user (i) better understand the variance intrinsic to the process they are studying and (ii) understand the range of specific input $x$ that can be used to achieve output $y$. Following existing work which used a fiber bundle framework to better model many-to-one processes we describe how morphisms of fiber bundles provide a template for building models which naturally capture the structure of many-to-many processes.
Coda, Elizabeth D.↗
Calibration of stochastic agent-based model with Gaussian process surrogate and Stein variational inference
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Robertson, Connor James↗