Stochastic Source Term Research Articles

Computation of Green's functions for the acoustic scattering by an elastic structure excited by a turbulent flow in water

To model the hydrodynamic noise produced by an elastic ship hull or propeller excited by a turbulent boundary layer, we need an efficient method to compute the acoustic scattering by an elastic body surrounded by a fluid. In 3D, Boundary Element Methods (BEM) are used to reduce the computational costs, for both the fluid and the elastic body. A natural way to compute the boundary integral representation (BIR) of the sound pressure is to use formulations based on the free space acoustic and elastic Green's functions. However, since the turbulent flow along the elastic body is known only statistically, the use of these Green's functions would be too expensive. A remedy is to compute a Green's function adapted to the physical problem, thus satisfying the transmission conditions of the fluid-structure problem. This so-called "tailored Green's function" is determined by solving a coupled acoustic-elastic problem with the BEM, and leads to a simplified BIR of the sound pressure compatible with a stochastic source term. We first validate the computation of the tailored Green's function over a classic spherical geometry. Then we compare the scattering of multiple quadrupoles by elastic or rigid NACA0012 profiles.

The graft‐versus‐host problem for data‐driven gravity‐wave parameterizations in a one‐dimensional quasibiennial oscillation model

AbstractTwo key challenges in the development of data‐driven gravity‐wave parameterizations are generalization, how to ensure that a data‐driven scheme trained on the present‐day climate will continue to work in a new climate regime, and calibration, how to account for biases in the “host” climate model. Both problems depend fundamentally on the response to out‐of‐sample inputs compared with the training dataset, and are often conflicting. The ability to generalize to new climate regimes often goes hand in hand with sensitivity to model biases. To probe these challenges, we employ a one‐dimensional (1D) quasibiennial oscillation (QBO) model with a stochastic source term that represents convectively generated gravity waves in the Tropics with randomly varying strengths and spectra. We employ an array of machine‐learning models consisting of a fully connected feed‐forward neural network, a dilated convolutional neural network, an encoder–decoder, a boosted forest, and a support‐vector regression model. Our results demonstrate that data‐driven schemes trained on “observations” can be critically sensitive to model biases in the wave sources. While able to emulate accurately the stochastic source term on which they were trained, all of our schemes fail to simulate fully the expected QBO period or amplitude, even with the slightest perturbation to the wave sources. The main takeaway is that some measures will always be required to ensure the proper response to climate change and to account for model biases. We examine one approach based on the ideas of optimal transport, where the wave sources in the model are first remapped to the observed one before applying the data‐driven scheme. This approach is agnostic to the data‐driven method and guarantees that the model adheres to the observational constraints, making sure the model yields the right results for the right reasons.

Extreme pulses in gain-switched semiconductor lasers.

Semiconductor lasers subjected to strong current modulation produce gain-switched optical pulse trains. These lasers can also produce pulse trains at sub-harmonic repetition rates relative to the driving current modulation. We experimentally observe, and numerically model, that these pulse trains can be interrupted by single-cycle extreme pulses whose characteristics and statistics are similar to rogue waves. Modeling indicates that drops in the circulating optical power in the optical cavity precede the appearance of extreme pulses. At the single photon level, the stochastic source terms in the optical field equation dominate the circulating optical power.

Sensor configuration optimization based on the entropy of adjoint concentration distribution for stochastic source term estimation in urban environment

Sensor monitoring is foundational to source term estimation (STE). Recently, stochastic STE methods based mainly on Bayesian inference have been attracting attention. However, few studies have developed sensor configuration optimization (SCO) methods aiming to ensure good STE performance in the monitoring area, such that the posterior probability could aggregate around the truths while addressing most sources. This research proposes a deployment method by designing an objective function and applying a simulated annealing (SA) algorithm. The objective function is set as the information joint entropy of the adjoint concentration. The performance of the proposed method was assessed by Bayesian inference STE for 25 unknown sources based on the obtained optimal configuration in a regular block-arrayed building group model. The STE results were compared with those of uniform and random configurations. According to the results, because the optimal configuration can provide the most informative measurements, it yields the best estimations. If we quantize the STE errors and take the average for all unknown sources, the location and strength estimation errors of the optimal configuration significantly decrease by approximately 45% and 99%, respectively, when compared with the random configuration and 39% and 96%, respectively, when compared with the uniform configuration.

Trailing-Edge Noise Prediction by Solving Helmholtz Equation with Stochastic Source Term

A numerical approach to predict broadband trailing-edge noise for low Mach number flows is presented. It is based on the combination of a Helmholtz solver to propagate sound waves with a sound source term computed stochastically. The sound propagation is performed in the frequency domain using a high-order finite element solver. The stochastic approach is based on a random particle-mesh method. The performance of this numerical approach is first examined for a Gaussian source. The numerical approach is then applied to a NACA0012 airfoil for flow Mach numbers of 0.11 and 0.16 and to a controlled-diffusion airfoil for a Mach number of 0.047. The predicted sound levels are compared with experimental data and acoustic results obtained from the acoustic perturbations equations. The mean flow does not significantly modify the acoustic propagation. Using a no-flow propagation model like the Helmholtz equation is therefore a valid approach for low Mach numbers. The reduced computational cost of a Helmholtz calculation, together with the speed of the random-particle mesh turbulence synthesis, allows for fast predictions. While this approach provides relative assessments between different configurations (in particular the spectrum shape), it often requires the inclusion of calibration factors determined from reference measurement or numerical data.

Implementation of a probabilistic surface density volume of fluid approach for spray atomisation

Eulerian Stochastic methods in Large Eddy Simulation context have been successfully used in reactive flows. They have been recently applied to spray atomisation, using surface and liquid volume as independent variables. These methods involve the solutions of Stochastic Partial Differential Equations, which are continuous in space and discontinuous in time. The implementation of these equations into a conventional finite volume solver can be challenging due to both the numerical diffusion of the scalar fields and the numerical integration of the stochastic source terms. This work presents the implementation and validation of the Eulerian Stochastic Fields surface/liquid volume atomisation method within the OpenFOAM framework. The paper discusses how algebraic compression methods can be used for the stochastic fields to minimise the numerical diffusion and the effects of the stochastic source term implementation. The model targets a-priori spray atomisation at large Weber numbers, although the implementation recovers the correct solution at low Weber model and captures correctly the capillary effects on liquid–gas interfaces. A turbulent liquid jet is used for validation and discussion of the accuracy and relative cost of the model implementation. The approach is then applied to a multi-hole gasoline direct injector where the agreement with experimental data showcases the capabilities of the solver for realistic injectors and atomisation regimes.

Modelling the asymmetries of the Sun’s radial p-mode line profiles

Context. The advent of space-borne missions has substantially increased the number and quality of the measured power spectrum of solar-like oscillators. It now allows for the p-mode line profiles to be resolved and facilitates an estimation of their asymmetry. The fact that this asymmetry can be measured for a variety of stars other than the Sun calls for a revisiting of acoustic mode asymmetry modelling. This asymmetry has been shown to be related to a highly localised source of stochastic driving in layers just beneath the surface. However, existing models assume a very simplified, point-like source of excitation. Furthermore, mode asymmetry could also be impacted by a correlation between the acoustic noise and the oscillating mode. Prior studies have modelled this impact, but only in a parametrised fashion, which deprives them of their predictive power. Aims. In this paper, we aim to develop a predictive model for solar radial p-mode line profiles in the velocity spectrum. Unlike the approach favoured by prior studies, this model is not described by free parameters and we do not use fitting procedures to match the observations. Instead, we use an analytical turbulence model coupled with constraints extracted from a 3D hydrodynamic simulation of the solar atmosphere. We then compare the resulting asymmetries with their observationally derived counterpart. Methods. We model the velocity power spectral density by convolving a realistic stochastic source term with the Green’s function associated with the radial homogeneous wave equation. We compute the Green’s function by numerically integrating the wave equation and we use theoretical considerations to model the source term. We reconstruct the velocity power spectral density and extract the line profile of radial p-modes as well as their asymmetry. Results. We find that stochastic excitation localised beneath the mode upper turning point generates negative asymmetry for ν < νmax and positive asymmetry for ν > νmax. On the other hand, stochastic excitation localised above this limit generates negative asymmetry throughout the p-mode spectrum. As a result of the spatial extent of the source of excitation, both cases play a role in the total observed asymmetries. By taking this spatial extent into account and using a realistic description of the spectrum of turbulent kinetic energy, both a qualitative and quantitative agreement can be found with solar observations performed by the GONG network. We also find that the impact of the correlation between acoustic noise and oscillation is negligible for mode asymmetry in the velocity spectrum.

Heterogeneous traffic flow modeling using an extended Payne model with stochastic source term

In this paper, we are extending Payne model with stochastic source term. The new model differs from that of Payne’s in such a way that the traffic pressure and hysteresis are formulate with driver decision. The traditional traffic pressure and hysteresis depend respectively on the traffic density and drivers decision (aggressive-timid, acceleration-deceleration). Our traffic is on an inclined road. In the context of a developing country where the traffic is very heterogeneous and there is a lack of discipline on the highway, the new traffic pressure must take care of the heterogeneity of the traffic. The road traffic hysteresis must take care of the dispersion of the vehicle on the highway. Moreover, the anticipation term is a multi-valued function depending on the vehicle class. The numerical solution of the problem was obtained using a wave analysis. The new model is a hyperbolic one in which the traffic information travels faster than the traffic and the wave propagate downstream leading to a jam formation.

Large Prandtl number asymptotics in randomly forced turbulent convection

We establish the convergence of statistically invariant states for the stochastic Boussinesq equations in the infinite Prandtl number limit and in particular demonstrate the convergence of the Nusselt number (a measure of heat transport in the fluid). This is a singular parameter limit significant in mantle convection and for gasses under high pressure. The equations are subject to a both temperature gradient on the boundary and internal heating in the bulk driven by a stochastic, white in time, gaussian forcing. Here, the stochastic source terms have a strong physical motivation for example as a model of radiogenic heating. Our approach uses mixing properties of the formal limit system to reduce the convergence of invariant states to an analysis of the finite time asymptotics of solutions and parameter-uniform moment bounds. Here, it is notable that there is a phase space mismatch between the finite Prandtl system and the limit equation, and we implement methods to lift both finite and infinite time convergence results to an extended phase space which includes velocity fields. For the infinite Prandtl stochastic Boussinesq equations, we show that the associated invariant measure is unique and that the dual Markovian dynamics are contractive in an appropriate Kantorovich–Wasserstein metric. We then address the convergence of solutions on finite time intervals, which is still a singular perturbation. In the process we derive well-posed equations which accurately approximate the dynamics up to the initial time when the Prandtl number is large.

Interference-exact radiative transfer equation

The Purcell effect, i.e., the modification of the spontaneous emission rate by optical interference, profoundly affects the light-matter coupling in optical resonators. Fully describing the optical absorption, emission, and interference of light hence conventionally requires combining the full Maxwell’s equations with stochastic or quantum optical source terms accounting for the quantum nature of light. We show that both the nonlocal wave and local particle features associated with interference and emission of propagating fields in stratified geometries can be fully captured by local damping and scattering coefficients derived from the recently introduced quantized fluctuational electrodynamics (QFED) framework. In addition to describing the nonlocal optical interference processes as local directionally resolved effects, this allows reformulating the well known and widely used radiative transfer equation (RTE) as a physically transparent interference-exact model that extends the useful range of computationally efficient and quantum optically accurate interference-aware optical models from simple structures to full optical devices.

Simulating the superheating of nanomaterials due to latent heat release in surface reconstruction

Surface reconstruction and phase transformation in nanomaterials can result in a significant rise in temperature due to the release of stored energy called “latent heat.” To simulate this behavior, we propose a hybrid continuum partial differential equation for non-Fourier heat transfer with a stochastic source term modeled using a kinetic Monte Carlo (KMC) algorithm for time-dependent rates to account for the latent heat release as the temperature is changing. The parameters required for the method are obtained through independent atomistic calculations. This includes energy barriers for KMC rates obtained using nudged elastic band calculations, and the non-Fourier thermal parameters obtained using a novel thermal parameter identification scheme described in this paper. As a demonstration of the approach, we study the superheating of silicon nanobeams observed in non-equilibrium molecular dynamics (NEMD) simulations. In these simulations, silicon (001) surfaces undergo a reconstruction from the ideal diamond crystalline surface to a reconstructed structure involving the formation of rows of dimers along a 〈110〉 direction. This reconstruction is accompanied by latent heat release that causes the nanobeam to dramatically superheat. The evolution of the nonuniform temperature profile along the nanobeam predicted by the continuum–KMC method is in excellent agreement with the NEMD results. A logarithmic dependence of the superheating temperature on nanobeam length is observed and theoretically discussed.

Self-organized criticality in a two-dimensional cellular automaton model of a magnetic flux tube with background flow

We investigate the transition to Self Organized Criticality in a two-dimensional model of a flux tube with a background flow. The magnetic induction equation, represented by a partial differential equation with a stochastic source term, is discretized and implemented on a two dimensional cellular automaton. The energy released by the automaton during one relaxation event is the magnetic energy. As a result of the simulations we obtain the time evolution of the energy release, of the system control parameter, of the event lifetime distribution and of the event size distribution, respectively, and we establish that a Self Organized Critical state is indeed reached by the system. Moreover, energetic initial impulses in the magnetohydrodynamic flow can lead to one dimensional signatures in the magnetic two dimensional system, once the Self Organized Critical regime is established. The applications of the model for the study of Gamma Ray Bursts is briefly considered, and it is shown that some astrophysical parameters of the bursts, like the light curves, the maximum released energy, and the number of peaks in the light curve can be reproduced and explained, at least on a qualitative level, by working in a framework in which the systems settles in a Self Organized Critical state via magnetic reconnection processes in the magnetized Gamma Ray Burst fireball.

Numerical simulations of the inviscid burgers equation with periodic boundary conditions and stochastic forcing

We perform numerical simulations in the one-dimensional torus for the first order Burgers equation forced by a stochastic source term with zero spatial integral. We suppose that this source term is a white noise in time, and consider various regularities in space. For the numerical tests, we apply a finite volume scheme combining the Godunov numerical flux with the Euler-Maruyama integrator in time. Our Monte-Carlo simulations are analyzed in bounded time intervals as well as in the large time limit, for various regularities in space. The empirical mean always converges to the space-average of the (deterministic) initial condition as t → ∞, just as the solution of the deterministic problem without source term, even if the stochastic source term is very rough. The empirical variance also stablizes for large time, towards a limit which depends on the space regularity and on the intensity of the noise.

Cosmological perturbations for an inflaton field coupled to radiation

Within the framework of the interacting fluid formalism, we provide the numerical solution to the Boltzmann equation describing the evolution of an inflaton field coupled to radiation. We study the behavior of the system during the slow-roll regime, in the case in which an additional stochastic source term is included in the set of equations, and we recover the expression for the cosmological perturbations previously obtained in the Warm inflation scenarios.

Variational Multiscale Analysis: The Fine-Scale Green's Function for Stochastic Partial Differential Equations

We present the variational multiscale (VMS) method for partial differential equations (PDEs) with stochastic coefficients and source terms. We use it as a method for generating accurate coarse-scale solutions while accounting for the effect of the unresolved fine scales through a model term that contains a fine-scale stochastic Green's function. For a natural choice of an “optimal” coarse-scale solution and $L^2$-orthogonal stochastic basis functions, we demonstrate that the fine-scale stochastic Green's function is intimately linked to its deterministic counterpart. In particular, (i) we demonstrate that whenever the deterministic fine-scale function vanishes, the stochastic fine-scale function satisfies a weaker and discrete notion of vanishing stochastic coefficients, and (ii) we derive an explicit formula for the fine-scale stochastic Green's function that only involves quantities needed to evaluate the fine-scale deterministic Green's function. We present numerical results that support our claims about the physical support of the stochastic fine-scale function and demonstrate the benefit of using the VMS method when the fine-scale Green's function is approximated by an easier to implement element Green's function.

Numerical simulation of incoherent optical wave propagation in nonlinear fibers

The present work concerns the study of pulsed laser systems containing a fiber amplifier for boosting optical output power. In this paper, this fiber amplification device is included into a MOPFA laser, a master oscillator coupled with fiber amplifier, usually a cladding-pumped high-power amplifier often based on an ytterbium-doped fiber. An experimental study has established that the observed nonlinear effects (such as Kerr effect, four waves mixing, Raman effect) could behave very differently depending on the characteristics of the optical source emitted by the master laser. However, it has not yet been possible to determine from the experimental data if the statistics of the photons is alone responsible for the various nonlinear scenarios observed. Therefore, we have developed a numerical simulation software for solving the generalized nonlinear Schrodinger equation with a stochastic source term in order to validate the hypothesis that the coherence properties of the master laser are mainly liable for the behavior of the observed nonlinear effects.

An inverse random source problem in quantifying the elastic modulus of nanomaterials

Nanotechnology deals with structures sized between 1 and 100 nanometer and involves manipulating and controlling materials or devices within that scale. Due to its small size, it is difficult to quantify the mechanical properties of nanomaterials which significantly rely on measurement techniques, conditions and environment. In this paper, to describe the elastic deformation of nanobelts obtained from an atomic force microscope (AFM) under contact mode, we propose a novel model based on a Euler–Bernoulli equation with a stochastic source term accounting for effects of initial bending, surface roughness and white noise during measurements. With the random source, the forward problem is demonstrated to have a unique and explicit path-wise solution. Furthermore, the inverse problem consists of identifying the elastic modulus of nanobelts and reconstructing the random source structure, i.e. the mean and the variance. Based upon explicit formulae of direct problems, the corresponding inverse problem can be reduced into a first kind of Fredholm-type integral equation where two kinds of regularization techniques are applied to obtain stable solutions, i.e. Tikhonov regularization (TR) for smooth solutions and a statistical regularization method from Bayes' formula and maximum likelihood estimation for discontinuous solutions, respectively. Numerical examples are presented to illustrate the validity and effectiveness of the proposed methods.

Numerical solution of the Stratonovich- and Ito–Euler equations: Application to the stochastic piston problem

We consider a piston with a velocity perturbed by Brownian motion moving into a straight tube filled with a perfect gas at rest. The shock generated ahead of the piston can be located by solving the one-dimensional Euler equations driven by white noise using the Stratonovich or Ito formulations. We approximate the Brownian motion with its spectral truncation and subsequently apply stochastic collocation using either sparse grid or the quasi-Monte Carlo (QMC) method. In particular, we first transform the Euler equations with an unsteady stochastic boundary into stochastic Euler equations over a fixed domain with a time-dependent stochastic source term. We then solve the transformed equations by splitting them up into two parts, i.e., a ‘deterministic part’ and a ‘stochastic part’. Numerical results verify the Stratonovich–Euler and Ito–Euler models against stochastic perturbation results, and demonstrate the efficiency of sparse grid and QMC for small and large random piston motions, respectively. The variance of shock location of the piston grows cubically in the case of white noise in contrast to colored noise reported in [1], where the variance of shock location grows quadratically with time for short times and linearly for longer times.

A parton picture of de Sitter space during slow-roll inflation

It is well-known that expectation values in de Sitter space are afflicted by infra-red divergences. Long ago, Starobinsky proposed that infra-red effects in de Sitter space could be accommodated by evolving the long-wavelength part of the field according to the classical equations of motion plus a stochastic source term. I argue that — when quantum-mechanical loop corrections are taken into account — the separate universe picture of superhorizon evolution in de Sitter space is equivalent, in a certain leading-logarithm approximation, to Starobinsky's stochastic approach. In particular the time evolution of a box of de Sitter space can be understood in exact analogy with the DGLAP evolution of partons within a hadron, which describes a slow logarithmic evolution in the distribution of the hadron's constituent partons with the energy scale at which they are probed.

Operator Splitting for an Immunology Model Using Reaction-Diffusion Equations with Stochastic Source Terms

When immune cells detect foreign molecules, they secrete soluble factors that attract other immune cells to the site of the infection. In this paper, I study numerical solutions to a model of this behavior proposed by Kepler. In this model the soluble factors are governed by a system of reaction-diffusion equations with sources that are centered on the cells. The motion of the model cells is a Langevin process that is biased toward the gradient of the soluble factors. I have shown that the solution to this system exists for all time and remains positive, the supremum is a priori bounded, and the derivatives are bounded for finite time. I have also developed a first order split scheme for solving the reaction-diffusion stochastic system. This allows us to make use of known first order schemes for solving the diffusion, the reaction, and the stochastic differential equations separately.