Stochastic Vector Field Research Articles

An implementation of Hasselmann’s paradigm for stochastic climate modelling based on stochastic Lie transport *

A generic approach to stochastic climate modelling is developed for the example of an idealised Atmosphere-Ocean model that rests upon Hasselmann’s paradigm for stochastic climate models. Namely, stochasticity is incorporated into the fast moving atmospheric component of an idealised coupled model by means of stochastic Lie transport, while the slow moving ocean model remains deterministic. More specifically the stochastic model stochastic advection by Lie transport (SALT) is constructed by introducing stochastic transport into the material loop in Kelvin’s circulation theorem. The resulting stochastic model preserves circulation, as does the underlying deterministic climate model. A variant of SALT called Lagrangian-averaged (LA)-SALT is introduced in this paper. In LA-SALT, we replace the drift velocity of the stochastic vector field by its expected value. The remarkable property of LA-SALT is that the evolution of its higher moments are governed by deterministic equations. Our modelling approach is substantiated by establishing local existence results, first, for the deterministic climate model that couples compressible atmospheric equations to incompressible ocean equation, and second, for the two stochastic SALT and LA-SALT models.

Polarization singularities: Topological and dynamical aspects

The review describes general principles and characteristics of inhomogeneously polarized paraxial optical fields and, especially, the polarization singularities (PSs). Main parameters of the optical vector waves are discussed, with the emphasis on the physical relevance and topological distinctiveness of the PSs. Special features of the stochastic vector fields are considered in the context of the PSs’ genericity and structural stability. A detailed attention is paid to interrelations between the PSs and the phase singularities of scalar fields formed by the orthogonal polarization projections of the total field, and their derivatives (complex Stokes fields, phase-difference fields, etc.). On this base, the practical approaches are discussed for the experimental PS identification and characterization. A particular examination of the internal energy flows associated with the PSs, and accompanying distributions of the optical momentum and angular momentum, reveals meaningful dynamical features of PSs and supplies additional physically transparent and informative means for their studies and characterization.

A novel eigenvalue-based iterative simulation method for multi-dimensional homogeneous non-Gaussian stochastic vector fields

The generation of sample functions of stochastic fields or processes is a prerequisite to use the Monte Carlo method in stochastic mechanics and structural reliability analysis. When it comes to non-Gaussian characteristics such as the spatial variability of material or structural properties, the simulation of multi-dimensional non-Gaussian stochastic vector fields becomes necessary. However, the conventional Cholesky decomposition-based iterative simulation method requires three times of decomposition operations at each iteration and a complex procedure of random shuffle, which inevitably augment the model complexity. To this end, this study develops a novel eigenvalue-based iterative simulation method which only requires one decomposition operation in the whole simulation process. Central to this method is to represent the cross power spectral density matrix with its eigenvalue matrix during the iteration procedure, which avoids the cumbersome Cholesky decomposition and random shuffle procedure. Meanwhile, it can yield unique convergence result and present a powerful ability in addressing cases with the complex-valued spectral matrix. Numerical example demonstrates that the proposed method has a similar accuracy as that of the Cholesky decomposition-based iterative simulation method. Meanwhile, the estimated cumulative distribution function from one simulated sample and the estimated ensemble correlation function from 1000 simulated samples are all consistent with their respective targets. In addition, the sensitive analysis is performed to demonstrate that the proposed method has a relatively wide applicability. Therefore, this method may have a great potential to simulate multi-dimensional homogeneous non-Gaussian stochastic vector fields in practice.

Color‐mapped noise vector fields for generating procedural micro‐patterns

AbstractStochastic micro‐patterns successfully enhance the realism of virtual scenes. Procedural models using noise combined with transfer functions are extremely efficient. However, most patterns produced today employ 1D transfer functions, which assign color, transparency, or other material attributes, based solely on the single scalar quantity of noise. Multi‐dimensional transfer functions have received widespread attention in other fields, such as scientific volume rendering. But their potential has not yet been well explored for modeling micro‐patterns in the field of procedural texturing. We propose a new procedural model for stochastic patterns, defined as the composition of a bi‐dimensional transfer function (a.k.a. color‐map) with a stochastic vector field. Our model is versatile, as it encompasses several existing procedural noises, including Gaussian noise and phasor noise. It also generates a much larger gamut of patterns, including locally structured patterns which are notoriously difficult to reproduce. We leverage the Gaussian assumption and a tiling and blending algorithm to provide real‐time generation and filtering. A key contribution is a real‐time approximation of the second order statistics over an arbitrary pixel footprint, which enables, in addition, the filtering of procedural normal maps. We exhibit a wide variety of results, including Gaussian patterns, profiled waves, concentric and non‐concentric patterns.

Spatio-temporal correlation functions in scalar turbulence from functional renormalization group

We provide the leading behavior at large wavenumbers of the two-point correlation function of a scalar field passively advected by a turbulent flow. We first consider the Kraichnan model, in which the turbulent carrier flow is modeled by a stochastic vector field with a Gaussian distribution, and then a scalar advected by a homogeneous and isotropic turbulent flow described by the Navier–Stokes equation, under the assumption that the scalar is passive, i.e., that it does not affect the carrier flow. We show that at large wavenumbers, the two-point correlation function of the scalar in the Kraichnan model decays as an exponential in the time delay, in both the inertial and dissipation ranges. We establish the expression, both from a perturbative and from a nonperturbative calculation, of the prefactor, which is found to be always proportional to k2. For a real scalar, the decay is Gaussian in t at small time delays, and it crosses over to an exponential only at large t. The assumption of delta-correlation in time of the stochastic velocity field in the Kraichnan model, hence, significantly alters the statistical temporal behavior of the scalar at small times.

Ambit Field Modelling of Isotropic, Homogeneous, Divergence-Free and Skewed Vector Fields in Two Dimensions

We discuss the application of ambit fields to the construction of stochastic vector fields in two dimensions that are divergence-free and statistically homogeneous and isotropic but are not invariant under the parity operation. These vector fields are derived from a stochastic stream function defined as a weighted integral with respect to a Lévy basis. By construction, the stream function is homogeneous and isotropic and the corresponding vector field is, in addition, divergence-free. From a decomposition of the kernel in the Lévy-based integral, necessary conditions for the violation of parity symmetry are inferred. In particular, we focus on such fields that allow for skewness of projected increments, which is one of the cornerstones of the Kraichnan–Leith–Bachelor theory of two-dimensional turbulence.

Synthesizing General Electromagnetic Partially Coherent Sources from Random, Correlated Complex Screens

We present a method to generate any genuine electromagnetic partially coherent source (PCS) from correlated, stochastic complex screens. The method described here can be directly implemented on existing spatial-light-modulator-based vector beam generators and can be used in any application which utilizes electromagnetic PCSs. Our method is based on the genuine cross-spectral density matrix criterion. Applying that criterion, we show that stochastic vector field realizations (corresponding to a desired electromagnetic PCS) can be generated by passing correlated Gaussian random numbers through “filters” with space-variant transfer functions. We include step-by-step instructions on how to generate the electromagnetic PCS field realizations. As an example, we simulate the synthesis of a new electromagnetic PCS. Using Monte Carlo analysis, we compute statistical moments from independent optical field realizations and compare those to the corresponding theory. We find that our method produces the desired source—the correct shape, polarization, and coherence properties—within 600 field realizations.

Variational principles for stochastic soliton dynamics.

We develop a variational method of deriving stochastic partial differential equations whose solutions follow the flow of a stochastic vector field. As an example in one spatial dimension, we numerically simulate singular solutions (peakons) of the stochastically perturbed Camassa-Holm (CH) equation derived using this method. These numerical simulations show that peakon soliton solutions of the stochastically perturbed CH equation persist and provide an interesting laboratory for investigating the sensitivity and accuracy of adding stochasticity to finite dimensional solutions of stochastic partial differential equations. In particular, some choices of stochastic perturbations of the peakon dynamics by Wiener noise (canonical Hamiltonian stochastic deformations, CH-SD) allow peakons to interpenetrate and exchange order on the real line in overtaking collisions, although this behaviour does not occur for other choices of stochastic perturbations which preserve the Euler-Poincaré structure of the CH equation (parametric stochastic deformations, P-SD), and it also does not occur for peakon solutions of the unperturbed deterministic CH equation. The discussion raises issues about the science of stochastic deformations of finite-dimensional approximations of evolutionary partial differential equation and the sensitivity of the resulting solutions to the choices made in stochastic modelling.

Markov-Type Vector Field for endless surface animation of water stream

In this paper, we propose a hybrid (physical-stochastic) model of surface element (surfel) fluctuations for the visual simulation of an endlessly running water surface. This model comprises two main phases: preprocessing and endless animation phases. First, we simulate a physics-based method for a specific period of time during the preprocessing phase. We construct a stochastic vector field in the simulation, referred to as a Markov-Type Vector Field (MTVF), using only the surface values of the fluid flow. Next, we import the MTVF data into the main endless animation phase and create a surface fluctuation animation by surfels and temporary velocity field modeling of the MTVF using a random sample. In our approach, the surfel edges that cover the fluid flow domain are transferred simply via a temporary single velocity and the new flow surface is determined directly based on their positions. MTVF allows us to generate a water surface animation endlessly in real time without the time-consuming processes of solving the corresponding physical equations. We describe the MTVF construction method and the endless surface animation steps, as well as present the results of experiments that demonstrate the plausibility of our method.

Camera Motion Estimation Using a Novel Online Vector Field Model in Particle Filters

In this paper, a novel algorithm for parametric camera motion estimation is introduced. More particularly, a novel stochastic vector field model is proposed, which can handle smooth motion patterns derived from long periods of stable camera motion and can also cope with rapid camera motion changes and periods when the camera remains still. The stochastic vector field model is established from a set of noisy measurements, such as motion vectors derived, e.g., from block matching techniques, in order to provide an estimation of the subsequent camera motion in the form of a motion vector field. A set of rules for a robust and online update of the camera motion model parameters is also proposed, based on the expectation maximization algorithm. The proposed model is embedded in a particle filters framework in order to predict the future camera motion based on current and prior observations. We estimate the subsequent camera motion by finding the optimum affine transform parameters so that, when applied to the current video frame, the resulting motion vector field to approximate the one estimated by the stochastic model. Extensive experimental results verify the usefulness of the proposed scheme in camera motion pattern classification and in the accurate estimation of the 2D affine camera transform motion parameters. Moreover, the camera motion estimation has been incorporated into an object tracker in order to investigate if the new schema improves its tracking efficiency, when camera motion and tracked object motion are combined.

EFFECTS OF SPATIAL VARIABILITY OF SOIL PROPERTIES ON SURFACE GROUND MOTION

The effects of spatial variability of soil properties on the behaviour of a cohesive and a cohesionless soil profile subjected to seismic excitation are analysed. The input data for the random fields are derived from existing, extensive soil investigations, and Monte Carlo simulation technique, combining digital generation of non-Gaussian stochastic vector fields with dynamic, equivalent linear and nonlinear finite element analyses, is used for this purpose. It is found that the variability of shear wave velocity has small effect on the ground surface response spectrum. The motion intensity and the variation of soil properties in the deterministic description of the profile considerably affect the standard deviation of the surface spectral acceleration, which is successively compared to the uncertainty introduced in attenuation relationships, used for the construction of hazard-consistent design spectra.

On a class of stochastic partial differential equations related to turbulent transport

We consider the Cauchy problem for the mass density ρ of particles which diffuse in an incompressible fluid. The dynamical behaviour of ρ is modeled by a linear, uniformly parabolic differential equation containing a stochastic vector field. This vector field is interpreted as the velocity field of the fluid in a state of turbulence. Combining a contraction method with techniques from white noise analysis we prove an existence and uniqueness result for the solution ρ∈C 1,2([0,T]×ℝ d ,(S)*), which is a generalized random field. For a subclass of Cauchy problems we show that ρ actually is a classical random field, i.e. ρ(t,x) is an L 2-random variable for all time and space parameters (t,x)∈[0,T]×ℝ d .

Simulation of homogeneous nonGaussian stochastic vector fields

A spectral representation-based simulation methodology is proposed to generate sample functions of a multi-variate, multi-dimensional, nonGaussian stochastic vector field, according to a prescribed cross-spectral density matrix and prescribed (nonGaussian) marginal probability distribution functions. The proposed methodology starts by generating a Gaussian vector field that is then transformed into the desired nonGaussian one using a memoryless nonlinear transformation in conjunction with an iterative scheme. The generation of the Gaussian vector field is performed taking advantage of the Fast Fourier Transform technique for great computational efficiency. The special case of simulation of nonGaussian vector fields modeling material properties is examined, mainly from the point of view of certain simplifying assumptions that can be made for such random media. Finally, a numerical example involving a tri-variate, two-dimensional, nonGaussian stochastic vector field is presented in order to demonstrate the capabilities and the efficiency of the proposed methodology.

Effects of spatial variability on soil liquefaction: some design recommendations

The effects of spatial variability of soil properties on the behaviour of saturated soil deposits subjected to seismic excitation are analysed, and their implications for geotechnical design are discussed. A Monte Carlo simulation methodology, combining digital generation of non-Gaussian stochastic vector fields with dynamic, nonlinear finite element analyses, is used for that purpose. The proposed procedure is applied to assess the soil liquefaction potential, which is found to be considerably affected by the inherent spatial variability of soil properties. Parametric studies, involving the degree of soil liquefaction and the frequency content of the seismic input, are performed. Finally, a characteristic percentile value of soil strength, which will predict a response equivalent to that provided by the more expensive Monte Carlo simulations, is proposed for use in deterministic design.

Stochastic spin-one massive field

Stochastic quantization schemes of Nelson and Parisi and Wu are applied to a spin-one massive field. Unlike the scalar case Nelson's stochastic spin-one massive field cannot be identified with the corresponding euclidean field even if the fourth component of the euclidean coordinate is taken as equal to the real physical time. In the Parisi-Wu quantization scheme the stochastic Proca vector field has a similar property as the scalar field; which has an asymptotically stationary part and a transient part. The large equal-time limit of the expectation values of the stochastic Proca field are equal to the expectation values of the corresponding euclidean field. In the Stueckelberg formalism the Parisi-Wu scheme gives rise to a stochastic vector field which differs from the massless gauge field in that the gauge cannot be fixed by the choice of boundary condition.