Random Forcing Term Research Articles

Numerical simulations of a stochastic dynamics leading to cascades and loss of regularity: Applications to fluid turbulence and generation of fractional Gaussian fields

Motivated by the modeling of the spatial structure of the velocity field of three-dimensional turbulent flows, and the phenomenology of cascade phenomena, a linear dynamics was recently proposed that can generate high velocity gradients from a smooth-in-space forcing term. It is based on a linear partial differential equation stirred by an additive random forcing term that is δ-correlated in time. The underlying proposed deterministic mechanism corresponds to a transport in Fourier space that aims to transfer energy injected at large scales towards small scales. The key role of the random forcing is to realize these transfers in a statistically homogeneous way. Whereas at finite times and positive viscosity the solutions are smooth, a loss of regularity is observed for the statistically stationary state in the inviscid limit. We present here simulations, based on finite volume methods in the Fourier domain and a splitting method in time, which are more accurate than the pseudospectral simulations. We show that our algorithm is able to reproduce accurately the expected local and statistical structure of the predicted solutions. We conduct numerical simulations in one, two, and three spatial dimensions, and we display the solutions both in physical and Fourier space. We additionally display key statistical quantities such as second-order structure functions and power spectral densities at various viscosities. Published by the American Physical Society 2024

Deep Latent Regularity Network for Modeling Stochastic Partial Differential Equations

Stochastic partial differential equations (SPDEs) are crucial for modelling dynamics with randomness in many areas including economics, physics, and atmospheric sciences. Recently, using deep learning approaches to learn the PDE solution for accelerating PDE simulation becomes increasingly popular. However, SPDEs have two unique properties that require new design on the models. First, the model to approximate the solution of SPDE should be generalizable over both initial conditions and the random sampled forcing term. Second, the random forcing terms usually have poor regularity whose statistics may diverge (e.g., the space-time white noise). To deal with the problems, in this work, we design a deep neural network called \emph{Deep Latent Regularity Net} (DLR-Net). DLR-Net includes a regularity feature block as the main component, which maps the initial condition and the random forcing term to a set of regularity features. The processing of regularity features is inspired by regularity structure theory and the features provably compose a set of basis to represent the SPDE solution. The regularity features are then fed into a small backbone neural operator to get the output. We conduct experiments on various SPDEs including the dynamic $\Phi^4_1$ model and the stochastic 2D Navier-Stokes equation to predict their solutions, and the results demonstrate that the proposed DLR-Net can achieve SOTA accuracy compared with the baselines. Moreover, the inference time is over 20 times faster than the traditional numerical solver and is comparable with the baseline deep learning models.

Generalized Langevin equationwith a nonlinear potential of mean force and nonlinear memory friction from a hybrid projection scheme.

We introduce a hybrid projection scheme that combines linear Mori projection and conditional Zwanzig projection techniques and use it to derive a generalized Langevin equation(GLE) for a general interacting many-body system. The resulting GLE includes (i) explicitly the potential of mean force (PMF) that describes the equilibrium distribution of the system in the chosen space of reaction coordinates, (ii) a random force term that explicitly depends on the initial state of the system, and (iii) a memory friction contribution that splits into two parts: a part that is linear in the past reaction-coordinate velocity and a part that is in general nonlinear in the past reaction coordinates but does not depend on velocities. Our hybrid scheme thus combines all desirable properties of the Zwanzig and Mori projection schemes. The nonlinear memory friction contribution is shown to be related to correlations between the reaction-coordinate velocity and the random force. We present a numerical method to compute all parameters of our GLE, in particular the nonlinear memory friction function and the random force distribution, from a trajectory in reaction coordinate space. We apply our method on the dihedral-angle dynamics of a butane molecule in water obtained from atomistic molecular dynamics simulations. For this example, we demonstrate that nonlinear memory friction is present and that the random force exhibits significant non-Gaussian corrections. We also present the derivation of the GLE for multidimensional reaction coordinates that are general functions of all positions in the phase-space of the underlying many-body system; this corresponds to a systematic coarse-graining procedure that preserves not only the correct equilibrium behavior but also the correct dynamics of the coarse-grained system.

Non-intrusive reduced-order modeling for uncertainty quantification of space–time-dependent parameterized problems

We propose a non-intrusive reduced-order modeling method for space–time-dependent parameterized problems in the context of uncertainty quantification. In the offline stage, proper orthogonal decomposition (POD) is used to extract the spatial modes based on a set of high-fidelity time-parameter-dependent snapshots and then to extract the temporal modes of the projection coefficients of the spatial modes. Finally, parameter-dependent combination coefficients are approximated using polynomial chaos expansions (PCEs). Cubic spline interpolation is used to evaluate the temporal modes at any other given time. In the online stage, a fast evaluation is provided at any given time and parameter by simply estimating the values of the polynomials and temporal modes. To validate the numerical performance of the proposed method, three time-dependent parameterized problems are tested: a one-dimensional (1-D) forced Burger’s equation with a random force term, a 1-D diffusion–reaction equation with a random field force term, and a two-dimensional incompressible fluid flow over a cylinder with a random inflow boundary condition. The results indicate that the proposed method is able to approximate the full-order model very inexpensively with a reasonable loss of accuracy for problems with uncorrelated or correlated input parameters. Furthermore, the proposed method is effective in estimating low-order moments, indicating that it has great potential for use in uncertainty quantification analysis of space–time-dependent problems.

Lp-Solution to the Random Linear Delay Differential Equation with a Stochastic Forcing Term

This paper aims at extending a previous contribution dealing with the random autonomous-homogeneous linear differential equation with discrete delay τ > 0 , by adding a random forcing term f ( t ) that varies with time: x ′ ( t ) = a x ( t ) + b x ( t − τ ) + f ( t ) , t ≥ 0 , with initial condition x ( t ) = g ( t ) , − τ ≤ t ≤ 0 . The coefficients a and b are assumed to be random variables, while the forcing term f ( t ) and the initial condition g ( t ) are stochastic processes on their respective time domains. The equation is regarded in the Lebesgue space L p of random variables with finite p-th moment. The deterministic solution constructed with the method of steps and the method of variation of constants, which involves the delayed exponential function, is proved to be an L p -solution, under certain assumptions on the random data. This proof requires the extension of the deterministic Leibniz’s integral rule for differentiation to the random scenario. Finally, we also prove that, when the delay τ tends to 0, the random delay equation tends in L p to a random equation with no delay. Numerical experiments illustrate how our methodology permits determining the main statistics of the solution process, thereby allowing for uncertainty quantification.

A Multi-scale Limit of a Randomly Forced Rotating 3-D Compressible Fluid

We study a singular limit of a scaled compressible Navier–Stokes–Coriolis system driven by both a deterministic and stochastic forcing terms in three dimensions. If the Mach number is comparable to the Froude number with both proportional to say \(\varepsilon \ll 1\), whereas the Rossby number scales like \(\varepsilon ^m\) for \(m>1\) large, then we show that any family of weak martingale solution to the 3-D randomly forced rotating compressible equation (under the influence of a deterministic centrifugal force) converges in probability, as \(\varepsilon \rightarrow 0\), to the 2-D incompressible Navier–Stokes system with a corresponding random forcing term.

On the Theory of a Nonlinear Dynamic Circuit Filtering

Stochastic Differential Equations (SDEs) describe physical systems to account for random forcing terms in the evolution of the state trajectory. The noisy sampling mixer, a component of digital wireless communications, can be regarded as a potential case from the dynamical systems’ viewpoint. The universality of the noisy sampling mixer is attributed to the fact that it adopts the structure of a nonlinear SDE and its linearized version becomes a time-varying bilinear SDE. This paper develops a mathematical theory for the nonlinear noisy sampling mixer from the filtering viewpoint. Since the filtering of stochastic systems hinges on the structure of dynamical systems and observation equation set up, we consider three ‘filtering models’. The first model, accounts for a nonlinear SDE coupled with a nonlinear observation equation. In the second model, we consider a bilinear SDE with a linear observation equation to achieve the nonlinear sampling filtering. Note that the bilinear SDE coupled with the linear observation is a consequence of the Carleman linearization to the nonlinear SDE and the nonlinear observation equation. In the third model, we consider a Stratonovich SDE coupled with a nonlinear observation equation. The filtering equation of this paper can be further utilized to guide the design process of the noisy sampling mixer.

Random time step probabilistic methods for uncertainty quantification in chaotic and geometric numerical integration

A novel probabilistic numerical method for quantifying the uncertainty induced by the time integration of ordinary differential equations (ODEs) is introduced. Departing from the classical strategy to randomize ODE solvers by adding a random forcing term, we show that a probability measure over the numerical solution of ODEs can be obtained by introducing suitable random time-steps in a classical time integrator. This intrinsic randomization allows for the conservation of geometric properties of the underlying deterministic integrator such as mass conservation, symplecticity or conservation of first integrals. Weak and mean-square convergence analysis are derived. We also analyse the convergence of the Monte Carlo estimator for the proposed random time step method and show that the measure obtained with repeated sampling converges in the mean-square sense independently of the number of samples. Numerical examples including chaotic Hamiltonian systems, chemical reactions and Bayesian inferential problems illustrate the accuracy, robustness and versatility of our probabilistic numerical method.

Stability of strong solutions for a model of incompressible two–phase flow under thermal fluctuations

We consider a model of a two–phase flow based on the phase field approach, where the fluid bulk velocity obeys the standard Navier–Stokes system while the concentration difference of the two fluids plays a role of order parameter governed by the Allen–Cahn equations. Possible thermal fluctuations are incorporated through a random forcing term in the Allen–Cahn equation. We show that suitable dissipative martingale solutions satisfy a stochastic version of the relative energy inequality. This fact is used for showing the weak–strong uniqueness principle both pathwise and in law.

Equations of motion for position-dependent coarse-grain mappings obtained with Mori-Zwanzig theory.

A position-dependent transformation is introduced for mapping a system of atomistic particles to a system of coarse-grained (CG) variables, which under some circumstances might be considered particles. This CG mapping allows atomistic particles to simultaneously contribute to more than a single CG particle and to change in time the CG particle they are associated with. That is, the CG mapping is dynamic. Mori-Zwanzig theory is then used to obtain the equations of motion for this CG mapping, resulting in conservative, dissipative, and random force terms in generalized, non-Markovian Langevin equations. In addition to the usual forces arising from the effective CG potential derived from atomistic interactions, new forces arise from the dynamic changes in the CG mapping itself. These new forces effectively account for changes arising from fluxes of atomistic particles into and out of CG ones as time progresses. Several examples are given showing the range of problems that can be addressed with this new CG mapping. These range from the usual case where atomistic particles are grouped into large molecular-like chunks, with mappings that remain fixed in time and for which an atomistic particle is part of only a single CG one, to the case where CG particles resemble fluid elements, containing many hundreds of independent atomistic particles. The new CG mapping also allows for hybrid descriptions, in which a part of the system remains atomistic or molecular-like and a part is highly coarse-grained to mesoscopic fluid element-like particles, for example. In the latter case, the equations of motion then provide the correct formalism for determining the forces, beyond the usual conservative ones. This provides a theoretical foundation upon which approximate equations of motion can be formulated to thus build numerical algorithms for expanded applications of accurate CG molecular dynamics.

Some non-existence results for a class of stochastic partial differential equations

Consider the following stochastic partial differential equation,∂tut(x)=Lut(x)+σ(ut(x))F˙(t,x)t>0andx∈Rd. The operator L is the generator of a strictly stable process and F˙ is the random forcing term which is assumed to be Gaussian. Under some additional conditions, most notably on σ and the initial condition, we show non-existence of global random field solutions. Our results complement those of P. Chow in [5] and [6] among others.

Melnikov processes and chaos in randomly perturbed dynamical systems

We consider a wide class of randomly perturbed systems subjected to stationary Gaussian processes and show that chaotic orbits exist almost surely under some nondegenerate condition, no matter how small the random forcing terms are. This result is very contrasting to the deterministic forcing case, in which chaotic orbits exist only if the influence of the forcing terms overcomes that of the other terms in the perturbations. To obtain the result, we extend Melnikov’s method and prove that the corresponding Melnikov functions, which we call the Melnikov processes, have infinitely many zeros, so that infinitely many transverse homoclinic orbits exist. In addition, a theorem on the existence and smoothness of stable and unstable manifolds is given and the Smale–Birkhoff homoclinic theorem is extended in an appropriate form for randomly perturbed systems. We illustrate our theory for the Duffing oscillator subjected to the Ornstein–Uhlenbeck process parametrically.

Langevin equation elucidates the mechanism of the Rayleigh-Bénard instability by coupling molecular motions and macroscopic fluctuations.

It is well known that Brownian motion can be described using Langevin equation. In this paper we extend the application of the Langevin equation to the Rayleigh-Bénard (RB) flow, assuming that each molecule in the system is a Brownian particle colliding with its surrounding molecules. The phenomenon of thermal instability, changing from a conductive to a convective state, is well reproduced by Langevin dynamics simulations. The roles of the drag force and the random force terms in the Langevin equation in triggering thermal instability are elucidated via numerical tests. Furthermore, we demonstrate that the strength of the fluctuation correlations increases as the Rayleigh number approaches the critical value, and the characteristics of the fluctuation correlations below the onset of instability foreshadow the form of the convective patterns emerging above the critical point. The Langevin equation, together with the form of the fluctuation correlations, sheds new light on the mechanism of the RB instability.

Idealized numerical simulations of Mesoscale Convective Systems and their implications for forecast error

Deep convective mass transfer in a rotating, stratified fluid is capable of generating characteristic potential vorticity features and, if of sufficient spatial scale, these may affect baroclinic wave development and hence weather predictability. In current global numerical prediction models, it is still necessary to include convection parametrization even though these algorithms are ill‐suited to representing singular, mesoscale convective events. Some stochastic representations of the non‐equilibrium convective process in ensemble prediction systems perturb the vertical component of vorticity in order to crudely represent the upscale impact of deep convection (i.e. kinetic energy backscatter schemes). These use ad hoc assumptions about the structure of random wind errors associated with convection. The idealized, explicit convection simulations described here form the basis of a study of these convectively generated vorticity perturbations, and their implications for stochastic convective forcing in ensemble prediction systems is considered. Cases with no initial flow and constant shear are examined with emphasis on the impact on vertical vorticity within the mesoscale cloud shield. The results are interpreted in the wider context of operational forecast error in the simulated infrared radiance for cases of intense convective activity over the North American Great Plains. A Met Office Unified Model forecast made with 2.2 km horizontal resolution during the Hazardous Weather Testbed project provides additional support for the idealized model findings. The Stochastic Convective Backscatter scheme proposed by the author is integrated forward in time using the simulated convective mass transport as a driver for the random forcing term. In its ensemble forecast model implementation, this term uses parametrized convective mass fluxes. The results suggest a retuning of the scheme's filter scales in order to obtain a representation of model wind error more consistent with the simulations.

Approximation of solutions to a delay equation with a random forcing term and non local conditions

The existence and approximation of a solution to a delay equation with a random forcing term and non local conditions is studied by using a stochastic version of the well-known Banach fixed point theorem and semigroup theory. Moreover, the convergence of Faedo-Galerkin approximations of the solution is shown. An example is given which illustrates the results.

Multidimensional Potential Burgers Turbulence

We consider the multidimensional generalised stochastic Burgers equation in the space-periodic setting: $$\frac{\partial \mathbf{u}}{\partial t}+(\nabla f(\mathbf{u}) \cdot \nabla) \mathbf{u}-\nu \Delta \mathbf{u}= \nabla \eta, \quad t \geq 0, \, \mathbf{x} \in\mathbb{T}^d=(\mathbb{R}/ \mathbb{Z})^d,$$ under the assumption that u is a gradient. Here f is strongly convex and satisfies a growth condition, ν is small and positive, while η is a random forcing term, smooth in space and white in time. For solutions u of this equation, we study Sobolev norms of u averaged in time and in ensemble: each of these norms behaves as a given negative power of ν. These results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence, namely the averages of the increments and of the energy spectrum. These quantities behave as a power of the norm of the relevant parameter, which is respectively the separation l in the physical space and the wavenumber k in the Fourier space. Our bounds do not depend on the initial condition and hold uniformly in \({\nu}\). We generalise the results obtained for the one-dimensional case in [10], confirming the physical predictions in [4, 30]. Note that the form of the estimates does not depend on the dimension: the powers of \({\nu, |{\mathbf{k}}|, \ell}\) are the same in the one- and the multi-dimensional setting.

A stochastic collocation method for stochastic Volterra equations of the second kind

This work describes and analyzes a stochastic collocation method for stochastic Volterra integral equations (SVIEs) of the second kind with random forcing terms. A collocation method is used in temporal direction, and a spectral collocation method is used in the stochastic dimension, which lead to an uncoupled linear system associated with the collocation points. The convergence analysis is carried out and the optimal error estimates are obtained. Numerical experiments are conducted to verify the error estimates and demonstrate the effectiveness of the proposed numerical method.

First-principles calculations of internal friction of a Gaussian chain

We consider internal friction of a Gaussian chain placed in vacuum. Dynamics of a single chain can be described by Langevin equation where the random force term comes from the random collisions of monomers among themselves. To nd the moment of these random forces, Gaussian distribution for chain conformation is used . Since collisions between monomers cause these forces, random forces on two monomers can be equal and opposite at the same time. Such correlation is exploited to nd friction constant for Fourier mode p. We show that this same mechanism of internal friction is also applicable for polymers in solution.

Statistical analysis and simulation of random shocks in stochastic Burgers equation

We study the statistical properties of random shock waves in stochastic Burgers equation subject to random space–time perturbations and random initial conditions. By using the response–excitation probability density function (PDF) method and the Mori–Zwanzig (MZ) formulation of irreversible statistical mechanics, we derive exact reduced-order equations for the one-point and two-point PDFs of the solution field. In particular, we compute the statistical properties of random shock waves in the inviscid limit by using an adaptive (shock-capturing) discontinuous Galerkin method in both physical and probability spaces. We consider stochastic flows generated by high-dimensional random initial conditions and random additive forcing terms, yielding multiple interacting shock waves collapsing into clusters and settling down to a similarity state. We also address the question of how random shock waves in space and time manifest themselves in probability space. The proposed new mathematical framework can be applied to different conservation laws, potentially leading to new insights into high-dimensional stochastic dynamical systems and more efficient computational algorithms.

A MULTI-FIDELITY STOCHASTIC COLLOCATION METHOD FOR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS WITH RANDOM INPUT DATA

Over the last few years there have been dramatic advances in the area of uncertainty quantification. In particular, we have seen a surge of interest in developing efficient, scalable, stable, and convergent computational methods for solving differential equations with random inputs. Stochastic collocation (SC) methods, which inherit both the ease of implementation of sampling methods like Monte Carlo and the robustness of nonsampling ones like stochastic Galerkin to a great deal, have proved extremely useful in dealing with differential equations driven by random inputs. In this work we propose a novel enhancement to stochastic collocation methods using deterministic model reduction techniques. Linear parabolic partial differential equations with random forcing terms are analysed. The input data are assumed to be represented by a finite number of random variables. A rigorous convergence analysis, supported by numerical results, shows that the proposed technique is not only reliable and robust but also efficient.