Stochastic Variational Principle Research Articles

Variational Integrators for Stochastic Mechanical Hybrid Systems

This paper introduces stochastic variational impact integrators for the class of hybrid mechanical systems that incorporate random noise. The governing equations are obtained by the application of the variational principle to the stochastic action integral, where both the continuous-time dynamics as well as the discrete transitions are considered. Furthermore, structure-preserving geometric integrators are derived through the discretization of the stochastic variational principle. This ensures the consistency in comparison to the continuous versions of the Euler-Lagrange or Hamilton’s equations. The effectiveness of the proposed methods in capturing the long-term energy behavior of a stochastic mechanical hybrid system is illustrated by numerical examples.

Wasserstein Hamiltonian Flow with Common Noise on Graph

We study the Wasserstein Hamiltonian flow with a common noise on the density manifold of a finite graph. Under the framework of the stochastic variational principle, we first develop the formulation of stochastic Wasserstein Hamiltonian flow and show the local existence of a unique solution. We also establish a sufficient condition for the global existence of the solution. Consequently, we obtain the global well-posedness for the nonlinear Schrödinger equations with common noise on a graph. In addition, using Wong–Zakai approximation of common noise, we prove the existence of the minimizer for an optimal control problem with common noise. We show that its minimizer satisfies the stochastic Wasserstein Hamiltonian flow on a graph as well.

Numerical integration of stochastic contact Hamiltonian systems via stochastic Herglotz variational principle

In this work, we establish a stochastic contact variational integrator and its discrete version via stochastic Herglotz variational principle for stochastic contact Hamiltonian systems. A general structure-preserving stochastic contact method is provided to seek the stochastic contact variational integrators. Numerical experiments are performed to verify the validity of this approach.

Transmission and reflection of energy at the boundary of a random two-component composite

A half-space x 2 > 0 is occupied by a two-component statistically uniform random composite with specified volume fractions and two-point correlation. It is bonded to a uniform half-space x 2 < 0 from which a plane wave is incident. The problem requires the specification of the properties of three media: those of the two constituents of the composite and those of the homogeneous half-space. The complexity of the problem is minimized by considering a model acoustic-wave problem in which the two media comprising the composite have the same modulus but different densities. The homogeneous half-space can have any chosen modulus and density. An approximate formulation based on a stochastic variational principle is formulated and solved explicitly in the particular case of an exponentially decaying correlation function, generalizing a previous solution in which the homogeneous medium had the same modulus as the composite. The main novelty of the present work is that the mean energy fluxes in the composite and reflected from the boundary are calculated, demonstrating explicitly the large backscatter associated with the mean-zero component of the reflected signal and the systematic transfer of energy from the decaying mean transmitted waves into the mean-zero disturbance as distance from the boundary increases.

Stochastic Variational Principles for Dissipative Equations with Advected Quantities

This paper presents symmetry reduction for material stochastic Lagrangian systems with advected quantities whose configuration space is a Lie group. Such variational principles yield deterministic as well as stochastic constrained variational principles for dissipative equations of motion in spatial representation. The general theory is presented for the finite-dimensional situation. In infinite dimensions we obtain partial differential equations and stochastic partial differential equations. When the Lie group is, for example, a diffeomorphism group, the general result is not directly applicable but the setup and method suggest rigorous proofs valid in infinite dimensions which lead to similar results. We apply this technique to the compressible Navier–Stokes equation and to magnetohydrodynamics for charged viscous compressible fluids. A stochastic Kelvin–Noether theorem is presented. We derive, among others, the classical deterministic dissipative equations from purely variational and stochastic principles, without any appeal to thermodynamics.

Stochastic variational principles for the collisional Vlasov-Maxwell and Vlasov-Poisson equations.

In this work, we recast the collisional Vlasov–Maxwell and Vlasov–Poisson equations as systems of coupled stochastic and partial differential equations, and we derive stochastic variational principles which underlie such reformulations. We also propose a stochastic particle method for the collisional Vlasov–Maxwell equations and provide a variational characterization of it, which can be used as a basis for a further development of stochastic structure-preserving particle-in-cell integrators.

Stochastic effects of waves on currents in the ocean mixed layer

This paper introduces an energy-preserving stochastic model for studying wave effects on currents in the ocean mixing layer. The model is called stochastic forcing by Lie transport (SFLT). The SFLT model is derived here from a stochastic constrained variational principle, so it has a Kelvin circulation theorem. The examples of SFLT given here treat 3D Euler fluid flow, rotating shallow water dynamics, and the Euler–Boussinesq equations. In each example, one sees the effect of stochastic Stokes drift and material entrainment in the generation of fluid circulation. We also present an Eulerian averaged SFLT model based on decomposing the Eulerian solutions of the energy-conserving SFLT model into sums of their expectations and fluctuations.

Semi-martingale driven variational principles

Spearheaded by the recent efforts to derive stochastic geophysical fluid dynamics models, we present a general framework for introducing stochasticity into variational principles through the concept of a semi-martingale driven variational principle and constraining the component variables to be compatible with the driving semi-martingale. Within this framework and the corresponding choice of constraints, the Euler–Poincaré equation can be easily deduced. We show that the deterministic theory is a special case of this class of stochastic variational principles. Moreover, this is a natural framework that enables us to correctly characterize the pressure term in incompressible stochastic fluid models. Other general constraints can also be incorporated as long as they are compatible with the driving semi-martingale.

The effect of a noise on the stochastic modified Camassa–Holm equation

We study the effect of noise on the stochastic modified Camassa–Holm equation. We first derive the stochastic modified Camassa–Holm equation by the stochastic variational principle. Then, we prove the well-posedness of the stochastic modified Camassa–Holm equation by the iterative process. Under the condition of the small noise intensity, we can also get the regularization of the solution from the parabolic term.

Circulation and Energy Theorem Preserving Stochastic Fluids

AbstractSmooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier–Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin–Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called thestochastic Euler–Poincaréandstochastic Navier–Stokes–Poincaréequations respectively. The stochastic Euler–Poincaré equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems.

Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers

We show that the Navier-Stokes as well as a random perturbation of this equation can be derived from a stochastic variational principle where the pressure is introduced as a Lagrange multiplier. Moreover we describe how to obtain corresponding constants of the motion.

Stochastic Geometric Models with Non-stationary Spatial Correlations in Lagrangian Fluid Flows

Inspired by spatiotemporal observations from satellites of the trajectories of objects drifting near the surface of the ocean in the National Oceanic and Atmospheric Administration’s “Global Drifter Program”, this paper develops data-driven stochastic models of geophysical fluid dynamics (GFD) with non-stationary spatial correlations representing the dynamical behaviour of oceanic currents. Three models are considered. Model 1 from Holm (Proc R Soc A 471:20140963, 2015) is reviewed, in which the spatial correlations are time independent. Two new models, called Model 2 and Model 3, introduce two different symmetry breaking mechanisms by which the spatial correlations may be advected by the flow. These models are derived using reduction by symmetry of stochastic variational principles, leading to stochastic Hamiltonian systems, whose momentum maps, conservation laws and Lie–Poisson bracket structures are used in developing the new stochastic Hamiltonian models of GFD.

A stochastic variational approach to viscous Burgers equations

We consider viscous Burgers equations in one dimension of space and derive their solutions from stochastic variational principles on the corresponding group of homeomorphisms. The metrics considered on this group are Lp metrics. The velocity corresponds to the drift of some stochastic Lagrangian processes. Existence of minima is proved in some cases by direct methods. We also give a representation of the solutions of viscous Burgers equations in terms of stochastic forward-backward systems.

A stochastic variational approach to the viscous Camassa–Holm and Leray-alpha equations

We derive the (d-dimensional) periodic incompressible and viscous Camassa–Holm equation as well as the Leray-alpha equations via stochastic variational principles. We discuss the existence of solution for these equations in the space H1 using the probabilistic characterization. The underlying Lagrangian flows are diffusion processes living in the group of diffeomorphisms of the torus. We study in detail these diffusions.

SI “Deterministic and Stochastic Variational Principles and Applications”. December 2015

Variational methods are important tools for deriving optimality conditions and corresponding algorithms for solving optimization problems. Ekeland’s variational principle is a deep assertion about the existence of an exact solution of a perturbed optimization problem in a neighborhood of an approximate solution of the original problem. The aim of this special issue was to discuss new variational approaches from the theoretical as well as computational point of view. Furthermore, we are interested in applications of the variational methods for deterministic and stochastic models, especially in approximation theory and optimal control. In the special issue the importance of variational methods in optimization and stochastics is described.We present newdevelopments on the field of deterministic and stochastic variational methods including applications in mathematics and economics. Most of the papers in the special issue are dealing with optimal control and state estimates problems in the deterministic as well as stochastic case and maximum principles (Abdelmadjid Abba: On Mean-field Partial Information Maximum Principle of Optimal Control for Stochastic Systems with Levy Processes, Vo Anh: Least-Squares Estimation of Multifractional Random Fields in a Hilbert-Valued Context, Brigitte Breckner: Multiple solutions of Dirichlet problems on the Sierpinski gasket, Ioana Ciotir: A variational approach to Neumann stochastic semi-linear equations modelling the thermostatic control, CsabaVarga, Hannelore Lisei: Amultiplicity result for a class of elliptic problems on a compact Riemannian manifold, Tina Engler: On Investment Consumption Modeling with Jump Process Extensions for Productive Sectors, Diana

Stochastic variational quantization and maximum entropy principle

In this note, we discuss the stochastic variational method (SVM) to derive the action principle of Schrodinger equation. In this formalism, the wave function is introduced as one of convenient ways to express the action. This work is a reformulation of Nelson—Yasue’s stochastic quantization scheme and its variational formulation by Yasue, and will furnish possible way to deal with more complicated system such as quantum field theory, suggesting an origin of quantum mechanics.

Variational principles for stochastic fluid dynamics.

This paper derives stochastic partial differential equations (SPDEs) for fluid dynamics from a stochastic variational principle (SVP). The paper proceeds by taking variations in the SVP to derive stochastic Stratonovich fluid equations; writing their Itô representation; and then investigating the properties of these stochastic fluid models in comparison with each other, and with the corresponding deterministic fluid models. The circulation properties of the stochastic Stratonovich fluid equations are found to closely mimic those of the deterministic ideal fluid models. As with deterministic ideal flows, motion along the stochastic Stratonovich paths also preserves the helicity of the vortex field lines in incompressible stochastic flows. However, these Stratonovich properties are not apparent in the equivalent Itô representation, because they are disguised by the quadratic covariation drift term arising in the Stratonovich to Itô transformation. This term is a geometric generalization of the quadratic covariation drift term already found for scalar densities in Stratonovich's famous 1966 paper. The paper also derives motion equations for two examples of stochastic geophysical fluid dynamics; namely, the Euler–Boussinesq and quasi-geostropic approximations.

Multiscale stochastic finite element method on random field modeling of geotechnical problems — a fast computing procedure

The Green-function-based multiscale stochastic finite element method (MSFEM) has been formulated based on the stochastic variational principle. In this study a fast computing procedure based on the MSFEM is developed to solve random field geotechnical problems with a typical coefficient of variance less than 1. A unique fast computing advantage of the procedure enables computation performed only on those locations of interest, therefore saving a lot of computation. The numerical example on soil settlement shows that the procedure achieves significant computing efficiency compared with Monte Carlo method.

Stochastic roots of growth phenomena

We show that the Gompertz equation describes the evolution in time of the median of a geometric stochastic process. Therefore, we induce that the process itself generates the growth. This result allows us further to exploit a stochastic variational principle to take account of self-regulation of growth through feedback of relative density variations. The conceptually well defined framework so introduced shows its usefulness by suggesting a form of control of growth by exploiting external actions.

Hierarchical stochastic finite element method for structural analysis

In this paper, the hierarchical approach is adopted for series representation of the stochastic nodal displacement vector using the hierarchical basis vectors, while the Karhunen-Loève series expansion technique is employed to discretize the random field into a set of random variables. A set of hierarchical basis vectors are defined to approximate the stochastic response quantities. The stochastic variational principle instead of the projection scheme is adopted to develop a hierarchical stochastic finite element method (HSFEM) for stochastic structures under stochastic loads. Simplified expressions of coefficients of governing equations and the first two statistical moments of the response quantities in the schemes of the HSFEM are developed, so that the time consumed for computation can be greatly reduced. Investigation in this paper suggests that the HSFEM yields a series of stiffness equations with similar dimensionality as the perturbation stochastic finite element method (PSFEM). Two examples are presented for numerical study on the performance of the HSFEM in elastic structural problems with stochastic Young's Modulus and external loads. Results show that the proposed method can achieve higher accuracy than the PSFEM for cases with large coefficients of variation, and yield results agreeing well with those obtained by the Monte Carlo simulation (MCS).