Orbit Classification and Sensitivity Analysis in Dynamical Systems Using Surrogate Models

Dynamics of many classical physics systems are described in terms of Hamilton’s equations. Commonly, initial conditions are only imperfectly known. The associated volume in phase space is preserved over time due to the symplecticity of the Hamiltonian flow. Here we study the propagation of uncertain initial conditions through dynamical systems using symplectic surrogate models of Hamiltonian flow maps. This allows fast sensitivity analysis with respect to the distribution of initial conditions and an estimation of local Lyapunov exponents (LLE) that give insight into local predictability of a dynamical system. In Hamiltonian systems, LLEs permit a distinction between regular and chaotic orbits. Combined with Bayesian methods we provide a statistical analysis of local stability and sensitivity in phase space for Hamiltonian systems. The intended application is the early classification of regular and chaotic orbits of fusion alpha particles in stellarator reactors. The degree of stochastization during a given time period is used as an estimate for the probability that orbits of a specific region in phase space are lost at the plasma boundary. Thus, the approach offers a promising way to accelerate the computation of fusion alpha particle losses.

THE GENERALIZED GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM`

Geometric Quantum Mechanics is a version of quantum theory that has been formulated in terms of Hamiltonian phase-space dynamics. The states in this framework belong to points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is described by the Schr{\"o}dinger equation. Besides, one has demonstrated that the stronger version of the uncertainty relation, namely the Robertson-Schr{\"o}dinger uncertainty relation, may be stated using symplectic form and Riemannian metric. In this research, the generalized Robertson-Schr{\"o}dinger uncertainty principle for spin $\frac{1}{2}$ system has been constructed by considering the operators corresponding to arbitrary direction.

THE DYNAMICAL EVOLUTION OF GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM

Geometric Quantum Mechanics is a formulation that demonstrates how quantum theory may be casted in the language of Hamiltonian phase-space dynamics. In this framework, the states are referring to points in complex projective Hilbert space, the observables are real valued functions on the space and the Hamiltonian flow is defined by Schr{\"o}dinger equation. Recently, the effort to cast uncertainty principle in terms of geometrical language appeared to become the subject of intense study in geometric quantum mechanics. One has shown that the stronger version of uncertainty relation i.e. the Robertson-Schr{\"o}dinger uncertainty relation can be expressed in terms of the symplectic form and Riemannian metric. In this paper, we investigate the dynamical behavior of the uncertainty relation for spin $\frac{1}{2}$ system based on this formulation. We show that the Robertson-Schr{\"o}dinger uncertainty principle is not invariant under Hamiltonian flow. This is due to the fact that during evolution process, unlike symplectic area, the Riemannian metric is not invariant under the flow.

A Novel Hybrid Monte Carlo Algorithm for Sampling Path Space

To sample from complex, high-dimensional distributions, one may choose algorithms based on the Hybrid Monte Carlo (HMC) method. HMC-based algorithms generate nonlocal moves alleviating diffusive behavior. Here, I build on an already defined HMC framework, hybrid Monte Carlo on Hilbert spaces (Beskos, et al. Stoch. Proc. Applic. 2011), that provides finite-dimensional approximations of measures π, which have density with respect to a Gaussian measure on an infinite-dimensional Hilbert (path) space. In all HMC algorithms, one has some freedom to choose the mass operator. The novel feature of the algorithm described in this article lies in the choice of this operator. This new choice defines a Markov Chain Monte Carlo (MCMC) method that is well defined on the Hilbert space itself. As before, the algorithm described herein uses an enlarged phase space Π having the target π as a marginal, together with a Hamiltonian flow that preserves Π. In the previous work, the authors explored a method where the phase space π was augmented with Brownian bridges. With this new choice, π is augmented by Ornstein–Uhlenbeck (OU) bridges. The covariance of Brownian bridges grows with its length, which has negative effects on the acceptance rate in the MCMC method. This contrasts with the covariance of OU bridges, which is independent of the path length. The ingredients of the new algorithm include the definition of the mass operator, the equations for the Hamiltonian flow, the (approximate) numerical integration of the evolution equations, and finally, the Metropolis–Hastings acceptance rule. Taken together, these constitute a robust method for sampling the target distribution in an almost dimension-free manner. The behavior of this novel algorithm is demonstrated by computer experiments for a particle moving in two dimensions, between two free-energy basins separated by an entropic barrier.

The essential coexistence phenomenon in Hamiltonian dynamics

We construct an example of a Hamiltonian flow $f^t$ on a four-dimensional smooth manifold $\mathcal {M}$ which after being restricted to an energy surface $\mathcal {M}_e$ demonstrates essential coexistence of regular and chaotic dynamics, that is, there is an open and dense $f^t$ -invariant subset $U\subset \mathcal {M}_e$ such that the restriction $f^t|U$ has non-zero Lyapunov exponents in all directions (except for the direction of the flow) and is a Bernoulli flow while, on the boundary $\partial U$ , which has positive volume, all Lyapunov exponents of the system are zero.

A Novel Hybrid Monte Carlo Algorithm for Sampling Path Space

To sample from complex, high-dimensional distributions, one may choose algorithms based on the Hybrid Monte Carlo (HMC) method. HMC-based algorithms generate nonlocal moves alleviating diffusive behavior. Here, I build on an already defined HMC framework, Hybrid Monte Carlo on Hilbert spaces [A. Beskos, F.J. Pinski, J.-M. Sanz-Serna and A.M. Stuart, Stoch. Proc. Applic. 121, 2201 - 2230 (2011); doi:10.1016/j.spa.2011.06.003] that provides finite-dimensional approximations of measures π which have density with respect to a Gaussian measure on an infinite-dimensional Hilbert (path) space. In all HMC algorithms, one has some freedom to choose the mass operator. The novel feature of the algorithm described in this article lies in the choice of this operator. This new choice defines a Markov Chain Monte Carlo (MCMC) method which is well defined on the Hilbert space itself. As before, the algorithm described herein uses an enlarged phase space Π having the target π as a marginal, together with a Hamiltonian flow that preserves Π. In the previous method, the phase space π was augmented with Brownian bridges. With the new choice for the mass operator, π is augmented with Ornstein-Uhlenbeck (OU) bridges. The covariance of Brownian bridges grows with its length, which has negative effects on the Metropolis-Hasting acceptance rate. This contrasts with the covariance of OU bridges which is independent of the path length. The ingredients of the new algorithm include the definition of the mass operator, the equations for the Hamiltonian flow, the (approximate) numerical integration of the evolution equations, and finally the Metropolis-Hastings acceptance rule. Taken together, these constitute a robust method for sampling the target distribution in an almost dimension-free manner. The behavior of this novel algorithm is demonstrated by computer experiments for a particle moving in two dimensions, between two free-energy basins separated by an entropic barrier.

Weyl Law on Asymptotically Euclidean Manifolds

We study the asymptotic behaviour of the eigenvalue counting function for self-adjoint elliptic linear operators defined through classical weighted symbols of order (1, 1), on an asymptotically Euclidean manifold. We first prove a two-term Weyl formula, improving previously known remainder estimates. Subsequently, we show that under a geometric assumption on the Hamiltonian flow at infinity, there is a refined Weyl asymptotics with three terms. The proof of the theorem uses a careful analysis of the flow behaviour in the corner component of the boundary of the double compactification of the cotangent bundle. Finally, we illustrate the results by analysing the operator $$Q=(1+|x|^2)(1-\varDelta )$$ Q = ( 1 + | x | 2 ) ( 1 - Δ ) on $$\mathbb {R}^d$$ R d .

Adiabatic Invariants for the FPUT and Toda Chain in the Thermodynamic Limit

We consider the Fermi–Pasta–Ulam–Tsingou (FPUT) chain composed by $$N \gg 1$$ N ≫ 1 particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature $$\beta ^{-1}$$ β - 1 . Given a fixed $${1\le m \ll N}$$ 1 ≤ m ≪ N , we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order $$\beta $$ β , for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics.

Global Geometry of Bayesian Statistics

In the previous work of the author, a non-trivial symmetry of the relative entropy in the information geometry of normal distributions was discovered. The same symmetry also appears in the symplectic/contact geometry of Hilbert modular cusps. Further, it was observed that a contact Hamiltonian flow presents a certain Bayesian inference on normal distributions. In this paper, we describe Bayesian statistics and the information geometry in the language of current geometry in order to spread our interest in statistics through general geometers and topologists. Then, we foliate the space of multivariate normal distributions by symplectic leaves to generalize the above result of the author. This foliation arises from the Cholesky decomposition of the covariance matrices.

Symplectic/Contact Geometry Related to Bayesian Statistics

In the previous work, the author gave the following symplectic/contact geometric description of the Bayesian inference of normal means: The space H of normal distributions is an upper halfplane which admits two operations, namely, the convolution product and the normalized pointwise product of two probability density functions. There is a diffeomorphism F of H that interchanges these operations as well as sends any e-geodesic to an e-geodesic. The product of two copies of H carries positive and negative symplectic structures and a bi-contact hypersurface N. The graph of F is Lagrangian with respect to the negative symplectic structure. It is contained in the bi-contact hypersurface N. Further, it is preserved under a bi-contact Hamiltonian flow with respect to a single function. Then the restriction of the flow to the graph of F presents the inference of means. The author showed that this also works for the Student t-inference of smoothly moving means and enables us to consider the smoothness of data smoothing. In this presentation, the space of multivariate normal distributions is foliated by means of the Cholesky decomposition of the covariance matrix. This provides a pair of regular Poisson structures, and generalizes the above symplectic/contact description to the multivariate case. The most of the ideas presented here have been described at length in a later article of the author.