Towards deterministic subspace identification for autonomous nonlinear systems

The problem of identifying deterministic autonomous linear and nonlinear systems is studied. A specific version of the theory of deterministic subspace identification for discrete-time autonomous linear systems is developed in continuous time. By combining the subspace approach to linear identification and the differential-geometric approach to nonlinear control systems, a novel identification framework for continuous-time autonomous nonlinear systems is developed

A generic construction for high order approximation schemes of semigroups using random grids

Our aim is to construct high order approximation schemes for general semigroups of linear operators $P_{t},t\geq 0$. In order to do it, we fix a time horizon $T$ and the discretization steps $h_{l}=\frac{T}{n^{l}},l\in \mathbb{N}$ and we suppose that we have at hand some short time approximation operators $Q_{l}$ such that $P_{h_{l}}=Q_{l}+O(h_{l}^{1+\alpha })$ for some $\alpha >0$. Then, we consider random time grids $\Pi (\omega )=\{t_0(\omega )=0<t_{1}(\omega )<...<t_{m}(\omega )=T\}$ such that for all $1\le k\le m$, $t_{k}(\omega )-t_{k-1}(\omega )=h_{l_{k}}$ for some $l_{k}\in \mathbb{N}$, and we associate the approximation discrete semigroup $P_{T}^{\Pi (\omega )}=Q_{l_{n}}...Q_{l_{1}}.$ Our main result is the following: for any approximation order $\nu$, we can construct random grids $\Pi_{i}(\omega )$ and coefficients $c_{i}$, with $i=1,...,r$ such that $$P_{t}f=\sum_{i=1}^{r}c_{i}\mathbb{E}(P_{t}^{\Pi _{i}(\omega )}f(x))+O(n^{-\nu})$$% with the expectation concerning the random grids $\Pi _{i}(\omega ).$ Besides, $\text{Card}(\Pi _{i}(\omega ))=O(n)$ and the complexity of the algorithm is of order $n$, for any order of approximation $\nu$. The standard example concerns diffusion processes, using the Euler approximation for~$Q_l$. In this particular case and under suitable conditions, we are able to gather the terms in order to produce an estimator of $P_tf$ with finite variance. However, an important feature of our approach is its universality in the sense that it works for every general semigroup $P_{t}$ and approximations. Besides, approximation schemes sharing the same $\alpha$ lead to the same random grids $\Pi_{i}$ and coefficients $c_{i}$. Numerical illustrations are given for ordinary differential equations, piecewise deterministic Markov processes and diffusions

Noise corrections to stochastic trace formulas

We review studies of an evolution operator L for a discrete Langevin equation with a strongly hyperbolic classical dynamics and a Gaussian noise. The leading eigenvalue of L yields a physically measurable property of the dynamical system, the escape rate from the repeller. The spectrum of the evolution operator L in the weak noise limit can be computed in several ways. A method using a local matrix representation of the operator allows to push the corrections to the escape rate up to order eight in the noise expansion parameter. These corrections then appear to form a divergent series. Actually, via a cumulant expansion, they relate to analogous divergent series for other quantities, the traces of the evolution operators L^n. Using an integral representation of the evolution operator L, we then investigate the high order corrections to the latter traces. Their asymptotic behavior is found to be controlled by sub-dominant saddle points previously neglected in the perturbative expansion, and to be ultimately described by a kind of trace formula.Comment: 21 pages, 4 figures,corrected typo

Postprocessed integrators for the high order integration of ergodic SDEs

The concept of effective order is a popular methodology in the deterministic literature for the construction of efficient and accurate integrators for differential equations over long times. The idea is to enhance the accuracy of a numerical method by using an appropriate change of variables called the processor. We show that this technique can be extended to the stochastic context for the construction of new high order integrators for the sampling of the invariant measure of ergodic systems. The approach is illustrated with modifications of the stochastic $\theta$-method applied to Brownian dynamics, where postprocessors achieving order two are introduced. Numerical experiments, including stiff ergodic systems, illustrate the efficiency and versatility of the approach.Comment: 21 pages, to appear in SIAM J. Sci. Compu

Deterministic models of quantum fields

Deterministic dynamical models are discussed which can be described in quantum mechanical terms. -- In particular, a local quantum field theory is presented which is a supersymmetric classical model. The Hilbert space approach of Koopman and von Neumann is used to study the classical evolution of an ensemble of such systems. Its Liouville operator is decomposed into two contributions, with positive and negative spectrum, respectively. The unstable negative part is eliminated by a constraint on physical states, which is invariant under the Hamiltonian flow. Thus, choosing suitable variables, the classical Liouville equation becomes a functional Schroedinger equation of a genuine quantum field theory. -- We briefly mention an U(1) gauge theory with ``varying alpha'' or dilaton coupling where a corresponding quantized theory emerges in the phase space approach. It is energy-parity symmetric and, therefore, a prototype of a model in which the cosmological constant is protected by a symmetry.Comment: 6 pages; synopsis of hep-th/0510267, hep-th/0503069, hep-th/0411176 . Talk at Constrained Dynamics and Quantum Gravity - QG05, Cala Gonone (Sardinia, Italy), September 12-16, 2005. To appear in the proceeding