Shadowing Lemma and Chaotic Orbit Determination

Orbit determination is possible for a chaotic orbit of a dynamical system, given a finite set of observations, provided the initial conditions are at the central time. In a simple discrete model, the standard map, we tackle the problem of chaotic orbit determination when observations extend beyond the predictability horizon. If the orbit is hyperbolic, a shadowing orbit is computed by the least squares orbit determination. We test both the convergence of the orbit determination iterative procedure and the behaviour of the uncertainties as a function of the maximum number $n$ of map iterations observed. When the initial conditions belong to a chaotic orbit, the orbit determination is made impossible by numerical instability beyond a computability horizon, which can be approximately predicted by a simple formula. Moreover, the uncertainty of the results is sharply increased if a dynamical parameter is added to the initial conditions as parameter to be estimated. The uncertainty of the dynamical parameter decreases like $n^a$ with $a<0$ but not large (of the order of unity). If only the initial conditions are estimated, their uncertainty decreases exponentially with $n$. If they belong to a non-chaotic orbit the computational horizon is much larger, if it exists at all, and the decrease of the uncertainty is polynomial in all parameters, like $n^a$ with $a\simeq 1/2$. The Shadowing Lemma does not dictate what the asymptotic behaviour of the uncertainties should be. These phenomena have significant implications, which remain to be studied, in practical problems of orbit determination involving chaos, such as the chaotic rotation state of a celestial body and a chaotic orbit of a planet-crossing asteroid undergoing many close approaches

Non-Turing computations via Malament-Hogarth space-times

We investigate the Church-Kalm\'ar-Kreisel-Turing Theses concerning theoretical (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical general relativity theory. We argue that (i) there are several distinguished Church-Turing-type Theses (not only one) and (ii) validity of some of these theses depend on the background physical theory we choose to use. In particular, if we choose classical general relativity theory as our background theory, then the above mentioned limitations (predicted by these Theses) become no more necessary, hence certain forms of the Church-Turing Thesis cease to be valid (in general relativity). (For other choices of the background theory the answer might be different.) We also look at various ``obstacles'' to computing a non-recursive function (by relying on relativistic phenomena) published in the literature and show that they can be avoided (by improving the ``design'' of our future computer). We also ask ourselves, how all this reflects on the arithmetical hierarchy and the analytical hierarchy of uncomputable functions.Comment: Final, published version: 25 pages, LaTex with two eps-figures, journal reference adde

Adaptive Galerkin approximation algorithms for partial differential equations in infinite dimensions

Space-time variational formulations of infinite-dimensional Fokker-Planck (FP) and Ornstein-Uhlenbeck (OU) equations for functions on a separable Hilbert space $H$ are developed. The well-posedness of these equations in the Hilbert space $L^{2}(H,\mu)$ of functions on $H$, which are square-integrable with respect to a Gaussian measure $\mu$ on $H$, is proved. Specifically, for the infinite-dimensional FP equation, adaptive space-time Galerkin discretizations, based on a tensorized Riesz basis, built from biorthogonal piecewise polynomial wavelet bases in time and the Hermite polynomial chaos in the Wiener-Itô decomposition of $L^{2}(H,\mu)$, are introduced and are shown to converge quasioptimally with respect to the nonlinear, best $N$-term approximation benchmark. As a consequence, the proposed adaptive Galerkin solution algorithms perform quasioptimally with respect to the best $N$-term approximation in the finite-dimensional case, in particular. All constants in our error and complexity bounds are shown to be independent of the number of "active" coordinates identified by the proposed adaptive Galerkin approximation algorithms