43,199 research outputs found
A probabilistic algorithm to test local algebraic observability in polynomial time
The following questions are often encountered in system and control theory.
Given an algebraic model of a physical process, which variables can be, in
theory, deduced from the input-output behavior of an experiment? How many of
the remaining variables should we assume to be known in order to determine all
the others? These questions are parts of the \emph{local algebraic
observability} problem which is concerned with the existence of a non trivial
Lie subalgebra of the symmetries of the model letting the inputs and the
outputs invariant. We present a \emph{probabilistic seminumerical} algorithm
that proposes a solution to this problem in \emph{polynomial time}. A bound for
the necessary number of arithmetic operations on the rational field is
presented. This bound is polynomial in the \emph{complexity of evaluation} of
the model and in the number of variables. Furthermore, we show that the
\emph{size} of the integers involved in the computations is polynomial in the
number of variables and in the degree of the differential system. Last, we
estimate the probability of success of our algorithm and we present some
benchmarks from our Maple implementation.Comment: 26 pages. A Maple implementation is availabl
A Geometric Index Reduction Method for Implicit Systems of Differential Algebraic Equations
This paper deals with the index reduction problem for the class of
quasi-regular DAE systems. It is shown that any of these systems can be
transformed to a generically equivalent first order DAE system consisting of a
single purely algebraic (polynomial) equation plus an under-determined ODE
(that is, a semi-explicit DAE system of differentiation index 1) in as many
variables as the order of the input system. This can be done by means of a
Kronecker-type algorithm with bounded complexity
Fast computation of power series solutions of systems of differential equations
We propose new algorithms for the computation of the first N terms of a
vector (resp. a basis) of power series solutions of a linear system of
differential equations at an ordinary point, using a number of arithmetic
operations which is quasi-linear with respect to N. Similar results are also
given in the non-linear case. This extends previous results obtained by Brent
and Kung for scalar differential equations of order one and two
Variational-Wavelet Approach to RMS Envelope Equations
We present applications of variational-wavelet approach to nonlinear
(rational) rms envelope equations. We have the solution as a multiresolution
(multiscales) expansion in the base of compactly supported wavelet basis. We
give extension of our results to the cases of periodic beam motion and
arbitrary variable coefficients. Also we consider more flexible variational
method which is based on biorthogonal wavelet approach.Comment: 21 pages, 8 figures, LaTeX2e, presented at Second ICFA Advanced
Accelerator Workshop, UCLA, November, 199
Spatially Adaptive Stochastic Multigrid Methods for Fluid-Structure Systems with Thermal Fluctuations
In microscopic mechanical systems interactions between elastic structures are
often mediated by the hydrodynamics of a solvent fluid. At microscopic scales
the elastic structures are also subject to thermal fluctuations. Stochastic
numerical methods are developed based on multigrid which allow for the
efficient computation of both the hydrodynamic interactions in the presence of
walls and the thermal fluctuations. The presented stochastic multigrid approach
provides efficient real-space numerical methods for generating the required
stochastic driving fields with long-range correlations consistent with
statistical mechanics. The presented approach also allows for the use of
spatially adaptive meshes in resolving the hydrodynamic interactions. Numerical
results are presented which show the methods perform in practice with a
computational complexity of O(N log(N))
Relative parametrization of linear multidimensional systems
In the last chapter of his book "The Algebraic Theory of Modular Systems "
published in 1916, F. S. Macaulay developped specific techniques for dealing
with " unmixed polynomial ideals " by introducing what he called " inverse
systems ". The purpose of this paper is to extend such a point of view to
differential modules defined by linear multidimensional systems, that is by
linear systems of ordinary differential (OD) or partial differential (PD)
equations of any order, with any number of independent variables, any number of
unknowns and even with variable coefficients in a differential field. The first
and main idea is to replace unmixed polynomial ideals by " pure differential
modules ". The second idea is to notice that a module is 0-pure if and only if
it is torsion-free and thus if and only if it admits an " absolute
parametrization " by means of arbitrary potential like functions, or,
equivalently, if it can be embedded into a free module by means of an "
absolute localization ". The third idea is to refer to a difficult theorem of
algebraic analysis saying that an r-pure module can be embedded into a module
of projective dimension equal to r, that is a module admitting a projective
resolution with exactly r operators. The fourth and final idea is to establish
a link between the use of extension modules for such a purpose and specific
formal properties of the underlying multidimensional system through the use of
involution and a "relative localization " leading to a "relative
parametrization ", that is to the use of potential-like functions satisfying a
kind of "minimum differential constraint " limiting, in some sense, the number
of independent variables appearing in these functions, in a way similar to the
situation met in the Cartan-K\"ahler theorem of analysis. The paper is written
in a rather effective self-contained way and we provide many explicit examples
that should become test examples for a future use of computer algebra.Comment: Presented for publication in the Springer journal
MSSP:Multidimensional Systems and Signal Processin
- …