3,008 research outputs found
Order reduction approaches for the algebraic Riccati equation and the LQR problem
We explore order reduction techniques for solving the algebraic Riccati
equation (ARE), and investigating the numerical solution of the
linear-quadratic regulator problem (LQR). A classical approach is to build a
surrogate low dimensional model of the dynamical system, for instance by means
of balanced truncation, and then solve the corresponding ARE. Alternatively,
iterative methods can be used to directly solve the ARE and use its approximate
solution to estimate quantities associated with the LQR. We propose a class of
Petrov-Galerkin strategies that simultaneously reduce the dynamical system
while approximately solving the ARE by projection. This methodology
significantly generalizes a recently developed Galerkin method by using a pair
of projection spaces, as it is often done in model order reduction of dynamical
systems. Numerical experiments illustrate the advantages of the new class of
methods over classical approaches when dealing with large matrices
Numerical integration of asymptotic solutions of ordinary differential equations
Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration
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The numerical solution of stefan problems with front-tracking and smoothing methods
Computer Algebra Solving of First Order ODEs Using Symmetry Methods
A set of Maple V R.3/4 computer algebra routines for the analytical solving
of 1st. order ODEs, using Lie group symmetry methods, is presented. The set of
commands includes a 1st. order ODE-solver and routines for, among other things:
the explicit determination of the coefficients of the infinitesimal symmetry
generator; the construction of the most general invariant 1st. order ODE under
given symmetries; the determination of the canonical coordinates of the
underlying invariant group; and the testing of the returned results.Comment: 14 pages, LaTeX, submitted to Computer Physics Communications.
Soft-package (On-Line Help) and sample MapleV session available at:
http://dft.if.uerj.br/symbcomp.htm or ftp://dft.if.uerj.br/pdetool
Order reduction methods for solving large-scale differential matrix Riccati equations
We consider the numerical solution of large-scale symmetric differential
matrix Riccati equations. Under certain hypotheses on the data, reduced order
methods have recently arisen as a promising class of solution strategies, by
forming low-rank approximations to the sought after solution at selected
timesteps. We show that great computational and memory savings are obtained by
a reduction process onto rational Krylov subspaces, as opposed to current
approaches. By specifically addressing the solution of the reduced differential
equation and reliable stopping criteria, we are able to obtain accurate final
approximations at low memory and computational requirements. This is obtained
by employing a two-phase strategy that separately enhances the accuracy of the
algebraic approximation and the time integration. The new method allows us to
numerically solve much larger problems than in the current literature.
Numerical experiments on benchmark problems illustrate the effectiveness of the
procedure with respect to existing solvers
Integrals of Motion in the Two Killing Vector Reduction of General Relativity
We apply the inverse scattering method to the midi-superspace models that are
characterized by a two-parameter Abelian group of motions with two spacelike
Killing vectors. We present a formulation that simplifies the construction of
the soliton solutions of Belinski\v i and Zakharov. Furthermore, it enables us
to obtain the zero curvature formulation for these models. Using this, and
imposing periodic boundary conditions corresponding to the Gowdy models when
the spatial topology is a three torus , we show that the equation of
motion for the monodromy matrix is an evolution equation of the Heisenberg
type. Consequently, the eigenvalues of the monodromy matrix are the generating
functionals for the integrals of motion. Furthermore, we utilise a suitable
formulation of the transition matrix to obtain explicit expressions for the
integrals of motion. This involves recursion relations which arise in solving
an equation of Riccati type. In the case when the two Killing vectors are
hypersurface orthogonal the integrals of motion have a particularly simple
form.Comment: 20 pages, plain TeX, SU-GP-93/7-8, UM-P-93/7
The Painlev\'e methods
This short review is an introduction to a great variety of methods, the
collection of which is called the Painlev\'e analysis, intended at producing
all kinds of exact (as opposed to perturbative) results on nonlinear equations,
whether ordinary, partial, or discrete.Comment: LaTex 2e, subject index, Nonlinear integrable systems: classical and
quantum, ed. A. Kundu, Special issue, Proceedings of Indian Science Academy,
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