2,285 research outputs found
Bounds for algorithms in differential algebra
We consider the Rosenfeld-Groebner algorithm for computing a regular
decomposition of a radical differential ideal generated by a set of ordinary
differential polynomials in n indeterminates. For a set of ordinary
differential polynomials F, let M(F) be the sum of maximal orders of
differential indeterminates occurring in F. We propose a modification of the
Rosenfeld-Groebner algorithm, in which for every intermediate polynomial system
F, the bound M(F) is less than or equal to (n-1)!M(G), where G is the initial
set of generators of the radical ideal. In particular, the resulting regular
systems satisfy the bound. Since regular ideals can be decomposed into
characterizable components algebraically, the bound also holds for the orders
of derivatives occurring in a characteristic decomposition of a radical
differential ideal.
We also give an algorithm for converting a characteristic decomposition of a
radical differential ideal from one ranking into another. This algorithm
performs all differentiations in the beginning and then uses a purely algebraic
decomposition algorithm.Comment: 40 page
Canonical Characteristic Sets of Characterizable Differential Ideals
We study the concept of canonical characteristic set of a characterizable
differential ideal. We propose an efficient algorithm that transforms any
characteristic set into the canonical one. We prove the basic properties of
canonical characteristic sets. In particular, we show that in the ordinary case
for any ranking the order of each element of the canonical characteristic set
of a characterizable differential ideal is bounded by the order of the ideal.
Finally, we propose a factorization-free algorithm for computing the canonical
characteristic set of a characterizable differential ideal represented as a
radical ideal by a set of generators. The algorithm is not restricted to the
ordinary case and is applicable for an arbitrary ranking.Comment: 26 page
Sum-of-Squares approach to feedback control of laminar wake flows
A novel nonlinear feedback control design methodology for incompressible
fluid flows aiming at the optimisation of long-time averages of flow quantities
is presented. It applies to reduced-order finite-dimensional models of fluid
flows, expressed as a set of first-order nonlinear ordinary differential
equations with the right-hand side being a polynomial function in the state
variables and in the controls. The key idea, first discussed in Chernyshenko et
al. 2014, Philos. T. Roy. Soc. 372(2020), is that the difficulties of treating
and optimising long-time averages of a cost are relaxed by using the
upper/lower bounds of such averages as the objective function. In this setting,
control design reduces to finding a feedback controller that optimises the
bound, subject to a polynomial inequality constraint involving the cost
function, the nonlinear system, the controller itself and a tunable polynomial
function. A numerically tractable approach to the solution of such optimisation
problems, based on Sum-of-Squares techniques and semidefinite programming, is
proposed.
To showcase the methodology, the mitigation of the fluctuation kinetic energy
in the unsteady wake behind a circular cylinder in the laminar regime at
Re=100, via controlled angular motions of the surface, is numerically
investigated. A compact reduced-order model that resolves the long-term
behaviour of the fluid flow and the effects of actuation, is derived using
Proper Orthogonal Decomposition and Galerkin projection. In a full-information
setting, feedback controllers are then designed to reduce the long-time average
of the kinetic energy associated with the limit cycle. These controllers are
then implemented in direct numerical simulations of the actuated flow. Control
performance, energy efficiency, and physical control mechanisms identified are
analysed. Key elements, implications and future work are discussed
Approaches to the automatic generation and control of finite element meshes
The algorithmic approaches being taken to the development of finite element mesh generators capable of automatically discretizing general domains without the need for user intervention are discussed. It is demonstrated that because of the modeling demands placed on a automatic mesh generator, all the approaches taken to date produce unstructured meshes. Consideration is also given to both a priori and a posteriori mesh control devices for automatic mesh generators as well as their integration with geometric modeling and adaptive analysis procedures
The Development of Intersection Homology Theory
This historical introduction is in two parts. The first is reprinted with
permission from ``A century of mathematics in America, Part II,'' Hist. Math.,
2, Amer. Math. Soc., 1989, pp.543-585. Virtually no change has been made to the
original text. In particular, Section 8 is followed by the original list of
references. However, the text has been supplemented by a series of endnotes,
collected in the new Section 9 and followed by a second list of references. If
a citation is made to the first list, then its reference number is simply
enclosed in brackets -- for example, [36]. However, if a citation is made to
the second list, then its number is followed by an `S' -- for example, [36S].
Further, if a subject in the reprint is elaborated on in an endnote, then the
subject is flagged in the margin by the number of the corresponding endnote,
and the endnote includes in its heading, between parentheses, the page number
or numbers on which the subject appears in the reprint below. Finally, all
cross-references appear as hypertext links in the dvi and pdf copies.Comment: 58 pages, hypertext links added; appeared in Part 3 of the special
issue of Pure and Applied Mathematics Quarterly in honor of Robert
MacPherson. However, the flags in the margin were unfortunately (and
inexplicably) omitted from the published versio
Algorithmic Semi-algebraic Geometry and Topology -- Recent Progress and Open Problems
We give a survey of algorithms for computing topological invariants of
semi-algebraic sets with special emphasis on the more recent developments in
designing algorithms for computing the Betti numbers of semi-algebraic sets.
Aside from describing these results, we discuss briefly the background as well
as the importance of these problems, and also describe the main tools from
algorithmic semi-algebraic geometry, as well as algebraic topology, which make
these advances possible. We end with a list of open problems.Comment: Survey article, 74 pages, 15 figures. Final revision. This version
will appear in the AMS Contemporary Math. Series: Proceedings of the Summer
Research Conference on Discrete and Computational Geometry, Snowbird, Utah
(June, 2006). J.E. Goodman, J. Pach, R. Pollack Ed
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
Research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis and computer science
This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis, and computer science during the period April l, 1988 through September 30, 1988
Nonlinear Integer Programming
Research efforts of the past fifty years have led to a development of linear
integer programming as a mature discipline of mathematical optimization. Such a
level of maturity has not been reached when one considers nonlinear systems
subject to integrality requirements for the variables. This chapter is
dedicated to this topic.
The primary goal is a study of a simple version of general nonlinear integer
problems, where all constraints are still linear. Our focus is on the
computational complexity of the problem, which varies significantly with the
type of nonlinear objective function in combination with the underlying
combinatorial structure. Numerous boundary cases of complexity emerge, which
sometimes surprisingly lead even to polynomial time algorithms.
We also cover recent successful approaches for more general classes of
problems. Though no positive theoretical efficiency results are available, nor
are they likely to ever be available, these seem to be the currently most
successful and interesting approaches for solving practical problems.
It is our belief that the study of algorithms motivated by theoretical
considerations and those motivated by our desire to solve practical instances
should and do inform one another. So it is with this viewpoint that we present
the subject, and it is in this direction that we hope to spark further
research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G.
Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50
Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art
Surveys, Springer-Verlag, 2009, ISBN 354068274
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