3,567 research outputs found
Computing the state difference equations for discrete overdetermined linear mD systems
We derive an algorithm that computes the state difference equations for a given set of poles of linear discrete overdetermined autonomous mD systems. These difference equations allow the realization of the dynamical system by means of delay, multiplication and addition elements in simulation diagrams. In doing so we generalize the classical Cayley–Hamilton theorem to multivariate polynomial ideals and provide a system theoretic interpretation to the notion of polynomial ideals, leading monomials and Gröbner bases. Furthermore, we extend the problem to include poles at infinity and so arrive at a description of overdetermined descriptor systems. This results in a new state space description of autonomous mD descriptor systems. In addition, we discuss the separation of the state variables of singular mD systems into a regular and singular part. A sufficient condition under which these two state vector parts can be interpreted as a forward evolving regular part and a backward evolving singular part is given. The robustness and efficiency of the developed algorithms are demonstrated via numerical experiments.postprin
The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows
The Gauss--Newton with approximated tensors (GNAT) method is a nonlinear
model reduction method that operates on fully discretized computational models.
It achieves dimension reduction by a Petrov--Galerkin projection associated
with residual minimization; it delivers computational efficency by a
hyper-reduction procedure based on the `gappy POD' technique. Originally
presented in Ref. [1], where it was applied to implicit nonlinear
structural-dynamics models, this method is further developed here and applied
to the solution of a benchmark turbulent viscous flow problem. To begin, this
paper develops global state-space error bounds that justify the method's design
and highlight its advantages in terms of minimizing components of these error
bounds. Next, the paper introduces a `sample mesh' concept that enables a
distributed, computationally efficient implementation of the GNAT method in
finite-volume-based computational-fluid-dynamics (CFD) codes. The suitability
of GNAT for parameterized problems is highlighted with the solution of an
academic problem featuring moving discontinuities. Finally, the capability of
this method to reduce by orders of magnitude the core-hours required for
large-scale CFD computations, while preserving accuracy, is demonstrated with
the simulation of turbulent flow over the Ahmed body. For an instance of this
benchmark problem with over 17 million degrees of freedom, GNAT outperforms
several other nonlinear model-reduction methods, reduces the required
computational resources by more than two orders of magnitude, and delivers a
solution that differs by less than 1% from its high-dimensional counterpart
New Algorithms for Solving Tropical Linear Systems
The problem of solving tropical linear systems, a natural problem of tropical
mathematics, has already proven to be very interesting from the algorithmic
point of view: it is known to be in but no polynomial time
algorithm is known, although counterexamples for existing pseudopolynomial
algorithms are (and have to be) very complex.
In this work, we continue the study of algorithms for solving tropical linear
systems. First, we present a new reformulation of Grigoriev's algorithm that
brings it closer to the algorithm of Akian, Gaubert, and Guterman; this lets us
formulate a whole family of new algorithms, and we present algorithms from this
family for which no known superpolynomial counterexamples work. Second, we
present a family of algorithms for solving overdetermined tropical systems. We
show that for weakly overdetermined systems, there are polynomial algorithms in
this family. We also present a concrete algorithm from this family that can
solve a tropical linear system defined by an matrix with maximal
element in time , and this time matches the complexity of the best of
previously known algorithms for feasibility testing.Comment: 17 page
Bounds on the number of connected components for tropical prevarieties
For a tropical prevariety in Rn given by a system of k tropical polynomials in n variables with degrees at most d, we prove that its number of connected components is less than k+7n−
Development of numerical algorithms for practical computation of nonlinear normal modes
When resorting to numerical algorithms, we show that nonlinear normal mode (NNM) computation is possible
with limited implementation effort, which paves the way to a practical method for determining the NNMs
of nonlinear mechanical systems. The proposed method relies on two main techniques, namely a shooting
procedure and a method for the continuation of NNM motions. In addition, sensitivity analysis is used to
reduce the computational burden of the algorithm. A simplified discrete model of a nonlinear bladed disk is
considered to demonstrate the developments
Sparse Gr\"obner Bases: the Unmixed Case
Toric (or sparse) elimination theory is a framework developped during the
last decades to exploit monomial structures in systems of Laurent polynomials.
Roughly speaking, this amounts to computing in a \emph{semigroup algebra},
\emph{i.e.} an algebra generated by a subset of Laurent monomials. In order to
solve symbolically sparse systems, we introduce \emph{sparse Gr\"obner bases},
an analog of classical Gr\"obner bases for semigroup algebras, and we propose
sparse variants of the and FGLM algorithms to compute them. Our prototype
"proof-of-concept" implementation shows large speed-ups (more than 100 for some
examples) compared to optimized (classical) Gr\"obner bases software. Moreover,
in the case where the generating subset of monomials corresponds to the points
with integer coordinates in a normal lattice polytope and under regularity assumptions, we prove complexity bounds which depend
on the combinatorial properties of . These bounds yield new
estimates on the complexity of solving -dim systems where all polynomials
share the same Newton polytope (\emph{unmixed case}). For instance, we
generalize the bound on the maximal degree in a Gr\"obner
basis of a -dim. bilinear system with blocks of variables of sizes
to the multilinear case: . We also propose
a variant of Fr\"oberg's conjecture which allows us to estimate the complexity
of solving overdetermined sparse systems.Comment: 20 pages, Corollary 6.1 has been corrected, ISSAC 2014, Kobe : Japan
(2014
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