1,561 research outputs found
Analysis of Reaction Network Systems Using Tropical Geometry
We discuss a novel analysis method for reaction network systems with
polynomial or rational rate functions. This method is based on computing
tropical equilibrations defined by the equality of at least two dominant
monomials of opposite signs in the differential equations of each dynamic
variable. In algebraic geometry, the tropical equilibration problem is
tantamount to finding tropical prevarieties, that are finite intersections of
tropical hypersurfaces. Tropical equilibrations with the same set of dominant
monomials define a branch or equivalence class. Minimal branches are
particularly interesting as they describe the simplest states of the reaction
network. We provide a method to compute the number of minimal branches and to
find representative tropical equilibrations for each branch.Comment: Proceedings Computer Algebra in Scientific Computing CASC 201
Finding unstable periodic orbits: A hybrid approach with polynomial optimization
We present a novel method to compute unstable periodic orbits (UPOs) that optimize the infinite-time average of a given quantity for polynomial ODE systems. The UPO search procedure relies on polynomial optimization to construct nonnegative polynomials whose sublevel sets approximately localize parts of the optimal UPO, and that can be used to implement a simple yet effective control strategy to reduce the UPO's instability. Precisely, we construct a family of controlled ODE systems, parameterized by a scalar k, such that the original ODE system is recovered for k=0 and such that the optimal orbit is less unstable, or even stabilized, for k>0. Periodic orbits for the controlled system can often be more easily converged with traditional methods, and numerical continuation in k allows one to recover optimal UPOs for the original system. The effectiveness of this approach is illustrated on three low-dimensional ODE systems with chaotic dynamics
Stokes, Gibbs and volume computation of semi-algebraic sets
We consider the problem of computing the Lebesgue volume of compact basic
semi-algebraic sets. In full generality, it can be approximated as closely as
desired by a converging hierarchy of upper bounds obtained by applying the
Moment-SOS (sums of squares) methodology to a certain innite-dimensional linear
program (LP). At each step one solves a semidenite relaxation of the LP which
involves pseudo-moments up to a certain degree. Its dual computes a polynomial
of same degree which approximates from above the discon-tinuous indicator
function of the set, hence with a typical Gibbs phenomenon which results in a
slow convergence of the associated numerical scheme. Drastic improvements have
been observed by introducing in the initial LP additional linear moment
constraints obtained from a certain application of Stokes' theorem for
integration on the set. However and so far there was no rationale to explain
this behavior. We provide a rened version of this extended LP formulation. When
the set is the smooth super-level set of a single polynomial, we show that the
dual of this rened LP has an optimal solution which is a continuous function.
Therefore in this dual one now approximates a continuous function by a
polynomial, hence with no Gibbs phenomenon, which explains and improves the
already observed drastic acceleration of the convergence of the hierarchy.
Interestingly, the technique of proof involves classical results on Poisson's
partial dierential equation (PDE)
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Boundary integral methods in high frequency scattering
In this article we review recent progress on the design, analysis and implementation of numerical-asymptotic boundary integral methods for the computation of frequency-domain acoustic scattering in a homogeneous unbounded medium by a bounded obstacle. The main aim of the methods is to allow computation of scattering at arbitrarily high frequency with finite computational resources
Efficient solution of parabolic equations by Krylov approximation methods
Numerical techniques for solving parabolic equations by the method of lines is addressed. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of a very small dimension to a known vector which is, in turn, computed accurately by exploiting well-known rational approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms
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