1,851 research outputs found
Quadratic Zonotopes:An extension of Zonotopes to Quadratic Arithmetics
Affine forms are a common way to represent convex sets of using
a base of error terms . Quadratic forms are an
extension of affine forms enabling the use of quadratic error terms .
In static analysis, the zonotope domain, a relational abstract domain based
on affine forms has been used in a wide set of settings, e.g. set-based
simulation for hybrid systems, or floating point analysis, providing relational
abstraction of functions with a cost linear in the number of errors terms.
In this paper, we propose a quadratic version of zonotopes. We also present a
new algorithm based on semi-definite programming to project a quadratic
zonotope, and therefore quadratic forms, to intervals. All presented material
has been implemented and applied on representative examples.Comment: 17 pages, 5 figures, 1 tabl
Correctly Rounded Newton-Cotes Quadrature
Numerical integration is an important operation for scientific computations. Although the different quadrature methods have been well studied from a mathematical point of view, the analysis of the actual error when performing the quadrature on a computer is often neglected. This step is however required for certified arithmetics. We study the Newton-Cotes quadrature scheme and give enough details on the algorithms and the error bounds to enable software developers to write a correctly-rounded Newton-Cotes quadrature
Rigorous numerics for PDEs with indefinite tail: existence of a periodic solution of the Boussinesq equation with time-dependent forcing
We consider the Boussinesq PDE perturbed by a time-dependent forcing. Even
though there is no smoothing effect for arbitrary smooth initial data, we are
able to apply the method of self-consistent bounds to deduce the existence of
smooth classical periodic solutions in the vicinity of 0. The proof is
non-perturbative and relies on construction of periodic isolating segments in
the Galerkin projections
A Verified Certificate Checker for Finite-Precision Error Bounds in Coq and HOL4
Being able to soundly estimate roundoff errors of finite-precision
computations is important for many applications in embedded systems and
scientific computing. Due to the discrepancy between continuous reals and
discrete finite-precision values, automated static analysis tools are highly
valuable to estimate roundoff errors. The results, however, are only as correct
as the implementations of the static analysis tools. This paper presents a
formally verified and modular tool which fully automatically checks the
correctness of finite-precision roundoff error bounds encoded in a certificate.
We present implementations of certificate generation and checking for both Coq
and HOL4 and evaluate it on a number of examples from the literature. The
experiments use both in-logic evaluation of Coq and HOL4, and execution of
extracted code outside of the logics: we benchmark Coq extracted unverified
OCaml code and a CakeML-generated verified binary
Period Doubling Renormalization for Area-Preserving Maps and Mild Computer Assistance in Contraction Mapping Principle
It has been observed that the famous Feigenbaum-Coullet-Tresser period
doubling universality has a counterpart for area-preserving maps of {\fR}^2.
A renormalization approach has been used in a "hard" computer-assisted proof of
existence of an area-preserving map with orbits of all binary periods in
Eckmann et al (1984). As it is the case with all non-trivial universality
problems in non-dissipative systems in dimensions more than one, no analytic
proof of this period doubling universality exists to date.
In this paper we attempt to reduce computer assistance in the argument, and
present a mild computer aided proof of the analyticity and compactness of the
renormalization operator in a neighborhood of a renormalization fixed point:
that is a proof that does not use generalizations of interval arithmetics to
functional spaces - but rather relies on interval arithmetics on real numbers
only to estimate otherwise explicit expressions. The proof relies on several
instance of the Contraction Mapping Principle, which is, again, verified via
mild computer assistance
Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)
Oxford, UK, 26 August 200
An elementary way to rigorously estimate convergence to equilibrium and escape rates
We show an elementary method to have (finite time and asymptotic) computer
assisted explicit upper bounds on convergence to equilibrium (decay of
correlations) and escape rate for systems satisfying a Lasota Yorke inequality.
The bounds are deduced by the ones of suitable approximations of the system's
transfer operator. We also present some rigorous experiment showing the
approach and some concrete result.Comment: 14 pages, 6 figure
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