368 research outputs found
Certified Roundoff Error Bounds using Bernstein Expansions and Sparse Krivine-Stengle Representations
Floating point error is an inevitable drawback of embedded systems
implementation. Computing rigorous upper bounds of roundoff errors is
absolutely necessary to the validation of critical software. This problem is
even more challenging when addressing non-linear programs. In this paper, we
propose and compare two new methods based on Bernstein expansions and sparse
Krivine-Stengle representations, adapted from the field of the global
optimization to compute upper bounds of roundoff errors for programs
implementing polynomial functions. We release two related software package
FPBern and FPKiSten, and compare them with state of the art tools. We show that
these two methods achieve competitive performance, while computing accurate
upper bounds by comparison with other tools.Comment: 20 pages, 2 table
RealCertify: a Maple package for certifying non-negativity
Let (resp. ) be the field of rational (resp. real)
numbers and be variables. Deciding the non-negativity
of polynomials in over or over semi-algebraic
domains defined by polynomial constraints in is a classical
algorithmic problem for symbolic computation.
The Maple package \textsc{RealCertify} tackles this decision problem by
computing sum of squares certificates of non-negativity for inputs where such
certificates hold over the rational numbers. It can be applied to numerous
problems coming from engineering sciences, program verification and
cyber-physical systems. It is based on hybrid symbolic-numeric algorithms based
on semi-definite programming.Comment: 4 pages, 2 table
Certified Roundoff Error Bounds Using Semidefinite Programming.
Roundoff errors cannot be avoided when implementing numerical programs with finite precision. The ability to reason about rounding is especially important if one wants to explore a range of potential representations, for instance for FPGAs or custom hardware implementation. This problem becomes challenging when the program does not employ solely linear operations as non-linearities are inherent to many interesting computational problems in real-world applications. Existing solutions to reasoning are limited in the presence of nonlinear correlations between variables, leading to either imprecise bounds or high analysis time. Furthermore, while it is easy to implement a straightforward method such as interval arithmetic, sophisticated techniques are less straightforward to implement in a formal setting. Thus there is a need for methods which output certificates that can be formally validated inside a proof assistant. We present a framework to provide upper bounds on absolute roundoff errors. This framework is based on optimization techniques employing semidefinite programming and sums of squares certificates, which can be formally checked inside the Coq theorem prover. Our tool covers a wide range of nonlinear programs, including polynomials and transcendental operations as well as conditional statements. We illustrate the efficiency and precision of this tool on non-trivial programs coming from biology, optimization and space control. Our tool produces more precise error bounds for 37 percent of all programs and yields better performance in 73 percent of all programs
A Sums-of-Squares Extension of Policy Iterations
In order to address the imprecision often introduced by widening operators in
static analysis, policy iteration based on min-computations amounts to
considering the characterization of reachable value set of a program as an
iterative computation of policies, starting from a post-fixpoint. Computing
each policy and the associated invariant relies on a sequence of numerical
optimizations. While the early research efforts relied on linear programming
(LP) to address linear properties of linear programs, the current state of the
art is still limited to the analysis of linear programs with at most quadratic
invariants, relying on semidefinite programming (SDP) solvers to compute
policies, and LP solvers to refine invariants.
We propose here to extend the class of programs considered through the use of
Sums-of-Squares (SOS) based optimization. Our approach enables the precise
analysis of switched systems with polynomial updates and guards. The analysis
presented has been implemented in Matlab and applied on existing programs
coming from the system control literature, improving both the range of
analyzable systems and the precision of previously handled ones.Comment: 29 pages, 4 figure
Formal Proofs for Nonlinear Optimization
We present a formally verified global optimization framework. Given a
semialgebraic or transcendental function and a compact semialgebraic domain
, we use the nonlinear maxplus template approximation algorithm to provide a
certified lower bound of over . This method allows to bound in a modular
way some of the constituents of by suprema of quadratic forms with a well
chosen curvature. Thus, we reduce the initial goal to a hierarchy of
semialgebraic optimization problems, solved by sums of squares relaxations. Our
implementation tool interleaves semialgebraic approximations with sums of
squares witnesses to form certificates. It is interfaced with Coq and thus
benefits from the trusted arithmetic available inside the proof assistant. This
feature is used to produce, from the certificates, both valid underestimators
and lower bounds for each approximated constituent. The application range for
such a tool is widespread; for instance Hales' proof of Kepler's conjecture
yields thousands of multivariate transcendental inequalities. We illustrate the
performance of our formal framework on some of these inequalities as well as on
examples from the global optimization literature.Comment: 24 pages, 2 figures, 3 table
Certification of inequalities involving transcendental functions: combining SDP and max-plus approximation
We consider the problem of certifying an inequality of the form ,
, where is a multivariate transcendental function, and
is a compact semialgebraic set. We introduce a certification method, combining
semialgebraic optimization and max-plus approximation. We assume that is
given by a syntaxic tree, the constituents of which involve semialgebraic
operations as well as some transcendental functions like , ,
, etc. We bound some of these constituents by suprema or infima of
quadratic forms (max-plus approximation method, initially introduced in optimal
control), leading to semialgebraic optimization problems which we solve by
semidefinite relaxations. The max-plus approximation is iteratively refined and
combined with branch and bound techniques to reduce the relaxation gap.
Illustrative examples of application of this algorithm are provided, explaining
how we solved tight inequalities issued from the Flyspeck project (one of the
main purposes of which is to certify numerical inequalities used in the proof
of the Kepler conjecture by Thomas Hales).Comment: 7 pages, 3 figures, 3 tables, Appears in the Proceedings of the
European Control Conference ECC'13, July 17-19, 2013, Zurich, pp. 2244--2250,
copyright EUCA 201
Perfluoroalkylation of aromatic systems
I. We have concentrated on the synthesis of aromatic systems containing CH(_2)R(_F) groups, where R(_F) is a bulky group derived from hexafluoropropene dimer, using fluoride ion induced methodology. A variety of procedures have been explored and methods of synthesis are described. We have made a major advance in this area by the simple expedient of extracting the polyfluoroalkylated products directly into commercially available perfluorinated fluids: FC-84 (bp 8O C), enabling simple purification. II. This new class of perfluoroalkylated derivatives has allowed us to explore some further chemistry. Synthesis and reactions are presented and the results are discussed in terms of electronic effects of the perfluoroalkyl group on the benzene ring. The unusual miscibility properties of fluorocarbon solvents have also prompted us to investigate the potential utility of these systems for dyes and catalysis. III. Synthetically useful perfluoroalkylated-s-triazines have been synthesised from 2,4,6-trifluoro-s-triazine, by fluoride ion induced reaction with hexafluoropropene. These systems so produced were extensively studied, and particular attention was paid to reactions with oxygen-centered nucleophiles. Model chemistry to demonstrate that attachment of perfluorocarbon groups to surfaces could be a useful form of treatment by using this methodology and was explored
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