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Documents  Schuster, Peter | enregistrements trouvés : 7

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Research talks;Computer Science;Logic and Foundations

A serious problem common to all interval algorithms is that they suffer from wrapping effects, i.e. unnecessary growth of approximations during a computation. This is essentially connected to functional dependencies inside vectors of data computed from the same inputs. Reducing these effects is an important issue in interval arithmetic, where the most successful approach uses Taylor models.
In TTE Taylor models have not been considered explicitly, as they use would not change the induced computability, already established using ordinary interval computations. However for the viewpoint of efficiency, they lead to significant improvements.
In the talk we report on recent improvements on the iRRAM software for exact real arithmetic (ERA) based on Taylor models. The techniques discussed should also easily be applicable to other software for exact real computations as long as they also are based on interval arithmetic.
As instructive examples we consider the one-dimensional logistic map and a few further discrete dynamical systems of higher dimensions
Joint work with Franz Brauße, Trier, and Margarita Korovina, Novosibirsk.
A serious problem common to all interval algorithms is that they suffer from wrapping effects, i.e. unnecessary growth of approximations during a computation. This is essentially connected to functional dependencies inside vectors of data computed from the same inputs. Reducing these effects is an important issue in interval arithmetic, where the most successful approach uses Taylor models.
In TTE Taylor models have not been considered ...

68Q25 ; 03D60 ; 65Y15

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- 316 p.
ISBN 978-1-4020-0152-9

Synthese library , 0306

Localisation : Colloque 1er étage (VENI)

logique mathatique # analyse non standard # analyse constructive # histoire # philosophie # épistémologie # continuum # mathématique constructive # mathématique non standard

00B25 ; 03-06 ; 03F60 ; 03H05

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Proof and computation in Coq | H

Multi angle

Research talks;Computer Science

In this talk, we are going to show on some elementary examples how computation can easily be incorporated inside proof in a proof system like Coq.

68N30 ; 68Q60 ; 68T15

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Research talks;Computer Science

From a (partial) differential equation to an actual program is a long road. This talk will present the formal verification of all the steps of this journey. This includes the mathematical error due to the numerical scheme (method error), that is usually bounded by pen-and-paper proofs. This also includes round-off errors due to the floating-point computations.
The running example will be a C program that implements a numerical scheme for the resolution of the one-dimensional acoustic wave equation. This program is annotated to specify both method error and round-off error, and formally verified using interactive and automatic provers. Some work in progress about the finite element method will also be presented.
From a (partial) differential equation to an actual program is a long road. This talk will present the formal verification of all the steps of this journey. This includes the mathematical error due to the numerical scheme (method error), that is usually bounded by pen-and-paper proofs. This also includes round-off errors due to the floating-point computations.
The running example will be a C program that implements a numerical scheme for the ...

68N30 ; 68Q60 ; 68N15 ; 65Y04 ; 65G50

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Research talks;Computer Science

Numerical software, common in scientific computing or embedded systems, inevitably uses an approximation of the real arithmetic in which most algorithms are designed. Finite-precision arithmetic, such as fixed-point or floating-point, is a common and efficient choice, but introduces an uncertainty on the computed result that is often very hard to quantify. We need adequate tools to estimate the errors introduced in order to choose suitable approximations which satisfy the accuracy requirements.
I will present a new programming model where the scientist writes his or her numerical program in a real-valued specification language with explicit error annotations. It is then the task of our verifying compiler to select a suitable floating-point or fixed-point data type which guarantees the needed accuracy. I will show how a combination of SMT theorem proving, interval and affine arithmetic and function derivatives yields an accurate, sound and automated error estimation which can handle nonlinearity, discontinuities and certain classes of loops.
Additionally, finite-precision arithmetic is not associative so that different, but mathematically equivalent, orders of computation often result in different magnitudes of errors. We have used this fact to not only verify but actively improve the accuracy by combining genetic programming with our error computation with encouraging results.
Numerical software, common in scientific computing or embedded systems, inevitably uses an approximation of the real arithmetic in which most algorithms are designed. Finite-precision arithmetic, such as fixed-point or floating-point, is a common and efficient choice, but introduces an uncertainty on the computed result that is often very hard to quantify. We need adequate tools to estimate the errors introduced in order to choose suitable ...

68Q60 ; 65G50 ; 68N30 ; 68T20

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Research talks;Computer Science

This talk is about verified numerical algorithms in Isabelle/HOL, with a focus on guaranteed enclosures for solutions of ODEs. The enclosures are represented by zonotopes, arising from the use of affine arithmetic. Enclosures for solutions of ODEs are computed by set-based variants of the well-known Runge-Kutta methods.
All of the algorithms are formally verified with respect to a formalization of ODEs in Isabelle/HOL: The correctness proofs are carried out for abstract algorithms, which are specified in terms of real numbers and sets. These abstract algorithms are automatically refined towards executable specifications based on lists, zonotopes, and software floating point numbers. Optimizations for low-dimensional, nonlinear dynamics allow for an application highlight: the computation of an accurate enclosure for the Lorenz attractor. This contributes to an important proof that originally relied on non-verified numerical computations.
This talk is about verified numerical algorithms in Isabelle/HOL, with a focus on guaranteed enclosures for solutions of ODEs. The enclosures are represented by zonotopes, arising from the use of affine arithmetic. Enclosures for solutions of ODEs are computed by set-based variants of the well-known Runge-Kutta methods.
All of the algorithms are formally verified with respect to a formalization of ODEs in Isabelle/HOL: The correctness proofs are ...

68T15 ; 34-04 ; 34A12 ; 37D45 ; 65G20 ; 65G30 ; 65G50 ; 65L70 ; 68N15 ; 68Q60 ; 68N30 ; 65Y04

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- 350 p.
ISBN 978-0-19-856651-9

Oxford logic guides , 0048

Localisation : Ouvrages RdC (From)

logique # théorie des ensembles # analyse constructive # mathématiques constructives # mathématiques intuitives

03-06 ; 00B25 ; 03F55 ; 03F60 ; 03F65

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