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Documents  68Q15 | enregistrements trouvés : 4

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The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral geometry. The most well known example is Smale's 17th problem, which asks to find a solution of a given system of n complex homogeneous polynomial equations in $n$ + 1 unknowns. This problem can be solved in average (and even smoothed) polynomial time.
In the course we will explain the concepts necessary to state and solve Smale's 17th problem. We also show how these ideas lead to new numerical algorithms for computing eigenpairs of matrices that provably run in average polynomial time. Making these algorithms more efficient or adapting them to structured settings are challenging and rewarding research problems. We intend to address some of these issues at the end of the course.
The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral ...

65F35 ; 65K05 ; 68Q15 ; 15A12 ; 65F10 ; 90C51 ; 65H10

Approximation methods and probabilistic algorithms are two important ways to obtain efficient algorithms giving approximate solutions to hard problems. We give some examples from optimization, counting and verification problems. Property testing is also a very efficient method to approximate verification problems.
complexity - difficult problem - approximation - probabilistic approximation schemes - optimization - counting
verification - property testing
Approximation methods and probabilistic algorithms are two important ways to obtain efficient algorithms giving approximate solutions to hard problems. We give some examples from optimization, counting and verification problems. Property testing is also a very efficient method to approximate verification problems.
complexity - difficult problem - approximation - probabilistic approximation schemes - optimization - counting
verification - ...

68Q15 ; 68Q17 ; 68Q19 ; 68W20 ; 68W25

Quelle est la puissance des machines de calculs analogiques (vs digitales)? Que peut-on calculer avec des équations différentielles ? Que cela nous apprend-t'il sur la physique et ses modèles de notre monde physique ?

03Dxx ; 68Q05 ; 68Q15

The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral geometry. The most well known example is Smale's 17th problem, which asks to find a solution of a given system of n complex homogeneous polynomial equations in $n$ + 1 unknowns. This problem can be solved in average (and even smoothed) polynomial time.
In the course we will explain the concepts necessary to state and solve Smale's 17th problem. We also show how these ideas lead to new numerical algorithms for computing eigenpairs of matrices that provably run in average polynomial time. Making these algorithms more efficient or adapting them to structured settings are challenging and rewarding research problems. We intend to address some of these issues at the end of the course.
The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways.
A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to fascinating problems of geometric probability and integral ...

65F35 ; 65K05 ; 68Q15 ; 15A12 ; 65F10 ; 90C51 ; 65H10

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