Study of dynamical systems from the viewpoint of complexity and computational capabilities

Gladkikh, A. A.; Малинецкий, Г Г

doi:10.1134/s0012266116070090

Search citation statements

Order By: Relevance

Paper Sections

Select...

Solving Various Problems Using Dynamical Systems1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2018

2020

Publication Types

Select...

Other1

Article1

Relationship

Self Cite0

Independent2

Authors

Journals

Cited by 2 publications

(2 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Many examples in this spirit are discussed in details in [203]. More recently a system of nonlinear differential equations of a similar form to sort numbers fed to the input has been investigated [183].…”

Section: Solving Various Problems Using Dynamical Systemsmentioning

confidence: 99%

A Survey on Analog Models of Computation

Bournez¹,

Pouly²

2018

Preprint

View full text Add to dashboard Cite

show abstract

Section: Solving Various Problems Using Dynamical Systemsmentioning

confidence: 99%

A Survey on Analog Models of Computation

Bournez¹,

Pouly²

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…In particular, algorithms for eigenvalue problems have been studied in this context, with the relation to integrable systems (Lax-pair formulation) [40]. Studying the connection between algorithms and dynamcal systems led to many surprising results, such as the definition of systems that can "sort numbers" [4], or, more generally, solve optimization problems with constraints [14]. Chaotic dynamical systems that "paint", i.e., have stationary distributions according to the distribution of paint in an image, have been studied in relation to Markov Chain Monte Carlo algorithms [35].…”

mentioning

confidence: 99%

On the Koopman Operator of Algorithms

Dietrich¹,

Thiem²,

Kevrekidis³

2020

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

A systematic mathematical framework for the study of numerical algorithms would allow comparisons, facilitate conjugacy arguments, as well as enable the discovery of improved, accelerated, data-driven algorithms. Over the course of the last century, the Koopman operator has provided a mathematical framework for the study of dynamical systems, which facilitates conjugacy arguments and can provide efficient reduced descriptions. More recently, numerical approximations of the operator have made it possible to analyze dynamical systems in a completely data-driven, essentially equation-free pipeline. Discrete or continuous time numerical algorithms (integrators, nonlinear equation solvers, optimization algorithms) are themselves dynamical systems. In this paper, we use the Koopman operator framework in the data-driven study of such algorithms and discuss benefits for analysis and acceleration of numerical computation. For algorithms acting on high-dimensional spaces by quickly contracting them towards low-dimensional manifolds, we demonstrate how basis functions adapted to the data help to construct efficient reduced representations of the operator.Our illustrative examples include the gradient descent and Nesterov optimization algorithms, as well as the Newton-Raphson algorithm.

show abstract

Study of dynamical systems from the viewpoint of complexity and computational capabilities

Cited by 2 publications

References 4 publications

A Survey on Analog Models of Computation

A Survey on Analog Models of Computation

On the Koopman Operator of Algorithms

Contact Info

Product

Resources

About