2019
DOI: 10.1016/j.matpur.2019.04.012
|View full text |Cite
|
Sign up to set email alerts
|

Abstract: Global dynamics in nonlinear stochastic systems is often difficult to analyze rigorously. Yet, many excellent numerical methods exist to approximate these systems. In this work, we propose a method to bridge the gap between computation and analysis by introducing rigorous validated computations for stochastic systems. The first step is to use analytic methods to reduce the stochastic problem to one solvable by a deterministic algorithm and to numerically compute a solution. Then one uses fixed-point arguments,… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
2
1
1

Citation Types

0
13
0

Year Published

2020
2020
2022
2022

Publication Types

Select...
5
1

Relationship

2
4

Authors

Journals

citations
Cited by 9 publications
(13 citation statements)
references
References 80 publications
0
13
0
Order By: Relevance
“…The only thing that remains to be discussed before we present the results of the computerassisted proofs is the important choice of the weights in the norm || • || Xν ,η we use on the space X ν . Actually, the crucial part for this problem is the careful choice of η ∈ (0, ∞) 10 , while ν can be chosen a bit more carelessly. Indeed, ν must be strictly larger than 1 because we need terms like (53) to be small, and not too large because we do not want the various…”
Section: Implementation Results and Commentsmentioning
confidence: 99%
See 3 more Smart Citations
“…The only thing that remains to be discussed before we present the results of the computerassisted proofs is the important choice of the weights in the norm || • || Xν ,η we use on the space X ν . Actually, the crucial part for this problem is the careful choice of η ∈ (0, ∞) 10 , while ν can be chosen a bit more carelessly. Indeed, ν must be strictly larger than 1 because we need terms like (53) to be small, and not too large because we do not want the various…”
Section: Implementation Results and Commentsmentioning
confidence: 99%
“…Therefore, considering Y , Z 0 , Z 1 and Z 2 as function of η, where the dependency in η is explicit (see (46), (54) or (55)) it is cheap to numerically optimize for η according to our needs. A possible optimization criteria, which has been successfully used in the past [10] and amounts to the computation of a Perron-Frobenius eigenvector, is to take η such that Z 1 is minimal. However, for our current problem such a choice often leads to Z 2 being too large, and thus to the second condition in (39) no longer being satisfied.…”
Section: Implementation Results and Commentsmentioning
confidence: 99%
See 2 more Smart Citations
“…The extra 1 could even be removed by choosing an appropriately weighted norm on the product space ℓ 1 ν × ℓ 1 ν (see e.g. the discussion about the weights in [7]). Therefore the only thing that we lost, at least concerning the crucial Z tail 1 estimate, by going through a system of differential-algebraic equation is that σR ′ ( ū) ν is replaced by σ ν R ′ ( ū) ν , which in most cases is perfectly acceptable.…”
Section: Derivation Of the Boundsmentioning
confidence: 99%