A class of new Magnus-type methods for semi-linear non-commutative Itô stochastic differential equations

Yang, Guoguo; Burrage, Kevin; Komori, Yoshio; Burrage, Pamela; Ding, Xiaohua

doi:10.1007/s11075-021-01089-7

Cited by 9 publications

(4 citation statements)

References 19 publications

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“…Theorem 1 has been used in the recent paper [34] (cf. Lemma 1) where a semi-linear noncommuative Itô-SDEs is studied and Euler, Milstein and derivative-free numerical schemes are developed, with a convergence analysis for those schemes.…”

Section: Description Of the Main Resultsmentioning

confidence: 99%

On the Stochastic Magnus Expansion and Its Application to SPDEs

2021

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We derive the stochastic version of the Magnus expansion for linear systems of stochastic differential equations (SDEs). The main novelty with respect to the related literature is that we consider SDEs in the Itô sense, with progressively measurable coefficients, for which an explicit Itô-Stratonovich conversion is not available. We prove convergence of the Magnus expansion up to a stopping time $$\tau $$ τ and provide a novel asymptotic estimate of the cumulative distribution function of $$\tau $$ τ . As an application, we propose a new method for the numerical solution of stochastic partial differential equations (SPDEs) based on spatial discretization and application of the stochastic Magnus expansion. A notable feature of the method is that it is fully parallelizable. We also present numerical tests in order to asses the accuracy of the numerical schemes.

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Section: Description Of the Main Resultsmentioning

confidence: 99%

On the Stochastic Magnus Expansion and Its Application to SPDEs

2021

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“…The Magnus type methods proposed by Yang et al [30] for noncommutative SDEs (9) belong to this class of exponential integrators. For these methods, construction of the random coefficients is discussed in [20].…”

Section: β(•mentioning

confidence: 99%

B-series for SDEs with application to exponential integrators for non-autonomous semi-linear problems

Arara,

Debrabant,

Kværnø

2024

JCD

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In this paper a set of previous general results for the development of B-series for a broad class of stochastic differential equations has been collected. The applicability of these results is demonstrated by the derivation of B-series for non-autonomous semi-linear SDEs and exponential Runge-Kutta methods applied to this class of SDEs, which is a significant generalization of existing theory on such methods.

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“…Notice that and are complex variables, thus the Itô SDE is a dimensional SDE when considering their real and imaginary components. Carrying out the parametric settings given in Yang et al (2021), the discretized Manakov system is a 396-dimensioal Itô SDE. Of course, it is impossible to solve the joint PDF in such high dimension.…”

Section: Illustrated Scenarios For Typical High-dimensional Systemsmentioning

confidence: 99%

DR-PDEE for engineered high-dimensional nonlinear stochastic systems: A physically-driven equation providing theoretical basis for data-driven approaches

Chen,

Sun,

Lyu

2024

Preprint

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For over half a century, the analysis, control, and optimization design of high-dimensional nonlinear stochastic dynamical systems have posed long-standing challenges in the fields of science and engineering. Emerging scientific ideas and powerful technologies, such as big data and artificial intelligence (AI), offer new opportunity for addressing this problem. Data-driven techniques and AI methods are beginning to empower the research on stochastic dynamics. However, what is the physical essence, theoretical foundation, and effective applicable spectrum of data-driven and AI-aided (DDAA) stochastic dynamics? Answering this question has become important and urgent for advancing research in stochastic dynamics more solidly and effectively. This paper will provide a perspective on answering this question from the viewpoint of system dimensionality reduction. In the DDAA framework, the dimension of observed data of the studied system, such as the dimension of the complete state variables of the system, is fundamentally unknown. Thus, it can be considered that the stochastic dynamical systems under the DDAA framework are dimension-reduced subsystems of real-world systems. Therefore, a question of interest is: To what extent can the probability information predicted by the dimension-reduced subsystem characterize the probability information of the real-world system and serve as a decision basis? The paper will discuss issues such as the dimension-reduced probability density evolution equation (DR-PDEE) satisfied by the probability density function (PDF) of path-continuous non-Markov responses in general high-dimensional systems, the dimension-reduced partial integro-differential equation satisfied by the PDF of path-discontinuous responses, and the non-exchangeability of dimension reduction and imposition of absorbing boundary conditions. These studies suggest that the DR-PDEE and the dimension-reduced partial integro-differential equation can serve as important theoretical bases for the effectiveness and applicability boundaries of the DDAA framework.

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A class of new Magnus-type methods for semi-linear non-commutative Itô stochastic differential equations

Cited by 9 publications

References 19 publications

On the Stochastic Magnus Expansion and Its Application to SPDEs

On the Stochastic Magnus Expansion and Its Application to SPDEs

B-series for SDEs with application to exponential integrators for non-autonomous semi-linear problems

DR-PDEE for engineered high-dimensional nonlinear stochastic systems: A physically-driven equation providing theoretical basis for data-driven approaches

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