Analysis of a splitting scheme for a class of nonlinear stochastic Schrödinger equations

Bréhier, Charles-Édouard; Cohen, David E.

doi:10.1016/j.apnum.2023.01.002

Cited by 11 publications

(18 citation statements)

References 72 publications

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“…Boundarypreserving methods have received a lot of attention the last two decades. Without being exhaustive, we mention the following articles [1,3,4,6,12,13,16,17,25,26,28,29,30,31]. Applications include, for example, population dynamics models [6,29,30,31] and heat flow models [3,4], defined e.g.…”

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confidence: 99%

“…Without being exhaustive, we mention the following articles [1,3,4,6,12,13,16,17,25,26,28,29,30,31]. Applications include, for example, population dynamics models [6,29,30,31] and heat flow models [3,4], defined e.g. in [0, 1] and defined in (0, ∞), respectively.…”

mentioning

confidence: 99%

“…The literature on schemes based on the Lamperti transform and on time splitting schemes is extensive. Without being exhaustive, we mention the following articles [6,13,16,26,30,31] on schemes based on the Lamperti transform and the following references [3,4,5,11,12,13,16,23,25] on time splitting schemes for differential equations. To the best of our knowledge, only the two recent articles [13,16] combines these two approaches to construct a positivity-preserving scheme for the Aït-Sahalia model and for the Cox-Ingersoll-Ross (CIR) process, respectively.…”

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confidence: 99%

“…We propose a scheme based on utilising the Lamperti transform associated with the considered SDE in equation ( 3) followed by a Lie-Trotter splitting strategy of the resulting transformed SDE in equation (3).…”

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confidence: 99%

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Boundary-preserving Lamperti-splitting schemes for some stochastic differential equations

Ulander

2024

JCD

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We propose and analyse boundary-preserving schemes for the strong approximations of some scalar SDEs with non-globally Lipschitz drift and diffusion coefficients whose state-space is bounded. The schemes consists of a Lamperti transform followed by a Lie-Trotter splitting. We prove L p (Ω)convergence of order 1, for every p ≥ 1, of the schemes and exploit the Lamperti transform to confine the numerical approximations to the state-space of the considered SDE. We provide numerical experiments that confirm the theoretical results and compare the proposed Lamperti-splitting schemes to other numerical schemes for SDEs.

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Boundary-preserving Lamperti-splitting schemes for some stochastic differential equations

Ulander

2024

JCD

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“…Bréhier and Cohen [27] investigated the qualitative characteristics and convergence order of a splitting scheme for the stochastic Schrödinger equations with additive noise. They showed the stochastic symplecticity of the numerical solution and consistently preserved the expected mass.…”

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Core Content Selection and Textbook Design for High School Mathematics

Zhang¹,

Zhou²,

Wang³

et al. 2021

School Mathematics Textbooks in China

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This paper is devoted to studying the following fractional L 2 -critical nonlinear Schrödinger equation≥ 0 is an external potential. We obtain normalized L 2 -norm solutions of the above equation by solving the associated constraint minimization problem (1.4). It shows that there is a threshold a * > 0 such that (1.4) has minimizers for 0 < a < a * , and minimizers do not exist for any a > a * . For the case of a = a * , it gives a fact that the existence and non-existence of minimizers depend strongly on the value of V (0). Especially for V (0) = 0, we prove that minimizers occur blow-up behavior and the mass of minimizers concentrates at the origin as a ր a * . Applying implicit function theorem, the uniqueness of minimizers is also proved for a > 0 small enough.

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Temporal approximation of stochastic evolution equations with irregular nonlinearities

Klioba,

Veraar

2024

J. Evol. Equ.

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In this paper, we prove convergence for contractive time discretisation schemes for semi-linear stochastic evolution equations with irregular Lipschitz nonlinearities, initial values, and additive or multiplicative Gaussian noise on 2-smooth Banach spaces X. The leading operator A is assumed to generate a strongly continuous semigroup S on X, and the focus is on non-parabolic problems. The main result concerns convergence of the uniform strong error$$\begin{aligned} \textrm{E}_{k}^{\infty } {:}{=}\Big (\mathbb {E}\sup _{j\in \{0, \ldots , N_k\}} \Vert U(t_j) - U^j\Vert _X^p\Big )^{1/p} \rightarrow 0\quad (k \rightarrow 0), \end{aligned}$$ E k ∞ : = ( E sup j ∈ { 0 , … , N k } ‖ U ( t j ) - U j ‖ X p ) 1 / p → 0 ( k → 0 ) , where $$p \in [2,\infty )$$ p ∈ [ 2 , ∞ ) , U is the mild solution, $$U^j$$ U j is obtained from a time discretisation scheme, k is the step size, and $$N_k = T/k$$ N k = T / k for final time $$T>0$$ T > 0 . This generalises previous results to a larger class of admissible nonlinearities and noise, as well as rough initial data from the Hilbert space case to more general spaces. We present a proof based on a regularisation argument. Within this scope, we extend previous quantified convergence results for more regular nonlinearity and noise from Hilbert to 2-smooth Banach spaces. The uniform strong error cannot be estimated in terms of the simpler pointwise strong error$$\begin{aligned} \textrm{E}_k {:}{=}\bigg (\sup _{j\in \{0,\ldots ,N_k\}}\mathbb {E}\Vert U(t_j) - U^{j}\Vert _X^p\bigg )^{1/p}, \end{aligned}$$ E k : = ( sup j ∈ { 0 , … , N k } E ‖ U ( t j ) - U j ‖ X p ) 1 / p , which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schrödinger equation, for which previous convergence results were not applicable.

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Analysis of a splitting scheme for a class of nonlinear stochastic Schrödinger equations

Cited by 11 publications

References 72 publications

Boundary-preserving Lamperti-splitting schemes for some stochastic differential equations

Boundary-preserving Lamperti-splitting schemes for some stochastic differential equations

Core Content Selection and Textbook Design for High School Mathematics

Temporal approximation of stochastic evolution equations with irregular nonlinearities

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