论文标题
随机线性schrödinger方程的符合性离散的大偏差原则
Large deviations principles for symplectic discretizations of stochastic linear Schrödinger Equation
论文作者
论文摘要
在本文中，我们考虑了随机线性schrödinger方程及其符合性离散的大偏差原理（LDP）。这些数值离散化是基于光谱Galerkin方法的空间半差异化，以及沿时间方向沿互合式方案的进一步完全离散化。首先，通过抽象的gärtner-ellis定理，我们证明可观察到的$ b_t = \ frac {u（t）} {t} $，$ t> 0 $的精确解决方案$ u $的$ u $的$ u $呈指数紧密，并满足$ ldp on $ l^2（0，f^2（0，π; \ mathbb c）$。然后，我们为两个$ \ {b^m_t \} _ {t> 0} $的LDP介绍了空间离散化$ \ {u^m \ \} _ {m \ in \ mathbb n} $和$ \ {b^m_n \} $ \ {u^m_n \} _ {m，n \ in \ mathbb n} $，其中$ b^m_t = \ frac {u^m（t）} {t} $ and $ b^m_n = \ frac {u^m_n} {u^m_n} {n frac {n frac {nτ} {nτ} $是$ nist $ nist $ nist $ b__t。此外，我们表明，两种半散制$ \ {u^m \ \} _ {m \ in \ mathbb n} $和完整的离散化$ \ {u^m_n \} _ {m {m，n \ in \ Mathbb n} $基于时间符号symplectic symplectic symplectic simplectic sisterve as persive as ysymptery Assmptecreationally Assmptsever assymptery assymptery assymptely assimptyly Assmptecreationally assymptecreationally assymptecreationally assympte and $ \ {b_t \} _ {t> 0} $。这些结果表明，符号离散化保留随机线性\ XDE方程的不动质量的能力，并首先提供了一种有效的方法来基于数值离散化的无限尺寸空间中的LDP速率函数。
In this paper, we consider the large deviations principles (LDPs) for the stochastic linear Schrödinger equation and its symplectic discretizations. These numerical discretizations are the spatial semi-discretization based on spectral Galerkin method, and the further full discretizations with symplectic schemes in temporal direction. First, by means of the abstract Gärtner--Ellis theorem, we prove that the observable $B_T=\frac{u(T)}{T}$, $T>0$ of the exact solution $u$ is exponentially tight and satisfies an LDP on $L^2(0, π; \mathbb C)$. Then, we present the LDPs for both $\{B^M_T\}_{T>0}$ of the spatial discretization $\{u^M\}_{M\in\mathbb N}$ and $\{B^M_N\}_{N\in \mathbb N}$ of the full discretization $\{u^M_N\}_{M,N\in\mathbb N}$, where $B^M_T=\frac{u^M(T)}{T}$ and $B^M_N=\frac{u^M_N}{Nτ}$ are the discrete approximations of $B_T$. Further, we show that both the semi-discretization $\{u^M\}_{M\in \mathbb N}$ and the full discretization $\{u^M_N\}_{M,N\in \mathbb N}$ based on temporal symplectic schemes can weakly asymptotically preserve the LDP of $\{B_T\}_{T>0}$. These results show the ability of symplectic discretizations to preserve the LDP of the stochastic linear \xde equation, and first provide an effective approach to approximating the LDP rate function in infinite dimensional space based on the numerical discretizations.
