In this work, we first derive the one-point large deviations principle (LDP) for both the stochastic Cahn–Hilliard equation with small noise and its spatial finite difference method (FDM). Then, we focus on giving th...In this work, we first derive the one-point large deviations principle (LDP) for both the stochastic Cahn–Hilliard equation with small noise and its spatial finite difference method (FDM). Then, we focus on giving the convergence of the one-point large deviations rate function (LDRF) of the spatial FDM, which is about the asymptotical limit of a parametric variational problem. The main idea for proving the convergence of the LDRF of the spatial FDM is via the Γ-convergence of objective functions. This relies on the qualitative analysis of skeleton equations of the original equation and the numerical method. In order to overcome the difficulty that the drift coefficient is not one-sided Lipschitz continuous, we derive the equivalent characterization of the skeleton equation of the spatial FDM and the discrete interpolation inequality to obtain the uniform boundedness of the solution to the underlying skeleton equation. These play important roles in deriving the T-convergence of objective functions.展开更多
In this paper, we consider two variational models for speckle reduction of ultrasound images. By employing the F-convergence argument we show that the solution of the SO model coincides with the minimizer of the JY mo...In this paper, we consider two variational models for speckle reduction of ultrasound images. By employing the F-convergence argument we show that the solution of the SO model coincides with the minimizer of the JY model. Furthermore, we incorporate the split Bregman technique to propose a fast alterative algorithm to solve the JY model. Some numericalexperiments are presented to illustrate the efficiency of the proposed algorithm.展开更多
基金supported by the National Natural Science Foundation of China(12201228,12171047)the Fundamental Research Funds for the Central Universities(3034011102)supported by National Key R&D Program of China(2020YFA0713701).
文摘In this work, we first derive the one-point large deviations principle (LDP) for both the stochastic Cahn–Hilliard equation with small noise and its spatial finite difference method (FDM). Then, we focus on giving the convergence of the one-point large deviations rate function (LDRF) of the spatial FDM, which is about the asymptotical limit of a parametric variational problem. The main idea for proving the convergence of the LDRF of the spatial FDM is via the Γ-convergence of objective functions. This relies on the qualitative analysis of skeleton equations of the original equation and the numerical method. In order to overcome the difficulty that the drift coefficient is not one-sided Lipschitz continuous, we derive the equivalent characterization of the skeleton equation of the spatial FDM and the discrete interpolation inequality to obtain the uniform boundedness of the solution to the underlying skeleton equation. These play important roles in deriving the T-convergence of objective functions.
基金Supported by the National Natural Science Foundation of China under Grants No.11671004 and 91330101Natural Science Foundation for Colleges and Universities in Jiangsu Province under Grants No.15KJB110018 and 14KJB110020
文摘In this paper, we consider two variational models for speckle reduction of ultrasound images. By employing the F-convergence argument we show that the solution of the SO model coincides with the minimizer of the JY model. Furthermore, we incorporate the split Bregman technique to propose a fast alterative algorithm to solve the JY model. Some numericalexperiments are presented to illustrate the efficiency of the proposed algorithm.
