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
The angular deviations and influential factors of Burgers orientation relationship(BOR)in Ti-6Al-4V and Ti-6.5Al-2Zr-1Mo-1V alloys were investigated by optical microscope(OM),scanning electron microscope(SEM),electron...The angular deviations and influential factors of Burgers orientation relationship(BOR)in Ti-6Al-4V and Ti-6.5Al-2Zr-1Mo-1V alloys were investigated by optical microscope(OM),scanning electron microscope(SEM),electron backscattered diffraction(EBSD)and high-angle annular dark-field scanning transmission electron microscope(HAADF-STEM).A spherical center angle model was introduced to calculate the angular deviations from the ideal BOR between α and β phases.The results indicate thatαand β phases in α colonies of both alloys do not follow the perfect BOR during β→α phase transformation,with angular deviation values less than 3°.Through detailed microstructure characterization,the broad face of α/β interfaces viewed along two different electron incident directions shows the atomic-scale terrace-ledge structure,and many dislocations are observed within α and β phases and near α/β interfaces.Further studies reveal that the angular deviations mainly originate from lattice distortions caused by dislocations in α and β phases and lattice mismatches at α/β interfaces.展开更多
In this paper, we use sample average approximation with adaptive multiple importance sampling to explore moderate deviations for the optimal values. Utilizing the moderate deviation principle for martingale difference...In this paper, we use sample average approximation with adaptive multiple importance sampling to explore moderate deviations for the optimal values. Utilizing the moderate deviation principle for martingale differences and an appropriate Delta method, we establish a moderate deviation principle for the optimal value. Moreover, for a functional form of stochastic programming, we obtain a functional moderate deviation principle for its optimal value.展开更多
In this work, the sample path large deviations for independent, identically distributed random variables under sub-linear expectations are established. The results obtained in sublinear expectation spaces extend the c...In this work, the sample path large deviations for independent, identically distributed random variables under sub-linear expectations are established. The results obtained in sublinear expectation spaces extend the corresponding ones in probability space.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(Nos.51971009,12002013,51831006)the Natural Science Foundation of Zhejiang Province,China(No.LZ23E010004).
文摘The angular deviations and influential factors of Burgers orientation relationship(BOR)in Ti-6Al-4V and Ti-6.5Al-2Zr-1Mo-1V alloys were investigated by optical microscope(OM),scanning electron microscope(SEM),electron backscattered diffraction(EBSD)and high-angle annular dark-field scanning transmission electron microscope(HAADF-STEM).A spherical center angle model was introduced to calculate the angular deviations from the ideal BOR between α and β phases.The results indicate thatαand β phases in α colonies of both alloys do not follow the perfect BOR during β→α phase transformation,with angular deviation values less than 3°.Through detailed microstructure characterization,the broad face of α/β interfaces viewed along two different electron incident directions shows the atomic-scale terrace-ledge structure,and many dislocations are observed within α and β phases and near α/β interfaces.Further studies reveal that the angular deviations mainly originate from lattice distortions caused by dislocations in α and β phases and lattice mismatches at α/β interfaces.
基金Supported by the National Natural Science Foundation of China(Grant No.12071175)。
文摘In this paper, we use sample average approximation with adaptive multiple importance sampling to explore moderate deviations for the optimal values. Utilizing the moderate deviation principle for martingale differences and an appropriate Delta method, we establish a moderate deviation principle for the optimal value. Moreover, for a functional form of stochastic programming, we obtain a functional moderate deviation principle for its optimal value.
基金Doctoral Scientific Research Starting Foundation of Jingdezhen Ceramic University (Grant No. 102/01003002031)Academic Achievement Re-cultivation Project of Jingdezhen Ceramic University (Grant No. 215/205062777)the Science and Technology Research Project of Jiangxi Provincial Department of Education of China (Grant No. GJJ2201041)。
文摘In this work, the sample path large deviations for independent, identically distributed random variables under sub-linear expectations are established. The results obtained in sublinear expectation spaces extend the corresponding ones in probability space.
