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Yiping Zhang

Publications and source records attributed to Yiping Zhang.

24 records · Page 2Linked to original sources
arXiv2020

Estimates of eigenvalues and eigenfunctions in elliptic homogenization with rapidly oscillating potentials

In this paper, for a family of second-order elliptic equations with rapidly oscillating periodic coefficients and rapidly oscillating periodic potentials, we are interested in the $H^1$ convergence rates and the Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The $H^1$ convergence rates rely on the Dirichlet correctors and the first-order corrector for the oscillating potentials. And the bound results rely on an $O(\varepsilon)$ estimate in $H^1$ for solutions with Dirichlet condition.

math.AP↗
arXiv2020

Approximate Two-Sphere One-Cylinder Inequality in Parabolic Periodic Homogenization

In this paper, for a family of second-order parabolic equation with rapidly oscillating and time-dependent periodic coefficients, we are interested in an approximate two-sphere one-cylinder inequality for these solutions in parabolic periodic homogenization, which implies an approximate quantitative propagation of smallness. The proof relies on the asymptotic behavior of fundamental solutions and the Lagrange interpolation technique.

math.AP↗
arXiv2019

Error Estimates of Reiterated Stokes Systems via Fourier Transform Methods

In this paper, we are interested in the error estimates of the reiterated Stokes systems in a bounded $C^{1,1}$ domain with Dirichlet boundary conditions. And we have obtained the $O(\varepsilon)$ error estimates for the velocity term and $O(\varepsilon^{1/2})$ error estimates for the pressure term. Compared to the general homogenization of Stokes systems problems, the difficulty in the reiterated homogenization is that we need to handle the different scales of $x$. To overcome this difficulty, we use the Fourier transform methods which was firstly introduced by the author in [10] to separate these different scales. We also note that this method may be adapted to a more general multi-scale homogenization problem.

math.AP↗
arXiv2019

Quantitative Estimates on Reiterated Homogenization of Linear Elliptic operators Using Fourier Transform Methods

In this paper, we are interested in the reiterated homogenization of linear elliptic equations of the form $-\frac{\partial}{\partial x_{i}} \left(a_{i j} \left(\frac{x}{\varepsilon}, \frac{x}{\varepsilon^{2}}\right) \frac{\partial u_\varepsilon}{\partial x_{j}}\right)=f$ in $Ω$ with Dirichlet boundary conditions. We obtain error estimates $O(\varepsilon)$ for a bounded $C^{1,1}$ domain for this equation as well as the interior Lipschitz estimates at (very) large scale. Compared to the general homogenization problems, the difficulty in the reiterated homogenization is that we need to handle different scales of $x$. To overcome this difficulty, we firstly introduce the Fourier transform in the homogenization theory to separate these different scales. We also note that this method may be adapted to the following reiterated homogenization problem: $-\frac{\partial}{\partial x_{i}} \left(a_{i j} \left(\frac{x}{\varepsilon},\cdots, \frac{x}{\varepsilon^{N}}\right) \frac{\partial u_\varepsilon}{\partial x_{j}} \right) = f$ in $Ω$ with Dirichlet boundary conditions. Moreover, our results may be extended to the related Neumann boundary problems without any real difficulty.

math.AP↗
arXiv2018

A Boosting Method to Face Image Super-resolution

Recently sparse representation has gained great success in face image super-resolution. The conventional sparsity-based methods enforce sparse coding on face image patches and the representation fidelity is measured by $\ell_{2}$-norm. Such a sparse coding model regularizes all facial patches equally, which however ignores distinct natures of different facial patches for image reconstruction. In this paper, we propose a new weighted-patch super-resolution method based on AdaBoost. Specifically, in each iteration of the AdaBoost operation, each facial patch is weighted automatically according to the performance of the model on it, so as to highlight those patches that are more critical for improving the reconstruction power in next step. In this way, through the AdaBoost training procedure, we can focus more on the patches (face regions) with richer information. Various experimental results on standard face database show that our proposed method outperforms state-of-the-art methods in terms of both objective metrics and visual quality.

cs.CV↗