Lei Zhang

Houman Owhadi, Lei Zhang

We construct finite-dimensional approximations of solution spaces of divergence form operators with $L^\infty$-coefficients. Our method does not rely on concepts of ergodicity or scale-separation, but on the property that the solution space of these operators is compactly embedded in $H^1$ if source terms are in the unit ball of $L^2$ instead of the unit ball of $H^{-1}$. Approximation spaces are generated by solving elliptic PDEs on localized sub-domains with source terms corresponding to approximation bases for $H^2$. The $H^1$-error estimates show that $\mathcal{O}(h^{-d})$-dimensional spaces with basis elements localized to sub-domains of diameter $\mathcal{O}(h^α\ln \frac{1}{h})$ (with $α\in [1/2,1)$) result in an $\mathcal{O}(h^{2-2α})$ accuracy for elliptic, parabolic and hyperbolic problems. For high-contrast media, the accuracy of the method is preserved provided that localized sub-domains contain buffer zones of width $\mathcal{O}(h^α\ln \frac{1}{h})$ where the contrast of the medium remains bounded. The proposed method can naturally be generalized to vectorial equations (such as elasto-dynamics).

Lei Zhang, Mingliang Wang, Xin Zhou et al.

Delivery Time Estimation (DTE) is a crucial component of the e-commerce supply chain that predicts delivery time based on merchant information, sending address, receiving address, and payment time. Accurate DTE can boost platform revenue and reduce customer complaints and refunds. However, the imbalanced nature of industrial data impedes previous models from reaching satisfactory prediction performance. Although imbalanced regression methods can be applied to the DTE task, we experimentally find that they improve the prediction performance of low-shot data samples at the sacrifice of overall performance. To address the issue, we propose a novel Dual Graph Multitask framework for imbalanced Delivery Time Estimation (DGM-DTE). Our framework first classifies package delivery time as head and tail data. Then, a dual graph-based model is utilized to learn representations of the two categories of data. In particular, DGM-DTE re-weights the embedding of tail data by estimating its kernel density. We fuse two graph-based representations to capture both high- and low-shot data representations. Experiments on real-world Taobao logistics datasets demonstrate the superior performance of DGM-DTE compared to baselines.