Jiayu Han

Yidu Yang, Jiayu Han, Hai Bi

The transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering theory. In this paper, for the Helmholtz transmission eigenvalue problem, we give a weak formulation which is a nonselfadjoint linear eigenvalue problem. Based on the weak formulation, we first discuss the non-conforming finite element approximation, and prove the error estimates of the discrete eigenvalues obtained by the Adini element, Morley-Zienkiewicz element, modified-Zienkiewicz element et. al. And we report some numerical examples to validate the efficiency of our approach for solving transmission eigenvalue problem.

Jiayu Han, Yidu Yang, Hai Bi

Numerical methods for the transmission eigenvalue problems are hot topics in recent years. Based on the work of Lin and Xie [Math. Comp., 84(2015), pp. 71-88], we build a multigrid method to solve the problems. With our method, we only need to solve a series of primal and dual eigenvalue problems on a coarse mesh and the associated boundary value problems on the finer and finer meshes. Theoretical analysis and numerical results show that our method is simple and easy to implement and is efficient for computing real and complex transmission eigenvalues.

Yidu Yang, Jiayu Han, Hai Bi

In this paper, using the linearization technique we write the Helmholtz transmission eigenvalue problem as an equivalent nonselfadjoint linear eigenvalue problem whose left-hand side term is a selfadjoint, continuous and coercive sesquilinear form. To solve the resulting nonselfadjoint eigenvalue problem, we give an $H^{2}$ conforming finite element discretization and establish a two grid discretization scheme. We present a complete error analysis for both discretization schemes, and theoretical analysis and numerical experiments show that the methods presented in this paper can efficiently compute real and complex transmission eigenvalues.

Jiayu Han, Yidu Yang

In this paper we develop an $H^m$-conforming ($m\ge1$) spectral element method on multi-dimensional domain associated with the partition into multi-dimensional rectangles. We construct a set of basis functions on the interval $[-1,1]$ that is made up of the generalized Jacobi polynomials (GJPs) and the nodal basis functions. So the basis functions on multi-dimensional rectangles consist of the tensorial product of the basis functions on the interval $[-1,1]$. Then we construct the spectral element interpolation operator and prove the associated interpolation error estimates. Finally we apply the $H^2$-conforming spectral element method to the Helmholtz transmission eigenvalues that is a hot topic in the field engineering and mathematics.

David K. Yi, James V. Bruno, Jiayu Han et al.

We investigate the extent to which verb alternation classes, as described by Levin (1993), are encoded in the embeddings of Large Pre-trained Language Models (PLMs) such as BERT, RoBERTa, ELECTRA, and DeBERTa using selectively constructed diagnostic classifiers for word and sentence-level prediction tasks. We follow and expand upon the experiments of Kann et al. (2019), which aim to probe whether static embeddings encode frame-selectional properties of verbs. At both the word and sentence level, we find that contextual embeddings from PLMs not only outperform non-contextual embeddings, but achieve astonishingly high accuracies on tasks across most alternation classes. Additionally, we find evidence that the middle-to-upper layers of PLMs achieve better performance on average than the lower layers across all probing tasks.

Jiayu Han

In this paper two types of multgrid methods, i.e., the Rayleigh quotient iteration and the inverse iteration with fixed shift, are developed for solving the Maxwell eigenvalue problem with discontinuous relative magnetic permeability and electric permittivity. With the aid of the mixed form of source problem associated with the eigenvalue problem, we prove the uniform convergence of the discrete solution operator to the solution operator in $L^2(Ω)$ using discrete compactness of edge element space. Then we prove the asymptotically optimal error estimates for both multigrid methods. Numerical experiments confirm our theoretical analysis.

Zhiwei Liu, Lei Zheng, Jiawei Zhang et al.

Cross-domain recommendation can alleviate the data sparsity problem in recommender systems. To transfer the knowledge from one domain to another, one can either utilize the neighborhood information or learn a direct mapping function. However, all existing methods ignore the high-order connectivity information in cross-domain recommendation area and suffer from the domain-incompatibility problem. In this paper, we propose a \textbf{J}oint \textbf{S}pectral \textbf{C}onvolutional \textbf{N}etwork (JSCN) for cross-domain recommendation. JSCN will simultaneously operate multi-layer spectral convolutions on different graphs, and jointly learn a domain-invariant user representation with a domain adaptive user mapping module. As a result, the high-order comprehensive connectivity information can be extracted by the spectral convolutions and the information can be transferred across domains with the domain-invariant user mapping. The domain adaptive user mapping module can help the incompatible domains to transfer the knowledge across each other. Extensive experiments on $24$ Amazon rating datasets show the effectiveness of JSCN in the cross-domain recommendation, with $9.2\%$ improvement on recall and $36.4\%$ improvement on MAP compared with state-of-the-art methods. Our code is available online ~\footnote{https://github.com/JimLiu96/JSCN}.

Jiayu Han, Zhimin Zhang

Using newly developed ${\bf H}(\mathrm{curl}^2)$ conforming elements, we solve the Maxwell's transmission eigenvalue problem. Both real and complex eigenvalues are considered. Based on the fixed-point weak formulation with reasonable assumptions, the optimal error estimates for numerical eigenvalues and eigenfunctions (in the ${\bf H}(\mathrm{curl}^2)$-norm and ${\bf H}(\mathrm{curl})$-semi-norm) are established.

Ridong Han, Tao Peng, Jiayu Han et al.

Wrong-labeling problem and long-tail relations severely affect the performance of distantly supervised relation extraction task. Many studies mitigate the effect of wrong-labeling through selective attention mechanism and handle long-tail relations by introducing relation hierarchies to share knowledge. However, almost all existing studies ignore the fact that, in a sentence, the appearance order of two entities contributes to the understanding of its semantics. Furthermore, they only utilize each relation level of relation hierarchies separately, but do not exploit the heuristic effect between relation levels, i.e., higher-level relations can give useful information to the lower ones. Based on the above, in this paper, we design a novel Recursive Hierarchy-Interactive Attention network (RHIA) to further handle long-tail relations, which models the heuristic effect between relation levels. From the top down, it passes relation-related information layer by layer, which is the most significant difference from existing models, and generates relation-augmented sentence representations for each relation level in a recursive structure. Besides, we introduce a newfangled training objective, called Entity-Order Perception (EOP), to make the sentence encoder retain more entity appearance information. Substantial experiments on the popular (NYT) dataset are conducted. Compared to prior baselines, our RHIA-EOP achieves state-of-the-art performance in terms of precision-recall (P-R) curves, AUC, Top-N precision and other evaluation metrics. Insightful analysis also demonstrates the necessity and effectiveness of each component of RHIA-EOP.

Yidu Yang, Jiayu Han

Based on the work of Xu and Zhou [Math.Comput., 69(2000), pp.881-909], we establish new three-level and multilevel finite element discretizations by local defect-correction technique. Theoretical analysis and numerical experiments show that the schemes are simple and easy to carry out, and can be used to solve singular nonsymmetric eigenvalue problems efficiently. We also discuss the local error estimates of finite element approximations; it's a new feature here that the estimates apply to the local domains containing corner points.