Weinan E, Zhongyi Huang
A new class of matching condition between the atomistic and continuum regions is presented for the multi-scale modeling of crystals. They ensure the accurate passage of large scale information between the atomistic and continuum regions and at the same time minimize the reflection of phonons at the interface. These matching conditions can be made adaptive if we choose appropriate weight functions. Applications to dislocation dynamics and friction between two-dimensional atomically flat crystal surfaces are described.
Weinan E, Zhongyi Huang
We present a coupled atomistic-continuum method for the modeling of defects and interface dynamics of crystalline materials. The method uses atomistic models such as molecular dynamics near defects and interfaces, and continuum models away from defects and interfaces. We propose a new class of matching conditions between the atomistic and continuum regions. These conditions ensure the accurate passage of large scale information between the atomistic and continuum regions and at the same time minimize the reflection of phonons at the atomistic-continuum interface. They can be made adaptive if we choose appropriate weight functions. We present applications to dislocation dynamics, friction between two-dimensional crystal surfaces and fracture dynamics. We compare results of the coupled method and the detailed atomistic model.
Paolo Antonelli, Agissilaos Athanassoulis, Zhongyi Huang, Peter A. Markowich
We study a nonlinear Schrödinger equation which arises as an effective single particle model in X-ray Free Electron Lasers (XFEL). This equation appears as a first-principles model for the beam-matter interactions that would take place in an XFEL molecular imaging experiment in \cite{frat1}. Since XFEL is more powerful by several orders of magnitude than more conventional lasers, the systematic investigation of many of the standard assumptions and approximations has attracted increased attention. In this model the electrons move under a rapidly oscillating electromagnetic field, and the convergence of the problem to an effective time-averaged one is examined. We use an operator splitting pseudo-spectral method to investigate numerically the behaviour of the model versus its time-averaged version in complex situations, namely the energy subcritical/mass supercritical case, and in the presence of a periodic lattice. We find the time averaged model to be an effective approximation, even close to blowup, for fast enough oscillations of the external field. This work extends previous analytical results for simpler cases \cite{xfel1}.
Zhizhang Wu, Zhongyi Huang
In this paper, we consider the numerical solution of the one-dimensional Schrödinger equation with a periodic lattice potential and a random external potential. This is an important model in solid state physics where the randomness is involved to describe some complicated phenomena that are not exactly known. Here we generalize the Bloch decomposition-based time-splitting pseudospectral method to the stochastic setting using the generalize polynomial chaos with a Galerkin procedure so that the main effects of dispersion and periodic potential are still computed together. We prove that our method is unconditionally stable and numerical examples show that it has other nice properties and is more efficient than the traditional method. Finally, we give some numerical evidence for the well-known phenomenon of Anderson localization.
Hao Wu, Zhongyi Huang, Shi Jin, Dongsheng Yin
The Dirac equation is an important model in relativistic quantum mechanics. In the semi-classical regime $ε\ll1$, even a spatially spectrally accurate time splitting method \cite{HuJi:05} requires the mesh size to be $O(ε)$, which makes the direct simulation extremely expensive. In this paper, we present the Gaussian beam method for the Dirac equation. With the help of an eigenvalue decomposition, the Gaussian beams can be independently evolved along each eigenspace and summed to construct an approximate solution of the Dirac equation. Moreover, the proposed Eulerian Gaussian beam keeps the advantages of constructing the Hessian matrices by simply using level set functions' derivatives. Finally, several numerical examples show the efficiency and accuracy of the method.
Zhongyi Huang, Shi Jin, Peter Markowich, Christof Sparber, Chunxiong Zheng
We present a time-splitting spectral scheme for the Maxwell-Dirac system and similar time-splitting methods for the corresponding asymptotic problems in the semi-classical and the non-relativistic regimes. The scheme for the Maxwell-Dirac system conserves the Lorentz gauge condition, is unconditionally stable and highly efficient as our numerical examples show. In particular we focus in our examples on the creation of positronic modes in the semi-classical regime and on the electron-positron interaction in the non-relativistic regime. Furthermore, in the non-relativistic regime, our numerical method exhibits uniform convergence in the small parameter $\dt$, which is the ratio of the characteristic speed and the speed of light.
Zhongyi Huang, Shi Jin, Peter Markowich, Christof Sparber
We present a new numerical method for accurate computations of solutions to (linear) one dimensional Schrödinger equations with periodic potentials. This is a prominent model in solid state physics where we also allow for perturbations by non-periodic potentials describing external electric fields. Our approach is based on the classical Bloch decomposition method which allows to diagonalize the periodic part of the Hamiltonian operator. Hence, the dominant effects from dispersion and periodic lattice potential are computed together, while the non-periodic potential acts only as a perturbation. Because the split-step communicator error between the periodic and non-periodic parts is relatively small, the step size can be chosen substantially larger than for the traditional splitting of the dispersion and potential operators. Indeed it is shown by the given examples, that our method is unconditionally stable and more efficient than the traditional split-step pseudo spectral schemes. To this end a particular focus is on the semiclassical regime, where the new algorithm naturally incorporates the adiabatic splitting of slow and fast degrees of freedom.
Haiying Xia, Zhongyi Huang, Yumei Tan, Shuxiang Song
Music emotion recognition is a key task in symbolic music understanding (SMER). Recent approaches have shown promising results by fine-tuning large-scale pre-trained models (e.g., MIDIBERT, a benchmark in symbolic music understanding) to map musical semantics to emotional labels. While these models effectively capture distributional musical semantics, they often overlook tonal structures, particularly musical modes, which play a critical role in emotional perception according to music psychology. In this paper, we investigate the representational capacity of MIDIBERT and identify its limitations in capturing mode-emotion associations. To address this issue, we propose a Mode-Guided Enhancement (MoGE) strategy that incorporates psychological insights on mode into the model. Specifically, we first conduct a mode augmentation analysis, which reveals that MIDIBERT fails to effectively encode emotion-mode correlations. We then identify the least emotion-relevant layer within MIDIBERT and introduce a Mode-guided Feature-wise linear modulation injection (MoFi) framework to inject explicit mode features, thereby enhancing the model's capability in emotional representation and inference. Extensive experiments on the EMOPIA and VGMIDI datasets demonstrate that our mode injection strategy significantly improves SMER performance, achieving accuracies of 75.2% and 59.1%, respectively. These results validate the effectiveness of mode-guided modeling in symbolic music emotion recognition.
Hao Shi, Zhengyi Jiang, Zhongyi Huang, Bo Bai, Hanxu Hou
As a special class of array codes, $(n,k,m)$ piggybacking codes are MDS codes (i.e., any $k$ out of $n$ nodes can retrieve all data symbols) that can achieve low repair bandwidth for single-node failure with low sub-packetization $m$. In this paper, we propose two new piggybacking codes that have lower repair bandwidth than the existing piggybacking codes given the same parameters. Our first piggybacking codes can support flexible sub-packetization $m$ with $2\leq m\leq n-k$, where $n - k > 3$. We show that our first piggybacking codes have lower repair bandwidth for any single-node failure than the existing piggybacking codes when $n - k = 8,9$, $m = 6$ and $30\leq k \leq 100$. Moreover, we propose second piggybacking codes such that the sub-packetization is a multiple of the number of parity nodes (i.e., $(n-k)|m$), by jointly designing the piggyback function for data node repair and transformation function for parity node repair. We show that the proposed second piggybacking codes have lowest repair bandwidth for any single-node failure among all the existing piggybacking codes for the evaluated parameters $k/n = 0.75, 0.8, 0.9$ and $n-k\geq 4$.
Xianliang Xu, Zhongyi Huang
The Wasserstein distance has been an attractive tool in many fields. But due to its high computational complexity and the phenomenon of the curse of dimensionality in empirical estimation, various extensions of the Wasserstein distance have been proposed to overcome the shortcomings such as the Sliced Wasserstein distance. It enjoys a low computational cost and dimension-free sample complexity, but there are few distributional limit results of it. In this paper, we focus on Sliced 1-Wasserstein distance and its variant max-Sliced 1-Wasserstein distance. We utilize the central limit theorem in Banach space to derive the limit distribution for the Sliced 1-Wasserstein distance. Through viewing the empirical max-Sliced 1-Wasserstein distance as a supremum of an empirical process indexed by some function class, we prove that the function class is P-Donsker under mild moment assumption. Moreover, for computing Sliced p-Wasserstein distance based on Monte Carlo method, we explore that how many random projections that can make sure the error small in high probability. We also provide upper bound of the expected max-Sliced 1-Wasserstein between the true and the empirical probability measures under different conditions and the concentration inequalities for max-Sliced 1-Wasserstein distance are also presented. As applications of the theory, we utilize them for two-sample testing problem.
Zhiquan Tan, Dingli Yuan, Zihao Wang, Zhongyi Huang
This paper considers the problem of calculating the matrix multiplication of two massive matrices $\mathbf{A}$ and $\mathbf{B}$ distributedly. We provide a modulo technique that can be applied to coded distributed matrix multiplication problems to reduce the recovery threshold. This technique exploits the special structure of interpolation points and can be applied to many existing coded matrix designs. Recently studied discrete Fourier transform based code achieves a smaller recovery threshold than the optimal MatDot code with the expense that it cannot resist stragglers. We also propose a distributed matrix multiplication scheme based on the idea of locally repairable code to reduce the recovery threshold of MatDot code and provide resilience to stragglers. We also apply our constructions to a type of matrix computing problems, where generalized linear models act as a special case.
Hao Shi, Zhengyi Jiang, Zhongyi Huang, Bo Bai, Gong Zhang, Hanxu Hou
Maximum distance separable (MDS) codes facilitate the achievement of elevated levels of fault tolerance in storage systems while incurring minimal redundancy overhead. Reed-Solomon (RS) codes are typical MDS codes with the sub-packetization level being one, however, they require large repair bandwidth defined as the total amount of symbols downloaded from other surviving nodes during single-node failure/repair. In this paper, we present the {\em set transformation}, which can transform any MDS code into set transformed code such that (i) the sub-packetization level is flexible and ranges from 2 to $(n-k)^{\lfloor\frac{n}{n-k}\rfloor}$ in which $n$ is the number of nodes and $k$ is the number of data nodes, (ii) the new code is MDS code, (iii) the new code has lower repair bandwidth for any single-node failure. We show that our set transformed codes have both lower repair bandwidth and lower field size than the existing related MDS array codes, such as elastic transformed codes \cite{10228984}. Specifically, our set transformed codes have $2\%-6.6\%$ repair bandwidth reduction compared with elastic transformed codes \cite{10228984} for the evaluated typical parameters.
Xianliang Xu, Ye Li, Zhongyi Huang
In this paper, we derive refined generalization bounds for the Deep Ritz Method (DRM) and Physics-Informed Neural Networks (PINNs). For the DRM, we focus on two prototype elliptic partial differential equations (PDEs): Poisson equation and static Schrödinger equation on the $d$-dimensional unit hypercube with the Neumann boundary condition. Furthermore, sharper generalization bounds are derived based on the localization techniques under the assumptions that the exact solutions of the PDEs lie in the Barron spaces or the general Sobolev spaces. For the PINNs, we investigate the general linear second order elliptic PDEs with Dirichlet boundary condition using the local Rademacher complexity in the multi-task learning setting. Finally, we discuss the generalization error in the setting of over-parameterization when solutions of PDEs belong to Barron space.
Yili Deng, Jie Fan, Jiguang He, Baojia Luo, Miaomiao Dong, Zhongyi Huang
Real-time, high-precision localization in large-scale wireless networks faces two primary challenges: clock offsets caused by network asynchrony and non-line-of-sight (NLoS) conditions. To tackle these challenges, we propose a low-complexity real-time algorithm for joint synchronization and NLoS identification-based localization. For precise synchronization, we resolve clock offsets based on accumulated time-of-arrival measurements from all the past time instances, modeling it as a large-scale linear least squares (LLS) problem. To alleviate the high computational burden of solving this LLS, we introduce the blockwise recursive Moore-Penrose inverse (BRMP) technique, a generalized recursive least squares approach, and derive a simplified formulation of BRMP tailored specifically for the real-time synchronization problem. Furthermore, we formulate joint NLoS identification and localization as a robust least squares regression (RLSR) problem and address it by using an efficient iterative approach. Simulations show that the proposed algorithm achieves sub-nanosecond synchronization accuracy and centimeter-level localization precision, while maintaining low computational overhead.
Deheng Yuan, Tao Guo, Zhongyi Huang, Shi Jin
We define a graph-based rate optimization problem and consider its computation, which provides a unified approach to the computation of various theoretical limits, including the (conditional) graph entropy, rate-distortion functions and capacity-cost functions with side information. Compared with their classical counterparts, theoretical limits with side information are much more difficult to compute since their characterizations as optimization problems have larger and more complex feasible regions. Following the unified approach, we develop effective methods to resolve the difficulty. On the theoretical side, we derive graph characterizations for rate-distortion and capacity-cost functions with side information and simplify the characterizations in special cases by reducing the number of decision variables. On the computational side, we design an efficient alternating minimization algorithm for the graph-based problem, which deals with the inequality constraint by a flexible multiplier update strategy. Moreover, simplified graph characterizations are exploited and deflation techniques are introduced, so that the computing time is greatly reduced. Theoretical analysis shows that the algorithm converges to an optimal solution. By numerical experiments, the accuracy and efficiency of the algorithm are illustrated and its significant advantage over existing methods is demonstrated.
Weijia Huang, Zhongyi Huang, Wenli Yang, Wei Zhu
In this paper, we propose image restoration models using optimal transport (OT) and total variation regularization. We present theoretical results of the proposed models based on the relations between the dual Lipschitz norm from OT and the G-norm introduced by Yves Meyer. We design a numerical method based on the Primal-Dual Hybrid Gradient (PDHG) algorithm for the Wasserstain distance and the augmented Lagrangian method (ALM) for the total variation, and the convergence analysis of the proposed numerical method is established. We also consider replacing the total variation in our model by one of its modifications developed in \cite{zhu}, with the aim of suppressing the stair-casing effect and preserving image contrasts. Numerical experiments demonstrate the features of the proposed models.
Panpan Niu, Yuhao Liu, Teng Fu, Jie Fan, Chaowen Deng, Zhongyi Huang
We investigate the performance of a Bayesian statistician tasked with recovering a rank-\(k\) signal matrix \(\bS \bS^{\top} \in \mathbb{R}^{n \times n}\), corrupted by element-wise additive Gaussian noise. This problem lies at the core of numerous applications in machine learning, signal processing, and statistics. We derive an analytic expression for the asymptotic mean-square error (MSE) of the Bayesian estimator under mismatches in the assumed signal rank, signal power, and signal-to-noise ratio (SNR), considering both sphere and Gaussian signals. Additionally, we conduct a rigorous analysis of how rank mismatch influences the asymptotic MSE. Our primary technical tools include the spectrum of Gaussian orthogonal ensembles (GOE) with low-rank perturbations and asymptotic behavior of \(k\)-dimensional spherical integrals.
Kaili Qi, Wenli Yang, Ye Li, Zhongyi Huang
Traditional image segmentation methods, such as variational models based on partial differential equations (PDEs), offer strong mathematical interpretability and precise boundary modeling, but often suffer from sensitivity to parameter settings and high computational costs. In contrast, deep learning models such as UNet, which are relatively lightweight in parameters, excel in automatic feature extraction but lack theoretical interpretability and require extensive labeled data. To harness the complementary strengths of both paradigms, we propose Variational Model Based Tailored UNet (VM_TUNet), a novel hybrid framework that integrates the fourth-order modified Cahn-Hilliard equation with the deep learning backbone of UNet, which combines the interpretability and edge-preserving properties of variational methods with the adaptive feature learning of neural networks. Specifically, a data-driven operator is introduced to replace manual parameter tuning, and we incorporate the tailored finite point method (TFPM) to enforce high-precision boundary preservation. Experimental results on benchmark datasets demonstrate that VM_TUNet achieves superior segmentation performance compared to existing approaches, especially for fine boundary delineation.
Xingpeng Xu, Zhiming Fang, Rui Ye, Zhongyi Huang, Yao Lu
Currently, there is an increasing number of super high-rise buildings in urban cities, the issue of evacuation in emergencies from such buildings comes to the fore. An evacuation experiment was carried out by our group in Shanghai Tower, it was found that the evacuation speed of pedestrians evacuated from the 126th floor was always slower than that of those from the 117th floor. Therefore, we propose a hypothesis that the expected evacuation distance will affect pedestrians' movement speed. In order to verify our conjecture, we conduct an experiment in a 12-story office building, that is, to study whether there would be an influence and what kind of influence would be caused on speed by setting the evacuation distance for participants in advance. According to the results, we find that with the increase of expected evacuation distance, the movement speed of pedestrians will decrease, which confirms our hypothesis. At the same time, we give the relation between the increase rate of evacuation distance and the decrease rate of speed. It also can be found that with the increase of expected evacuation distance, the speed decrease rate of the male is greater than that for female. In addition, we study the effects of actual evacuation distance, gender, BMI on evacuation speed. Finally, we obtain the correlation between heart rate and speed during evacuation. The results in this paper are beneficial to the study of pedestrian evacuation in super high-rise buildings.
Yixiao Hu, Lihui Chai, Zhongyi Huang, Xu Yang
Seismic tomography solves high-dimensional optimization problems to image subsurface structures of Earth. In this paper, we propose to use random batch methods to construct the gradient used for iterations in seismic tomography. Specifically, we use the frozen Gaussian approximation to compute seismic wave propagation, and then construct stochastic gradients by random batch methods. The method inherits the spirit of stochastic gradient descent methods for solving high-dimensional optimization problems. The proposed idea is general in the sense that it does not rely on the usage of the frozen Gaussian approximation, and one can replace it with any other efficient wave propagation solvers, e.g., Gaussian beam methods and spectral element methods. We prove the convergence of the random batch method in the mean-square sense, and show the numerical performance of the proposed method by two-dimensional and three-dimensional examples of wave-equation-based travel-time inversion and full-waveform inversion, respectively. As a byproduct, we also prove the convergence of the accelerated full-waveform inversion using dynamic mini-batches and spectral element methods.