Aihui Zhou

Huajie Chen, Xiaoying Dai, Xingao Gong et al.

The Kohn-Sham equation is a powerful, widely used approach for computation of ground state electronic energies and densities in chemistry, materials science, biology, and nanosciences. In this paper, we study the adaptive finite element approximations for the Kohn-Sham model. Based on the residual type a posteriori error estimators proposed in this paper, we introduce an adaptive finite element algorithm with a quite general marking strategy and prove the convergence of the adaptive finite element approximations. Using D{\" o}rfler's marking strategy, we then get the convergence rate and quasi-optimal complexity. We also carry out several typical numerical experiments that not only support our theory,but also show the robustness and efficiency of the adaptive finite element computations in electronic structure calculations.

Xiaoying Dai, Lianhua He, Aihui Zhou

In this paper, we study an adaptive finite element method for multiple eigenvalue problems of a class of second order elliptic equations. By using some eigenspace approximation technology and its crucial property which is also presented in this paper, we extend the results in \cite{dai-xu-zhou08} to multiple eigenvalue problems, we obtain both convergence rate and quasi-optimal complexity of the adaptive finite element eigenvalue approximation.

Huajie Chen, Xingao Gong, Lianhua He et al.

In this paper, we study finite dimensional approximations of Kohn-Sham models, which are widely used in electronic structure calculations. We prove the convergence of the finite dimensional approximations and derive the a priori error estimates for ground state energies and solutions. We also provide numerical simulations for several molecular systems that support our theory.

Yan Pan, Xiaoying Dai, Stefano de Gironcoli et al.

Motivated by the recently proposed parallel orbital-updating approach in real space method, we propose a parallel orbital-updating based plane-wave basis method for electronic structure calculations, for solving the corresponding eigenvalue problems. In addition, we propose two new modified parallel orbital-updating methods. Compared to the traditional plane-wave methods, our methods allow for two-level parallelization, which is particularly interesting for large scale parallelization. Numerical experiments show that these new methods are more reliable and efficient for large scale calculations on modern supercomputers

Yuzhi Zhou, Huajie Chen, Aihui Zhou

We propose a novel numerical algorithm for computing the electronic structure related eigenvalue problem of incommensurate systems. Unlike the conventional practice that approximates the system by a large commensurate supercell, our algorithm directly discretizes the eigenvalue problem under the framework of a plane wave method. The emerging ergodicity and the interpretation from higher dimensions give rise to many unique features compared to what we have been familiar with in the periodic system. The numerical results of 1D and 2D quantum eigenvalue problems are presented to show the reliability and efficiency of our scheme. Furthermore, the extension of our algorithm to full Kohn-Sham density functional theory calculations are discussed.

Xiaoying Dai, Zhuang Liu, Xin Zhang et al.

In this paper, we propose a parallel optimization method for electronic structure calculations based on a single orbital-updating approximation. It is shown by our numerical experiments that the method is efficient and reliable for atomic and molecular systems of large scale over supercomputers.

Shengyue Wang, Aihui Zhou

Computing many eigenpairs of the Schr{ö}dinger operator presents a computational bottleneck in large-scale quantum simulations due to the global communication overhead of explicit orthogonalization. To address this issue, we propose a quasi-orthogonal evolution model utilizing inverse operators and develop a corresponding discrete numerical scheme. Instead of forcing explicit orthogonalization, the proposed framework confines the numerical approximations within a quasi-Stiefel set, ensuring the iterates maintain full column rank without requiring $\left\langle U, U \right\rangle=I_N$. Moreover, the method naturally absorbs orthogonality errors and asymptotically converges to the exact eigenfunctions, even when initialized with non-orthogonal random data. The scheme guarantees monotonic dissipation of the target energy functional, with exponential convergence rates rigorously established for the discrete energy, gradient, and eigenfunction approximations. Furthermore, infinite-dimensional analysis proves that the admissible time step size is independent of the spatial discretization. This property overcomes the mesh-dependent stability constraints typical of conventional explicit or semi-implicit schemes, permitting larger time increments to accelerate global convergence. Numerical experiments validate the theoretical findings.

Jun Fang, Xingyu Gao, Aihui Zhou

In this paper, we propose a decomposition approach for eigenvalue problems with spatial symmetries, including the formulation, discretization as well as implementation. This approach can handle eigenvalue problems with either Abelian or non-Abelian symmetries, and is friendly for grid-based discretizations such as finite difference, finite element or finite volume methods. With the formulation, we divide the original eigenvalue problem into a set of subproblems and require only a smaller number of eigenpairs for each subproblem. We implement the decomposition approach with finite elements and parallelize our code in two levels. We show that the decomposition approach can improve the efficiency and scalability of iterative diagonalization. In particular, we apply the approach to solving Kohn--Sham equations of symmetric molecules consisting of hundreds of atoms.

Wei Huang, Yinggui Wang, Anda Cheng et al.

The distributed (federated) LLM is an important method for co-training the domain-specific LLM using siloed data. However, maliciously stealing model parameters and data from the server or client side has become an urgent problem to be solved. In this paper, we propose a secure distributed LLM based on model slicing. In this case, we deploy the Trusted Execution Environment (TEE) on both the client and server side, and put the fine-tuned structure (LoRA or embedding of P-tuning v2) into the TEE. Then, secure communication is executed in the TEE and general environments through lightweight encryption. In order to further reduce the equipment cost as well as increase the model performance and accuracy, we propose a split fine-tuning scheme. In particular, we split the LLM by layers and place the latter layers in a server-side TEE (the client does not need a TEE). We then combine the proposed Sparsification Parameter Fine-tuning (SPF) with the LoRA part to improve the accuracy of the downstream task. Numerous experiments have shown that our method guarantees accuracy while maintaining security.

Junming Ma, Chaofan Yu, Aihui Zhou et al.

We present S3ML, a secure serving system for machine learning inference in this paper. S3ML runs machine learning models in Intel SGX enclaves to protect users' privacy. S3ML designs a secure key management service to construct flexible privacy-preserving server clusters and proposes novel SGX-aware load balancing and scaling methods to satisfy users' Service-Level Objectives. We have implemented S3ML based on Kubernetes as a low-overhead, high-available, and scalable system. We demonstrate the system performance and effectiveness of S3ML through extensive experiments on a series of widely-used models.

Xiaoying Dai, Zhuang Liu, Liwei Zhang et al.

In this paper, we study a conjugate gradient method for electronic structure calculations. We propose a Hessian based step size strategy, which together with three orthogonality approaches yields three algorithms for computing the ground state energy of atomic and molecular systems. Under some mild assumptions, we prove that our algorithms converge locally. It is shown by our numerical experiments that the conjugate gradient method is efficient.

Xiaoying Dai, Xingao Gong, Aihui Zhou et al.

In this paper, we propose an orbital iteration based parallel approach for electronic structure calculations. This approach is based on our understanding of the single-particle equations of independent particles that move in an effective potential. With this new approach, the solution of the single-particle equation is reduced to some solutions of independent linear algebraic systems and a small scale algebraic problem. It is demonstrated by our numerical experiments that this new approach is quite efficient for full-potential calculations for a class of molecular systems.

Lianhua He, Aihui Zhou

In this paper, we study adaptive finite element approximations in a perturbation framework, which makes use of the existing adaptive finite element analysis of a linear symmetric elliptic problem. We prove the convergence and complexity of adaptive finite element methods for a class of elliptic partial differential equations. For illustration, we apply the general approach to obtain the convergence and complexity of adaptive finite element methods for a nonsymmetric problem, a nonlinear problem as well as an unbounded coefficient eigenvalue problem.