# Registered Data

## [00969] Eigenvalue Problems in Electronic Structure Calculations

**Session Date & Time**:- 00969 (1/3) : 4C (Aug.24, 13:20-15:00)
- 00969 (2/3) : 4D (Aug.24, 15:30-17:10)
- 00969 (3/3) : 4E (Aug.24, 17:40-19:20)

**Type**: Proposal of Minisymposium**Abstract**: The first principles electronic structure calculations have become important tools for studying the material mechanism, understanding and predicting the material properties, and have achieved great success. The key mathematical models for electronic structure calculations are eigenvalue problems or equivalent forms. There are still many challenges on the design of highly efficient and highly accurate computational methods for dealing these eigenvalue problems or equivalent forms, especially for larger system. The purpose of this mini-symposium is to provide a platform for exchanging the recent developments on the numerical methods and theories for eigenvalue problems or equivalent forms arising in electronic structure calculations, and exploring the topic of further research and collaborations.**Organizer(s)**: Huajie Chen (Beijing Normal University), Xiaoying Dai (Academy of Mathematics and Systems Science, CAS), Xin Liu (Academy of Mathematics and Systems Science, CAS), Yuzhi Zhou ( Institute of Applied Physics and Computational Mathematics)**Classification**:__35Q55__,__65N25__,__81Q05__,**Speakers Info**:- Zhaojun Bai (University of California, Davis )
- Stefano de Gironcoli (International School for Advanced Studies)
- Muhammad Hassan (Sorbonne University)
- Guanghui Hu (University of Macau)
- Defeng Sun (The Hong Kong Polytechnic University)
- Meiyue Shao (Fudan University )
- Benjamin Stamm (University of Stuttgart)
- Yanfei Wang (Institute of Geology and Geophysics, Chinese Academy of Sciences)
- Aihui Zhou (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
- Yuzhi Zhou (Institute of Applied Physics and Computational Mathematics)
- Liwei Zhang (University of British Columbia)
- Yukuan Hu (Chinese Academy of Sciences)

**Talks in Minisymposium**:**[03145] A mixed precision LOBPCG algorithm****Author(s)**:- Daniel Kressner (EPF Lausanne)
- Yuxin Ma (Fudan University)
**Meiyue Shao**(Fudan University)

**Abstract**: The LOBPCG algorithm is a popular approach for computing a few smallest eigenvalues of a large Hermitian positive definite matrix. We propose a mixed precision variant of LOBPCG that uses a (sparse) Cholesky factorization computed in lower precision as the preconditioner. We carry out a rounding error and convergence analysis of PINVIT, a simplified variant of LOBPCG. Our theoretical results predict and our numerical experiments confirm that the impact on convergence remains marginal.

**[03997] Sampling-based approaches for multimarginal optimal transport problems with Coulomb cost****Author(s)**:**Yukuan Hu**(Academy of Mathematics and Systems Science, Chinese Academy of Sciences)- Mengyu Li (Renmin University of China)
- Xin Liu (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
- Cheng Meng (Renmin University of China)

**Abstract**: The multimarginal optimal transport problem with Coulomb cost find applications in understanding strongly correlated systems. We develop for its Monge-like reformulation novel methods that favor highly scalable subiteration schemes and avoid the full matrix multiplications in the existing ones. Convergence properties are built on the random matrix theory. For large-scale global resolution, we embed the proposed methods into a grid refinements-based framework. The numerical results corroborate the effectiveness and better scalability of our approach.

**[04173] Model and data driven electromagnetic inverse problems with optimal transport****Author(s)**:**Yanfei Wang**(Institute of Geology and Geophysics, Chinese Academy of Sciences)

**Abstract**: Electromagnetic inverse problems have important applications in non-destructive testing and evaluation of materials. By using electromagnetic measurements to probe the properties of materials like metals and composites, researchers can gain insight into the structural integrity, conductivity, and other important properties of these materials. In this study, we consider application of electromagnetic inverse problems is in geophysics, i.e., using electromagnetic measurements to study the composition and structure of the Earth's subsurface. We propose a new attempt to use the probability metric (Wasserstein metric) for electromagnetic inversion. This lays the foundation for the future application of probability metric type of methods to large-scale electromagnetic inversion. In addition, data driven electromagnetic inverse problems will be also addressed.

**[04226] Grassmann Extrapolation of Density Matrices for Born–Oppenheimer Molecular Dynamics****Author(s)**:**Benjamin Stamm**(University of Stuttgart)

**Abstract**: Born–Oppenheimer molecular dynamics (BOMD) is a powerful but expensive technique. We show how converged densities from previous DFT-calculations in the trajectory can be used to extrapolate a new guess for the SCF-iterations. We apply the method to real-life, multiscale, polarizable QM/MM BOMD simulations, showing that sizeable performance gains can be achieved. This is joint-work with É. Polack, G. Dusson and F. Lipparini.

**[04280] Numerical Analysis of the Operator Modification Approach for the Calculation of Band Diagrams of Crystalline Materials****Author(s)**:- Eric Cancès (CERMICS, École des Ponts and Inria Paris)
**Muhammad Hassan**(Laboratoire Jacques-Louis Lions, Sorbonne Université)- Laurent Vidal (CERMICS, École des Ponts and Inria Paris)

**Abstract**: In solid-state physics, electronic properties of crystalline materials are often described by the spectrum of periodic Schrödinger operators. Due to Bloch’s theorem, the numerical computation of quantities of interest involves computing integrals over the Brillouin zone of energy bands, which are piecewise smooth, periodic functions obtained by solving a parametrized elliptic eigenvalue problem. Classic discretization strategies for resolving these eigenvalue problems produce approximate energy bands that are either non-periodic or discontinuous, both of which cause difficulties when employing numerical quadrature. We present here an alternative discretization strategy based on an ad hoc operator modification approach. We derive a priori error estimates for the resulting energy bands and we show that these bands are periodic and can be made arbitrarily smooth (away from band crossings) by adjusting suitable parameters in the operator modification approach. We also present numerical experiments involving a toy model in 1D, graphene in 2D, and silicon in 3D to validate our theoretical results and showcase the efficiency of the operator modification approach.

**[04317] An extended plane wave framework for the electronic structure calculations of twisted bilayer material systems****Author(s)**:**Yuzhi Zhou**- Xiaoying Dai (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
- Aihui Zhou (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)

**Abstract**: In this talk, we introduce extensions of our PW framework for the practical electronic calculations of twisted bilayer material systems in the following aspects: (1) a tensor-producted basis set with PWs in the incommensurate dimensions and localized functions in z direction, (2) the practical application of our newly developed cutoff techniques, and (3) a quasi-band structure picture under the small twisted angles and weak interlayer coupling limits. With (1) and (2), we have remarkably reduced the dimensions of hamiltonian matrix, which makes the electronic structure calculations of twisted bilayer 2D material systems affordable to most modern computers. And (3) helps us better organize the calculations as well as understand results. We further use the linear TGB system with magic twisted angles as numerical examples. We have reproduced the famous flat bands with key features in good quantitative with other theoretical and experimental results. In terms of efficiency, our framework has much less computational cost compared to the commensurate cell approximations. While it is also more extendable compared to the traditional model hamiltonians and tight binding calculations. Lastly, nonlinear terms like Hartree energy and exchange-correlation energy can be readily included in the framework thus more effective and accurate DFT calculations of incommensurate 2D material systems can be expected in the near future.

**[04337] Kohn-Sham GGA Models and Their Approximations****Author(s)**:**Aihui Zhou**(Academy of Mathematics and Systems Science, Chinese Academy of Sciences)

**Abstract**: In this presentation, I will talk about the finite dimensional approximations of Kohn-Sham GGA models, which are often used in electronic structure calculations. I will show the convergence of the finite dimensional approximations and present the a priori error estimates for ground state energy and solution approximations.

**[04472] Recent Advances in Self-Consistent-Field Iterations for Solving Eigenvector-Dependent Nonlinear Eigenvalue Problems****Author(s)**:**Zhaojun Bai**(University of California, Davis)

**Abstract**: Much like the power method for solving linear eigenvalue problems, self-Consistent-Field (SCF) iteration is a gateway algorithm to solve eigenvector-dependent nonlinear eigenvalue problems such as ones arising from electronic structure calculations. The SCF was introduced in computational physics back in the 1950s. In this talk, from numerical linear algebra perspective, we present recent advances in the SCF, such as sharp estimation of convergence rate and geometry interpretation of the SCF for a class of NEPv.

**[04929] Applications of Atomic Cluster Expansion in Electronic Structure Calculations****Author(s)**:**Liwei Zhang**(University of British Columbia)

**Abstract**: Nonlinear eigenvalue problems are quite typical in the field of electronic structure calculations. For decades, people tended to solve them by the so-called self-consistent field iterations method, which suffers from both convergence and numerical efficiency. In this talk, we will introduce a generalized Atomic Cluster Expansion (ACE) framework, which provides a complete and symmetry-preserving basis for approximating equivariant properties, to give rise to a way to skip the self-consistent procedure. We will also cover some potential applications of ACE in the context of post-DFT electronic structure calculation models.

**[05006] An efficient LOBPCG solver for Kohn-Sham solution****Author(s)**:**Guanghui Hu**(University of Macau)

**Abstract**: In this talk, an efficient implementation of the LOBPCG solver in the self-consistent field iteration for the Kohn-Sham solution is introduced. It is found that in an $h$-adaptive finite element framework, the precondition in the LOBPCG method plays an important role to guarantee the fast convergence of the solver. Several choices are introduced, and the comparison of the performance among those choices will be demonstrated in detail.