Physics-informed Reduced-Order Learning from the First Principles for Simulation of Quantum Nanostructures

Multi-dimensional direct numerical simulation (DNS) of the Schr\"odinger equation is needed for design and analysis of quantum nanostructures that offer numerous applications in biology, medicine, materials, electronic/photonic devices, etc. In large-scale nanostructures, extensive computational effort needed in DNS may become prohibitive due to the high degrees of freedom (DoF). This study employs a reduced-order learning algorithm, enabled by the first principles, for simulation of the Schr\"odinger equation to achieve high accuracy and efficiency. The proposed simulation methodology is applied to investigate two quantum-dot structures; one operates under external electric field, and the other is influenced by internal potential variation with periodic boundary conditions. The former is similar to typical operations of nanoelectronic devices, and the latter is of interest to simulation and design of nanostructures and materials, such as applications of density functional theory. Using the proposed methodology, a very accurate prediction can be realized with a reduction in the DoF by more than 3 orders of magnitude and in the computational time by 2 orders, compared to DNS. The proposed physics-informed learning methodology is also able to offer an accurate prediction beyond the training conditions, including higher external field and larger internal potential in untrained quantum states.Comment: 18 pages, 11 figures. An additional demonstration using Fourier-based plan waves in the revised versio

ChASE: Chebyshev Accelerated Subspace iteration Eigensolver for sequences of Hermitian eigenvalue problems

Solving dense Hermitian eigenproblems arranged in a sequence with direct solvers fails to take advantage of those spectral properties which are pertinent to the entire sequence, and not just to the single problem. When such features take the form of correlations between the eigenvectors of consecutive problems, as is the case in many real-world applications, the potential benefit of exploiting them can be substantial. We present ChASE, a modern algorithm and library based on subspace iteration with polynomial acceleration. Novel to ChASE is the computation of the spectral estimates that enter in the filter and an optimization of the polynomial degree which further reduces the necessary FLOPs. ChASE is written in C++ using the modern software engineering concepts which favor a simple integration in application codes and a straightforward portability over heterogeneous platforms. When solving sequences of Hermitian eigenproblems for a portion of their extremal spectrum, ChASE greatly benefits from the sequence's spectral properties and outperforms direct solvers in many scenarios. The library ships with two distinct parallelization schemes, supports execution over distributed GPUs, and it is easily extensible to other parallel computing architectures.Comment: 33 pages. Submitted to ACM TOM

Correlations in sequences of generalized eigenproblems arising in Density Functional Theory

Density Functional Theory (DFT) is one of the most used ab initio theoretical frameworks in materials science. It derives the ground state properties of a multi-atomic ensemble directly from the computation of its one-particle density n(r). In DFT-based simulations the solution is calculated through a chain of successive self-consistent cycles: in each cycle a series of coupled equations (Kohn-Sham) translates to a large number of generalized eigenvalue problems whose eigenpairs are the principal means for expressing n(r). A simulation ends when n(r) has converged to the solution within the required numerical accuracy. This usually happens after several cycles, resulting in a process calling for the solution of many sequences of eigenproblems. In this paper, the authors report evidence showing unexpected correlations between adjacent eigenproblems within each sequence. By investigating the numerical properties of the sequences of generalized eigenproblems it is shown that the eigenvectors undergo an "evolution" process. At the same time it is shown that the Hamiltonian matrices exhibit a similar evolution and manifest a specific pattern in the information they carry. Correlation between eigenproblems within a sequence is of capital importance: information extracted from the simulation at one step of the sequence could be used to compute the solution at the next step. Although they are not explored in this work, the implications could be manifold: from increasing the performance of material simulations, to the development of an improved iterative solver, to modifying the mathematical foundations of the DFT computational paradigm in use, thus opening the way to the investigation of new materials. (C) 2012 Elsevier B.V. All rights reserved