Bootstrapped block Lanczos for large-dimension eigenvalue problems
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Zbikowski, Ryan M. (ORCID:0009000238544196)↗
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A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networks
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Li, Haoya (ORCID:0000000170767600)↗
Solution methodology for nonlinear eigenvalue problems from plasma instabilities
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Castillo, Andres [Stanford University]↗
Multigrid Algorithms with Projection and Prolongation over Elements of the Phase Space for K-Eigenvalue Transport Problems
This paper describes new multilevel acceleration methods for solving the multigroup neu- tron transport eigenvalue problems. These multilevel algorithms use different projection and prolongation operators in the phase space. The Nonlinear Diffusion Acceleration (NDA) method with multiple grids in energy is formulated with the prolongation oper- ator based on multiplication iterative correction and linear-in-energy mapping. Another multilevel NDA method uses the projection operator with coarsening in energy between the high-order transport and low-order NDA equations. The third algorithm is formu- lated with the partial-current based CMFD low-order equations and applies projection operators in space and energy. The numerical results are presented.
Cornejo, Luke↗
Pseudodiagonalization Method for Accelerating Nonlinear Subspace Diagonalization in Density Functional Theory
In density functional theory, each self-consistent field (SCF) nonlinear step updates the discretized Kohn-Sham orbitals by solving a linear eigenvalue problem. The concept of pseudodiagonalization is to solve this linear eigenvalue problem approximately, and specifically utilizing a method involving a small number of Jacobi rotations that takes advantage of the good initial guess to the solution given by the approximation to the orbitals from the previous SCF iteration. The approximate solution to the linear eigenvalue problem can be very rapid, particularly for those steps near SCF convergence. Here, we adapt pseudodiagonalization to finite-temperature and metallic systems, where partially-occupied orbitals must be individually resolved with some accuracy. We apply pseudodiagonalization to the subspace eigenvalue problem that arises in Chebyshev-filtered subspace iteration. In tests on metallic and other systems for a range of temperatures, we show that pseudodiagonalization achieves similar rates of SCF convergence to exact diagonalization.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
A Shift Selection Strategy for Parallel Shift-invert Spectrum Slicing in Symmetric Self-consistent Eigenvalue Computation
The central importance of large-scale eigenvalue problems in scientific computation necessitates the development of massively parallel algorithms for their solution. Recent advances in dense numerical linear algebra have enabled the routine treatment of eigenvalue problems with dimensions on the order of hundreds of thousands on the world’s largest supercomputers. In cases where dense treatments are not feasible, Krylov subspace methods offer an attractive alternative due to the fact that they do not require storage of the problem matrices. However, demonstration of scalability of either of these classes of eigenvalue algorithms on computing architectures capable of expressing massive parallelism is non-trivial due to communication requirements and serial bottlenecks, respectively. In this work, we introduce the SISLICE method: a parallel shift-invert algorithm for the solution of the symmetric self-consistent field (SCF) eigenvalue problem. The SISLICE method drastically reduces the communication requirement of current parallel shift-invert eigenvalue algorithms through various shift selection and migration techniques based on density of states estimation and k-means clustering, respectively. This work demonstrates the robustness and parallel performance of the SISLICE method on a representative set of SCF eigenvalue problems and outlines research directions that will be explored in future work.
97 MATHEMATICS AND COMPUTING↗
Efficient Mixed-Precision Matrix Factorization of the Inverse Overlap Matrix in Electronic Structure Calculations with AI-Hardware and GPUs
In recent years, a new kind of accelerated hardware has gained popularity in the artificial intelligence (AI) community which enables extremely high-performance tensor contractions in reduced precision for deep neural network calculations. In this article, we exploit Nvidia Tensor cores, a prototypical example of such AI-hardware, to develop a mixed precision approach for computing a dense matrix factorization of the inverse overlap matrix in electronic structure theory, S –1 . This factorization of S –1 , written as ZZT = S –1 , is used to transform the general matrix eigenvalue problem into a standard matrix eigenvalue problem. Here we present a mixed precision iterative refinement algorithm where Z is given recursively using matrix–matrix multiplications and can be computed with high performance on Tensor cores. To understand the performance and accuracy of Tensor cores, comparisons are made to GPU-only implementations in single and double precision. Additionally, we propose a nonparametric stopping criteria which is robust in the face of lower precision floating point operations. The algorithm is particularly useful when we have a good initial guess to Z, for example, from previous time steps in quantum-mechanical molecular dynamics simulations or from a previous iteration in a geometry optimization.
36 MATERIALS SCIENCE↗
Computation of high-frequency magnetoelastic waves in layered materials
Here, the direct calculation of magnetoelastic wave dispersion in layered media is presented using an efficient, accurate computational technique. The governing, coupled equations for elasticity and magnetism, the Navier and Landau-Lifshitz equations, respectively, are linearized to form a quadratic eigenvalue problem that determines a complex web of wave-number–frequency dispersion branches and their corresponding mode profiles. Numerical discretization of the eigenvalue problem via a spectral collocation method (SCM) is employed to determine the complete dispersion maps for both a single, finite-thickness magnetic layer and a finite magnetic-nonmagnetic double-layer. The SCM, previously used to study elastic waves in nonmagnetic media, is fast, accurate, and adaptable to a variety of sample configurations and geometries. Emphasis is placed on the extremely high-frequency regimes being accessed in ultrafast magnetism experiments. The dispersion maps and modes provide insight into how energy propagates through the coupled system, including how energy can be transferred between elastic- and magnetic-dominated waves as well as between different layers. The numerical computations for a single layer are further understood by a simplified analytical calculation in the high-frequency, exchange-dominated regime where the resonance condition required for energy exchange (an anticrossing) between quasi-elastic and quasi-magnetic dispersion branches is determined. Nonresonant interactions are shown to be well approximated by the dispersion of uncoupled elastic and magnetic waves. The methods and results provide fundamental theoretical tools to model and understand current and future magnetic devices powering spintronic innovation.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Generalized Mercier stability criterion for stellarators
The Mercier criterion is a well-known stability criterion for tokamaks. It is derived from a 2 × 2 matrix eigenvalue problem arising from the expansion of resonant solutions about a singular surface where m−nq=0, with m and n being the poloidal and toroidal mode numbers, respectively, and q being the safety factor. The stability criterion is that the eigenvalues must be real, otherwise, the solution oscillates, violating the Newcomb crossing criterion. Because of the non-axisymmetry of stellarators, different toroidal as well as poloidal harmonics couple to each other. It follows that each singular surface can have multiple resonant harmonics, with multiplicity M≥1. The corresponding matrix eigenvalue problem involves a 2M×2M matrix, resulting in M pairs of positive and negative eigenvalues. The generalized stability criterion is that all eigenvalues must be real. While the original Mercier criterion can be expressed in terms of quadratures of equilibrium quantities over the singular surface, which can be evaluated anywhere, the generalized Mercier criterion can only be evaluated on rational q surfaces with a given set of resonant harmonics.
Physics↗
A universal variational quantum eigensolver for non-Hermitian systems
Abstract Many quantum algorithms are developed to evaluate eigenvalues for Hermitian matrices. However, few practical approach exists for the eigenanalysis of non-Hermintian ones, such as arising from modern power systems. The main difficulty lies in the fact that, as the eigenvector matrix of a general matrix can be non-unitary, solving a general eigenvalue problem is inherently incompatible with existing unitary-gate-based quantum methods. To fill this gap, this paper introduces a Variational Quantum Universal Eigensolver (VQUE), which is deployable on noisy intermediate scale quantum computers. Our new contributions include: (1) The first universal variational quantum algorithm capable of evaluating the eigenvalues of non-Hermitian matrices—Inspired by Schur’s triangularization theory, VQUE unitarizes the eigenvalue problem to a procedure of searching unitary transformation matrices via quantum devices; (2) A Quantum Process Snapshot technique is devised to make VQUE maintain the potential quantum advantage inherited from the original variational quantum eigensolver—With additional $$O(log_{2}{N})$$ O ( l o g 2 N ) quantum gates, this method efficiently identifies whether a unitary operator is triangular with respect to a given basis; (3) Successful deployment and validation of VQUE on a real noisy quantum computer, which demonstrates the algorithm’s feasibility. We also undertake a comprehensive parametric study to validate VQUE’s scalability, generality, and performance in realistic applications.
97 MATHEMATICS AND COMPUTING↗
Study of recirculating liquid fuel in a 1D critical stationary system
Several studies have been conducted to investigate the physics of liquid fuel reactors, showing also applications with molten salts. A liquid nuclear fuel implies changes in the neutron balance equation to take into account the precursors' displacement and the emission of delayed neutrons in a different position than at the original fission. This requires to upgrade the computer codes normally used to calculate nuclear reactors using only solid fuel. In this work, we revisit a simple problem with liquid fuel, which is proposed for the verification of the numerical solutions obtained by advanced computer codes. This problem studies criticality with constant coefficients, thus neglecting thermal feedback. We elaborate on the analytical solution of the problem, deriving also a generalized eigenvalue problem by finite-volume integration over the cells of a discretized mesh to study the evolution of the dominance ratio with fuel velocity. Finally, we investigate the influence of the fuel velocity on the reactivity of the system. (authors)
21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
Enhancing scalability of a matrix-free eigensolver for studying many-body localization
We propose several techniques to enhance the parallel scalability of a matrix-free eigensolver designed for studying many-body localization (MBL) of quantum spin chain models with nearest-neighbor interactions and on-site disorder. This type of problem is computationally challenging because the dimension of the associated Hamiltonian matrix grows exponentially with respect to the number of spins L, and we need to average over different realizations of the random disorder to obtain relevant statistical behavior. For each disorder realization, we need to compute eigenvalues from different regions of the spectrum and their corresponding eigenvectors. In previous work, the interior eigenstates for a single eigenvalue problem are computed via the shift-and-invert Lanczos algorithm. Due to the extremely high memory footprint of the LU factorizations, this technique is not well suited for large L’s. For example, we need thousands of compute nodes on modern high performance computing infrastructures to go beyond L = 24. The matrix-free approach does not suffer from this memory bottleneck, however, its scalability is limited by a computation and communication load imbalance. To reduce this imbalance and to significantly enhance the scalability of the matrix-free eigensolver, we reorder the matrix and leverage the consistent space runtime, CSPACER. We also show its efficiency in managing irregular communication patterns at scale compared to optimized MPI non-blocking two-sided and one-sided RMA implementation variants. This effort enables us to study MBL for spin chains with a larger number of spins. The efficiency and effectiveness of the proposed algorithm is demonstrated by computing eigenstates on a massively parallel many-core high performance computer.
METIS↗
Nuclear Materials Packaging, Transportation, and Systems Analysis Group Software Quality Assurance Plan: ANSYS Mechanical Finite Element Analysis Software Version 2023R1
ANSYS Inc. develops and markets engineering simulation software and services used in the aerospace, automotive, manufacturing, electronics, biomedical, energy, defense, and many other industries. ANSYS is dedicated to engineering simulation and is the world’s leading software provider. ANSYS was founded in 1970 and is headquartered in Canonsburg, Pennsylvania. ANSYS provides an engineering analysis tool combining structural, thermal, computational fluid dynamics, acoustic, and electromagnetic simulation capabilities. ANSYS has two main programs, which use the same solvers: (1) Mechanical APDL (ANSYS Design Parametric Language), a Fortran-based coding platform, and (2) ANSYS Workbench, which uses a graphical user interface to aid in finite element analysis implementation. This plan covers both APDL and Workbench. The ANSYS computer program is a large-scale, multipurpose finite element program that can be used to solve several classes of engineering analyses. The analysis capabilities of ANSYS include the ability to solve static and dynamic structural analyses, steady-state and transient heat transfer problems, mode-frequency and buckling eigenvalue problems, static or time-varying magnetic analyses, and various types of field and coupled-field applications. The program contains many special features that allow nonlinearities or secondary effects such as plasticity, large strain, hyperelasticity, creep, swelling, large deflections, contact, stress stiffening, temperature dependency, material anisotropy, and radiation to be included in the solution. As ANSYS has been developed, other special capabilities such as substructuring, submodeling, random vibration, kinetostatics, kinetodynamics, free convection fluid analysis, acoustics, magnetics, piezoelectrics, coupled-field analysis, and design optimization have been added to the program. These capabilities contribute further to making ANSYS a multipurpose analysis tool for varied engineering disciplines. The ANSYS program has been in commercial use for over 50 years and has been used extensively in the aerospace, automotive, construction, electronic, energy services, manufacturing, nuclear, plastics, oil, and steel industries. Additionally, many consulting firms and hundreds of universities have used ANSYS for analysis, research, and educational purposes. ANSYS is recognized worldwide as one of the most widely used and capable programs of its type. Ansys design analysis software is the first created within a quality system with ISO 9001 certification, the internationally accepted quality standard. Product development, testing, maintenance and support processes also meet the United States Nuclear Regulatory Commission's quality requirements, as they have for nearly four decades. The Quality Assurance Service Agreement is suitable for the customers working in the nuclear industry who need to meet specific federal regulations including 10CRF50 Appendix B and provisions of 10CFR21. ANSYS has retained its original International Organization for Standardization (ISO) 9001 accreditation certificate since1995-05-04, It’s current certificate is valid until 2027-05-29.
11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
Application of Lagrangian techniques for calculating the on-axis rotational transform
The Floquet exponents of periodic field lines are studied through the variations of the magnetic action on the magnetic axis, which is assumed to be elliptical. The near-axis formalism developed by Mercier, Solov'ev and Shafranov is combined with a Lagrangian approach. The on-axis Floquet exponent is shown to coincide with the on-axis rotational transform. A discrete solution suitable for numerical implementation is introduced, which gives the Floquet exponents as solutions to an eigenvalue problem. This discrete formalism expresses the exponents as the eigenvalues of a $6$ X $6$ matrix.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Unleashed from constrained optimization: quantum computing for quantum chemistry employing generator coordinate inspired method
Hybrid quantum-classical approaches offer potential solutions to quantum chemistry problems, yet they often manifest as constrained optimization problems. Here, we explore the interconnection between constrained optimization and generalized eigenvalue problems through the Unitary Coupled Cluster (UCC) excitation generators. Inspired by the generator coordinate method, we employ these UCC excitation generators to construct non-orthogonal, overcomplete many-body bases, projecting the system Hamiltonian into an effective Hamiltonian, which bypasses issues such as barren plateaus that heuristic numerical minimizers often encountered in standard variational quantum eigensolver (VQE). Diverging from conventional quantum subspace expansion methods, we introduce an adaptive scheme that robustly constructs the many-body basis sets from a pool of the UCC excitation generators. This scheme supports the development of a hierarchical ADAPT quantum-classical strategy, enabling a balanced interplay between subspace expansion and ansatz optimization to address complex, strongly correlated quantum chemical systems cost-effectively, setting the stage for more advanced quantum simulations in chemistry.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
An eigenvalue-based method for computing the relaxed pressure in compressible multiphase flow with N phases
The modeling of compressible multiphase flows is a decades-old area of study with many applications across various fields. Many of these application areas use stiff pressure relaxation. This process involves the solution of a nonlinear system with N + 1 equations and N + 1 unknowns, where N is the number of phases. The resolution of this system with general equations of state (EOSs) is difficult. Furthermore, nonlinear systems can admit multiple solutions, and current solution methods do not address this possibility. Very recently, a thermodynamic relaxation method was introduced, which effectively maps a relatively simple predictor equation of state onto a more complex target equation of state. In this context, the target EOSs are the chosen EOSs for the thermodynamic model. Furthermore, this thermodynamic relaxation has the benefit of simplifying the stiff pressure relaxation system of equations. In this article, we show this system reduces to a polynomial of degree N, which can be recast as an eigenvalue problem through the use of the associated companion matrix. We show that although this eigenvalue method is generally less efficient than Newton–Raphson iteration, it does not suffer from convergence issues and finds all N roots of the polynomial. Hence, the method provides a fail-safe for root-finding iterative methods and a way to address the issue of multiple solutions to the nonlinear system of equations in stiff pressure relaxation.
Eigenvalue algorithm↗
Quantum Krylov subspace algorithms for ground- and excited-state energy estimation
Quantum Krylov subspace diagonalization (QKSD) algorithms provide a low-cost alternative to the conventional quantum phase estimation algorithm for estimating the ground- and excited-state energies of a quantum many-body system. While QKSD algorithms typically rely on using the Hadamard test for estimating Krylov subspace matrix elements of the form $\langle \phi_i|e^{-\widehat{H}τ}|\phi_j\rangle$, the associated quantum circuits require an ancilla qubit with controlled multiqubit gates that can be quite costly for near-term quantum hardware. In this paper, we show that a wide class of Hamiltonians relevant to condensed-matter physics and quantum chemistry contain symmetries that can be exploited to avoid the use of the Hadamard test. We propose a multifidelity estimation protocol that can be used to compute such quantities, showing that our approach, when combined with efficient single-fidelity estimation protocols, provides a substantial reduction in circuit depth. In addition, here we develop a unified theory of quantum Krylov subspace algorithms and present three quantum-classical algorithms for the ground- and excited-state energy estimation problems, where each algorithm provides various advantages and disadvantages in terms of total number of calls to the quantum computer, gate depth, classical complexity, and stability of the generalized eigenvalue problem within the Krylov subspace.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗