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DOE OSTI2020

The tensor t‐function: A definition for functions of third‐order tensors

Summary A definition for functions of multidimensional arrays is presented. The definition is valid for third‐order tensors in the tensor t‐product formalism, which regards third‐order tensors as block circulant matrices. The tensor function definition is shown to have similar properties as standard matrix function definitions in fundamental scenarios. To demonstrate the definition's potential in applications, the notion of network communicability is generalized to third‐order tensors and computed for a small‐scale example via block Krylov subspace methods for matrix functions. A complexity analysis for these methods in the context of tensors is also provided.

Lund, Kathryn↗

DOE OSTI2020

Approximate inverse-based block preconditioners in poroelasticity

We focus on the fully implicit solution of the linear systems arising from a three-field mixed finite element approximation of Biot’s poroleasticity equations. The objective is to develop algebraic block preconditioners for the efficient solution of such systems by Krylov subspace methods. In this work, we investigate the use of approximate inverse-based techniques to decouple the native system of equations and obtain explicit sparse approximations of the Schur complements related to the physics-based partitioning of the unknowns by field type. Here, the proposed methods are tested in various numerical experiments including real-world applications dealing with petroleum and geotechnical engineering.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2025

Leveraging operator learning to accelerate convergence of the preconditioned conjugate gradient method

We propose a new deflation strategy to accelerate the convergence of the preconditioned conjugate gradient (PCG) method for solving parametric large-scale linear systems of equations. Unlike traditional deflation techniques that rely on eigenvector approximations or recycled Krylov subspaces, we generate the deflation subspaces using operator learning, specifically the Deep Operator Network (DeepONet). To this aim, we introduce two complementary approaches for assembling the deflation operators. The first approach approximates near-null space vectors of the discrete PDE operator using the basis functions learned by the DeepONet. The second approach directly leverages solutions predicted by the DeepONet. To further enhance convergence, we also propose several strategies for prescribing the sparsity pattern of the deflation operator. Here, a comprehensive set of numerical experiments encompassing steady-state, time-dependent, scalar, and vector-valued problems posed on both structured and unstructured geometries is presented and demonstrates the effectiveness of the proposed DeepONet-based deflated PCG method, as well as its generalization across a wide range of model parameters and problem resolutions.

Deflation↗

DOE OSTI2020

A low-rank solver for the stochastic unsteady Navier–Stokes problem

Here we study a low-rank iterative solver for the unsteady Navier–Stokes equations for incompressible flows with a stochastic viscosity. The equations are discretized using the stochastic Galerkin method, and we consider an all-at-once formulation where the algebraic systems at all the time steps are collected and solved simultaneously. The problem is linearized with Picard’s method. To efficiently solve the linear systems at each step, we use low-rank tensor representations within the Krylov subspace method, which leads to significant reductions in storage requirements and computational costs. Combined with effective mean-based preconditioners and the idea of inexact solve, we show that only a small number of linear iterations are needed at each Picard step. The proposed algorithm is tested with a model of flow in a two-dimensional symmetric step domain with different settings to demonstrate the computational efficiency.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

N 5 -Scaling Excited-State-Specific Perturbation Theory

In this paper, we show that by working in a basis similar to that of the natural transition orbitals and using a modified zeroth-order Hamiltonian, the cost of a recently introduced perturbative correction to excited-state mean field theory can be reduced from seventh to fifth order in the system size. The (occupied) 2 (virtual) 3 asymptotic scaling matches that of ground-state second-order Møller-Plesset theory but with a significantly higher prefactor because the bottleneck is iterative: it appears in the Krylov-subspace-based solution of the linear equation that yields the first-order wave function. Here, we discuss the details of the modified zeroth-order Hamiltonian we use to reduce the cost and the automatic code generation process we used to derive and verify the cost scaling of the different terms. Overall, we find that our modifications have little impact on the method's accuracy, which remains competitive with singles and doubles equation-of-motion coupled cluster.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Semi-Empirical Shadow Molecular Dynamics: A PyTorch Implementation

Here, extended Lagrangian Born–Oppenheimer molecular dynamics (XL-BOMD) in its most recent shadow potential energy version has been implemented in the semiempirical PyTorch-based software PySeQM. The implementation includes finite electronic temperatures, canonical density matrix perturbation theory, and an adaptive Krylov subspace approximation for the integration of the electronic equations of motion within the XL-BOMB approach (KSA-XL-BOMD). The PyTorch implementation leverages the use of GPU and machine learning hardware accelerators for the simulations. The new XL-BOMD formulation allows studying more challenging chemical systems with charge instabilities and low electronic energy gaps. The current public release of PySeQM continues our development of modular architecture for large-scale simulations employing semi-empirical quantum-mechanical treatment. Applied to molecular dynamics, simulation of 840 carbon atoms, one integration time step executes in 4 s on a single Nvidia RTX A6000 GPU.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Graph-based quantum response theory and shadow Born–Oppenheimer molecular dynamics

Graph-based linear scaling electronic structure theory for quantum-mechanical molecular dynamics simulations [A. M. N. Niklasson et al., J. Chem. Phys. 144, 234101 (2016)] is adapted to the most recent shadow potential formulations of extended Lagrangian Born–Oppenheimer molecular dynamics, including fractional molecular-orbital occupation numbers [A. M. N. Niklasson, J. Chem. Phys. 152, 104103 (2020) and A. M. N. Niklasson, Eur. Phys. J. B 94, 164 (2021)], which enables stable simulations of sensitive complex chemical systems with unsteady charge solutions. The proposed formulation includes a preconditioned Krylov subspace approximation for the integration of the extended electronic degrees of freedom, which requires quantum response calculations for electronic states with fractional occupation numbers. For the response calculations, we introduce a graph-based canonical quantum perturbation theory that can be performed with the same natural parallelism and linear scaling complexity as the graph-based electronic structure calculations for the unperturbed ground state. Further, the proposed techniques are particularly well-suited for semi-empirical electronic structure theory, and the methods are demonstrated using self-consistent charge density-functional tight-binding theory both for the acceleration of self-consistent field calculations and for quantum-mechanical molecular dynamics simulations. Graph-based techniques combined with the semi-empirical theory enable stable simulations of large, complex chemical systems, including tens-of-thousands of atoms.

74 ATOMIC AND MOLECULAR PHYSICS↗

DOE OSTI2021

Exponential Time Differencing Schemes for Fuel Depletion and Transport in Molten Salt Reactors: Theory and Implementation

A numerical framework for modeling depletion and mass transport in liquid-fueled molten salt reactions is presented based on exponential time differencing. The solution method involves using the finite volume method to transform the system of partial differential equations (PDEs) into a much larger system of ordinary differential equations. The key part of this method involves solving for the exponential of a matrix. We explore six different algorithms to compute the exponential in a series of progression problems that explore physical transport phenomena in molten salt reactors. This framework shows good results for solving linear parabolic PDEs with each of the six matrix exponential algorithms. For large problems, the series solvers such as Padé and Taylor have large run times, which can be mitigated by using the Krylov subspace.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2025

Accelerating resonant spectroscopy simulations using multishifted biconjugate gradient

Resonant spectroscopies, which involve intermediate states with finite lifetimes, provide important insights into collective excitations in quantum materials that are otherwise inaccessible. However, theoretical understanding in this area is often limited by the numerical challenges of solving Kramers-Heisenberg-type response functions for large-scale systems. To address this, we introduce a multishifted biconjugate gradient algorithm that exploits the shared structure of Krylov subspaces across spectra with varying incident energies, effectively reducing the computational complexity to that of linear spectroscopies. Both mathematical proofs and numerical benchmarks confirm that this algorithm substantially accelerates spectral simulations, achieving constant complexity independent of the number of incident energies, while ensuring accuracy and stability. This development provides a scalable, versatile framework for simulating advanced spectroscopies in quantum materials.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2023

Stochastic quantum Krylov protocol with double-factorized Hamiltonians

Here we propose a class of randomized quantum Krylov diagonalization (rQKD) algorithms capable of solving the eigenstate estimation problem with modest quantum resource requirements. Compared to previous real-time evolution quantum Krylov subspace methods, our approach expresses the time evolution operator e –i$\widehat{H}$$\tau$ as a linear combination of unitaries and subsequently uses a stochastic sampling procedure to reduce circuit depth requirements. While our methodology applies to any Hamiltonian with fast-forwardable subcomponents, we focus on its application to the explicitly double-factorized electronic-structure Hamiltonian. To demonstrate the potential of the proposed rQKD algorithm on near-term quantum devices, we provide numerical benchmarks for a variety of molecular systems with circuit-based state-vector simulators including the effects of sampling noise, achieving ground-state energy errors of less than 1 kcal mol -1 with circuit depths orders of magnitude shallower than those required for low-rank deterministic Trotter-Suzuki decompositions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Systematic input scheme for many-boson Hamiltonians with applications to the two-dimensional 𝜙 4 theory

We develop a novel, systematic input scheme for many-boson Hamiltonians in order to solve field theory problems within the light-front Hamiltonian formalism via quantum computing. We present our discussion of this input scheme based on the light-front Hamiltonian of the two-dimensional ϕ 4 theory. In our input scheme, we employ a set of quantum registers, where each register encodes the occupation of a distinct boson mode as binaries. We squeeze the boson operators of each mode and present the Hamiltonian in terms of unique combinations of the squeezed-boson operators. We design the circuit modules for these unique combinations. Based on these circuit modules, we block encode the many-boson Hamiltonian utilizing the idea of quantum walk. For demonstration purposes, we present the spectral calculations of the Hamiltonian utilizing the hybrid quantum-classical symmetry-adapted quantum Krylov subspace diagonalization algorithm based on our input scheme, where the quantum computations are performed with the IBM Qiskit quantum simulator. The results of the hybrid calculations agree with exact results. Here, we can incorporate the input scheme in this work with the input scheme for many-fermion Hamiltonians; they jointly offer new pathways to solving the structure and dynamics of more general field theory problems on future fault-tolerant quantum computers.

Ab initio calculations↗

DOE OSTI2024

Sparsity-Independent Lyapunov Exponent in the Sachdev-Ye-Kitaev Model

The saturation of a recently proposed universal bound on the Lyapunov exponent has been conjectured to signal the existence of a gravity dual. This saturation occurs in the low-temperature limit of the dense Sachdev-Ye-Kitaev (SYK) model, N Majorana fermions with q body ( q > 2 ) infinite-range interactions. We calculate certain out-of-time-order correlators (OTOCs) for N ≤ 64 fermions for a highly sparse SYK model and find no significant dependence of the Lyapunov exponent on sparsity up to near the percolation limit where the Hamiltonian breaks up into blocks. This provides strong support to the saturation of the Lyapunov exponent in the low-temperature limit of the sparse SYK. A key ingredient to reaching N = 64 is the development of a novel quantum spin model simulation library that implements highly optimized matrix-free Krylov subspace methods on graphical processing units. This leads to a significantly lower simulation time as well as vastly reduced memory usage over previous approaches, while using modest computational resources. Strong sparsity-driven statistical fluctuations require both the use of a much larger number of disorder realizations with respect to the dense limit and a careful finite size scaling analysis. The saturation of the bound in the sparse SYK points to the existence of a gravity analog that would enlarge substantially the number of field theories with this feature. Published by the American Physical Society 2024

Physics↗

DOE OSTI2025

Efficient truncations of SU( N c ) lattice gauge theory for quantum simulation

Quantum simulations of lattice gauge theories offer the potential to directly study the nonperturbative dynamics of quantum chromodynamics, but naive analyses suggest that they require large computational resources. Large 𝑁 𝑐 expansions are performed to order 1/𝑁 𝑐 to simplify the Hamiltonian of pure SU⁡(𝑁𝑐) lattice gauge theories. A reformulation of the electric basis is introduced with a truncation strategy based on the construction of local Krylov subspaces with plaquette operators. Numerical simulations show that these truncated Hamiltonians are consistent with traditional lattice calculations at relatively small couplings. It is shown that the computational resources required for quantum simulation of time evolution generated by these Hamiltonians is 17–19 orders of magnitude smaller than previous approaches, provided that the truncations in this work can reach lattice spacings in three-dimensional simulations comparable to the two-dimensional simulations performed.

Lattice QCD↗

DOE OSTI2024

Modifying the Asynchronous Jacobi Method for Data Corruption Resilience

Moving scientific computation from high-performance computing (HPC) and cloud computing (CC) environments to devices on the edge, i.e., physically near instruments of interest, has received tremendous interest in recent years. Such edge computing environments can operate on data in situ, offering enticing benefits over data aggregation to HPC and CC facilities that include avoiding costs of transmission, increased data privacy, and real-time data analysis. Because of the inherent unreliability of edge computing environments, new fault-tolerant approaches must be developed before the benefits of edge computing can be realized. Motivated by algorithm-based fault tolerance, a variant of the asynchronous Jacobi (ASJ) method is developed that achieves resilience to data corruption by rejecting solution approximations from neighbor devices according to a bound derived from convergence theory. Numerical results on a two-dimensional Poisson problem show that the new rejection criterion, along with a novel approximation to the shortest path length on which the criterion depends, restores convergence for the ASJ variant in the presence of certain types data corruption. Numerical results are obtained for when the singular values in the analytic bound are approximated. Additional linear systems are also explored, one with a more dense sparsity pattern and one that includes advection. All results indicate that successful resilience to data corruption depends on whether the bound tightens fast enough to reject corrupted data before the iteration evolution deviates significantly from that predicted by the convergence theory defining the bound. This observation generalizes to future work on algorithm-based fault tolerance for other asynchronous algorithms, including upcoming approaches that leverage Krylov subspaces.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

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↗

DOE OSTI2023

PERKS: a Locality-Optimized Execution Model for Iterative Memory-bound GPU Applications

Iterative memory-bound solvers commonly occur in HPC codes. Typical GPU implementations have a loop on the host side that invokes the GPU kernel as much as time/algorithm steps there are. The termination of each kernel implicitly acts the barrier required after advancing the solution every time step. We propose an execution model for running memory-bound iterative GPU kernels: PERsistent KernelS (PERKS). In this model, the time loop is moved inside persistent kernel, and device-wide barriers are used for synchronization. We then reduce the traffic to device memory by caching subset of the output in each time step in the unused registers and shared memory. PERKS can be generalized to any iterative solver: they largely independent of the solver's implementation. We explain the design principle of PERKS and demonstrate effectiveness of PERKS for a wide range of iterative 2D/3D stencil benchmarks (geomean speedup of 2.12x for 2D stencils and 1.24x for 3D stencils over state-of-art libraries), and a Krylov subspace conjugate gradient solver (geomean speedup of 4.86x in smaller SpMV datasets from SuiteSparse and 1.43x in larger SpMV datasets over a state-of-art library). All PERKS-based implementations available at: https://github.com/neozhang307/PERKS.

Zhang, Lingqi↗

DOE OSTI2025

A two-level GPU-accelerated incomplete LU preconditioner for general sparse linear systems

This paper presents a parallel preconditioning approach based on incomplete LU (ILU) factorizations in the framework of Domain Decomposition (DD) for general sparse linear systems. We focus on distributed memory parallel architectures, specifically, those that are equipped with graphic processing units (GPUs). In addition to block-Jacobi, we present general purpose two-level ILU Schur complement-based approaches, where different strategies are presented to solve the coarse-level reduced system. These strategies are combined with modified ILU methods in the construction of the coarse-level operator, in order to effectively remove smooth errors by targeting an algebraically smooth vector. We leverage available GPU-based sparse matrix kernels to accelerate the setup and the solve phases of the proposed ILU preconditioner. We evaluate the efficiency of the proposed methods as a smoother for algebraic multigrid (AMG) and as a preconditioner for Krylov subspace methods on challenging anisotropic diffusion problems and a collection of general sparse matrices.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Initial Exploration of the Use of Mixed Precision in Iterative Solvers (B639388 Subcontract Quarter I Report)

The first quarter of the project was primarily spent identifying potential projects at the intersection of finite precision analysis, mixed precision computation, and Krylov subspace methods. We summarize our findings in the remainder of the document. Other activities include attending biweekly xSDK meetings as well as contributing material to the technical report and journal versions of the multiprecision landscape paper. The subsequent quarter will be spent selecting a subset of the described projects to focus on, performing initial numerical experiments to evaluate the potential for the use of mixed precision, and developing initial theoretical analysis.

97 MATHEMATICS AND COMPUTING↗