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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

Parallel-in-Time Solution of Allen-Cahn Equations by Integrating Operator Learning into the Parareal Method

While recent advances in deep learning have shown promising efficiency gains in solving time-dependent partial differential equations (PDEs), matching the accuracy of conventional numerical solvers still remains a challenge. One strategy to improve the accuracy of deep learning-based solutions for time-dependent PDEs is to use the learned model as the coarse propagator in the Parareal method and a traditional numerical method as the fine solver. However, successful integration of deep learning into the Parareal method requires consistency between the coarse and fine solvers, particularly for PDEs exhibiting rapid changes such as sharp transitions. Here, to ensure this consistency, we propose using convolutional neural networks (CNNs) to learn the fully discrete time-stepping operator defined by the same numerical scheme employed as the fine solver. We demonstrate the effectiveness of the proposed method in solving the classical and mass-conservative Allen–Cahn (AC) equations. Through iterative updates in the Parareal algorithm, our approach achieves a significant computational speedup compared to traditional fine solvers while converging to high-accuracy solutions. Our results highlight that the proposed hybrid Parareal algorithm effectively accelerates simulations, particularly when implemented on multiple GPUs, and converges to the desired accuracy in only a few iterations. Another advantage of our method is that the CNN model is trained on trajectory-based data generated from random initial conditions, such that the trained model can be used to solve the AC equations with various initial conditions without retraining. This work demonstrates the potential of integrating neural network methods into parallel-in-time frameworks for efficient and accurate simulations of time-dependent PDEs.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Mixed-precision numerics in scientific applications: survey and perspectives

The explosive demand for artificial intelligence (AI) workloads has led to a significant increase in silicon area dedicated to lower-precision computations on recent high-performance computing hardware designs. However, mixed-precision capabilities, which can achieve performance improvements of up to 8x compared to double-precision in extreme compute-intensive workloads, remain largely untapped in most scientific applications. A growing number of efforts have shown that mixed-precision algorithmic innovations can deliver superior performance without sacrificing accuracy. These developments should prompt computational scientists to seriously consider whether their scientific modeling and simulation applications could benefit from the acceleration offered by new hardware and mixed-precision algorithms. In this survey, we (1) review progress across diverse scientific domains—fluid dynamics, weather and climate, quantum chemistry, and computational genomics—that have begun adopting mixed-precision strategies; (2) examine state-of-the-art algorithmic techniques such as iterative refinement, splitting and emulation schemes, and adaptive precision solvers; (3) assess their implications for accuracy, performance, and resource utilization; and (4) survey the emerging software ecosystem that enables mixed-precision methods at scale. We conclude with perspectives and recommendations on cross-cutting opportunities, domain-specific challenges, and the role of co-design between application scientists, numerical analysts, and computer scientists. Collectively, this survey underscores that mixed-precision numerics can reshape computational science by aligning algorithms with the evolving landscape of hardware capabilities.

Graphics processing units↗
DOE OSTI2012

PyTrilinos: Recent Advances in the Python Interface to Trilinos

PyTrilinos is a set of Python interfaces to compiled Trilinos packages. This collection supports serial and parallel dense linear algebra, serial and parallel sparse linear algebra, direct and iterative linear solution techniques, algebraic and multilevel preconditioners, nonlinear solvers and continuation algorithms, eigensolvers and partitioning algorithms. Also included are a variety of related utility functions and classes, including distributed I/O, coloring algorithms and matrix generation. PyTrilinos vector objects are compatible with the popular NumPy Python package. As a Python front end to compiled libraries, PyTrilinos takes advantage of the flexibility and ease of use of Python, and the efficiency of the underlying C++, C and Fortran numerical kernels. This paper covers recent, previously unpublished advances in the PyTrilinos package.

Spotz, William F.↗
DOE OSTI2023

Carbon Organisms Rhizosphere and Protection in Soil Environment model script and input data for soil moisture-respiration responses in tropical forests

Objectives: Climatic drying is predicted for many tropical forests, yet models remain poorly parameterized for tropical forests, hampering predictions of forest-climate feedbacks. We applied an integrated model–experiment approach, parameterizing an ecosystem model Carbon Organisms Rhizosphere and Protection in the Soil Environment (CORPSE) with tropical forest observational data, and comparing model predictions with a field drying manipulation. We hypothesized that drying would suppress soil CO2 fluxes (i.e., respiration) in already-drier tropical forests, but increases CO2 fluxes in wetter tropical forests by alleviating anaerobiosis. We measured soil CO2 fluxes, soil moisture, soil temperature, and forest floor biomass during wet-dry cycles (2015 – 2022) in four Panamanian forests that vary in rainfall and soil fertility. We used the field data to parameterize and run tests in the model.Results: Measured CO2 fluxes declined in the dry season and peaked in the early wet season ahead of peak soil moisture, resulting in a lower soil moisture optimum for respiration than previously modeled. We used this data to parameterize the model, which then predicted increased soil CO2 fluxes in wetter and fertile forests with drying, and decreased fluxes in drier, infertile forests. In contrast to model predictions, a chronic throughfall exclusion experiment in the forests initially suppressed soil CO2 fluxes across forests, with sustained suppression after four years in the wettest forest only (-28 ± 4% during the dry season), but elevated soil CO2 fluxes in a fertile forest after four years (+75 ± 28% during the late wet season), as predicted by the model. The unexpected negative drying effect in the wettest, most infertile forest could have resulted from reduced vertical flushing of nutrients into soils. Including hydro-nutrient interactions in ecosystem models could improve predictions of tropical forest-climate feedbacks (results presented in Cusack et al. 2023). Datasets included: Code files:CORPSE_array.py: Defines the equations of the CORPSE modelCORPSE_solvers: Functions for running the CORPSE model using either iterative or ordinary differential equation (ODE) solversrun_Panama_sims.py: Read in datasets and run the model simulations for this studyInput data:PanamaGradientEcosystemChem_BT_CPools_20152016CO2_DC_20190615.xlsx: Plot characteristics used in running model simulationsLiCor compiled surface flux only to 2020_03 DC_20200825.xlsx: Surface gas exchange fluxes used in model-data comparisonsPARCHED litterfall data for Ben Sulman LD 20200902.xlsx: Litterfall data used to drive model simulationsInitialization data:state_500y_20190823.csv: Initial state of model pools based on previous spinup runsOutput data:Outputs/prev_moisture_response.csv: Simulations of multiple sites using original model moisture response function.Outputs/updated_moisture_response.csv: Simulations of multiple sites using updated model moisture response function.Outputs/dry15_prev_moisture_response.csv: Simulations with soil moisture reduced by 15%, using original moisture response function.Outputs/dry15_updated_moisture_response.csv: Simulations with soil moisture reduced by 15%, using updated moisture response function.Outputs/dry30_prev_moisture_response.csv: Simulations with soil moisture reduced by 30%, using original moisture response function.Outputs/dry30_updated_moisture_response.csv: Simulations with soil moisture reduced by 30%, using updated moisture response function.Outputs/latestart_prev_moisture_response.csv: Simulations with extended dry season, using original moisture response function.Outputs/latestart_updated_moisture_response.csv: Simulations with extended dry season, using updated moisture response function.Outputs/[site name]_oneyear.csv: One-year simulation for each site in expanded site list using original moisture response function.Outputs/[site name]_oneyear_dried.csv: One-year simulation for each site in expanded site list using original moisture response function, with soil moisture reduced by 25%.Outputs/[site name]_oneyear_updated_moisture_response.csv: One-year simulation for each site in expanded site list using updated moisture response function.Outputs/[site name]_oneyear_updated_moisture_response_dried.csv: One-year simulation for each site in expanded site list using updated moisture response function, with soil moisture reduced by 25%.Field plot location data:There is also a .kml file that includes coordinates for all 32 plots included in the study of four forests (n = 4 throughfall reduction and n = 4 control plots per site).

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2020

Diffusion Synthetic Acceleration for Heterogeneous Domains, Compatible with Voids

A standard approach to solving the S N transport equations is to use source iteration with diffusion synthetic acceleration (DSA). Although this approach is widely used and effective on many problems, there remain some practical issues with DSA preconditioning, particularly on highly heterogeneous domains. For large-scale parallel simulation, it is critical that both (a) preconditioned source iteration converges rapidly and (b) the action of the DSA preconditioner can be applied using fast, scalable solvers, such as algebraic multigrid (AMG). For heterogeneous domains, these two interests can be at odds. In particular, there exist DSA diffusion discretizations that can be solved rapidly using AMG, but they do not always yield robust/fast convergence of the larger source iteration. Conversely, there exist robust DSA discretizations where source iteration converges rapidly on difficult heterogeneous problems, but fast parallel solvers like AMG tend to struggle applying the action of such operators. Moreover, very few current methods for the solution of deterministic transport are compatible with voids. This paper develops a new heterogeneous DSA preconditioner based on only preconditioning the optically thick subdomains. The resulting method proves robust on a variety of heterogeneous transport problems, including a linearized hohlraum mesh related to inertial confinement fusion. Moreover, the action of the preconditioner is easily computed using O(1) AMG iterations, convergence of the transport iteration typically requires 2 to 5× fewer iterations than current state-of-the-art “full” DSA, and the proposed method is trivially compatible with voids. On the hohlraum problem, rapid convergence is obtained by preconditioning less than 3% of the mesh elements with five to ten AMG iterations.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2018

Solving linear equations with messenger-field and conjugate gradient techniques: An application to CMB data analysis

We discuss linear system solvers invoking a messenger-field and compare them with (preconditioned) conjugate gradient approaches. We show that the messenger-field techniques correspond to fixed point iterations of an appropriately preconditioned initial system of linear equations. We then argue that a conjugate gradient solver applied to the same preconditioned system, or equivalently a preconditioned conjugate gradient solver using the same preconditioner and applied to the original system, will in general ensure at least a comparable and typically better performance in terms of the number of iterations to convergence and time-to-solution. We illustrate our conclusions with two common examples drawn from the cosmic microwave background (CMB) data analysis: Wiener filtering and map-making. In addition, and contrary to the standard lore in the CMB field, we show that the performance of the preconditioned conjugate gradient solver can depend significantly on the starting vector. In conclusion, this observation seems of particular importance in the cases of map-making of high signal-to-noise ratio sky maps and therefore should be of relevance for the next generation of CMB experiments.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2013

MICADO: Parallel implementation of a 2D-1D iterative algorithm for the 3D neutron transport problem in prismatic geometries

The large increase in computing power over the past few years now makes it possible to consider developing 3D full-core heterogeneous deterministic neutron transport solvers for reference calculations. Among all approaches presented in the literature, the method first introduced in [1] seems very promising. It consists in iterating over resolutions of 2D and ID MOC problems by taking advantage of prismatic geometries without introducing approximations of a low order operator such as diffusion. However, before developing a solver with all industrial options at EDF, several points needed to be clarified. In this work, we first prove the convergence of this iterative process, under some assumptions. We then present our high-performance, parallel implementation of this algorithm in the MICADO solver. Benchmarking the solver against the Takeda case shows that the 2D-1D coupling algorithm does not seem to affect the spatial convergence order of the MOC solver. As for performance issues, our study shows that even though the data distribution is suited to the 2D solver part, the efficiency of the ID part is sufficient to ensure a good parallel efficiency of the global algorithm. After this study, the main remaining difficulty implementation-wise is about the memory requirement of a vector used for initialization. An efficient acceleration operator will also need to be developed. (authors)

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

DS-TIDE: Harnessing Dynamical Systems for Efficient Time-Independent Differential Equation Solving

Time-Independent Differential Equations (TIDEs) are central to modeling equilibrium behavior across a wide range of scientific and engineering domains, from electrostatics to porous media flow. Conventional numerical solvers offer reliable solutions but incur significant computational costs due to fine-grained discretization and iterative procedures. Machine learning-based approaches address this by replacing iterative solving processes with one-time inference; however, their sophisticated models require extensive training resources that often exceed those of traditional solvers. Consequently, designing a TIDE solver that achieves high accuracy, broad applicability, and exceptional computational efficiency remains a fundamental challenge. In this paper, we propose DS-TIDE, a novel hardware solver that is inspired by, and subsequently leverages, the intrinsic connection between Dynamical Systems (DS) and Differential Equations (DEs) to efficiently and accurately solve TIDEs. DS-TIDE employs a CMOS-compatible DS-based processor, whose physical states evolve under carefully designed DE-driven dynamics and naturally converge to equilibrium -- the solution of the target TIDE -- within ~1µs on a ~1-watt DS-TIDE processor. To enhance expressivity, DS-TIDE incorporates Heterogeneous Dynamics with Temporal Layering (HDTL), which solves TIDEs through a three-stage DS evolution -- conditioning, solving, and decoding -- each governed by specialized dynamics. The entire evolution process is analogous to an infinitely deep neural network temporally unrolled, offering the system the capability of representing complex equations. Furthermore, DS-TIDE is equipped with an on-device DS-DE Auto-Alignment mechanism that dynamically adapts intrinsic hardware dynamics within milliseconds, effectively aligning the system’s dynamics to diverse target DEs. Experimental results across TIDEs from a wide range of scientific and engineering domains demonstrate that DS-TIDE achieves ~10^3× speedup, ~10^5× energy savings, and competitive or superior accuracy compared to state-of-the-art numerical and ML-based solvers.

Liu, Chuan↗
DOE OSTI2023

Combining Sparse Approximate Factorizations with Mixed-precision Iterative Refinement

The standard LU factorization-based solution process for linear systems can be enhanced in speed or accuracy by employing mixed-precision iterative refinement. Most recent work has focused on dense systems. We investigate the potential of mixed-precision iterative refinement to enhance methods for sparse systems based on approximate sparse factorizations. In doing so, we first develop a new error analysis for LU- and GMRES-based iterative refinement under a general model of LU factorization that accounts for the approximation methods typically used by modern sparse solvers, such as low-rank approximations or relaxed pivoting strategies. We then provide a detailed performance analysis of both the execution time and memory consumption of different algorithms, based on a selected set of iterative refinement variants and approximate sparse factorizations. Our performance study uses the multifrontal solver MUMPS, which can exploit block low-rank factorization and static pivoting. We evaluate the performance of the algorithms on large, sparse problems coming from a variety of real-life and industrial applications showing that mixed-precision iterative refinement combined with approximate sparse factorization can lead to considerable reductions of both the time and memory consumption.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A scalable multidimensional fully implicit solver for Hall magnetohydrodynamics

We propose an optimally performant fully implicit algorithm for the Hall magnetohydrodynamics (HMHD) equations based on multigrid-preconditioned Jacobian-free Newton-Krylov methods. HMHD is a challenging system to solve numerically because it supports stiff fast dispersive waves. The preconditioner is formulated using an operator-split approximate block factorization (Schur complement), informed by physics insight. We use a vector-potential formulation (instead of a magnetic field one) to allow a clean segregation of the problematic $\nabla$ x $\nabla$ x operator in the electron Ohm's law subsystem. This segregation allows the formulation of an effective damped block-Jacobi smoother for multigrid. We demonstrate by analysis that our proposed block-Jacobi iteration is convergent and has the smoothing property. The resulting HMHD solver is verified linearly with wave propagation examples, and nonlinearly with the GEM challenge reconnection problem by comparison against another HMHD code. We demonstrate the excellent algorithmic and parallel performance of the algorithm up to 16384 MPI tasks in two dimensions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2016

New preconditioning strategy for Jacobian-free solvers for variably saturated flows with Richards’ equation

We develop a new approach for solving the nonlinear Richards’ equation arising in variably saturated flow modeling. The growing complexity of geometric models for simulation of subsurface flows leads to the necessity of using unstructured meshes and advanced discretization methods. Typically, a numerical solution is obtained by first discretizing PDEs and then solving the resulting system of nonlinear discrete equations with a Newton-Raphson-type method. Efficiency and robustness of the existing solvers rely on many factors, including an empiric quality control of intermediate iterates, complexity of the employed discretization method and a customized preconditioner. We propose and analyze a new preconditioning strategy that is based on a stable discretization of the continuum Jacobian. We will show with numerical experiments for challenging problems in subsurface hydrology that this new preconditioner improves convergence of the existing Jacobian-free solvers 3-20 times. Furthermore, we show that the Picard method with this preconditioner becomes a more efficient nonlinear solver than a few widely used Jacobian-free solvers.

58 GEOSCIENCES↗
DOE OSTI2015

Comparison of multigrid algorithms for high‐order continuous finite element discretizations

Summary We present a comparison of different multigrid approaches for the solution of systems arising from high‐order continuous finite element discretizations of elliptic partial differential equations on complex geometries. We consider the pointwise Jacobi, the Chebyshev‐accelerated Jacobi, and the symmetric successive over‐relaxation smoothers, as well as elementwise block Jacobi smoothing. Three approaches for the multigrid hierarchy are compared: (1) high‐order h ‐multigrid, which uses high‐order interpolation and restriction between geometrically coarsened meshes; (2) p ‐multigrid, in which the polynomial order is reduced while the mesh remains unchanged, and the interpolation and restriction incorporate the different‐order basis functions; and (3) a first‐order approximation multigrid preconditioner constructed using the nodes of the high‐order discretization. This latter approach is often combined with algebraic multigrid for the low‐order operator and is attractive for high‐order discretizations on unstructured meshes, where geometric coarsening is difficult. Based on a simple performance model, we compare the computational cost of the different approaches. Using scalar test problems in two and three dimensions with constant and varying coefficients, we compare the performance of the different multigrid approaches for polynomial orders up to 16. Overall, both h ‐multigrid and p ‐multigrid work well; the first‐order approximation is less efficient. For constant coefficients, all smoothers work well. For variable coefficients, Chebyshev and symmetric successive over‐relaxation smoothing outperform Jacobi smoothing. While all of the tested methods converge in a mesh‐independent number of iterations, none of them behaves completely independent of the polynomial order. When multigrid is used as a preconditioner in a Krylov method, the iteration number decreases significantly compared with using multigrid as a solver. Copyright © 2015 John Wiley & Sons, Ltd.

Sundar, Hari↗
DOE OSTI2018

A multigrid waveform relaxation method for solving the poroelasticity equations

In this work, a multigrid waveform relaxation method is proposed for solving a collocated finite difference discretization of the linear Biot’s model. This gives rise to the first space–time multigrid solver for poroelasticity equations in the literature. The waveform relaxation iteration is based on a point-wise Vanka smoother that couples the pressure variable at a grid-point with the displacements around it. A semi-algebraic mode analysis is proposed to theoretically analyze the convergence of the multigrid waveform relaxation algorithm. This analysis is novel since it combines the semi-algebraic analysis, suitable for parabolic problems, with the non-standard analysis for overlapping smoothers. The practical utility of the method is illustrated through several numerical experiments in one and two dimensions.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

An iterative dynamic chemical stiffness removal method for reacting flow simulations

Abstract An iterative dynamic chemical stiffness removal method (IDCSR) based on quasi-steady-state approximation (QSSA) is proposed. The IDCSR method is built on a previously developed non-iterative method which has proved to work well for small timestep sizes. A novel iterative procedure is designed in IDCSR to enable explicit time integration of stiff chemistry at relatively large timestep sizes relevant to practical reacting flow simulations. The effectiveness of the iterative procedure is first demonstrated with a toy problem and homogeneous auto-ignition with fixed integration step sizes, showing that larger timestep sizes can be allowed for explicit time integration using IDCSR compared with the previous non-iterative method. IDCSR is then compared with existing explicit chemistry solvers for simulations of homogeneous auto-ignition and shows similar or lower computational cost but significantly higher accuracy across a wide range of timestep sizes. IDCSR is further combined with an automatic adaptive time-stepping scheme for simulations of 0-D homogeneous auto-ignition and a 2-D laminar lifted n -dodecane jet flame. For the 0-D auto-ignition simulations, IDCSR is shown to reduce both the error (by 43%–90%) and computational cost (by 6–15 times) compared with existing explicit solvers, while achieving speed-up factors of up to 400 compared with VODE for a wide range of timestep sizes and reaction mechanisms. For the 2-D jet flame simulations, speed-up factors of 15 and 31 for chemistry integration, and 5 and 9 for overall simulation, are achieved by IDCSR compared with CVODE with and without analytic Jacobian, respectively.

Xu, Chao (ORCID:0000000153074159)↗
DOE OSTI2020

Accelerating the solution of the S N equations with highly anisotropic scattering using the Fokker-Planck approximation

The discrete ordinates method can model forward-peaked transport problems accurately. However, convergence of discrete ordinates solution can become arbitrarily slow upon use of standard iterative procedures like source iteration. Standard zeroth and first moment-based acceleration methods like nonlinear diffusion acceleration and diffusion synthetic acceleration are ineffective in accelerating such problems because these methods do not correct higher order Legendre-moments of angular flux. In this paper, we explore the idea of using the Fokker-Planck approximation as a preconditioner to accelerate the convergence of solution of forward-peaked transport problems. Finally, we observe that such preconditioning can successfully reduce the iteration count by up to four orders of magnitude, and solver runtime by up to three orders of magnitude when compared to diffusion synthetic acceleration.

42 ENGINEERING↗
DOE OSTI2013

A high-order integral solver for scalar problems of diffraction by screens and apertures in three-dimensional space

We present a novel methodology for the numerical solution of problems of diffraction by infinitely thin screens in three-dimensional space. Our approach relies on new integral formulations as well as associated high-order quadrature rules. The new integral formulations involve weighted versions of the classical integral operators related to the thin-screen Dirichlet and Neumann problems as well as a generalization to the open-surface problem of the classical Calderón formulae. The high-order quadrature rules we introduce for these operators, in turn, resolve the multiple Green function and edge singularities (which occur at arbitrarily close distances from each other, and which include weakly singular as well as hypersingular kernels) and thus give rise to super-algebraically fast convergence as the discretization sizes are increased. When used in conjunction with Krylov-subspace linear algebra solvers such as GMRES, the resulting solvers produce results of high accuracy in small numbers of iterations for low and high frequencies alike. We demonstrate our methodology with a variety of numerical results for screen and aperture problems at high frequencies—including simulation of classical experiments such as the diffraction by a circular disc (featuring in particular the famous Poisson spot), evaluation of interference fringes resulting from diffraction across two nearby circular apertures, as well as solution of problems of scattering by more complex geometries consisting of multiple scatterers and cavities.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Joint Design of Multimodal Transit Networks and Shared Autonomous Mobility Fleets

Providing quality transit service to travelers in low-density areas, particularly travelers without personal vehicles, is a constant challenge for transit agencies. The advent of fully-autonomous vehicles (AVs) and their inclusion in mobility service fleets may allow transit agencies to offer better service and/or reduce their own capital and operational costs. This study focuses on the problem of allocating resources between transit patterns and operating (or subsidizing) shared-use AV mobility services (SAMSs) in a large metropolitan area. To address this question, a joint transit network redesign and SAMS fleet size determination problem (JTNR-SFSDP) is introduced, and a bi-level mathematical programming formulation and solution approach are presented. The upper-level problem modifies a transit network frequency setting problem (TNFSP) formulation via incorporating SAMS fleet size as a decision variable and allowing the removal of bus routes. The lower-level problem consists of a dynamic combined mode choice-traveler assignment problem (DCMC-TAP) formulation. The heuristic solution procedure involves solving the upper-level problem using a nonlinear programming solver and solving the lower-level problem using an iterative agent-based assignment-simulation approach. To illustrate the effectiveness of the modeling framework, this study uses traveler demand from Chicago along with the regions existing multimodal transit network. The computational results indicate significant traveler benefits, in terms of improved average traveler wait times, associated with optimizing the joint design of multimodal transit networks and SAMS fleets compared with the initial transit network design.

bi-level programming↗
DOE OSTI2020

A self-consistent field formulation of excited state mean field theory

In this work, we show that, as in Hartree-Fock theory, the orbitals for excited state mean field theory can be optimized via a self-consistent one-electron equation in which electron-electron repulsion is accounted for through mean field operators. In addition to showing that this excited state ansatz is sufficiently close to a mean field product state to admit a one-electron formulation, this approach brings the orbital optimization speed to within roughly a factor of two of ground state mean field theory. The approach parallels Hartree Fock theory in multiple ways, including the presence of a commutator condition, a one-electron mean-field working equation, and acceleration via direct inversion in the iterative subspace. When combined with a configuration interaction singles Davidson solver for the excitation coefficients, the self-consistent field formulation dramatically reduces the cost of the theory compared to previous approaches based on quasi-Newton descent.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗