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DOE OSTI2023

Data exchange and processing synchronization in distributed systems

Systems, methods, techniques and apparatuses of asynchronous communication is distributed systems are disclosed. One exemplary embodiment is a method determining, with a plurality of agent nodes structured to communicate asynchronously in a distributed system, a first set of iterations including an iteration determined by each of the plurality of agent nodes; determining, with a first agent node of the plurality of agent nodes, a local vector clock; receiving, with the first agent node, a first iteration of the first set of iterations and a remote vector clock determined based on the first iteration; updating, with the first agent node, the local vector clock based on the received remote vector clock; and determining a first iteration of a second set of iterations based on the first set of iterations after determining all iterations of the first set of iterations have been received based on the local vector clock.

Cintuglu, Mehmet H.↗

DOE OSTI2013

Atomic data of tungsten for current and future uses in fusion and plasma science

Atomic physics has played an important role throughout the history of experimental plasma physics. For example, accurate knowledge of atomic properties has been crucial for understanding the plasma energy balance and for diagnostic development. With the shift in magnetic fusion research toward high-temperature burning plasmas like those expected to be produced in the ITER tokamak, the atomic physics of tungsten has become important. Tungsten will be a constituent of ITER plasmas because of its use as a plasma-facing material able to withstand high heat loads with lower tritium retention than other possible materials. Already, ITER diagnostics are being developed based on using tungsten radiation. In particular, the ITER Core Imaging X-ray Spectrometer (CIXS), which is designed to measure the core ion temperature and bulk plasma motion, is being based on the x-ray emission of neonlike tungsten ions (W{sup 64+}). In addition, tungsten emission will at ITER be measured by extreme ultraviolet (EUV) and optical spectrometers to determine its concentration in the plasma and to assess power loss and tungsten sputtering rates. On present-day tokamaks tungsten measurements are therefore being performed in preparation of ITER. Tungsten has very complex spectra and most are still unknown. The WOLFRAM project at Livermore aims to produce data for tungsten in various spectral bands: Lshell x-ray emission for CIXS development, soft x-ray and EUV M- and N-shell tungsten emission for understanding the edge radiation from ITER plasmas as well as from contemporary tokamaks, and O-shell emission for developing spectral diagnostics of the ITER divertor.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

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 OSTI2019

Hybrid Fixed-Point Fixed-Stress Splitting Method for Linear Poroelasticity

Efficient and accurate poroelasticity models are critical in modeling geophysical problems such as oil exploration, gas-hydrate detection, and hydrogeology. We propose an efficient operator splitting method for Biot’s model of linear poroelasticity based on fixed-point iteration and constrained stress. In this method, we eliminate the constraint on time step via combining our fixed-point approach with a physics-based restraint between iterations. Three different cases are considered to demonstrate the stability and consistency of the method for constant and variable parameters. The results are validated against the results from the fully coupled approach. In case I, a single iteration is used for continuous coefficients. The relative error decreases with an increase in time. In case II, material coefficients are assumed to be linear. In the single iteration approach, the relative error grows significantly to 40% before rapidly decaying to zero. This is an artifact of the approximate solutions approaching the asymptotic solution. The error in the multiple iterations oscillates within 10 − 6 before decaying to the asymptotic solution. Nine iterations per time step are enough to achieve the relative error close to 10 − 7 . In the last case, the hybrid method with multiple iterations requires approximately 16 iterations to make the relative error 5 × 10 − 6 .

Delgado, Paul M.↗

DOE OSTI2016

TH-AB-BRA-09: Stability Analysis of a Novel Dose Calculation Algorithm for MRI Guided Radiotherapy

Purpose: To determine the iterative deterministic solution stability of the Linear Boltzmann Transport Equation (LBTE) in the presence of magnetic fields. Methods: The LBTE with magnetic fields under investigation is derived using a discrete ordinates approach. The stability analysis is performed using analytical and numerical methods. Analytically, the spectral Fourier analysis is used to obtain the convergence rate of the source iteration procedures based on finding the largest eigenvalue of the iterative operator. This eigenvalue is a function of relevant physical parameters, such as magnetic field strength and material properties, and provides essential information about the domain of applicability required for clinically optimal parameter selection and maximum speed of convergence. The analytical results are reinforced by numerical simulations performed using the same discrete ordinates method in angle, and a discontinuous finite element spatial approach. Results: The spectral radius for the source iteration technique of the time independent transport equation with isotropic and anisotropic scattering centers inside infinite 3D medium is equal to the ratio of differential and total cross sections. The result is confirmed numerically by solving LBTE and is in full agreement with previously published results. The addition of magnetic field reveals that the convergence becomes dependent on the strength of magnetic field, the energy group discretization, and the order of anisotropic expansion. Conclusion: The source iteration technique for solving the LBTE with magnetic fields with the discrete ordinates method leads to divergent solutions in the limiting cases of small energy discretizations and high magnetic field strengths. Future investigations into non-stationary Krylov subspace techniques as an iterative solver will be performed as this has been shown to produce greater stability than source iteration. Furthermore, a stability analysis of a discontinuous finite element space-angle approach (which has been shown to provide the greatest stability) will also be investigated. Dr. B Gino Fallone is a co-founder and CEO of MagnetTx Oncology Solutions (under discussions to license Alberta bi-planar linac MR for commercialization)

60 APPLIED LIFE SCIENCES↗

DOE OSTI2017

Improvement of transport-corrected scattering stability and performance using a Jacobi inscatter algorithm for 2D-MOC

The MPACT code, being developed collaboratively by the University of Michigan and Oak Ridge National Laboratory, is the primary deterministic neutron transport solver being deployed within the Virtual Environment for Reactor Applications (VERA) as part of the Consortium for Advanced Simulation of Light Water Reactors (CASL). In many applications of the MPACT code, transport-corrected scattering has proven to be an obstacle in terms of stability, and considerable effort has been made to try to resolve the convergence issues that arise from it. Most of the convergence problems seem related to the transport-corrected cross sections, particularly when used in the 2D method of characteristics (MOC) solver, which is the focus of this work. Here in this paper, the stability and performance of the 2-D MOC solver in MPACT is evaluated for two iteration schemes: Gauss-Seidel and Jacobi. With the Gauss-Seidel approach, as the MOC solver loops over groups, it uses the flux solution from the previous group to construct the inscatter source for the next group. Alternatively, the Jacobi approach uses only the fluxes from the previous outer iteration to determine the inscatter source for each group. Consequently for the Jacobi iteration, the loop over groups can be moved from the outermost loop$-$as is the case with the Gauss-Seidel sweeper$-$to the innermost loop, allowing for a substantial increase in efficiency by minimizing the overhead of retrieving segment, region, and surface index information from the ray tracing data. Several test problems are assessed: (1) Babcock & Wilcox 1810 Core I, (2) Dimple S01A-Sq, (3) VERA Progression Problem 5a, and (4) VERA Problem 2a. The Jacobi iteration exhibits better stability than Gauss-Seidel, allowing for converged solutions to be obtained over a much wider range of iteration control parameters. Additionally, the MOC solve time with the Jacobi approach is roughly 2.0-2.5× faster per sweep. While the performance and stability of the Jacobi iteration are substantially improved compared to the Gauss-Seidel iteration, it does yield a roughly 8$-$10% increase in the overall memory requirement.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Efficient smoothed particle radiation hydrodynamics I: Thermal radiative transfer

This work presents efficient solution techniques for radiative transfer in the smoothed particle hydrodynamics discretization. Two choices that impact efficiency are how the material and radiation energy are coupled, which determines the number of iterations needed to converge the emission source, and how the radiation diffusion equation is solved, which must be done in each iteration. The coupled material and radiation energy equations are solved using an inexact Newton iteration scheme based on nonlinear elimination, which reduces the number of Newton iterations needed to converge within each time step. During each Newton iteration, the radiation diffusion equation is solved using Krylov iterative methods with a multigrid preconditioner, which abstracts and optimizes much of the communication when running in parallel. The code is verified for an infinite medium problem, a one-dimensional Marshak wave, and a two and three-dimensional manufactured problem, and exhibits first-order convergence in time and second-order convergence in space. For these problems, the number of iterations needed to converge the inexact Newton scheme and the diffusion equation is independent of the number of spatial points and the number of processors.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2022

Low-synch Gram–Schmidt with delayed reorthogonalization for Krylov solvers

The parallel strong-scaling of iterative methods is often determined by the number of global reductions at each iteration. Low-synch Gram-Schmidt algorithms are applied here to the Arnoldi algorithm to reduce the number of global reductions and therefore to improve the parallel strong-scaling of iterative solvers for nonsymmetric matrices such as the GMRES and the Krylov-Schur iterative methods. In the Arnoldi context, the factorization is "left-looking" and processes one column at a time. Among the methods for generating an orthogonal basis for the Arnoldi algorithm, the classical Gram-Schmidt algorithm, with reorthogonalization (CGS2) requires three global reductions per iteration. A new variant of CGS2 that requires only one reduction per iteration is presented and applied to the Arnoldi algorithm. Delayed CGS2 (DCGS2) employs the minimum number of global reductions per iteration (one) for a one-column at-a-time algorithm. The main idea behind the new algorithm is to group global reductions by rearranging the order of operations. DCGS2 must be carefully integrated into an Arnoldi expansion or a GMRES solver. Numerical stability experiments assess robustness for Krylov-Schur eigenvalue computations. Performance experiments on the ORNL Summit supercomputer then establish the superiority of DCGS2 over CGS2.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2016

First demonstration of rapid shutdown using neon shattered pellet injection for thermal quench mitigation on DIII-D

Shattered pellet injection (SPI) is one of the prime candidates for the ITER disruption mitigation system because of its deeper penetration and larger particle flux than massive gas injection (MGI) using deuterium. Here, the ITER disruption mitigation system will likely use mostly high Z species such as neon because of more effective thermal mitigation and pumping constraints on the maximum amount of deuterium or helium that could be injected. An upgrade of the SPI on DIII-D enables ITER relevant injection characteristics in terms of quantities and gas species. This upgraded SPI system was used on DIII-D for the first time in 2014 for a direct comparison with MGI using identical quantities of neon.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Navigating the Dynamic Noise Landscape of Variational Quantum Algorithms with QISMET

In the Noisy Intermediate Scale Quantum (NISQ) era, the dynamic nature of quantum systems causes noise sources to constantly vary over time. Transient errors from the dynamic NISQ noise landscape are challenging to comprehend and are especially detrimental to classes of applications that are iterative and/or long-running, and therefore their timely mitigation is important for quantum advantage in real-world applications. The most popular examples of iterative long-running quantum applications are variational quantum algorithms (VQAs). Iteratively, VQA’s classical optimizer evaluates circuit candidates on an objective function and picks the best circuits towards achieving the application’s target. Noise fluctuation can cause a significant transient impact on the objective function estimation of the VQA iterations’ tuning candidates. This can severely affect VQA tuning and, by extension, its accuracy and convergence. Furthermore, this paper proposes QISMET: Quantum Iteration Skipping to Mitigate Error Transients, to navigate the dynamic noise landscape of VQAs. QISMET actively avoids instances of high fluctuating noise which are predicted to have a significant transient error impact on specific VQA iterations. To achieve this, QISMET estimates transient error in VQA iterations and designs a controller to keep the VQA tuning faithful to the transient-free scenario. By doing so, QISMET efficiently mitigates a large portion of the transient noise impact on VQAs and is able to improve the fidelity by 1.3x-3x over a traditional VQA baseline, with 1.6-2.4x improvement over alternative approaches, across different applications and machines.

Ravi, Gokul Subramanian↗

DOE OSTI2012

The Fusion Nuclear Science Pathways Assessment

With the strong commitment of the US to the success of the ITER burning plasma mission, and the project overall, it is prudent to consider how to take the most advantage of this investment. The production of energy from fusion has been a long sought goal, and the subject of several programmatic investigations and time line proposals. The nuclear aspects of fusion research have largely been avoided experimentally for practical reasons, resulting in a strong emphasis on plasma science. Meanwhile, ITER has brought into focus how the interface between the plasma and engineering/technology, presents the most challenging problems for design. In fact, this situation is becoming the rule and no longer the exception. ITER will demonstrate the deposition of 0.5 GW of neutron heating to the blanket, deliver a heat load of 10-20 MW/m 2 or more on the divertor, inject 50-100 MW of heating power to the plasma, all at the expected size scale of a power plant. However, in spite of this, and a number of other technologies relevant power plant, ITER will provide a low neutron exposure compared to the levels expected to a fusion power plant, and will purchase its tritium entirely from world reserves accumulated from decades of CANDU reactor operations. Such a decision for ITER is technically well founded, allowing the use of conventional materials and water coolant, avoiding the thick tritium breeding blankets required for tritium self-sufficiency, and allowing the concentration on burning plasma and plasma-engineering interface issues. The neutron fluence experienced in ITER over its entire lifetime will be ~ 0.3 MW-yr/m 2 , while a fusion power plant is expected to experience 120-180 MW-yr/m 2 over its lifetime. ITER utilizes shielding blanket modules, with no tritium breeding, except in test blanket modules (TBM) located in 3 ports on the midplane, which will provide early tests of the fusion nuclear environment with very low tritium production (a few g per year).

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2014

A complex guided spectral transform Lanczos method for studying quantum resonance states

A complex guided spectral transform Lanczos (cGSTL) algorithm is proposed to compute both bound and resonance states including energies, widths, and wavefunctions. The algorithm comprises of two layers of complex-symmetric Lanczos iterations. A short inner layer iteration produces a set of complex formally orthogonal Lanczos polynomials. They are used to span the guided spectral transform function determined by a retarded Green operator. An outer layer iteration is then carried out with the transform function to compute the eigen-pairs of the system. The guided spectral transform function is designed to have the same wavefunctions as the eigenstates of the original Hamiltonian in the spectral range of interest. Therefore, the energies and/or widths of bound or resonance states can be easily computed with their wavefunctions or by using a root-searching method from the guided spectral transform surface. The new cGSTL algorithm is applied to bound and resonance states of HO{sub 2}, and compared to previous calculations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2017

Gas Gun Model and Comparison to Experimental Performance of Pipe Guns Operating with Light Propellant Gases and Large Cryogenic Pellets

Injection of multiple large (~10 to 30 mm diameter) shattered pellets into ITER plasmas is presently part of the scheme planned to mitigate the deleterious effects of disruptions on the vessel components. To help in the design and optimize performance of the pellet injectors for this application, a model referred to as “the gas gun simulator” has been developed and benchmarked against experimental data. The computer code simulator is a Java program that models the gas-dynamics characteristics of a single-stage gas gun. Following a stepwise approach, the code utilizes a variety of input parameters to incrementally simulate and analyze the dynamics of the gun as the projectile is launched down the barrel. Using input data, the model can calculate gun performance based on physical characteristics, such as propellant-gas and fast-valve properties, barrel geometry, and pellet mass. Although the model is fundamentally generic, the present version is configured to accommodate cryogenic pellets composed of H2, D2, Ne, Ar, and mixtures of them and light propellant gases (H2, D2, and He). The pellets are solidified in situ in pipe guns that consist of stainless steel tubes and fast-acting valves that provide the propellant gas for pellet acceleration (to speeds ~200 to 700 m/s). The pellet speed is the key parameter in determining the response time of a shattered pellet system to a plasma disruption event. The calculated speeds from the code simulations of experiments were typically in excellent agreement with the measured values. With the gas gun simulator validated for many test shots and over a wide range of physical and operating parameters, it is a valuable tool for optimization of the injector design, including the fast valve design (orifice size and volume) for any operating pressure (~40 bar expected for the ITER application) and barrel length for any pellet size (mass, diameter, and length). Key design parameters and proposed values for the pellet injectors for the ITER disruption mitigation systems are discussed.

Combs, Stephen Kirk↗

DOE OSTI2015

MO-FG-204-03: Using Edge-Preserving Algorithm for Significantly Improved Image-Domain Material Decomposition in Dual Energy CT

Purpose: To significantly improve dual energy CT (DECT) imaging by establishing a new theoretical framework of image-domain material decomposition with incorporation of edge-preserving techniques. Methods: The proposed algorithm, HYPR-NLM, combines the edge-preserving non-local mean filter (NLM) with the HYPR-LR (Local HighlY constrained backPRojection Reconstruction) framework. Image denoising using HYPR-LR framework depends on the noise level of the composite image which is the average of the different energy images. For DECT, the composite image is the average of high- and low-energy images. To further reduce noise, one may want to increase the window size of the filter of the HYPR-LR, leading resolution degradation. By incorporating the NLM filtering and the HYPR-LR framework, HYPR-NLM reduces the boost material decomposition noise using energy information redundancies as well as the non-local mean. We demonstrate the noise reduction and resolution preservation of the algorithm with both iodine concentration numerical phantom and clinical patient data by comparing the HYPR-NLM algorithm to the direct matrix inversion, HYPR-LR and iterative image-domain material decomposition (Iter-DECT). Results: The results show iterative material decomposition method reduces noise to the lowest level and provides improved DECT images. HYPR-NLM significantly reduces noise while preserving the accuracy of quantitative measurement and resolution. For the iodine concentration numerical phantom, the averaged noise levels are about 2.0, 0.7, 0.2 and 0.4 for direct inversion, HYPR-LR, Iter- DECT and HYPR-NLM, respectively. For the patient data, the noise levels of the water images are about 0.36, 0.16, 0.12 and 0.13 for direct inversion, HYPR-LR, Iter-DECT and HYPR-NLM, respectively. Difference images of both HYPR-LR and Iter-DECT show edge effect, while no significant edge effect is shown for HYPR-NLM, suggesting spatial resolution is well preserved for HYPR-NLM. Conclusion: HYPR-NLM provides an effective way to reduce the generic magnified image noise of dual–energy material decomposition while preserving resolution. This work is supported in part by NIH grants 7R01HL111141 and 1R01-EB016777. This work is also supported by the Natural Science Foundation of China (NSFC Grant No. 81201091), Fundamental Research Funds for the Central Universities in China, and Fund Project for Excellent Abroad Scholar Personnel in Science and Technology.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2017

Improved Diffuse Foreground Subtraction with the ILC Method: CMB Map and Angular Power Spectrum Using Planck and WMAP Observations

We report an improved technique for diffuse foreground minimization from Cosmic Microwave Background (CMB) maps using a new multiphase iterative harmonic space internal-linear-combination (HILC) approach. Our method nullifies a foreground leakage that was present in the old and usual iterative HILC method. In phase 1 of the multiphase technique, we obtain an initial cleaned map using the single iteration HILC approach over the desired portion of the sky. In phase 2, we obtain a final CMB map using the iterative HILC approach; however, now, to nullify the leakage, during each iteration, some of the regions of the sky that are not being cleaned in the current iteration are replaced by the corresponding cleaned portions of the phase 1 map. We bring all input frequency maps to a common and maximum possible beam and pixel resolution at the beginning of the analysis, which significantly reduces data redundancy, memory usage, and computational cost, and avoids, during the HILC weight calculation, the deconvolution of partial sky harmonic coefficients by the azimuthally symmetric beam and pixel window functions, which in a strict mathematical sense, are not well defined. Using WMAP 9 year and Planck 2015 frequency maps, we obtain foreground-cleaned CMB maps and a CMB angular power spectrum for the multipole range 2⩽ℓ⩽2500. Our power spectrum matches the published Planck results with some differences at different multipole ranges. We validate our method by performing Monte Carlo simulations. Finally, we show that the weights for HILC foreground minimization have the intrinsic characteristic that they also tend to produce a statistically isotropic CMB map.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2013

Application of nonlinear Krylov acceleration to radiative transfer problems

The iterative solution technique used for radiative transfer is normally nested, with outer thermal iterations and inner transport iterations. We implement a nonlinear Krylov acceleration (NKA) method in the PDT code for radiative transfer problems that breaks nesting, resulting in more thermal iterations but significantly fewer total inner transport iterations. Using the metric of total inner transport iterations, we investigate a crooked-pipe-like problem and a pseudo-shock-tube problem. Using only sweep preconditioning, we compare NKA against a typical inner / outer method employing GMRES / Newton and find NKA to be comparable or superior. Finally, we demonstrate the efficacy of applying diffusion-based preconditioning to grey problems in conjunction with NKA. (authors)

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Anderson acceleration stability in NDA-accelerated k-eigenvalue problems

Anderson acceleration (AA) has been used to improve the stability and convergence rate of multiphysics iterative methods for reactor analysis. Most applications studied assume a tightly converged solution for the different physics problems, and AA is usually applied to state variables like temperature, density, and heat generation rate. In this paper, we study the theoretical performance of AA in NDA-accelerated k-eigenvalue problems. The problems and algorithms studied are simplified from the coupled iteration scheme adopted by MPACT and many other high-fidelity whole-core reactor codes. Compared to previous analyses of AA for these iteration schemes, we study the case with a partially converged neutronics solution and possibly partially converged nonlinear diffusion acceleration (NDA)/coarse mesh finite difference (CMFD) solutions. We observe that the performance of the iteration scheme with AA is very sensitive to the initial guess and is affected by the partially converged CMFD solutions. When the NDA solution is fully converged, using AA cannot achieve the optimal convergence rate in large-sized problems. Conversely, if the NDA solution is partially converged, the iteration scheme with AA can diverge or converge extremely slowly. It is found that the loss of robustness for AA is due to the fact that it is applied to the iterative subspace of state variables rather than the fundamental unknowns of the governing equations. To improve the robustness, the scalar flux should also be considered in the implementation of AA. After considering the residuals of flux, we observe that the stability is regardless of the partial convergence of NDA solutions. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2019

Approximate Green’s Function Coupled Cluster A Method Employing Effective Dimension Reduction

The Green’s function coupled cluster (GFCC) method, originally proposed in the early 1990s, is a powerful many-body tool for computing and analyzing the electronic structures of molecular structures and periodic systems, especially when electrons are strongly correlated. However, in order for the GFCC method to become an approach that can be routinely used in the electronic structure calculations, more robust linear algebraic techniques and approximations need to be employed to reduce its extremely high computational overhead. In our recent studies, we showed that the GFCC equation can be numerically solved by seeking solutions of multiple linear systems in the frequency regime of interest, which can be easily distributed in a massively parallel environment, and broaden the avenue for more state-of-the-art numerical linear algebraic tools associated with multiple linear systems to be kicked in to this field. In the present pursuit of this practice, we show a successful application of model order reduction (MOR) techniques in the GFCC framework. Briefly speaking, for a frequency regime of interest that requires a high resolution of spectral function, instead of solving GFCC linear equation in the original dimension for every single frequency point, an easily-solvable linear system model with reduced dimension can be built upon projecting the original GFCC linear system into a subspace, from which the interpolation and extrapolation can give a reasonable approximation to the results of the original GFCC linear equations. Here, we show that the subspace can be properly constructed in an iterative manner from the auxiliary vectors of the GFCC linear equations at some select frequencies inside the frequency regime. During the iterations, the quality of the subspace, as well as the linear system model, can be systematically improved. The approach is tested in this work in terms of the efficiency and accuracy of computing spectral functions for some typical molecular systems including carbon monoxide, 1,3-butadiene, benzene, and adenine molecules. As a byproduct, the high quality linear system model is also found to be able to provide high quality initial guess for the existing linear solver to improve the convergence performance.

Peng, Bo↗