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DOE OSTI2022

Root-N Krylov-space correction vectors for spectral functions with the density matrix renormalization group

In this work, we propose a method to compute spectral functions of generic Hamiltonians using the density matrix renormalization group (DMRG) algorithm directly in the frequency domain, based on a modified Krylov-space decomposition to compute the correction vectors. Our approach entails the calculation of the root-N (N=2 is the standard square root) of the Hamiltonian propagator using Krylov-space decomposition and repeating this procedure N times to obtain the actual correction vector. We show that our method greatly alleviates the burden of keeping a large bond dimension at large target frequencies, a problem found with conventional correction-vector DMRG, whereas achieving better computational performance at large N. We apply our method to spin and charge spectral functions of t-J and Hubbard models in the challenging two-leg ladder geometry and provide evidence that the root-N approach reaches a much improved spectral resolution compared to the conventional correction vector.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

NASA NTRS1990

Application of the Schwinger multichannel formulation to electron-impact excitation of the b 3Sigma(+) state of CO

The Chi 1Sigma - b 3Sigma(+) transition of CO is described using the Schwinger multichannel method in the two-state approximation, in an energy range from threshold to 20 eV. The resonance structure is analyzed by performing a partial-wave decomposition and the resonance positions are established based on the typical discontinuous behavior in the K-matrix elements as well as the pi radian change in the partial-wave eigenphases. The resonance behavior is related to the concept of core-excited resonances and the 'grandparent' model of resonances. The results are related to the formation of the negative ion by carrying out bound-state calculations on the (5 sigma)(3 s sigma)-squared core-excited Rydberg state of CO(-), and the position is found to agree well with the low-energy resonance positions.

Weatherford, Charles A.↗

NASA NTRS2009

New insights into the origin of magnetite crystals in ALH84001 carbonate disks

Martian meteorite ALH84001 preserves evidence of interaction with aqueous fluids while on Mars in the form of microscopic carbonate disks which are believed to have precipitated approximately 3.9 Ga ago at beginning of the Noachian epoch. Intimately associated within and throughout these carbonate disks are nanocrystal magnetites (Fe3O4) with unusual chemical and physical properties, whose origins have become the source of considerable debate. One group of hypotheses argues that these Fe3O4 are the product of partial thermal decomposition of the host carbonate. Alternatively, the origins of Fe3O4 and carbonate may be unrelated; that is, from the perspective of the carbonate the magnetite is allochthonous. We have sought to resolve between these hypotheses through the detailed characterized of the compositional and structural relationships of the carbonate disks and associated magnetites with the orthopyroxene matrix in which they are embedded. We focus this discussion on the composition of ALH84001 magnetites and then compare these observations with those from experimental thermal decomposition studies of sideritic carbonates under a range of plausible geological heating scenarios.

Thomas-Keprta, K. L.↗

NASA NTRS1994

Preconditioned domain decomposition scheme for three-dimensional aerodynamic sensitivity analysis

A discrete sensitivity analysis algorithm had previously been developed and applied to two-dimensional aerodynamic optimization problems, where the computational domains were discretized by using single grids. The sparse, unsymmetric systems of linear equations resulting from this algorithm were solved by a direct matrix inversion matrix. However, for large two-dimensional problems and, practically, all three-dimensional problems, direct inversion methods become inapplicable, primarily due to the prohibitive computer storage needed. In an attempt to alleviate such hindrances, the sensitivity analysis with domain decomposition (SADD) scheme was developed. This scheme divides the computational domain into smaller and nonoverlapping subdomains (multiblock grids) that are solved separately. Then, the final solution is constructed from the subdomain solutions. As the number of grid points in the interface boundaries of the subdomains becomes large, the computer memory required to store the effective coefficient matrix of these interface points starts to increase. Presented in this Technical Note is a preconditioned iterative procedure to overcome this particular problem.

Eleshaky, Mohamed E.↗

NASA NTRS1994

Time integration algorithms for the two-dimensional Euler equations on unstructured meshes

Explicit and implicit time integration algorithms for the two-dimensional Euler equations on unstructured grids are presented. Both cell-centered and cell-vertex finite volume upwind schemes utilizing Roe's approximate Riemann solver are developed. For the cell-vertex scheme, a four-stage Runge-Kutta time integration, a fourstage Runge-Kutta time integration with implicit residual averaging, a point Jacobi method, a symmetric point Gauss-Seidel method and two methods utilizing preconditioned sparse matrix solvers are presented. For the cell-centered scheme, a Runge-Kutta scheme, an implicit tridiagonal relaxation scheme modeled after line Gauss-Seidel, a fully implicit lower-upper (LU) decomposition, and a hybrid scheme utilizing both Runge-Kutta and LU methods are presented. A reverse Cuthill-McKee renumbering scheme is employed for the direct solver to decrease CPU time by reducing the fill of the Jacobian matrix. A comparison of the various time integration schemes is made for both first-order and higher order accurate solutions using several mesh sizes, higher order accuracy is achieved by using multidimensional monotone linear reconstruction procedures. The results obtained for a transonic flow over a circular arc suggest that the preconditioned sparse matrix solvers perform better than the other methods as the number of elements in the mesh increases.

Slack, David C.↗

NASA NTRS1976

Recursive partitioned inversion of large (1500 x 1500) symmetric matrices

A recursive algorithm was designed to invert large, dense, symmetric, positive definite matrices using small amounts of computer core, i.e., a small fraction of the core needed to store the complete matrix. The described algorithm is a generalized Gaussian elimination technique. Other algorithms are also discussed for the Cholesky decomposition and step inversion techniques. The purpose of the inversion algorithm is to solve large linear systems of normal equations generated by working geodetic problems. The algorithm was incorporated into a computer program called SOLVE. In the past the SOLVE program has been used in obtaining solutions published as the Goddard earth models.

Putney, B. H.↗

NASA NTRS1991

Turbulence modeling in supersonic combusting flows

To support the National Aerospace Plane project, the RPLUS3D CFD code has been developed at NASA Lewis. The code has the ability to solve three-dimensional flowfields with finite rate combustion of hydrogen and air. The combustion processes of the hydrogen-air system are simulated by an 18-reaction path, 8-species chemical kinetic mechanism. The code uses a Lower-Upper (LU) decomposition numerical algorithm as its basis, making it a very efficient and robust code. Except for the Jacobian matrix for the implicit chemistry source terms, there is no inversion of a matrix even though it uses a fully implicit numerical algorithm. A k-epsilon (two equation) turbulence model is incorporated into the RPLUS3D code.

Chitsomboon, Tawit↗

NASA NTRS1992

Implementation of a kappa-epsilon turbulence model to RPLUS3D code

The RPLUS3D code has been developed at the NASA Lewis Research Center to support the National Aerospace Plane (NASP) project. The code has the ability to solve three dimensional flowfields with finite rate combustion of hydrogen and air. The combustion process of the hydrogen-air system are simulated by an 18 reaction path, 8 species chemical kinetic mechanism. The code uses a Lower-Upper (LU) decomposition numerical algorithm as its basis, making it a very efficient and robust code. Except for the Jacobian matrix for the implicit chemistry source terms, there is no inversion of a matrix even though a fully implicit numerical algorithm is used. A k-epsilon turbulence model has recently been incorporated into the code. Initial validations have been conducted for a flow over a flat plate. Results of the validation studies are shown. Some difficulties in implementing the k-epsilon equations to the code are also discussed.

Chitsomboon, Tawit↗

NASA NTRS1996

A Feasibility Study on a Parallel Mechanism for Examining the Space Shuttle Orbiter Payload Bay Radiators

The goal of the project was to develop the necessary analysis tools for a feasibility study of a cable suspended robot system for examining the space shuttle orbiter payload bay radiators These tools were developed to address design issues such as workspace size, tension requirements on the cable, the necessary accuracy and resolution requirements and the stiffness and movement requirements of the system. This report describes the mathematical models for studying the inverse kinematics, statics, and stiffness of the robot. Each model is described by a matrix. The manipulator Jacobian was also related to the stiffness matrix, which characterized the stiffness of the system. Analysis tools were then developed based on the singular value decomposition (SVD) of the corresponding matrices. It was demonstrated how the SVD can be used to quantify the robot's performance and to provide insight into different design issues.

Roberts, Rodney G.↗

DOE OSTI2025

Wormholes, branes and finite matrices in sine dilaton gravity

We compute the double trumpet in sine dilaton gravity via WdW quantization. The wormhole size is discretized. The wormhole amplitude matches the spectral correlation of a finite-cut matrix integral, where matrices have large but finite dimensions. This strongly suggests an identification of the sine dilaton gravity theory with the q-deformed JT gravity matrix integral. At the very least, it captures all universal content of that matrix model. The disk decomposes into the physical (gauge invariant) solutions of the WdW equation, which are trumpets with discrete sizes. This decomposition modifies the usual no-boundary wavefunction to a normalizable one in sine dilaton gravity.

2D Gravity↗

NASA NTRS2010

Origin of Magnetite Crystals in Martian Meteorite ALH84001 Carbonate Disks

Martian meteorite ALH84001 preserves evidence of interaction with aqueous fluids while on Mars in the form of microscopic carbonate disks which are believed to have precipitated approx.3.9 Ga ago at beginning of the Noachian epoch. Intimately associated within and throughout these carbonate disks are nanocrystal magnetites (Fe3O4) with unusual chemical and physical properties, whose origins have become the source of considerable debate. One group of hypotheses argues that these Fe3O4 are the product of partial thermal decomposition of the host carbonate. Alternatively, the origins of Fe3O4 and carbonate may be unrelated; that is, from the perspective of the carbonate the magnetite is allochthonous. We have sought to resolve between these hypotheses through the detailed characterized of the compositional and structural relationships of the carbonate disks and associated magnetites with the orthopyroxene matrix in which they are embedded [1]. We focus this discussion on the composition of ALH84001 magnetites and then compare these observations with those from our thermal decomposition studies of sideritic carbonates under a range of plausible geological heating scenarios.

Thomas-Keprta, K.L.↗

DOE OSTI2021

Direct evidence for the role of microbial community composition in the formation of soil organic matter composition and persistence

The largest terrestrial carbon sink on earth is soil carbon stocks. As the climate changes, the rate at which the Earth’s climate warms depends in part on the persistence of soil organic carbon. Microbial turnover forms the backbone of soil organic matter (SOM) formation and it has been recently proposed that SOM molecular complexity is a key driver of stability. Despite this, the links between microbial diversity, chemical complexity and biogeochemical nature of SOM remain missing. Here we tested the hypotheses that distinct microbial communities shape the composition of SOM, and microbial-derived SOM has distinct decomposition potential depending on its community of origin. We inoculated microbial communities of varying diversities into a model soil matrix amended with simple carbon (cellobiose) and measured the thermal stability of the resultant SOM. Using a Rock-Eval ® ramped thermal analysis, we found that microbial community composition drives the chemical fingerprint of soil carbon. While diversity was not a driver of SOM composition, bacteria-only communities lead to more thermally labile soil C pools than communities with bacteria and fungi. Our results provide direct evidence for a link between microbial community structure, SOM composition, and thermal stability. This evidence demonstrates the relevance of soil microorganisms in building persistent SOM stocks.

Domeignoz-Horta, Luiz A. (ORCID:0000000346186253)↗

NASA NTRS1984

Finite elements and the method of conjugate gradients on a concurrent processor

An algorithm for the iterative solution of finite element problems on a concurrent processor is presented. The method of conjugate gradients is used to solve the system of matrix equations, which is distributed among the processors of a MIMD computer according to an element-based spatial decomposition. This algorithm is implemented in a two-dimensional elastostatics program on the Caltech Hypercube concurrent processor. The results of tests on up to 32 processors show nearly linear concurrent speedup, with efficiencies over 90% for sufficiently large problems.

Lyzenga, G. A.↗

NASA NTRS1985

Finite elements and the method of conjugate gradients on a concurrent processor

Lyzenga, G. A.↗

NASA NTRS1987

Growth process of gamma'-particles in a liquid-quenched Ni-15 at percent Al alloy

The growth of gamma-prime particles in a supersaturated Ni-15 at% Al alloy prepared by the liquid-quench method was studied using a magnetic mass balance, transmission electron microscopy, and field ion microscopy. Three stages in the precipitation and growth of these particles are demonstrated. In the first phase, the gamma-prime phase was produced by nucleation of spinodal decomposition of the alloy. In the second stage, spherical gamma-prime particles were arranged at random in the matrix with the size independent of aging time. Then, the Al concentration in the matrix decreased with the -1/3 power of the aging time. These results suggest that the degree of ordering increases with aging time. In the third stage, the particles were coarsened with aging time. They were arranged in the matrix (100) zone and were cubic, with the cube face lying on (100). The interfacial free energy was 28.4 mJ/sq m at 873 K and 22.4 mJ/sq m at 973 K.

Tachikake, A.↗

DOE OSTI2025

A Linear-Complexity Tensor Butterfly Algorithm for Compressing High-Dimensional Oscillatory Integral Operators

This paper presents a multilevel tensor compression algorithm called tensor butterfly algorithm for efficiently representing large-scale and high-dimensional oscillatory integral operators, including Green's functions for wave equations and integral transforms such as Radon transforms and Fourier transforms. The proposed algorithm leverages a tensor extension of the so-called complementary low-rank property of existing matrix butterfly algorithms. The algorithm partitions the discretized integral operator tensor into subtensors of multiple levels and factorizes each subtensor at the middle level as a Tucker-type interpolative decomposition, whose factor matrices are formed in a multilevel fashion. For a d-dimensional (d > 1) integral operator discretized into a 2d-mode tensor with n2d entries, the overall CPU time and memory requirement scale as O(nd), in stark contrast to the O(nd log n) complexity of existing matrix algorithms such as matrix butterfly algorithms and fast Fourier transforms (FFTs), where n is the number of points per direction. When comparing with other tensor algorithms such as quantized tensor train (QTT), the proposed algorithm also shows superior CPU and memory performance for tensor contraction. Remarkably, the tensor butterfly algorithm can efficiently model high-frequency Green's function interactions between two unit cubes, each spanning 512 wavelengths per direction, which represents problems of scale over 512× larger than that existing butterfly algorithms can handle, with the same amount of computation resources. On the other hand, for a problem representing 64 wavelengths per direction, which is the largest size existing algebraic matrix algorithms can handle, our tensor butterfly algorithm exhibits 200x speedups and 30× memory reduction compared with existing ones. Moreover, the tensor butterfly algorithm also permits O(nd)-complexity FFTs and Radon transforms up to d = 6 dimensions.

Kielstra, P Michael↗

DOE OSTI2021

Tensor force role in β decays analyzed within the Gogny-interaction shell model

The half-life of the famous C 14 β decay is anomalously long, with different mechanisms: the tensor force, cross-shell mixing, and three-body forces, proposed to explain the cancellations that lead to a small transition matrix element. In this study, we revisit and analyze the role of the tensor force for the β decay of C 14 as well as of neighboring isotopes. We add a tensor force to the Gogny interaction, and derive an effective Hamiltonian for shell-model calculations. The calculations were carried out in a p – s d model space to investigate cross-shell effects. Furthermore, we decompose the wave functions according to the total orbital angular momentum L in order to analyze the effects of the tensor force and cross-shell mixing. The inclusion of the tensor force significantly improves the shell-model calculations of the β -decay properties of carbon isotopes. In particular, the anomalously slow β decay of C 14 can be explained by the isospin T = 0 part of the tensor force, which changes the components of N 14 with the orbital angular momentum L = 0 , 1 , and results in a dramatic suppression of the Gamow-Teller transition strength. At the same time, the description of other nearby β decays are improved. Decomposition of wave function into L components illuminates how the tensor force modifies nuclear wave functions, in particular suppression of β -decay matrix elements. Cross-shell mixing also has a visible impact on the β -decay strength. Inclusion of the tensor force does not seem to significantly change, however, binding energies of the nuclei within the phenomenological interaction.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS1998

A Parallel Non-Overlapping Domain-Decomposition Algorithm for Compressible Fluid Flow Problems on Triangulated Domains

This paper considers an algebraic preconditioning algorithm for hyperbolic-elliptic fluid flow problems. The algorithm is based on a parallel non-overlapping Schur complement domain-decomposition technique for triangulated domains. In the Schur complement technique, the triangulation is first partitioned into a number of non-overlapping subdomains and interfaces. This suggests a reordering of triangulation vertices which separates subdomain and interface solution unknowns. The reordering induces a natural 2 x 2 block partitioning of the discretization matrix. Exact LU factorization of this block system yields a Schur complement matrix which couples subdomains and the interface together. The remaining sections of this paper present a family of approximate techniques for both constructing and applying the Schur complement as a domain-decomposition preconditioner. The approximate Schur complement serves as an algebraic coarse space operator, thus avoiding the known difficulties associated with the direct formation of a coarse space discretization. In developing Schur complement approximations, particular attention has been given to improving sequential and parallel efficiency of implementations without significantly degrading the quality of the preconditioner. A computer code based on these developments has been tested on the IBM SP2 using MPI message passing protocol. A number of 2-D calculations are presented for both scalar advection-diffusion equations as well as the Euler equations governing compressible fluid flow to demonstrate performance of the preconditioning algorithm.

Barth, Timothy J.↗