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All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Analysis of a New Implicit Solver for a Semiconductor Model

Here, we present and analyze a new iterative solver for implicit discretizations of a simplified Boltzmann--Poisson system. The algorithm builds on recent work that incorporated a sweeping algorithm for the Vlasov--Poisson equations as part of nested inner-outer iterative solvers for the Boltzmann--Poisson equations. The new method eliminates the need for nesting and requires only one transport sweep per iteration. It arises as a new fixed-point formulation of the discretized system which we prove to be contractive for a given electric potential. We also derive an accelerator to improve the convergence rate for systems in the drift-diffusion regime. We numerically compare the efficiency of the new solver, with and without acceleration, against a recently developed nested iterative solver.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2027

Nodal capacity expansion planning with flexible large-scale load siting

We propose explicitly incorporating large-scale load siting into a stochastic nodal power system capacity expansion planning model that concurrently co-optimizes generation, transmission, and storage expansion. The potential operational flexibility of some of these large loads is also taken into account by considering them as consisting of a set of tranches with different reliability requirements, which are modeled as a constraint on expected served energy across operational scenarios. We implement our model as a two-stage stochastic mixed-integer optimization problem with cross-scenario expectation constraints. To overcome the challenge of scalability, we build upon existing work to implement this model on a high performance computing platform and exploit scenario parallelization using an augmented Progressive Hedging Algorithm. The algorithm is implemented using the bounding features of mpisppy, which have shown to provide satisfactory provable optimality gaps despite the absence of theoretical guarantees of convergence. We test our approach and assess the value of this proactive planning framework on total system cost and reliability metrics using realistic testcases geographically assigned to San Diego and South Carolina, with datacenter and direct air capture facilities as large loads.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2026

Modeling the Effects of Trimethylsilanol on Syngas Combustion Kinetics

A perturbation analysis modeling approach based on a genetic algorithm was used to identify possible reaction pathways to explain previous experimental observations of the strong acceleration of syngas auto-ignition by trimethylsilanol (TMSO). Organosilicon reactions were taken from an existing chemical kinetic mechanism for tetramethylsilane pyrolysis which also contained TMSO reactions. The experimental targets for optimization were ignition delay times and peak OH radical concentrations in the range 1010–1070 K, 5 atm with dilute (4%) ϕ = 0.1 syngas mixtures in N 2 /Ar containing 0, 200, and 1000 ppm TMSO. Hundreds of thousands of trials resulted in three models which accurately predicted the experimental auto-ignition times and peak OH concentrations. Rate of production analyses indicated that TMSO forms dimethylsilanediol, which then reacts creating two “catalytic loops.” The net effect of the loops is to accelerate the reaction H 2 + O = OH + H at a faster effective rate than H 2 /O 2 kinetics alone, thus enhancing the reactivity of syngas. The formation of the loops may be attributable to the high Si–C bond energies in siloxanols, reducing rates of thermal decomposition and allowing dimethylsilanediol to react with H 2 for a longer duration compared with its alkane analog. Here, the results of the current work provide valuable direction for fundamental studies of these new hypothesized reaction pathways.

Mansfield, Andrew B. [Eastern Michigan University,↗
NASA NTRS1996

Structure Identification Within a Transitioning Swept-Wing Boundary Layer

Extensive measurements are made in a transitioning swept-wing boundary layer using hot-film, hot-wire and cross-wire anemometry. The crossflow-dominated flow contains stationary vortices that breakdown near mid-chord. The most amplified vortex wavelength is forced by the use of artificial roughness elements near the leading edge. Two-component velocity and spanwise surface shear-stress correlation measurements are made at two constant chord locations, before and after transition. Streamwise surface shear stresses are also measured through the entire transition region. Correlation techniques are used to identify stationary structures in the laminar regime and coherent structures in the turbulent regime. Basic techniques include observation of the spatial correlations and the spatially distributed auto-spectra. The primary and secondary instability mechanisms are identified in the spectra in all measured fields. The primary mechanism is seen to grow, cause transition and produce large-scale turbulence. The secondary mechanism grows through the entire transition region and produces the small-scale turbulence. Advanced techniques use Linear Stochastic Estimation (LSE) and Proper Orthogonal Decomposition (POD) to identify the spatio-temporal evolutions of structures in the boundary layer. LSE is used to estimate the instantaneous velocity fields using temporal data from just two spatial locations and the spatial correlations. Reference locations are selected using maximum RMS values to provide the best available estimates. POD is used to objectively determine modes characteristic of the measured flow based on energy. The stationary vortices are identified in the first laminar modes of each velocity component and shear component. Experimental evidence suggests that neighboring vortices interact and produce large coherent structures with spanwise periodicity at double the stationary vortex wavelength. An objective transition region detection method is developed using streamwise spatial POD solutions which isolate the growth of the primary and secondary instability mechanisms in the first and second modes, respectively. Temporal evolutions of dominant POD modes in all measured fields are calculated. These scalar POD coefficients contain the integrated characteristics of the entire field, greatly reducing the amount of data to characterize the instantaneous field. These modes may then be used to train future flow control algorithms based on neural networks.

Chapman, Keith↗
NASA NTRS2018

Polarization Decomposition and Temperature Bias Resolution for SMAP Passive Soil Moisture Retrieval Using Time Series Brightness Temperature Observations

In passive microwave remote sensing of soil moisture, the tau-omega (τ-ω) model has often been used to provide soil moisture estimates at a spatial scale representative of the satellite footprint dimensions. For modeling simplicity, model parameters such as the single scattering albedo (ω) and vegetation opacity (τ) that go into the geophysical inversion process are often assumed to be independent of polarizations. Although this absence of polarization dependence can often be justified in special cases as in low-frequency remote sensing or under dense vegetation conditions, it is not a robust assumption in general. Additional model parameterization errors arising from this assumption are possible, leading to degradation in soil moisture estimation accuracy. In this paper, we propose a time series approach to try to resolve the polarization dependence of several τ-ω model parameters as well as the temperature bias arising from the ancillary temperature data. The Version 4 of the Soil Moisture Active Passive (SMAP) Level 1B brightness temperature time series observations were used to illustrate the mechanics of this approach, with an emphasis on a comparison between resulting satellite soil moisture retrievals and in situ data collected at several core validation sites. It was found that this time series approach resulted in significant reduction of the dry bias exhibited in the current SMAP passive soil moisture data products, while retaining the same performance in other metrics of the current baseline passive soil moisture retrieval algorithm.

time series↗
DOE OSTI2017

Coupled land surface–subsurface hydrogeophysical inverse modeling to estimate soil organic carbon content and explore associated hydrological and thermal dynamics in the Arctic tundra

Quantitative characterization of soil organic carbon (OC) content is essential due to its significant impacts on surface–subsurface hydrological–thermal processes and microbial decomposition of OC, which both in turn are important for predicting carbon–climate feedbacks. While such quantification is particularly important in the vulnerable organic-rich Arctic region, it is challenging to achieve due to the general limitations of conventional core sampling and analysis methods, and to the extremely dynamic nature of hydrological–thermal processes associated with annual freeze–thaw events. In this study, we develop and test an inversion scheme that can flexibly use single or multiple datasets – including soil liquid water content, temperature and electrical resistivity tomography (ERT) data – to estimate the vertical distribution of OC content. Our approach relies on the fact that OC content strongly influences soil hydrological–thermal parameters and, therefore, indirectly controls the spatiotemporal dynamics of soil liquid water content, temperature and their correlated electrical resistivity. We employ the Community Land Model to simulate nonisothermal surface–subsurface hydrological dynamics from the bedrock to the top of canopy, with consideration of land surface processes (e.g., solar radiation balance, evapotranspiration, snow accumulation and melting) and ice–liquid water phase transitions. For inversion, we combine a deterministic and an adaptive Markov chain Monte Carlo (MCMC) optimization algorithm to estimate a posteriori distributions of desired model parameters. For hydrological–thermal-to-geophysical variable transformation, the simulated subsurface temperature, liquid water content and ice content are explicitly linked to soil electrical resistivity via petrophysical and geophysical models. We validate the developed scheme using different numerical experiments and evaluate the influence of measurement errors and benefit of joint inversion on the estimation of OC and other parameters. We also quantify the propagation of uncertainty from the estimated parameters to prediction of hydrological–thermal responses. We find that, compared to inversion of single dataset (temperature, liquid water content or apparent resistivity), joint inversion of these datasets significantly reduces parameter uncertainty. We find that the joint inversion approach is able to estimate OC and sand content within the shallow active layer (top 0.3 m of soil) with high reliability. Due to the small variations of temperature and moisture within the shallow permafrost (here at about 0.6 m depth), the approach is unable to estimate OC with confidence. However, if the soil porosity is functionally related to the OC and mineral content, which is often observed in organic-rich Arctic soil, the uncertainty of OC estimate at this depth remarkably decreases. Our study documents the value of the new surface–subsurface, deterministic–stochastic inversion approach, as well as the benefit of including multiple types of data to estimate OC and associated hydrological–thermal dynamics.

58 GEOSCIENCES↗
DOE OSTI2021

Optimal dimensionality selection for independent component analysis of transcriptomic data

Independent component analysis is an unsupervised machine learning algorithm that separates a set of mixed signals into a set of statistically independent source signals. Applied to high-quality gene expression datasets, independent component analysis effectively reveals both the source signals of the transcriptome as co-regulated gene sets, and the activity levels of the underlying regulators across diverse experimental conditions. Two major variables that affect the final gene sets are the diversity of the expression profiles contained in the underlying data, and the user-defined number of independent components, or dimensionality, to compute. Availability of high-quality transcriptomic datasets has grown exponentially as high-throughput technologies have advanced; however, optimal dimensionality selection remains an open question. We computed independent components across a range of dimensionalities for four gene expression datasets with varying dimensions (both in terms of number of genes and number of samples). We computed the correlation between independent components across different dimensionalities to understand how the overall structure evolves as the number of user-defined components increases. We then measured how well the resulting gene clusters reflected known regulatory mechanisms, and developed a set of metrics to assess the accuracy of the decomposition at a given dimension. We found that over-decomposition results in many independent components dominated by a single gene, whereas under-decomposition results in independent components that poorly capture the known regulatory structure. From these results, we developed a new method, called OptICA, for finding the optimal dimensionality that controls for both over- and under-decomposition. Specifically, OptICA selects the highest dimension that produces a low number of components that are dominated by a single gene. We show that OptICA outperforms two previously proposed methods for selecting the number of independent components across four transcriptomic databases of varying sizes. OptICA avoids both over-decomposition and under-decomposition of transcriptomic datasets resulting in the best representation of the organism’s underlying transcriptional regulatory network.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

Poisson-response Tensor-on-Tensor Regression and Applications

We introduce Poisson-response tensor-on-tensor regression (PToTR), a novel regression framework designed to handle tensor responses composed element-wise of random Poisson-distributed counts. Tensors, or multi-dimensional arrays, composed of counts are common data in fields such as inter national relations, social networks, epidemiology, and medical imaging, where events occur across multiple dimensions like time, location, and dyads. PToTR accommodates such tensor responses alongside tensor covariates, providing a versatile tool for multi dimensional data analysis. We propose algorithms for maximum likelihood estimation under a canonical polyadic (CP) structure on the regression coefficient tensor that satisfy the positivity of Poisson parameters and then provide an initial theoretical error analysis for PToTR estimators. We also demonstrate the utility of PToTR through three concrete applications: longitudinal data analysis of the Integrated Crisis Early Warning System database, positron emission tomography (PET) image reconstruction, and change-point detection of communication patterns in longitudinal dyadic data. These applications highlight the versatility of PToTR in addressing complex, structured count data across various domains.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2007

Persistent Nature of Secondary Diurnal Modes of Precipitation over Oceanic and Continental Regimes

This investigation seeks a better understanding of the assorted mechanisms controlling the global distribution of precipitation diurnal variability based on the use of Tropical Rainfall Measuring Mission (TRMM) microwave radiometer and radar data. The horizontal distributions of precipitation's diurnal cycle are derived from eight years of TRMM Microwave Imager (TMI) and Precipitation Radar (PR) measurements involving three TRMM standard rain rate retrieval algorithms -- the resultant distributions analyzed at various spatiotemporal scales. The results reveal the prominent and expected late-evening to early-morning (LE-EM) precipitation maxima over oceans and the counterpart prominent and expected mid- to late-afternoon (MLA) maxima over continents. Moreover, and not generally recognized, the results reveal a widespread distribution of secondary maxima occurring over both oceans and continents -- maxima which generally mirror their counterpart regime's behavior. That is, many ocean regions exhibit clearcut secondary MLA precipitation maxima while many continental regions exhibit just as evident secondary LE-EM maxima. This investigation is the first comprehensive study of these globally prevalent secondary maxima and their widespread nature, a type of study only made possible when the analysis procedure is applied to a high-quality global-scale precipitation dataset. The characteristics of the secondary maxima are mapped and described on global grids using an innovative clock-face format, while a current study to be published at a later date provides physically-based explanations of the seasonal-regional distributions of the secondary maxima. In addition to an "explicit" maxima identification scheme, a "Fourier decomposition" maxima identification scheme is used to examine the amplitude and phase properties of the primary and secondary maxima -- as well as tertiary and quaternary maxima. Accordingly, the advantages, ambiguities, and pitfalls resulting from use of Fourier harmonic analysis are explained.

Yang, S.↗
DOE OSTI2023

Parallelized domain decomposition for multi-dimensional Lagrangian random walk mass-transfer particle tracking schemes

Lagrangian particle tracking schemes allow a wide range of flow and transport processes to be simulated accurately, but a major challenge is numerically implementing the inter-particle interactions in an efficient manner. This article develops a multi-dimensional, parallelized domain decomposition (DDC) strategy for mass-transfer particle tracking (MTPT) methods in which particles exchange mass dynamically. We show that this can be efficiently parallelized by employing large numbers of CPU cores to accelerate run times. In order to validate the approach and our theoretical predictions we focus our efforts on a well-known benchmark problem with pure diffusion, where analytical solutions in any number of dimensions are well established. In this work, we investigate different procedures for “tiling” the domain in two and three dimensions (2-D and 3-D), as this type of formal DDC construction is currently limited to 1-D. An optimal tiling is prescribed based on physical problem parameters and the number of available CPU cores, as each tiling provides distinct results in both accuracy and run time. We further extend the most efficient technique to 3-D for comparison, leading to an analytical discussion of the effect of dimensionality on strategies for implementing DDC schemes. Increasing computational resources (cores) within the DDC method produces a trade-off between inter-node communication and on-node work. For an optimally subdivided diffusion problem, the 2-D parallelized algorithm achieves nearly perfect linear speedup in comparison with the serial run-up to around 2700 cores, reducing a 5 h simulation to 8 s, while the 3-D algorithm maintains appreciable speedup up to 1700 cores.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Implementation of CMFD Acceleration Scheme in PROTEUS-MOC

The PROTEUS-MOC code is a three-dimensional (3D) transport code, which is being developed at Argonne National Laboratory under the DOE-NE's Nuclear Energy Advanced Modeling and Simulation (NEAMS) program. It provides a fully consistent 3D discretization of the transport equation by combining the two-dimensional (2D) method of characteristics (MOC) for the planar direction with the discontinuous Galerkin method for the treatment of the axial variable. PROTEUS-MOC uses the unstructured finite element method (FEM) and thus allows the explicit representation of any extruded geometries. It also has a capability of massively parallel computation in energy group, angle, plane, and space using a top-level GMRES linear algebra solver. For parallel computation in space, it uses an unstructured domain decomposition by adapting METIS. The computational burden for direct heterogeneous core calculations based on unstructured geometries is excessive for practical reactor problems, and hence it is essential to implement an efficient acceleration scheme in PROTEUS-MOC. Recently, the coarse mesh finite difference (CMFD) method, which was originally applied to the nodal diffusion methods, has been successfully applied to the MOC transport solvers, resulting in a remarkable reduction in the number of fission source iterations. In addition, a modified CMFD method has been developed to overcome the limitation of the conventional CMFD that is applicable only to neutronics solvers with a conservative discretization. This CMFD method was derived to be consistent with the FEM, which is based on a nonconservative discretization. In this method, the neutron balance in each coarse mesh is enforced by introducing a pseudo absorption cross section (PAXS), and the well-established alternating solution process of CMFD and transport calculations is employed to accelerate source convergence. This modified CMFD acceleration scheme was adopted in PROTEUS-MOC to accelerate the fission source iteration. Pin-cell and assembly based CMFD schemes have been implemented by defining individual pin cells or assemblies as a coarse mesh. For this, an algorithm to map the finite elements to coarse meshes was developed, and the routines were developed to determine the coupling coefficients and PAXS of the CMFD equations from the transport solution. Preliminary performance tests were carried out using the C5G7 benchmark problem. This paper presents the main features of this CMFD acceleration scheme and the preliminary test results.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2022

Factorization of Binary Matrices: Rank Relations, Uniqueness and Model Selection of Boolean Decomposition

The application of binary matrices are numerous. Representing a matrix as a mixture of a small collection of latent vectors via low-rank decomposition is often seen as an advantageous method to interpret and analyze data. In this work, we examine the factorizations of binary matrices using standard arithmetic (real and nonnegative) and logical operations (Boolean and $\mathbb{Z}$ 2 ). We examine the relationships between the different ranks, and discuss when factorization is unique. In particular, we characterize when a Boolean factorization X = W$\land$H has a unique W, a unique H (for a fixed W), and when both W and H are unique, given a rank constraint. We introduce a method for robust Boolean model selection, called BMFk, and show on numerical examples that BMFk not only accurately determines the correct number of Boolean latent features but reconstruct the pre-determined factors accurately.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Microbial communities in the liver and brain are informative for postmortem submersion interval estimation in the late phase of decomposition: A study in mouse cadavers recovered from freshwater

Introduction Bodies recovered from water, especially in the late phase of decomposition, pose difficulties to the investigating authorities. Various methods have been proposed for postmortem submersion interval (PMSI) estimation and drowning identification, but some limitations remain. Many recent studies have proved the value of microbiota succession in viscera for postmortem interval estimation. Nevertheless, the visceral microbiota succession and its application for PMSI estimation and drowning identification require further investigation. Methods In the current study, mouse drowning and CO 2 asphyxia models were developed, and cadavers were immersed in freshwater for 0 to 14 days. Microbial communities in the liver and brain were characterized via 16S rDNA high-throughput sequencing. Results Only livers and brains collected from 5 to 14 days postmortem were qualified for sequencing. There was significant variation between microbiota from liver and brain. Differences in microbiota between the cadavers of mice that had drowned and those only subjected to postmortem submersion decreased over the PMSI. Significant successions in microbial communities were observed among the different subgroups within the late phase of the PMSI in livers and brains. Eighteen taxa in the liver which were mainly related to Clostridium_sensu_stricto and Aeromonas , and 26 taxa in the brain which were mainly belonged to Clostridium_sensu_stricto , Acetobacteroides , and Limnochorda , were selected as potential biomarkers for PMSI estimation based on a random forest algorithm. The PMSI estimation models established yielded accurate prediction results with mean absolute errors ± the standard error of 1.282 ± 0.189 d for the liver and 0.989 ± 0.237 d for the brain. Conclusions The present study provides novel information on visceral postmortem microbiota succession in corpses submerged in freshwater which sheds new light on PMSI estimation based on the liver and brain in forensic practice.

Wang, Linlin↗
DOE OSTI2021

Machine Learning Assisted Safety Modeling and Analysis of Advanced Reactors

With the advances in computational power and numerical methods, analysts can now rely on first-principle simulations to predict ultra-fine details in a variety of applications. Advances in machine learning (ML) have produced algorithms that can now learn high-level abstractions via hierarchical models. This project aims to leverage advances in ML techniques and the available high-resolution simulation data to develop a novel modeling and simulation (M\&S) methodology for reactor safety analysis. While application-agnostic ML techniques are available, complex physics constraints need to be incorporated into ML techniques to build ML-based closures for computationally efficient predictive simulations. This project intends to develop a physics-guided data-driven multi-scale methodology for M\&S of advanced reactors. The project focuses on thermal fluid (T/F) phenomena, which play major roles in advanced reactor safety. Specifically, we propose a data-driven coarse-mesh turbulence model based on local flow features for the transient analysis of thermal mixing and stratification in a sodium-cooled fast reactor (SFR). The model has a coarse-mesh setup to ensure computational efficiency, while it is trained by fine-mesh computational fluid dynamics (CFD) data with Reynolds-averaged Navier-Stokes (RANS) turbulence model to ensure accuracy. Three different neural networks are developed and tested for loss-of-flow transients in the hot pool of SFR, i.e. the densely connected convolutional neural network (DCNN), long-short-term-memory network based on proper orthogonal decomposition (POD-LSTM), and the DCNN informed by LSTM (DCNN-LSTM). The performances of these three neural networks are evaluated based on baseline models. The DCNN-LSTM model has been chosen for further hyperparameter optimization. Furthermore, based on a simplified two-dimensional case, uncertainty quantification (UQ) of the developed ML-based closure are investigated with three methods, i.e. Monte Carlo dropout, deep ensemble, and Bayesian neural network. The developed ML-based turbulent viscosity closure relation based on deep ensemble is then integrated into the system analysis module SAM and serves as a term in the conservation equations. Such a SAM-ML based procedure guarantees that the obtained results are consistent with the physical constraints of the thermal-fluid system. The SAM-ML simulation on the same loss-of-flow transient showed comparable accuracy with the CFD simulation but with a much coarser mesh setup. Last but not least, the ML-based closure improvement with the support of higher-fidelity data from large eddy simulation (LES) is discussed. As a first step towards this direction, a baseline LES simulation is performed to obtain comparable data with RANS results. Based on the early results, future investigation on further improving the ML-based closure is discussed. We believe the developed approach that combines scientific machine learning with nuclear system analysis code can benefit the advanced reactor community as more accurate safety analyses will better characterize reactor safety margins and reduce licensing efforts.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
NASA NTRS1994

VLSI neuroprocessors

Electronic and optoelectronic hardware implementations of highly parallel computing architectures address several ill-defined and/or computation-intensive problems not easily solved by conventional computing techniques. The concurrent processing architectures developed are derived from a variety of advanced computing paradigms including neural network models, fuzzy logic, and cellular automata. Hardware implementation technologies range from state-of-the-art digital/analog custom-VLSI to advanced optoelectronic devices such as computer-generated holograms and e-beam fabricated Dammann gratings. JPL's concurrent processing devices group has developed a broad technology base in hardware implementable parallel algorithms, low-power and high-speed VLSI designs and building block VLSI chips, leading to application-specific high-performance embeddable processors. Application areas include high throughput map-data classification using feedforward neural networks, terrain based tactical movement planner using cellular automata, resource optimization (weapon-target assignment) using a multidimensional feedback network with lateral inhibition, and classification of rocks using an inner-product scheme on thematic mapper data. In addition to addressing specific functional needs of DOD and NASA, the JPL-developed concurrent processing device technology is also being customized for a variety of commercial applications (in collaboration with industrial partners), and is being transferred to U.S. industries. This viewgraph p resentation focuses on two application-specific processors which solve the computation intensive tasks of resource allocation (weapon-target assignment) and terrain based tactical movement planning using two extremely different topologies. Resource allocation is implemented as an asynchronous analog competitive assignment architecture inspired by the Hopfield network. Hardware realization leads to a two to four order of magnitude speed-up over conventional techniques and enables multiple assignments, (many to many), not achievable with standard statistical approaches. Tactical movement planning (finding the best path from A to B) is accomplished with a digital two-dimensional concurrent processor array. By exploiting the natural parallel decomposition of the problem in silicon, a four order of magnitude speed-up over optimized software approaches has been demonstrated.

Kemeny, Sabrina E.↗
DOE OSTI2020

Exploring Hilbert space on a budget: Novel benchmark set and performance metric for testing electronic structure methods in the regime of strong correlation

This work explores the ability of classical electronic structure methods to efficiently represent (compress) the information content of full configuration interaction (FCI) wave functions. We introduce a benchmark set of four hydrogen model systems of different dimensionalities and distinctive electronic structures: a 1D chain, a 1D ring, a 2D triangular lattice, and a 3D close-packed pyramid. To assess the ability of a computational method to produce accurate and compact wave functions, we introduce the accuracy volume, a metric that measures the number of variational parameters necessary to achieve a target energy error. Using this metric and the hydrogen models, we examine the performance of three classical deterministic methods: (i) selected configuration interaction (sCI) realized both via an a posteriori (ap-sCI) and variational selection of the most important determinants, (ii) an a posteriori singular value decomposition (SVD) of the FCI tensor (SVD-FCI), and (iii) the matrix product state representation obtained via the density matrix renormalization group (DMRG). We find that the DMRG generally gives the most efficient wave function representation for all systems, particularly in the 1D chain with a localized basis. For the 2D and 3D systems, all methods (except DMRG) perform best with a delocalized basis, and the efficiency of sCI and SVD-FCI is closer to that of DMRG. For larger analogs of the models, the DMRG consistently requires the fewest parameters but still scales exponentially in 2D and 3D systems, and the performance of SVD-FCI is essentially equivalent to that of ap-sCI.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2019

Sparse low rank approximation of potential energy surfaces with applications in estimation of anharmonic zero point energies and frequencies

We introduce a method that exploits sparse representation of potential energy surfaces (PES) on a polynomial basis set selected by compressed sensing . The method is useful for studies involving large numbers of PES evaluations, such as the search for local minima, transition states, or integration. We apply this method for estimating zero point energies and frequencies of molecules using a three step approach. In the first step, we interpret the PES as a sparse tensor on polynomial basis and determine its entries by a compressed sensing based algorithm using only a few PES evaluations. Then, we establish a rank reduction strategy to compress this tensor in a suitable low-rank canonical tensor format using standard tensor compression tools. This allows representing a high dimensional PES as a small sum of products of one dimensional functions. Finally, a low dimensional Gauss–Hermite quadrature rule is used to integrate the product of sparse canonical low-rank representation of PES and Green’s function in the second-order diagrammatic vibrational many-body Green’s function theory (XVH2) for estimation of zero-point energies and frequencies. Numerical tests on molecules considered in this work indicate a more efficient scaling of computational cost with molecular size as compared to other methods.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Data-Driven Learning for the Mori--Zwanzig Formalism: A Generalization of the Koopman Learning Framework

A theoretical framework which unifies the conventional Mori--Zwanzig formalism and the approximate Koopman learning of deterministic dynamical systems from noiseless observation is presented. In this framework, the Mori--Zwanzig formalism, developed in statistical mechanics to tackle the hard problem of construction of reduced-order dynamics for high-dimensional dynamical systems, can be considered as a natural generalization of the Koopman description of the dynamical system. We next show that, similar to the approximate Koopman learning methods, data-driven methods can be developed for the Mori--Zwanzig formalism with Mori's linear projection operator. We have developed two algorithms to extract the key operators, the Markov and the memory kernel, using time series of a reduced set of observables in a dynamical system. We have adopted the Lorenz `96 system as a test problem and solved for the above operators. These operators exhibit complex behaviors, which are unlikely to be captured by traditional modeling approaches in Mori--Zwanzig analysis. The nontrivial generalized fluctuation-dissipation relationship, which relates the memory kernel with the two-time correlation statistics of the orthogonal dynamics, was numerically verified as a validation of the solved operators. Here we present numerical evidence that the generalized Langevin equation, a key construct in the Mori--Zwanzig formalism, is more advantageous in predicting the evolution of the reduced set of observables than the conventional approximate Koopman operators.

97 MATHEMATICS AND COMPUTING↗