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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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At least 739 records · Page 41
All sourcesNASA NTRSDOE OSTI
DOE OSTI2015

SU-E-T-20: A Correlation Study of 2D and 3D Gamma Passing Rates for Prostate IMRT Plans

Purpose: To investigate the correlation between the two-dimensional gamma passing rate (2D %GP) and three-dimensional gamma passing rate (3D %GP) in prostate IMRT quality assurance. Methods: Eleven prostate IMRT plans were randomly selected from the clinical database and were used to obtain dose distributions in the phantom and patient. Three types of delivery errors (MLC bank sag errors, central MLC errors and monitor unit errors) were intentionally introduced to modify the clinical plans through an in-house Matlab program. This resulted in 187 modified plans. The 2D %GP and 3D %GP were analyzed using different dose-difference and distance-toagreement (1%-1mm, 2%-2mm and 3%-3mm) and 20% dose threshold. The 2D %GP and 3D %GP were then compared not only for the whole region, but also for the PTVs and critical structures using the statistical Pearson’s correlation coefficient (γ). Results: For different delivery errors, the average comparison of 2D %GP and 3D %GP showed different conclusions. The statistical correlation coefficients between 2D %GP and 3D %GP for the whole dose distribution showed that except for 3%/3mm criterion, 2D %GP and 3D %GP of 1%/1mm criterion and 2%/2mm criterion had strong correlations (Pearson’s γ value >0.8). Compared with the whole region, the correlations of 2D %GP and 3D %GP for PTV were better (the γ value for 1%/1mm, 2%/2mm and 3%/3mm criterion was 0.959, 0.931 and 0.855, respectively). However for the rectum, there was no correlation between 2D %GP and 3D %GP. Conclusion: For prostate IMRT, the correlation between 2D %GP and 3D %GP for the PTV is better than that for normal structures. The lower dose-difference and DTA criterion shows less difference between 2D %GP and 3D %GP. Other factors such as the dosimeter characteristics and TPS algorithm bias may also influence the correlation between 2D %GP and 3D %GP.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

SU-F-T-394: Impact of PTV Margins With Taking Into Account Shape Variation On IMRT Plans For Prostate Cancer

Purpose: The purpose of this study was to investigate the impact of planning target volume (PTV) margins with taking into consideration clinical target volume (CTV) shape variations on treatment plans of intensity modulated radiation therapy (IMRT) for prostate cancer. Methods: The systematic errors and the random errors for patient setup errors in right-left (RL), anterior-posterior (AP), and superior-inferior (SI) directions were obtained from data of 20 patients, and those for CTV shape variations were calculated from 10 patients, who were weekly scanned using cone beam computed tomography (CBCT). The setup error was defined as the difference in prostate centers between planning CT and CBCT images after bone-based registrations. CTV shape variations of high, intermediate and low risk CTVs were calculated for each patient from variances of interfractional shape variations on each vertex of three-dimensional CTV point distributions, which were manually obtained from CTV contours on the CBCT images. PTV margins were calculated using the setup errors with and without CTV shape variations for each risk CTV. Six treatment plans were retrospectively made by using the PTV margins with and without CTV shape variations for the three risk CTVs of 5 test patients. Furthermore, the treatment plans were applied to CBCT images for investigating the impact of shape variations on PTV margins. Results: The percentages of population to cover with the PTV, which satisfies the CTV D98 of 95%, with and without the shape variations were 89.7% and 74.4% for high risk, 89.7% and 76.9% for intermediate risk, 84.6% and 76.9% for low risk, respectively. Conclusion: PTV margins taking into account CTV shape variation provide significant improvement of applicable percentage of population (P < 0.05). This study suggested that CTV shape variation should be taken consideration into determination of the PTV margins.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2013

Training and Research on Probabilistic Hydro-Thermo-Mechanical Modeling of Carbon Dioxide Geological Sequestration in Fractured Porous Rocks

Colorado School of Mines conducted research and training in the development and validation of an advanced CO 2 GS (Geological Sequestration) probabilistic simulation and risk assessment model. CO 2 GS simulation and risk assessment is used to develop advanced numerical simulation models of the subsurface to forecast CO 2 behavior and transport; optimize site operational practices; ensure site safety; and refine site monitoring, verification, and accounting efforts. As simulation models are refined with new data, the uncertainty surrounding the identified risks decrease, thereby providing more accurate risk assessment. The models considered the full coupling of multiple physical processes (geomechanical and fluid flow) and describe the effects of stochastic hydro-mechanical (H-M) parameters on the modeling of CO 2 flow and transport in fractured porous rocks. Graduate students were involved in the development and validation of the model that can be used to predict the fate, movement, and storage of CO 2 in subsurface formations, and to evaluate the risk of potential leakage to the atmosphere and underground aquifers. The main major contributions from the project include the development of: 1) an improved procedure to rigorously couple the simulations of hydro-thermo-mechanical (H-M) processes involved in CO 2 GS; 2) models for the hydro-mechanical behavior of fractured porous rocks with random fracture patterns; and 3) probabilistic methods to account for the effects of stochastic fluid flow and geomechanical properties on flow, transport, storage and leakage associated with CO 2 GS. The research project provided the means to educate and train graduate students in the science and technology of CO 2 GS, with a focus on geologic storage. Specifically, the training included the investigation of an advanced CO 2 GS simulation and risk assessment model that can be used to predict the fate, movement, and storage of CO 2 in underground formations, and the evaluation of the risk of potential CO 2 leakage to the atmosphere and underground aquifers.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2023

Using iterative random forest to find geospatial environmental and Sociodemographic predictors of suicide attempts

Despite a recent global decrease in suicide rates, death by suicide has increased in the United States. It is therefore imperative to identify the risk factors associated with suicide attempts to combat this growing epidemic. In this study, we aim to identify potential risk factors of suicide attempt using geospatial features in an Artificial intelligence framework. We use iterative Random Forest, an explainable artificial intelligence method, to predict suicide attempts using data from the Million Veteran Program. This cohort incorporated 405,540 patients with 391,409 controls and 14,131 attempts. Our predictive model incorporates multiple climatic features at ZIP-code-level geospatial resolution. We additionally consider demographic features from the American Community Survey as well as the number of firearms and alcohol vendors per 10,000 people to assess the contributions of proximal environment, access to means, and restraint decrease to suicide attempts. In total 1,784 features were included in the predictive model. Our results show that geographic areas with higher concentrations of married males living with spouses are predictive of lower rates of suicide attempts, whereas geographic areas where males are more likely to live alone and to rent housing are predictive of higher rates of suicide attempts. We also identified climatic features that were associated with suicide attempt risk by age group. Additionally, we observed that firearms and alcohol vendors were associated with increased risk for suicide attempts irrespective of the age group examined, but that their effects were small in comparison to the top features. Taken together, our findings highlight the importance of social determinants and environmental factors in understanding suicide risk among veterans.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2021

Self-Supervised Anomaly Detection via Neural Autoregressive Flows with Active Learning

Many self-supervised methods have been proposed with the target of image anomaly detection. These methods often rely on the paradigm of data augmentation with predefined transformations such as flipping, cropping, and rotations. However, it is not straightforward to apply these techniques for non-image data, such as time series or tabular data, while the performance of the existing deep approaches has been under our expectation on tasks beyond images. In this work, we propose a novel active learning (AL) scheme that relied on neural autoregressive flows (NAF) for self-supervised anomaly detection, specifically on small-scale data. Unlike other generative models such as GANs or VAEs, flow-based models allow to explicitly learn the probability density and thus can assign accurate likelihoods to normal data which makes it usable to detect anomalies. The proposed NAF-AL method is achieved by efficiently generating random samples from latent space and transforming them into feature space along with likelihoods via invertible mapping. The samples with lower likelihoods are selected and further checked by outlier detection using Mahalanobis distance. The augmented samples incorporating with normal samples are used for training a better detector so as to approach decision boundaries. Compared with random transformations, NAF-AL can be interpreted as a likelihood-oriented data augmentation that is more efficient and robust. Extensive experiments show that our approach outperforms existing baselines on multiple time series and tabular datasets, and a real-world application in advanced manufacturing, with significant improvement on anomaly detection accuracy and robustness over the state-of-the-art.

Zhang, Jiaxin↗
DOE OSTI2018

A Galerkin method with two-dimensional Haar basis functions for the computation of the Karhunen–Loève expansion

We study the numerical approximation of a homogeneous Fredholm integral equation of second kind associated with the Karhunen–Loève expansion of Gaussian random fields. We develop, validate, and discuss an algorithm based on the Galerkin method with two-dimensional Haar wavelets as basis functions. The shape functions are constructed from the orthogonal decomposition of tensor-product spaces of one-dimensional Haar functions, and a recursive algorithm is employed to compute the matrix of the discrete eigenvalue system without the explicit calculation of integrals, allowing the implementation of a fast and efficient algorithm that provides considerable reduction in CPU time, when compared with classical Galerkin methods. Numerical experiments confirm the convergence rate of the method and assess the approximation error and the sparsity of the eigenvalue system when the wavelet expansion is truncated. We illustrate the numerical solution of a diffusion problem with random input data with the present method. In this problem, accuracy was retained after dropping the coefficients below a threshold value that was numerically determined. A similar method with scaling functions rather than wavelet functions does not need a discrete wavelet transform and leads to eigenvalue systems with better conditioning but lower sparsity.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

A Pattern-Recognition-Based Ensemble Data Imputation Framework for Sensors from Building Energy Systems

Building operation data are important for monitoring, analysis, modeling, and control of building energy systems. However, missing data is one of the major data quality issues, making data imputation techniques become increasingly important. There are two key research gaps for missing sensor data imputation in buildings: the lack of customized and automated imputation methodology, and the difficulty of the validation of data imputation methods. In this paper, a framework is developed to address these two gaps. First, a validation data generation module is developed based on pattern recognition to create a validation dataset to quantify the performance of data imputation methods. Second, a pool of data imputation methods is tested under the validation dataset to find an optimal single imputation method for each sensor, which is termed as an ensemble method. The method can reflect the specific mechanism and randomness of missing data from each sensor. The effectiveness of the framework is demonstrated by 18 sensors from a real campus building. The overall accuracy of data imputation for those sensors improves by 18.2% on average compared with the best single data imputation method.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2025

Quantification and prediction of solidification textures under additive manufacturing conditions

Crystallographic textures are a major determinant of the macroscale anisotropic properties of polycrystalline metallic alloys produced in a wide range of additive manufacturing (AM) processes. Here, we introduce a statistical method that can accurately quantify the degree of orientational order of textures despite the large random fluctuations in the orientation of individual grains inherent in AM processes. The method, demonstrated for laser and resolidification of AlSi thin films, extends Z-scoring to a dynamical regime to assess the statistical significance of observed textures compared to randomly generated ones at different stages of solidification. We further show that, combined with phase-field modeling, this method can be used to infer fundamental anisotropic properties of the solid-liquid interface that are essential for texture prediction, and are compared here to the results of atomistic simulations. In addition, phase-field modeling reveals that, even at rapid AM solidification rates, the observed 〈110〉-dominated textures in the AlSi thin films are controlled predominantly by the anisotropy of the interface free-energy and sheds light on the physical mechanism of grain competition. These results significantly enhance both the existing tools for the quantification and prediction of AM crystallographic textures and our basic understanding of their formation.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Data-Efficient Methods for Determining Flory–Huggins χ Parameters in Multicomponent Polymer Formulations

Polymer formulations are essential in diverse applications including personal care products, coatings, paints, adhesives, and plastic materials. Designing these formulations requires navigating large, complex design spaces, where phase and self-assembly behavior critically impact performance. The Flory–Huggins χ parameter, which quantifies segmental miscibility, is widely used to parametrize the excess free energy of mixing in formulation models. In this work, we introduce two data-efficient, top-down methods for estimating χ parameters using the Random Phase Approximation (RPA): (i) Boundary Nonlinear Regression (Boundary-NLR), which fits theoretical spinodal boundaries to experimental phase boundaries, and (ii) Surrogate Model Inverse Parameter Estimation (SMIPE), which uses a Gaussian Process Classifier to fit sparse phase maps via a surrogate model. Both methods allow rapid parametrization of polymer field-theoretic models without the need for additional experiments. We evaluate these approaches on data sets involving polymer–solvent–nonsolvent ternary mixtures and block copolymer–solvent systems, demonstrating their robustness to experimental noise and their relevance for real-world formulation design.

copolymers↗
DOE OSTI2017

Developing reservoir monthly inflow forecasts using artificial intelligence and climate phenomenon information

Reservoirs are fundamental human–built infrastructures that collect, store, and deliver fresh surface water in a timely manner for many purposes. Efficient reservoir operation requires policy makers and operators to understand how reservoir inflows are changing under different hydrological and climatic conditions to enable forecast–informed operations. Over the last decade, the uses of Artificial Intelligence and Data Mining [AI & DM] techniques in assisting reservoir streamflow subseasonal to seasonal forecasts have been increasing. In this study, Random Forest [RF), Artificial Neural Network (ANN), and Support Vector Regression (SVR) are employed and compared with respect to their capabilities for predicting 1 month–ahead reservoir inflows for two headwater reservoirs in USA and China. Both current and lagged hydrological information and 17 known climate phenomenon indices, i.e., PDO and ENSO, etc., are selected as predictors for simulating reservoir inflows. Here, the results show three methods are capable of providing monthly reservoir inflows with satisfactory statistics; the results obtained by Random Forest have the best statistical performances compared with the other two methods; another advantage of Random Forest algorithm is its capability of interpreting raw model inputs; climate phenomenon indices are useful in assisting monthly or seasonal forecasts of reservoir inflow; and different climate conditions are autocorrelated with up to several months, and the climatic information and their lags are cross correlated with local hydrological conditions in our case studies.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2020

A mass-transfer particle-tracking method for simulating transport with discontinuous diffusion coefficients

The problem of a spatially discontinuous diffusion coefficient (D(x)) is one that may be encountered in hydrogeologic systems due to natural geological features or as a consequence of numerical discretization of flow properties. To date, mass-transfer particle-tracking (MTPT) methods, a family of Lagrangian methods in which diffusion is jointly simulated by random walk and diffusive mass transfers, have been unable to solve this problem. This manuscript presents a new mass-transfer (MT) algorithm that enables MTPT methods to accurately solve the problem of discontinuous D(x). To achieve this, we derive a semi-analytical solution to the discontinuous D(x) problem by employing a predictor-corrector approach, and we use this semi-analytical solution as the weighting function in a reformulated MT algorithm. Furthermore, this semi-analytical solution is generalized for cases with multiple 1D interfaces as well as for 2D cases, including a 2 × 2 tiling of 4 subdomains that corresponds to a numerically-generated diffusion field. The solutions generated by this new mass-transfer algorithm closely agree with an analytical 1D solution or, in more complicated cases, trusted numerical results, demonstrating the success of our proposed approach.

42 ENGINEERING↗
DOE OSTI2016

Self-consistent approach to beta decay and delayed neutron emission

A brief overview of the recent self-consistent studies of nuclear beta decay is given including the relativistic quasi-particle random-phase approximation or QRPA and Finite Amplitude Method. The results of our self-consistent continuum QRPA model based on the density functional description of the ground states are presented. They are in a good agreement with the recent experimental beta-decay half-lives and delayed neutron emission branchings for the nuclei approaching (and beyond) the neutron closed shells N = 50 near {sup 78}Ni and N = 82 near {sup 132}Sn. A comparison with the recent calculations from relativistic QRPA model, Finite Amplitude Method and semi-microscopic finite-range droplet model is performed. An importance of the quasi-particle phonon coupling is stressed for the description of the beta decay and delayed multi-neutron emission rates. A strategy of extending our approach to the deformed nuclei and the open problems are discussed.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2017

A Facile Method to Fine‐Tune Polymer Aggregation Properties and Blend Morphology of Polymer Solar Cells Using Donor Polymers with Randomly Distributed Alkyl Chains

Abstract The device performance of polymer solar cells (PSCs) is strongly dependent on the blend morphology. One of the strategies for improving PSC performance is side‐chain engineering, which plays an important role in controlling the aggregation properties of the polymers and thus the domain crystallinity/purity of the donor–acceptor blends. In particular, for a family of high‐performance donor polymers with strong temperature‐dependent aggregation properties, the device performances are very sensitive to the size of alkyl chains, and the best device performance can only be achieved with an optimized odd‐numbered alkyl chain. However, the synthetic route of odd‐numbered alkyl chains is costly and complicated, which makes it difficult for large‐scale synthesis. Here, this study presents a facile method to optimize the aggregation properties and blend morphology by employing donor polymers with a mixture of two even‐numbered, randomly distributed alkyl chains. In a model polymer system, this study suggests that the structural and electronic properties of the random polymers comprising a mixture of 2‐octyldodecyl and 2‐decyltetradecyl alkyl chains can be systematically tuned by varying the mixing ratio, and a high power conversion efficiency (11.1%) can be achieved. This approach promotes the scalability of donor polymers and thus facilitates the commercialization of PSCs.

Yao, Huatong↗
DOE OSTI2024

Static Subspace Approximation for Random Phase Approximation Correlation Energies: Implementation and Performance

Developing theoretical understanding of complex reactions and processes at interfaces requires using methods that go beyond semilocal density functional theory to accurately describe the interactions between solvent, reactants and substrates. Methods based on many-body perturbation theory, such as the random phase approximation (RPA), have previously been limited due to their computational complexity. However, this is now a surmountable barrier due to the advances in computational power available, in particular through modern GPU-based supercomputers. In this work, we describe the implementation of RPA calculations within BerkeleyGW and show its favorable computational performance on large complex systems relevant for catalysis and electrochemistry applications. Our implementation builds off of the static subspace approximation which, by employing a compressed representation of the frequency dependent polarizability, enables the evaluation of the RPA correlation energy with significant acceleration and systematically controllable accuracy. We find that the computational cost of calculating the RPA correlation energy scales only linearly with system size for systems containing up to 50 thousand bands, and is expected to scale quadratically thereafter. We also show excellent strong scaling results across several supercomputers, demonstrating the performance and portability of this implementation.

algorithmic development↗
DOE OSTI2022

Ch3MS-RF: a random forest model for chemical characterization and improved quantification of unidentified atmospheric organics detected by chromatography–mass spectrometry techniques

Abstract. The chemical composition of ambient organic aerosols plays a critical role in driving their climate and health-relevant properties and holds important clues to the sources and formation mechanisms of secondary aerosol material. In most ambient atmospheric environments, this composition remains incompletely characterized, with the number of identifiable species consistently outnumbered by those that have no mass spectral matches in the literature or the National Institute of Standards and Technology/National Institutes of Health/Environmental Protection Agency (NIST/NIH/EPA) mass spectral databases, making them nearly impossible to definitively identify. This creates significant challenges in utilizing the full analytical capabilities of techniques which separate and generate spectra for complex environmental samples. In this work, we develop the use of machine learning techniques to quantify and characterize novel, or unidentifiable, organic material. This work introduces Ch3MS-RF (Chemical Characterization by Chromatography–Mass Spectrometry Random Forest Modeling), an open-source, R-based software tool, for efficient machine-learning-enabled characterization of compounds separated in chromatography–mass spectrometry applications but not identifiable by comparison to mass spectral databases. A random forest model is trained and tested on a known 130 component representative external standard to predict the response factors of novel environmental organics based on position in volatility–polarity space and mass spectrum, enabling the reproducible, efficient, and optimized quantification of novel environmental species. Quantification accuracy on a reserved 20 % test set randomly split from the external standard compound list indicates that random forest modeling significantly outperforms the commonly used methods in both precision and accuracy, with a median response factor percent error of −2 %, for modeled response factors, compared to > 15 %, for typically used proxy assignment-based methods. Chemical properties modeling, evaluated on the same reserved 20 % test set and an extrapolation set of species identified in ambient organic aerosol samples collected in the Amazon rainforest, also demonstrate robust performance. Extrapolation set property prediction mean absolute errors for carbon number, oxygen to carbon ratio (O : C), average carbon oxidation state (OSc‾), and vapor pressure are 1.8, 0.15, 0.25, and 1.0 (log(atm)), respectively. Extrapolation set out-of-sample R2 for all properties modeled are above 0.75, with the exception of vapor pressure. While predictive performance for vapor pressure is less robust compared to the other chemical properties modeled, random-forest-based modeling was significantly more accurate than other commonly used methods of vapor pressure prediction, decreasing the mean vapor pressure prediction error to 0.24 (log(atm)) from 0.55 (log(atm)) (chromatography-based vapor pressure prediction) and 1.2 (log(atm)) (chemical formula-based vapor pressure prediction). The random forest model significantly advances an untargeted analysis of the full scope of chemical speciation yielded by two-dimensional gas chromatography (GCxGC-MS) techniques and can be applied to gas chromatography coupled with electron ionization mass spectrometry (GC-MS) as well. It enables the accurate estimation of key chemical properties commonly utilized in the atmospheric chemistry community, which may be used to more efficiently identify important tracers for further individual analysis and to characterize compound populations uniquely formed under specific ambient conditions.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2012

Randomized Controlled Trial of Forward-Planned Intensity Modulated Radiotherapy for Early Breast Cancer: Interim Results at 2 Years

Purpose: This single-center randomized trial was designed to investigate whether intensity-modulated radiotherapy (IMRT) reduces late toxicity in patients with early-stage breast cancer. Methods and Materials: The standard tangential plans of 1,145 nonselected patients were analyzed. The patients with inhomogeneous plans were randomized to a simple method of forward-planned IMRT or standard radiotherapy (RT). The primary endpoint was serial photographic assessment of breast shrinkage. Results: At 2 years, no significant difference was found in the development of any photographically assessed breast shrinkage between the patients randomized to the interventional or control group (odds ratio, 1.51; 95% confidence interval, 0.83-1.58; p = .41). The patients in the control group were more likely to develop telangiectasia than those in the IMRT group (odds ratio, 1.68; 95% confidence interval 1.13-2.40; p = .009). Poor baseline surgical cosmesis resulted in poor overall cosmesis at 2 years after RT. In patients who had good surgical cosmesis, those randomized to IMRT were less likely to deteriorate to a moderate or poor overall cosmesis than those in the control group (odds ratio, 0.63; 95% confidence interval, 0.39-1.03, p = .061). Conclusions: IMRT can lead to a significant reduction in telangiectasia at comparatively early follow-up of only 2 years after RT completion. An important component of breast induration and shrinkage will actually result from the surgery and not from the RT. Surgical cosmesis is an important determinant of overall cosmesis and could partially mask the longer term benefits of IMRT at this early stage.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2014

Multi-bit quantum random number generation by measuring positions of arrival photons

We report upon the realization of a novel multi-bit optical quantum random number generator by continuously measuring the arrival positions of photon emitted from a LED using MCP-based WSA photon counting imaging detector. A spatial encoding method is proposed to extract multi-bits random number from the position coordinates of each detected photon. The randomness of bits sequence relies on the intrinsic randomness of the quantum physical processes of photonic emission and subsequent photoelectric conversion. A prototype has been built and the random bit generation rate could reach 8 Mbit/s, with random bit generation efficiency of 16 bits per detected photon. FPGA implementation of Huffman coding is proposed to reduce the bias of raw extracted random bits. The random numbers passed all tests for physical random number generator.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2026

Advanced measurement techniques in quantum Monte Carlo: The permutation matrix representation approach

In a typical finite temperature quantum Monte Carlo (QMC) simulation, estimators for simple static observables such as specific heat and magnetization are known. With a great deal of system-specific manual labor, one can sometimes also derive more complicated non-local or even dynamic observable estimators. In contrast, we show that arbitrary static observables can be estimated within the permutation matrix representation (PMR) flavor for any Hamiltonian. We then generalize these results to general imaginary-time correlation functions and non-trivial integrated susceptibilities thereof. Finally, we demonstrate the practical versatility of our method by estimating various non-local, random observables for the transverse-field Ising model on a square lattice and a toy random model.

Permutation matrix representation↗