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At least 73 records · Page 4

LATTE: Los Alamos TravelTime package based on Eikonal equation

This Fortran code focuses on traveltime computation and tomography based on eikonal equation. Specifically, the package provides three major functionalities: (1) forward modeling of traveltime from single-point or ensemble source based on factorized eikonal equation, (2) adjoint-state first-arrival traveltime tomography based on picked first arrival traveltime using steepest descent, conjugate gradient, or limited-memory BFGS inversion scheme, and (3) adjoint-state joint transmission-reflection tomography based on picked first-arrival and reflection traveltimes. The package applies to forward modeling and tomography based on traveltime in 2D and 3D isotropic regular-grid models. We name this package LATTE – Los Alamos TravelTime package based on Eikonal equation. * The code is for accompanying a journal paper under preparation. The paper will be submitted via LA-UR separately later.

Gao, Kai↗

Pragmatic Uncertainty Quantification and Propagation in Inverse Estimation of Structural Dynamics Parameters given Material Property Uncertainties and Limited Sensor Data

In this report we demonstrate some relatively simple and inexpensive methods to effectively account for various sources of epistemic lack-of-knowledge type uncertainty in inverse problems. The demonstration problem involves inverse estimation of six parameters of a bolted joint that attaches a kettlebell shaped object to a thick plate. The parameters are efficiently inverted in a modal-based model calibration using gradient-based optimization. Two material properties of the kettlebell are treated as uncertain to within given epistemic uncertainty bounds. We apply and test interval and sparse-sample probabilistic approaches to account for uncertainty in the estimated parameters (and various scalar functionals of the parameters as generic quantities of interest, QOIs) due to uncertainties in the material properties. We also investigate the error effects of limited numbers of vibration sensors (accelerometers) on the kettlebell and plate, and therefore abbreviated excitation/response information in the parameter inversions. We propose and demonstrate a Leave-K-Sensors-Out “cross-prediction” UQ approach to estimate related uncertainties on the parameters and QOI functionals. We indicate how uncertainties from material properties and limited sensors are treated in a combined manner. The economical combined UQ approach involves just three to five samples (i.e. three to five inverse simulations), with no added complication or error/uncertainty from use of surrogate models for affordability. Finally, we describe a related economical UQ approach for handling potential parameter solution non-uniqueness and numerical optimization related precision uncertainties in the estimated parameter values. Indicated further research is identified.

36 MATERIALS SCIENCE↗

Gas hydrate quantification in Walker Ridge block 313, Gulf of Mexico, from full-waveform inversion of ocean-bottom seismic data

The Gulf of Mexico (GOM) Joint Industry Project Leg 2 logging-while-drilling data in Walker Ridge lease block 313 (WR313) in the GOM detected gas hydrate in coarse- and fine-grained sediments at sites WR313-G and WR313-H. The coarse-grained units are thin ([Formula: see text]) and highly saturated, whereas the fine-grained unit is thick (approximately 200 m) with low saturation and fracture-filling gas hydrate. Unlike its coarse-grained counterpart, the seismic character of the fine-grained unit does not clearly indicate the presence of gas hydrate, which would likely have remained undiscovered in the absence of drilling. In this paper, through frequency-domain acoustic full-waveform inversion (FWI) of ocean-bottom seismometer data along a 2D multichannel seismic transect near sites WR313-G and WR313-H, we detect and quantify gas hydrate in the fine-grained unit. Key results are as follows: First, the base of the gas hydrate stability zone, which is not obvious in the reflection profile, can be discerned in the FWI results. Second, the gas hydrate in the fine-grained unit is mainly confined to the area between two sets of opposite-dipping normal faults implying that the fault architecture may be partially responsible for this gas hydrate accumulation and distribution.

Geochemistry & Geophysics↗

A Multi-Scale Computational Platform for Predictive Modeling of Corrosion in Al-Steel Joints (Final Report)

The research team proposed to develop innovative multi-scale models to predict corrosion and the resulting mechanical performances in aluminum-steel joints. The methods of joining considered are resistance spot welding, self-piercing riveting, and rivet-welding, all suitable for mass production applications. The multi-scale models integrate high throughput first-principle calculations based on density functional theory (DFT), high throughput calculation of phase diagrams (CALPHAD) modeling, and finite element method (FEM) simulations. These models are to be validated through laboratory experiments. Furthermore, the models are available as open source so as to enable scientists and engineers in the community to adapt and contribute to the development and application. The approaches rely on the research team’s extensive experience on the prediction of properties of individual phases at finite temperatures and variable compositions through DFT calculations, and our broad expertise on dissimilar material joining and their corrosion. The proposed computational framework enables high throughput computations for improved predictions of corrosion and the associated mechanical performance in dissimilar material joints, resulting in significant reduction in computational time needed by the current state-of-the-art methods. With the participation of researchers from three universities, an auto manufacturer, two manufacturing technology/equipment suppliers, and a software developer/vendor, the interdisciplinary research team applies the technical development on both phase-based modeling and laboratory experiments into the automobile body joining processes for validation and technology demonstration. The global cost of corrosion was estimated at about 3.4% of the global GDP in 2013. By using available corrosion control practices, it is estimated a saving between 15-35% of the cost of corrosion. In the U.S., more than $276 billion is spent repairing corrosion damage. Prediction of the corrosion and its impact on performance of the dissimilar material joints is critical for reducing the massive number of the current corrosion-based recalls for automobiles. Thus, the project goal is to develop models to enable predictive maintenance and end-of-life planning of multi-metal joints with risk of corrosion under different conditions such as exposure to high temperatures in summer and salt solutions in winter, quantified through its pH. An academia-industry consortium led by the University of Michigan and including Pennsylvania State University, University of Illinois Urbana-Champaign, University of Georgia, General Motors Company, Livermore Software Technology Corporation, and Optimal Process Technologies, LLC. created multi-scale models for prediction of corrosion in aluminum-steel joint structures such of them used in vehicle subassemblies – chassis and transmission systems. Starting from the first principle calculations, the team developed mathematical and data-driven models to predict the metallic components, which are formed during joining of two metals, for example aluminum and steel - a lightweight multilateral system which is currently used in more than 60% car bodies. These models were used for simulating chemical reactions that are happening when the joining metallic components are exposed to high temperatures and different pH values. The team was able to predict how the corrosion installs on the metallic components and how they lead to a sudden failure of components in cars. Newly developed machine learning algorithms combining Science, Technology, Engineering and Math disciplines, advanced finite element simulation and experimental validations have been integrated in a platform for prediction of the corrosion evolution and prediction the failure of joints under mechanical loadings and fatigue. Moreover, based on machine learning and inverse analysis, the team proposed solutions for designing new metallic alloys less susceptible to corrosion when joining multi-material assembles. An average of 4% error compared with experiments was achieved for the most common joints that are used in vehicle subassemblies.

36 MATERIALS SCIENCE↗

The Collaborative Seismic Earth Model: Generation 2

Geological interpretations, earthquake source inversions and ground motion modeling, among other applications, require models that jointly resolve crustal and mantle structure. With the second generation of the Collaborative Seismic Earth Model (CSEM2), we present a global multi-resolution tomographic Earth model that serves this purpose. The model evolves through successive regional- and global-scale refinements. While the first generation aggregated regional models, with this study, we ensure consistency between all individual submodels, resulting in a model that accurately explains wave propagation across scales. Recent regional tomographic models were incorporated, comprising continental-scale inversions for Asia and Africa, as well as regional inversions for the Western US, Central Andes, Iran, and Southeast Asia. Across all regional refinements, over 793,000 source-receiver pairs contributed. Moreover, the long-wavelength Earth model (LOWE) introduces large-scale structures outside of pre-existing local refinements. A full-waveform inversion for global anisotropic P-and S-wave speed structure over a total of 194 iterations with a minimum period of 50 s on a large data set of 1 hr of waveform data from 2,423 earthquakes and over 6 million source-receiver pairs ensures that regional updates in the crust and uppermost mantle translate into updates of deeper, global-scale structure. To test the performance of CSEM2, we evaluate waveform fits between observed and synthetic seismograms at 50 s for an independent data set on the global scale, and on the regional scale for lower periods. We accurately simulate waveforms within and across regional refinements, maintaining the original resolution of the submodels embedded in the global framework.

58 GEOSCIENCES↗

On the Strong Convergence of Forward-Backward Splitting in Reconstructing Jointly Sparse Signals

We consider the problem of reconstructing an infinite set of sparse, finite-dimensional vectors, that share a common sparsity pattern, from incomplete measurements. This is in contrast to the work (Daubechies et al., Pure Appl. Math. 57(11), 1413–1457, 2004), where the single vector signal can be infinite-dimensional, and (Fornasier and Rauhut, SIAM J. Numer. Anal. 46(2), 577613, 2008), which extends the aforementioned work to the joint sparse recovery of finite number of infinite-dimensional vectors. In our case, to take account of the joint sparsity and promote the coupling of nonvanishing components, we employ a convex relaxation approach with mixed norm penalty ℓ 2,1 . This paper discusses the computation of the solutions of linear inverse problems with such relaxation by a forward-backward splitting algorithm. However, since the solution matrix possesses infinitely many columns, the arguments of Daubechies et al. (Pure Appl. Math. 57(11), 1413–1457, 2004) no longer apply. As such, we establish new strong convergence results for the algorithm, in particular when the set of jointly sparse vectors is infinite.

97 MATHEMATICS AND COMPUTING↗

GLAD-M35: a joint P and S global tomographic model with uncertainty quantification

We present our third and final generation joint P and S global adjoint tomography (GLAD) model, GLAD-M35, and quantify its uncertainty based on a low-rank approximation of the inverse Hessian. Starting from our second-generation model, GLAD-M25, we added 680 new earthquakes to the database for a total of 2160 events. New P-wave categories are included to compensate for the imbalance between P- and S-wave measurements, and we enhanced the window selection algorithm to include more major-arc phases, providing better constraints on the structure of the deep mantle and more than doubling the number of measurement windows to 40 million. Two stages of a Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi-Newton inversion were performed, each comprising five iterations. With this BFGS update history, we determine the model’s standard deviation and resolution length through randomized singular value decomposition.

58 GEOSCIENCES↗

Numerical evaluation of photosensitive tracers as a strategy for separating surface and subsurface transient storage in streams

For this work, we numerically evaluated photosensitive tracers as a potential strategy for separating the effects of surface and hyporheic storage zones (SSZs and HSZs, respectively) on stream corridor transport. Correctly separating HSZ and SSZ effects is critical to estimating the hydro-biogeochemical function of a stream because HSZs and SSZs expose solutes to significantly different biogeochemical conditions, like sunlight exposure, microbial processes, and oxygen availability. Our numerical experiments used a multiscale river-corridor transport model implemented in the ATS code, which accommodates multiple storage zones with distinct travel time distributions and biogeochemical reactions. For parameter inferences, we used Bayesian inverse modeling. We found that breakthrough curves for photo-decaying tracers from day and night injection can delineate surface and hyporheic transient storage contributions, but only when interpreted jointly through a two-storage zone model. Numerical experiments that used only daytime injection or interpreted breakthrough curves with a single storage zone model yielded good fit to breakthrough curves, but parameter estimates were biased and controlling processes misattributed, examples of good model fits for the wrong reasons. Using those biased parameter estimates in reactive transport simulations resulted in significantly different projections of denitrification, which underscores the potential for stream function to be mischaracterized if tracer tests are interpreted through an inappropriately simplified model for transient storage. More generally, this study highlights the role of modeling in evaluating the experimental design and identifying the potential of system mischaracterization and its implications.

54 ENVIRONMENTAL SCIENCES↗

Machine learning inversion from small-angle scattering for charged polymers

We develop Monte Carlo simulations for uniformly charged polymers and a machine learning algorithm to interpret the intra-polymer structure factor of the charged polymer system, which can be obtained from small-angle scattering experiments. The polymer is modeled as a chain of fixed-length bonds, where the connected bonds are subject to bending energy, and there is also a screened Coulomb potential for charge interaction between all joints. The bending energy is determined by the intrinsic bending stiffness, and the charge interaction depends on the interaction strength and screening length. All three contribute to the stiffness of the polymer chain and lead to longer and larger polymer conformations. The screening length also introduces a second length scale for the polymer besides the bending persistence length. To obtain the inverse mapping from the structure factor to these polymer conformation and energy-related parameters, we generate a large data set of structure factors by running simulations for a wide range of polymer energy parameters. We use principal component analysis to investigate the intra-polymer structure factors and determine the feasibility of the inversion using the nearest neighbor distance. We employ Gaussian process regression to achieve the inverse mapping and extract the characteristic parameters of polymers from the structure factor with low relative error.

36 MATERIALS SCIENCE↗

Characterizing the effects of drive asymmetries, component offsets, and joint gaps in double shell capsule implosions

This work provides a numerical study of how double shell capsule deformations caused by drive asymmetries and fabrication imperfections affect implosion symmetry and neutron yield. Hydrodynamics simulations are performed in two dimensions and focus on low-mode deformations that are caused by corresponding asymmetries in the Hohlraum drive, component offsets, and ablator joint gaps. By providing a parameter study of these features, our goal is to understand the dominant sources for inner shell deformation and yield degradation. The discussed capsules are composed of an aluminum ablator with a chromium inner shell. The latter encloses a carbon-deuterium foam ball that serves as fuel. We find that for clean capsules, even-numbered low-mode asymmetries in the drive are imprinted on the ablator and smoothly transferred to the inner shell during shell collision. The resulting deformation of the inner shell is more pronounced with larger fuel radius, while the yield is inversely proportional to the amplitude of the drive asymmetry and varies by factors ≤4 in comparison with clean simulations. Capsule component offsets in the vertical direction and ablator thickness nonuniformity result in p1-type deformations of the imploding inner shell. Finally, joint gaps have the largest effect in deforming the ablator and inner shell and degrading yield. While small gap widths (1 μm) result in prolate inner shells, larger gap widths (4 μm) cause an oblate deformation. More importantly, capsules with a small outer gap (1 μm) experience a dramatic drop in yield, typically <3% of a clean simulation.

Goodarzi, S. (ORCID:000000018556489X)↗

Development of a Comparison Framework for Evaluating Environmental Contours of Extreme Sea States

Environmental contours of extreme sea states are often utilized for the purposes of reliability-based offshore design. Many methods have been proposed to estimate environmental contours of extreme sea states, including, but not limited to, the traditional inverse first-order reliability method (I-FORM) and subsequent modifications, copula methods, and Monte Carlo methods. These methods differ in terms of both the methodology selected for defining the joint distribution of sea state parameters and in the method used to construct the environmental contour from the joint distribution. It is often difficult to compare the results of proposed methods to determine which method should be used for a particular application or geographical region. The comparison of the predictions from various contour methods at a single site and across many sites is important to making environmental contours of extreme sea states useful in practice. The goal of this paper is to develop a comparison framework for evaluating methods for developing environmental contours of extreme sea states. This paper develops generalized metrics for comparing the performance of contour methods to one another across a collection of study sites, and applies these metrics and methods to develop conclusions about trends in the wave resource across geographic locations, as demonstrated for a pilot dataset. These proposed metrics and methods are intended to judge the environmental contours themselves relative to other contour methods, and are thus agnostic to a specific device, structure, or field of application. The metrics developed and applied in this paper include measures of predictive accuracy, physical validity, and aggregated temporal performance that can be used to both assess contour methods and provide recommendations for the use of certain methods in various geographical regions. The application and aggregation of the metrics proposed in this paper outline a comparison framework for environmental contour methods that can be applied to support design analysis workflows for offshore structures. This comparison framework could be extended in future work to include additional metrics of interest, potentially including those to address issues pertinent to a specific application area or analysis discipline, such as metrics related to structural response across contour methods or additional physics-based metrics based on wave dynamics.

54 ENVIRONMENTAL SCIENCES↗

Topological band inversion in HgTe(001): Surface and bulk signatures from photoemission

HgTe is a versatile topological material and has enabled the realization of a variety of topological states, including two- and three-dimensional (3D) topological insulators and topological semimetals. Nevertheless, a quantitative understanding of its electronic structure remains challenging, in particular, due to coupling of the Te $5p$-derived valence electrons to Hg $5d$ core states at shallow binding energy. Here, we present a joint experimental and theoretical study of the electronic structure in strained HgTe(001) films in the 3D topological-insulator regime, based on angle-resolved photoelectron spectroscopy and density functional theory. The results establish detailed agreement in terms of: (i) electronic band dispersions and orbital symmetries, (ii) surface and bulk contributions to the electronic structure, and (iii) the importance of Hg $5d$ states in the valence-band formation. Supported by theory, our experiments directly image the paradigmatic band inversion in HgTe, underlying its nontrivial band topology.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Improving Subsurface Stress Characterization for Carbon Dioxide Storage Projects by Incorporating Machine Learning Techniques

The overall objective of this project is to develop a framework for reliable characterization and prediction of the state of stress in the overburden and underburden (including the basement) in CO 2 storage reservoirs using machine learning and integrated geomechanics and geophysical methods. Specifically, we propose to develop workflow encompassing of technologies and/or methods to predict stress and pressure changes due to CO 2 injection in an active tertiary recovery site and their impacts on subtle fault activation, fractures and occurrence of microseismic events and compare responses to field observations. In this project, we anticipate using dataset from the Farnsworth field Unit (FWU) which is operated by Purdure Petroleum. A novel elastic-waveform VSP inversion technique will be used to estimate high-resolution spatial and temporal changes of elastic moduli in CO 2 storage reservoirs, which will be combined with velocity-stress relationship derived from laboratory tests to obtain subsurface pressure and stress. Clustered microseismic data will be jointly inverted for improved focal mechanisms. Least-squares reverse-time migration of microseismic waveform data will be performed to directly image fracture/fault zones. Additionally, a deep neural network machine learning technique with convolutional and recurrent layers will be used for learning the spectro-temporal structures in microseismic waveforms. The results of this geotechnical data analysis will be integrated to develop a high-resolution 3D mechanical earth model extending from the overburden sealing formations to the underburden including the basement. Mechanical properties will be derived through integration of mechanical logs, tests, available results from chemo-mechanical laboratory tests, and elastic inversion of seismic data using a combination of Bayesian and stochastic methods as well as machine learning technique. Failure features (faults/fractures) will be represented and/or modeled based on seismic and core data analysis. A transient hydrodynamic-geomechanical model will be developed through coupling with the calibrated FWU reservoir simulation model. The full physics coupled model will be used to train a reduced order proxy model using machine learning algorithm for estimating stress which will then be used with appropriate constitutive relationships and forward seismological models to simulate pressure changes and induced microseismicity. An advanced optimization framework will be developed to perform a history match to minimize error between field observations and simulated. The history matched proxy model will be verified against the full-physics equivalent. The field observations that will be used in the coupled model calibration process include pressure/stress inverted from VSP, moment magnitude from microseismic analysis, real time downhole pressure measurements, production and injection data. Parameter sensitivity and uncertainty analysis will be performed to characterize the impact of model parameter uncertainty on stress estimates. The proposed project will have significant impact on future field implementation of the proposed technology. Because the project field site is an ongoing CO 2 EOR development, the value of the new technology will be demonstrated in an operational context and evaluated as a viable risk mitigation strategy. Cost/benefit will be evaluated together with the various commercial incentives for CO 2 sequestration available to oil and gas operators. The extensive available dataset and ongoing data acquisition under the SWP Phase III work plan provides flexibility for investigation of multiple approaches and reduces technical risk.

58 GEOSCIENCES↗

Emerging low-cloud feedback and adjustment in global satellite observations

From mid-2003 to mid-2024, a global decrease in low-cloud amount enhanced the absorption of solar radiation by 0.22±0.07 W m −2 per decade (±1σ range), accelerating the energy imbalance trend during that period (0.44 W m −2 per decade). Through controlling factor analysis, here we show that the low-cloud trend is due to a combination of cloud feedback and adjustments to greenhouse gases and aerosols (respectively 0.09±0.02, 0.05±0.03, and 0.03±0.03 W m −2 per decade), which jointly account for 74 % of the trend. The contribution of natural climate variability is weak but uncertain (0.01±0.08 W m −2 per decade), owing to a poorly constrained trend in boundary-layer inversion strength. Importantly, the observed low-cloud radiative trend lies well within the range of values simulated by contemporary global climate models under conditions close to present day. Any systematic model error in the representation of present-day global energy imbalance trends is thus likely to originate in processes unrelated to low clouds.

Geosciences↗

A New Probability Table Methodology with Covariances on Probability Tables

Through this work, we test a new methodology of evaluating probability tables informed by a combination of physics models and experimental data. This methodology uses generalized least squares to produce a joint mean and covariance for average cross sections and probability tables. This methodology has been tested and validated using total Monte Carlo. Attempts were made to parameterize the probability table using a formulation provided in. Although the normal inverse Gaussian fit was successful for the total cross section probability distribution, the rational-function fit to the capture, inelastic, and fission cross sections was unsuccessful due to time constraints.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

The significant impact of aerosol vertical structure on lower atmosphere stability and its critical role in aerosol–planetary boundary layer (PBL) interactions

The aerosol–planetary boundary layer (PBL) interaction was proposed as an important mechanism to stabilize the atmosphere and exacerbate surface air pollution. Despite the tremendous progress made in understanding this process, its magnitude and significance still have large uncertainties and vary largely with aerosol distribution and meteorological conditions. In this study, we focus on the role of aerosol vertical distribution in thermodynamic stability and PBL development by jointly using micropulse lidar, sun photometer, and radiosonde measurements taken in Beijing. Despite the complexity of aerosol vertical distributions, cloud-free aerosol structures can be largely classified into three types: well-mixed, decreasing with height, and inverse structures. The aerosol–PBL relationship and diurnal cycles of the PBL height and PM 2.5 associated with these different aerosol vertical structures show distinct characteristics. The vertical distribution of aerosol radiative forcing differs drastically among the three types, with strong heating in the lower, middle, and upper PBL, respectively. Such a discrepancy in the heating rate affects the atmospheric buoyancy and stability differently in the three distinct aerosol structures. Absorbing aerosols have a weaker effect of stabilizing the lower atmosphere under the decreasing structure than under the inverse structure. As a result, the aerosol–PBL interaction can be strengthened by the inverse aerosol structure and can be potentially neutralized by the decreasing structure. Moreover, aerosols can both enhance and suppress PBL stability, leading to both positive and negative feedback loops. This study attempts to improve our understanding of the aerosol–PBL interaction, showing the importance of the observational constraint of aerosol vertical distribution for simulating this interaction and consequent feedbacks.

54 ENVIRONMENTAL SCIENCES↗

Machine learning for detection of 3D features using sparse x-ray tomographic reconstruction

In many inertial confinement fusion (ICF) experiments, the neutron yield and other parameters cannot be completely accounted for with one and two dimensional models. This discrepancy suggests that there are three dimensional effects that may be significant. Sources of these effects include defects in the shells and defects in shell interfaces, the fill tube of the capsule, and the joint feature in double shell targets. Due to their ability to penetrate materials, x rays are used to capture the internal structure of objects. Methods such as computational tomography use x-ray radiographs from hundreds of projections, in order to reconstruct a three dimensional model of the object. In experimental environments, such as the National Ignition Facility and Omega-60, the availability of these views is scarce, and in many cases only consists of a single line of sight. Mathematical reconstruction of a 3D object from sparse views is an ill-posed inverse problem. These types of problems are typically solved by utilizing prior information. Neural networks have been used for the task of 3D reconstruction as they are capable of encoding and leveraging this prior information. We utilize half a dozen, different convolutional neural networks to produce different 3D representations of ICF implosions from the experimental data. Deep supervision is utilized to train a neural network to produce high-resolution reconstructions. These representations are used to track 3D features of the capsules, such as the ablator, inner shell, and the joint between shell hemispheres. Machine learning, supplemented by different priors, is a promising method for 3D reconstructions in ICF and x-ray radiography, in general.

Wolfe, Bradley T. (ORCID:0000000268301614)↗

A non-commutative Bayes' theorem

Using a diagrammatic reformulation of Bayes' theorem, we provide a necessary and sufficient condition for the existence of Bayesian inference in the setting of finite-dimensional C* -algebras. In other words, we prove an analogue of Bayes' theorem in the joint classical and quantum context. Our analogue is justified by recent advances in categorical probability theory, which have provided an abstract formulation of the classical Bayes' theorem. In the process, we further develop non-commutative almost everywhere equivalence and illustrate its important role in non-commutative Bayesian inversion. The construction of such Bayesian inverses, when they exist, involves solving a positive semidefinite matrix completion problem for the Choi matrix. This gives a solution to the open problem of constructing Bayesian inversion for completely positive unital maps acting on density matrices that do not have full support. In conclusion, we illustrate how the procedure works for several examples relevant to quantum information theory.

97 MATHEMATICS AND COMPUTING↗