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At least 577 records · Page 32

Role of non-Gaussian quantum fluctuations in neutrino entanglement

The flavor evolution of neutrinos in environments with large neutrino number densities is an open problem at the nexus of astrophysics and neutrino flavor physics. Among the many unanswered questions pertaining to this problem, it remains to be determined whether neutrino-neutrino coherent scattering can give rise to nontrivial quantum entanglement among neutrinos, and whether this can affect the flavor evolution in a meaningful way. To gain further insight into this question, here we study a simple system of two interacting neutrino beams and obtain the exact phase space explored by this system using the Husimi quasiprobability distribution. We observe that the entanglement induced by the coupling leads to strong delocalization in phase-space with largely non-Gaussian quantum fluctuations. The link between the neutrino entanglement and quantum fluctuations is illustrated using the one- and two-neutrino entropy. In addition, we propose an approximate phase-space method to describe the interacting neutrinos problem, where the exact evolution is replaced by a set of independent mean-field evolutions with a statistical sampling of the initial conditions. The phase-space approach provides a simple and accurate method to describe the gross features of the neutrino entanglement problem. Applications are shown using time-independent and time-dependent Hamiltonians in the nonadiabatic regime.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Role of non-gaussian quantum fluctuations in neutrino entanglement

The flavor evolution of neutrinos in environments with large neutrino number densities is an open problem at the nexus of astrophysics and neutrino flavor physics. Among the many unanswered questions pertaining to this problem, it remains to be determined whether neutrino-neutrino coherent scattering can give rise to nontrivial quantum entanglement among neutrinos, and whether this can affect the flavor evolution in a meaningful way. To gain further insight into this question, here we study a simple system of two interacting neutrino beams, and obtain the exact phase-space explored by this system using the Husimi quasi-probability distribution. We observe that the entanglement induced by the coupling leads to strong delocalization in phase-space with largely non-Gaussian quantum fluctuations. The link between the neutrino entanglement and quantum fluctuations is illustrated using the one- and two-neutrino entropy. In addition, we propose an approximate phase-space method to describe the interacting neutrinos problem, where the exact evolution is replaced by a set of independent mean-field evolutions with a statistical sampling of the initial conditions. The phase-space approach provides a simple and accurate method to describe the gross features of the neutrino entanglement problem. Applications are shown using time-independent and time-dependent Hamiltonians in the non-adiabatic regime.

Patwardhan, Amol↗

Application of finite-element-techniques to the interaction of conduction and radiation in an absorbing, scattering and emitting medium

In this paper, the authors demonstrate that a Galerkin finite element method of analysis, utilizing isoparametric elements, offers a viable means of solving continuum thermal radiation problems with conduction in a participating medium. The participating medium was considered to be a gray radiation medium exhibiting isotropic absorption, emission, and scattering characteristics and optical properties that are independent of temperature. The medium was considered to be bounded by infinite parallel opaque, gray surfaces with diffuse emission and reflection characteristics. In solving this problem, a finite element formulation was developed to describe a system in radiative equilibrium. Then the results of this first analysis were linked with a second finite element model which incorporated conduction into the analysis. The results of this study were found to be in good agreement with existing published data. The model offers the following advantageous features: geometric generality, a computational algorithm which is 'convenient' and 'computable', and a functional basis for extension of the radiation model to higher order approximation.

Wu, S. T.↗

An agentic artificially intelligent X-ray scientist

Executing experimental tasks in both normal research laboratories and large-scale scientific facilities often requires extensive human supervision and remains a key challenge on the path to fully autonomous, artificial intelligence (AI)-driven science. Here we demonstrate a large language model-driven agent that autonomously performs X-ray sample alignment on a synchrotron beamline by planning actions, executing instrumental commands, interpreting observations and iterating towards experimental goals. Based on existing large language models with structured tool-use via the model context protocol, our AI X-ray scientist was guided and tested using an in-house-built virtual experimental setup that mirrors a six-circle diffractometer at an operational synchrotron beamline. The agentic workflow developed in the virtual environment was directly deployed on a real beamline, where it correctly identified reference reflections and determined the orientation matrix, an essential first step in any type of single-crystal scattering experiment. Our AI X-ray scientist responded effectively to unexpected experimental conditions, demonstrating adaptive problem-solving and readiness for addressing practical experimental situations. Our study provides a step towards autonomous operation across diverse experimental environments at large-scale scattering facilities.

Chen, Zhantao (ORCID:0000000319543868)↗

Multimodal Nondestructive Dry Cask Basket Structure and Spent Fuel Evaluation

This NEUP-IRP assembled a team from diverse institutions and a diverse set of expertise to examine the problem of assessing the structural health of dry nuclear fuel storage casks, specifically the TN-32 design. With no physical access designed into these storage casks and the high cost and risk associated with opening casks for direct inspection, the technical difficulty of structural assessment of the basket structure and fuel assemblies is high. In response to this challenge, this IRP brought together experts in both traditional and novel NDE methodologies: active acoustics (University of Mississippi), emissions tomography (University of Florida), muon tomography (Oregon State University), and passive-active ultrasonics (University of South Carolina). Further, our industry partners in the Electric Power Research Institute (EPRI) and AREVA provided invaluable insight into technical design details for the cask, practical knowledge of operating parameters at a storage facility, and key relationships that enabled activities unforeseen at the time of the proposed work but were critical in guiding the path of the research. The team decided early in the project that we had a collective goal of laying the technical ground work for practical solutions that could be implemented in the field with reasonable cost, time, and reliability. A fortunate opportunity arose early in this IRP project to obtain access to a new TN-32 cask at Columbiana High Tech. Both acoustics teams were given access for testing concepts in the first year of the project. This ended up providing critical guidance in the design of subsequent laboratory experiments. Specifically, it became clear that the only reliable acoustic pathways to the interior were through the trunnions, base plate, and top ring of the cask. Based on these field tests and discussions about what would be of practical interest, the team as a whole decided a realistic goal was to detect catastrophic failure of a single fuel assembly, or one that was missing. It is clear from this work that the active acoustics / resonance and muon tomography methodologies would be capable of detecting interior structural integrity problems at this level. The other two areas of investigation (emissions tomography and passive/active ultrasonics) would need more testing to determine a successful implementation, but each may also have other applications such as long term monitoring and monitoring during transportation. This IRP project resulted in technical advances with applications that extend beyond the problem at hand, supported the training of 10 graduate students + 1 post doc, 21 publications, and 13 conference presentations. The remainder of this summary highlights significant activities and achievements in each of the methodologies and a brief discussion on how they can be combined to leverage strengths. Active Acoustics The active acoustics team at UM used both linear and non-linear structural resonance methods to infer information about the internal structure of a cask. Significant activities were a set of four field test measurements on a full scale TN-32 cask at the Columbiana Hi Tech facility in Greensboro, NC and the design and fabrication of a 1/6th scale model TN-32 cask for laboratory measurements. A significant set of acoustic data has been acquired on both the full scale and laboratory scale casks. It was determined early in the project that mechanical resonance techniques were not going to be sensitive to small scale (centimeter) cracks and structural defects in the basket structure or fuel assemblies. We were able to ascertain that both the energy and complexity of the resonance spectra were sensitive to “bulk state” of the fuel assemblies – an intact assembly versus a severely degraded assembly. By developing a novel combination of metrics, we are able to measure a statistically significant shift due to one out of the 32 fuel assemblies being either missing or degraded to a point of complete loss of structural coherence (i.e. “rubble”). We believe this combination of resonance metrics is new to the field and will have applications beyond the fuel storage problem. Emissions Tomography The Emissions Tomography team at the University of Florida explored the concept of using scattering of radiation emissions from the stored fuel as method of evaluating the internal structural health of the cask. Due to the need for emissive fuel assemblies and the difficulties of gaining access to storage facilities, this team primarily relied on simulations to inform feasibility. A key question was the flux estimates of neutrons with a small enough number of scattering events to retain spatial information. Some key conclusions from the transmission component of this work are that the number of scattering events falls rapidly with energy – setting a 11MeV threshold, the average number of scattering events is 8∓4. Rough estimates of scanning times using an ideal detector array would be 1 week. The team also explored the case of significant structural failure resulting in a shift of the source material (fuel). The result is that a +/- 2% mass shift results in a detectable (5.13%) shift in the correlated 2 neutron emission counts and thus may be a candidate as an initial survey. Finally, simulations for predicting the azimuthal dependence of the flux based on specific loading patterns were conducted. Due to self-shielding effects of the assemblies, a periodic pattern results and could potentially be used to identify missing or severely degraded assemblies. This work forms a basis for proper detector placement with the goal of optimizing sensitivity. Muon Tomography Muon tomography (MT) is a relatively novel method of interrogating interior structure on a large scale. One of the advantages of this technique is that the exciting energy source (cosmic muons) is ubiquitous and relatively stable in time. Challenging aspects are relatively low flux rates and appropriate detector technologies. The efforts of the muon group at Oregon State University focused on two areas: (1) simulation of various combinations of muon scattering events and detection schemes to assess sensitivities and time scales for image construction and (2) building a prototype muon detection system applicable to the dry storage cask problem as well as many other similar systems. Significant accomplishments in this area are: (1) demonstration that MT is a viable method for detecting missing or severely degraded rod assemblies, (2) a scintillation based detector with 4 panels and 32 channels of position data each were constructed, tested, and calibrated. Testing of the panels using natural muon background and no scattering objects yielded largely expected results – a straight line path for the muon through the four panels. However some anomalies (paths deviating from a line) were also detected. A redesign of the board improved the signal which greatly improved the muon tracking through the panels. Testing with lead bricks as scattering bodies showed a reduction in the coincidence rate with increasing bricks indicating increased scattering events. A refinement of the scintillator detector design resulted in a spatial image resolution of around 2 cm based on lead brick spacing tests lasting 24 hours. Passive/Active Ultrasonics This team at the University of South Carolina investigated the feasibility of combining passive ultrasonics, in which an elastic pulse is generated by a degradation process (such as a crack extending), and active ultrasonics, in which a known excitation is introduced into the structure and detected by an array of transducers. Significant accomplishments in this area include the development of elastic wave propagation simulation in relevant materials, development of diagnostic algorithms, and laboratory testing and validation using laser doppler vibrometry. This group demonstrated the applicability of various non-contact vibration detection methodologies such as air-coupled ultrasonic transducers and scanning laser doppler vibrometry. While passive acoustic emissions monitoring will likely have a role during the transportation of a fuel storage cask, the level of detailed information about the defect generating the event which can be ascertained from the detections in a complex structure remains for future work. Thermography The EPRI team contributed early in the project a series of thermography measurements in which various levels of water pooled in a mock-up canister were analyzed using commercially available infrared imaging technologies. While large levels of water (100 L, ~10 cm deep) were observable in the thermograms, more realistic levels of pooled water (1 L) were not detectable whether localized or distributed.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

Scalable quantum simulations of scattering in scalar field theory on 120 qubits

Simulations of collisions of fundamental particles on a quantum computer are expected to have an exponential advantage over classical methods and promise to enhance searches for new physics. Furthermore, scattering in scalar field theory has been shown to be bounded-error quantum polynomial time (BQP) complete, making it a representative problem for which quantum computation is efficient. As a step toward large-scale quantum simulations of collision processes, scattering of wave packets in one-dimensional scalar field theory is simulated using 120 qubits of IBM’s Heron superconducting quantum computer ibm_fez. Variational circuits compressing vacuum preparation, wave packet initialization, and time evolution are determined using classical resources. By leveraging physical properties of states in the theory, such as symmetries and locality, the variational quantum algorithm constructs scalable circuits that can be used to simulate arbitrarily large system sizes. A new strategy is introduced to mitigate errors in quantum simulations, which enables the extraction of meaningful results from circuits with up to 4924 two-qubit gates and two-qubit gate depths of 103. The effect of interactions is clearly seen, and is found to be in agreement with classical matrix product state simulations. Finally, the developments that will be necessary to simulate high-energy inelastic collisions on a quantum computer are discussed.

quantum circuits↗

Accelerating the solution of the S N equations with highly anisotropic scattering using the Fokker-Planck approximation

The discrete ordinates method can model forward-peaked transport problems accurately. However, convergence of discrete ordinates solution can become arbitrarily slow upon use of standard iterative procedures like source iteration. Standard zeroth and first moment-based acceleration methods like nonlinear diffusion acceleration and diffusion synthetic acceleration are ineffective in accelerating such problems because these methods do not correct higher order Legendre-moments of angular flux. In this paper, we explore the idea of using the Fokker-Planck approximation as a preconditioner to accelerate the convergence of solution of forward-peaked transport problems. Finally, we observe that such preconditioning can successfully reduce the iteration count by up to four orders of magnitude, and solver runtime by up to three orders of magnitude when compared to diffusion synthetic acceleration.

42 ENGINEERING↗

Ab initio nucleon-nucleus elastic scattering with chiral effective field theory uncertainties

Effective interactions for nucleon-nucleus elastic scattering from first principles require the use of the same nucleon-nucleon interaction in the structure and reaction calculations and a consistent treatment of the relevant operators at each order. Systematic investigations of the effect of truncation uncertainties of chiral nucleon-nucleon ( N N ) forces have been carried out for scattering observables in the two- and three-nucleons system as well as for bound-state properties of light nuclei. Here we extend this type of study to proton and neutron elastic scattering for 16 O and 12 C. Using the frameworks of the spectator expansion of multiple scattering theory as well as the no-core shell model, we employ one specific chiral interaction from the LENPIC collaboration and consistently calculate the leading-order effective nucleon-nucleus interaction up to the third chiral order, from which we extract elastic scattering observables. Then we apply pointwise as well as correlated uncertainty quantification for the estimation of the chiral truncation error. We calculate and analyze proton elastic scattering observables for 16 O and 12 C as well as neutron elastic scattering observables for 12 C between 65 and 185 MeV projectile kinetic energy. We find in all cases qualitatively similar results for the chiral truncation uncertainties as in few-body systems and assess them using similar diagnostic tools. The order-by-order convergence of the scattering observables for 16 O and 12 C is very reasonable around 100 MeV, while for higher energies the chiral expansion parameter becomes too large for convergence. Comparing proton and neutron differential cross sections reveals that their description is comparable up to around 100 MeV. Here, we also find a nearly perfect correlation between the differential cross section for neutron scattering and the N N Wolfenstein amplitudes for small momentum transfers. The diagnostic tools for studying order-by-order convergence in observables in few-body systems can be employed for observables in nucleon-nucleus scattering with only minor modifications provided the momentum scale in the problems is not too large. We also find that the chiral N N interaction on which our study is based gives a very good description of differential cross sections for 16 O and 12 C as low as 65-MeV projectile energy, particularly in the forward direction. In addition, the very forward direction of the neutron differential cross section mirrors the behavior of the N N interaction amazingly well.

6 ≤ A ≤ 19↗

Direct Calculation of the Scattering Amplitude Without Partial Wave Decomposition: Inclusion of Correlation Effects - III

In the first two papers in this series, we developed a method for studying electron-hydrogen scattering that does not use partial wave analysis. We constructed an ansatz for the wave function in both the static and static exchange approximations and calculated the full scattering amplitude. Here we go beyond the static exchange approximation, and include correlation in the wave function via a modified polarized orbital. This correlation function provides a significant improvement over the static exchange approximation: the resultant elastic scattering amplitudes are in very good agreement with fully converged partial wave calculations for electron-hydrogen scattering. A fully variational modification of this approach is discussed in the conclusion of the article Popular summary of Direct calculation of the scattering amplitude without partial wave expansion. III ....." by J. Shertzer and A. Temkin. In this paper we continue the development of In this paper we continue the development of a new approach to the way in which researchers have traditionally used to calculate the scattering cross section of (low-energy) electrons from atoms. The basic mathematical problem is to solve the Schroedinger Equation (SE) corresponding the above physical process. Traditionally it was always the case that the SE was reduced to a sequence of one-dimensional (ordinary) differential equations - called partial waves which were solved and from the solutions "phase shifts" were extracted, from which the scattering cross section was calculated.

Shertzer, Janine↗

The method of transition boundary for the solution of diffraction problems

A new method, which enables one to solve some diffraction problems, is put forth. The technique is based on a relation between the diffracted and scattered geometric optics waves at the transition boundaries. The concept of initial waves is also used instead of the incident field. The general algorithm of the method is outlined and applied to the diffraction problems of waves by perfect electric conductor half-screen, resistive half-plane and impedance sheet junction.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Effects of scattering on the field-induced T c enhancement in thin superconducting films in a parallel magnetic field

The problem of the normal-superconducting phase boundary for films in a parallel magnetic field, discussed in the classical paper by Ginzburg and Landau for temperatures close to the critical, is revisited with the help of the microscopic BCS theory for arbitrary temperatures taking pair-breaking and transport scattering into account. Although confirming experimental findings of the T c enhancement by the magnetic field, we find that the transport scattering pushes the phase transition curve to higher fields and higher temperatures for nearly all practical scattering rates. Still, the T c enhancement disappears in the dirty limit. Here, we also consider intriguing changes, such as reentrant superconductivity, caused to the phase boundary by pair-breaking magnetic ions spread on one of the film faces. These features await experimental verification.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

X-ray resonance scattering in a spherically symmetric coronal model

In the solar corona the opacities of some of the prominent X-ray emission lines are on the order of tau of about I over typical coronal path lengths. A particular solution of the radiative transfer problem involving an extended, spherically symmetric coronal shell radiating isotropic, homogeneous emission in which single-scattering also takes place is presented and discussed. Within the context of this simplified model, it is found that scattered radiation is an important contribution to the total emergent resonance line flux and that for the He-like family of resonance (r), intercombination (i), and forbidden (f) lines, the ratio G = (f + i)/r would decrease as a function of optical depth for disk-center emission in an extended spherically symmetric corona.

Haisch, B. M.↗

Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations

We developed a new class of physics-informed generative adversarial networks (PI-GANs) to solve forward, inverse, and mixed stochastic problems in a unified manner based on a limited number of scattered measurements. Unlike standard GANs relying solely on data for training, here we encoded into the architecture of GANs the governing physical laws in the form of stochastic differential equations (SDEs) using automatic differentiation. In particular, we applied Wasserstein GANs with gradient penalty (WGAN-GP) for its enhanced stability compared to vanilla GANs. We first tested WGAN-GP in approximating Gaussian processes of different correlation lengths based on data realizations collected from simultaneous reads at sparsely placed sensors. We obtained good approximation of the generated stochastic processes to the target ones even if there is a mismatch between the input noise dimensionality and the effective dimensionality of the target stochastic processes. We also studied the overfitting issue for both the discriminator and the generator, and we found that overfitting occurs also in the generator in addition to the discriminator as previously reported. Subsequently, we considered the solution of elliptic SDEs requiring approximations of three stochastic processes, namely the solution, the forcing, and the diffusion coefficient. Here again, we assumed data collected from simultaneous reads at a limited number of sensors for the multiple stochastic processes. Furthermore, three generators were used for the PI-GANs: two of them were feed forward deep neural networks (DNNs), while the other one was the neural network induced by the SDE. For the case where we have one group of data, we employed one feed forward DNN as the discriminator, while for the case of multiple groups of data we employed multiple discriminators in PI-GANs. We solved forward, inverse, and mixed problems without changing the framework of PI-GANs, obtaining both the means and the standard deviations of the stochastic solution and the diffusion coefficient in good agreement with benchmarks. In this work, we have demonstrated the effectiveness of PI-GANs in solving SDEs for about 120 dimensions. In principle, PI-GANs could tackle very high dimensional problems given more sensor data with low-polynomial growth in computational cost.

97 MATHEMATICS AND COMPUTING↗

The Measurement of Aerosol Optical Properties using Continuous Wave Cavity Ring-Down Techniques

Large uncertainties in the effects that aerosols have on climate require improved in situ measurements of extinction coefficient and single-scattering albedo. This paper describes the use of continuous wave cavity ring-down (CW-CRD) technology to address this problem. The innovations in this instrument are the use of CW-CRD to measure aerosol extinction coefficient, the simultaneous measurement of scattering coefficient, and small size suitable for a wide range of aircraft applications. Our prototype instrument measures extinction and scattering coefficient at 690 nm and extinction coefficient at 1550 nm. The instrument itself is small (60 x 48 x 15 cm) and relatively insensitive to vibrations. The prototype instrument has been tested in our lab and used in the field. While improvements in performance are needed, the prototype has been shown to make accurate and sensitive measurements of extinction and scattering coefficients. Combining these two parameters, one can obtain the single-scattering albedo and absorption coefficient, both important aerosol properties. The use of two wavelengths also allows us to obtain a quantitative idea of the size of the aerosol through the Angstrom exponent. Minimum sensitivity of the prototype instrument is 1.5 x 10(exp -6)/m (1.5 M/m). Validation of the measurement of extinction coefficient has been accomplished by comparing the measurement of calibration spheres with Mie calculations. This instrument and its successors have potential to help reduce uncertainty currently associated with aerosol optical properties and their spatial and temporal variation. Possible applications include studies of visibility, climate forcing by aerosol, and the validation of aerosol retrieval schemes from satellite data.

Strawa, Anthony W.↗

The Measurement of Aerosol Optical Properties Using Continuous Wave Cavity Ring-Down Techniques

Large uncertainties in the effects that aerosols have on climate require improved in-situ measurements of extinction coefficient and single-scattering albedo. This abstract describes the use of continuous wave cavity ring-down (CW-CRD) technology to address this problem. The innovations in this instrument are the use of CW-CRD to measure aerosol extinction coefficient, the simultaneous measurement of scattering coefficient, and small size suitable for a wide range of aircraft applications. Our prototype instrument measures extinction and scattering coefficient at 690 nm and extinction coefficient at 1550 nm. The instrument itself is small (60 x 48 x 15 cm) and relatively insensitive to vibrations. The prototype instrument has been tested in our lab and used in the field. While improvements in performance are needed, the prototype has been shown to make accurate and sensitive measurements of extinction and scattering coefficients. Combining these two parameters, one can obtain the single-scattering albedo and absorption coefficient, both important aerosol properties. The use of two wavelengths also allows us to obtain a quantitative idea of the size of the aerosol through the Angstrom exponent. Minimum sensitivity of the prototype instrument is 1.5 x 10(exp -6)/m (1.5/Mm). Validation of the measurement of extinction coefficient has been accomplished by comparing the measurement of calibration spheres with Mie calculations. This instrument and its successors have potential to help reduce uncertainty currently associated with aerosol optical properties and their spatial and temporal variation. Possible applications include studies of visibility, climate forcing by aerosol, and the validation of aerosol retrieval schemes from satellite data.

Strawa, A. W.↗

Bayesian Neural Network Variational Autoencoder Inverse Mapper (BNN-VAIM) and its application in Compton Form Factors extraction

Abstract We extend the Variational Autoencoder Inverse Mapper (VAIM) framework for the inverse problem of extracting Compton Form Factors (CFFs) from deeply virtual exclusive reactions, such as the unpolarized Deeply virtual exclusive scattering (DVCS) cross section. VAIM is an end-to-end deep learning framework to address the solution ambiguity issue in ill-posed inverse problems, which comprises of a forward mapper and a backward mapper to simulate the forward and inverse processes, respectively. In particular, we incorporate Bayesian Neural Network (BNN) into the VAIM architecture (BNN-VAIM) for uncertainty quantification. By sampling the weights and biases distributions of the BNN in the backward mapper of the VAIM, BNN-VAIM is able to estimate prediction uncertainty associated with each individual solution obtained for an ill-posed inverse problem. We first demonstrate the uncertainty quantification capability of BNN-VAIM in a toy inverse problem. Then, we apply BNN-VAIM to the inverse problem of extracting 8 CFFs from the unpolarized DVCS cross section.

Instruments & Instrumentation↗

Radiative transfer calculated from a Markov chain formalism

The theory of Markov chains is used to formulate the radiative transport problem in a general way by modeling the successive interactions of a photon as a stochastic process. Under the minimal requirement that the stochastic process is a Markov chain, the determination of the diffuse reflection or transmission from a scattering atmosphere is equivalent to the solution of a system of linear equations. This treatment is mathematically equivalent to, and thus has many of the advantages of, Monte Carlo methods, but can be considerably more rapid than Monte Carlo algorithms for numerical calculations in particular applications. We have verified the speed and accuracy of this formalism for the standard problem of finding the intensity of scattered light from a homogeneous plane-parallel atmosphere with an arbitrary phase function for scattering. Accurate results over a wide range of parameters were obtained with computation times comparable to those of a standard 'doubling' routine. The generality of this formalism thus allows fast, direct solutions to problems that were previously soluble only by Monte Carlo methods. Some comparisons are made with respect to integral equation methods.

Esposito, L. W.↗