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A Convex Optimization Approach to Improving Suboptimal Hyperparameters of Sliced Normal Distributions

Sliced Normal (SN) distributions are a generalization of Gaussian distributions where the quadratic argument of the exponential is replaced with a sum of squares polynomial. SNs may be used to represent the distribution of a diverse set of random variables including multi-modal, non-symmetric, and skewed distributions. Unfortunately, the likelihood function of a SN includes a normalization constant and the inclusion of this normalization constant makes the likelihood a non-convex function of the hyperparameters which define the SN. In previous work, suboptimal fitting of the hyperparameters was performed by transforming the given data into a higher dimensional monomial basis and selecting the optimal hyperparameters of a Gaussian fit in this space. However, this approach did not account for the effect of lifting on the normalization constant. Indeed, it was observed that as the number of monomials is increased the likelihood of the Sliced Normal can decrease. In this paper, we increase the likelihood of Sliced Normals found using the previous method by developing a convex formulation which scales the covariance matrix of the Gaussian fit such that the likelihood of the Sliced Normal is maximized. The result is significant improvements of the log likelihood of fitted SN distributions, including a significant increase, especially for problems with 500+ monomials.

Convex optimization approach to improving suboptim↗
DOE OSTI2015

Multiresolution quantum chemistry in multiwavelet bases: excited states from time-dependent Hartree–Fock and density functional theory via linear response

Using the fully numerical method for time-dependent Hartree–Fock and density functional theory (TD-HF/DFT) with the Tamm–Dancoff (TD) approximation we use a multiresolution analysis (MRA) approach to present our findings. From a reformulation with effective use of the density matrix operator, we obtain a general form of the HF/DFT linear response equation in the first quantization formalism. It can be readily rewritten as an integral equation with the bound-state Helmholtz (BSH) kernel for the Green's function. The MRA implementation of the resultant equation permits excited state calculations without virtual orbitals. Moreover, the integral equation is efficiently and adaptively solved using a numerical multiresolution solver with multiwavelet bases. Our implementation of the TD-HF/DFT methods is applied for calculating the excitation energies of H 2 , Be, N 2 , H 2 O, and C 2 H 4 molecules. The numerical errors of the calculated excitation energies converge in proportion to the residuals of the equation in the molecular orbitals and response functions. The energies of the excited states at a variety of length scales ranging from short-range valence excitations to long-range Rydberg-type ones are consistently accurate. It is shown that the multiresolution calculations yield the correct exponential asymptotic tails for the response functions, whereas those computed with Gaussian basis functions are too diffuse or decay too rapidly. Finally, we introduce a simple asymptotic correction to the local spin-density approximation (LSDA) so that in the TDDFT calculations, the excited states are correctly bound.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

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↗
DOE OSTI2025

Hardness of Observing Strong-to-Weak Symmetry Breaking

Spontaneous symmetry breaking (SSB) is the cornerstone of our understanding of quantum phases of matter. Recent works have generalized this concept to the domain of mixed states in open quantum systems, where symmetries can be realized in two distinct ways dubbed strong and weak. Novel intrinsically mixed phases of quantum matter can then be defined by the spontaneous breaking of strong symmetry down to weak symmetry. However, proposed order parameters for strong-to-weak SSB (based on mixed-state fidelities or purities) seem to require exponentially many copies of the state, raising the question: Is it possible to efficiently detect strong-to-weak SSB, in general? In this work, we answer this question negatively in the paradigmatic cases of ℤ2 and U(1) symmetries. We construct ensembles of pseudorandom mixed states that do not break the strong symmetry yet are computationally indistinguishable from states that do. This rules out the existence of efficient state-agnostic protocols to detect strong-to-weak SSB.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2012

Advanced Computational Methods for Security Constrained Financial Transmission Rights: Structure and Parallelism

Financial Transmission Rights (FTRs) help power market participants reduce price risks associated with transmission congestion. FTRs are issued based on a process of solving a constrained optimization problem with the objective to maximize the FTR social welfare under power flow security constraints. Security constraints for different FTR categories (monthly, seasonal or annual) are usually coupled and the number of constraints increases exponentially with the number of categories. Commercial software for FTR calculation can only provide limited categories of FTRs due to the inherent computational challenges mentioned above. In this paper, a novel non-linear dynamical system (NDS) approach is proposed to solve the optimization problem. The new formulation and performance of the NDS solver is benchmarked against widely used linear programming (LP) solvers like CPLEX™ and tested on large-scale systems using data from the Western Electricity Coordinating Council (WECC). The NDS is demonstrated to outperform the widely used CPLEX algorithms while exhibiting superior scalability. Furthermore, the NDS based solver can be easily parallelized which results in significant computational improvement.

Environmental Molecular Sciences Laboratory↗
DOE OSTI2016

Using full configuration interaction quantum Monte Carlo in a seniority zero space to investigate the correlation energy equivalence of pair coupled cluster doubles and doubly occupied configuration interaction

Over the past few years, pair coupled cluster doubles (pCCD) has shown promise for the description of strong correlation. This promise is related to its apparent ability to match results from doubly occupied configuration interaction (DOCI), even though the latter method has exponential computational cost. Here, by modifying the full configuration interaction quantum Monte Carlo algorithm to sample only the seniority zero sector of Hilbert space, we show that the DOCI and pCCD energies are in agreement for a variety of 2D Hubbard models, including for systems well out of reach for conventional configuration interaction algorithms. Our calculations are aided by the sign problem being much reduced in the seniority zero space compared with the full space. We present evidence for this and then discuss the sign problem in terms of the wave function of the system which appears to have a simplified sign structure.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

Gaussian processes with built-in dimensionality reduction: Applications to high-dimensional uncertainty propagation

Uncertainty quantification (UQ) tasks, such as model calibration, uncertainty propagation, and optimization under uncertainty, typically require several thousand evaluations of the underlying computer codes. To cope with the cost of simulations, one replaces the real response surface with a cheap surrogate based, e.g., on polynomial chaos expansions, neural networks, support vector machines, or Gaussian processes (GP). However, the number of simulations required to learn a generic multivariate response grows exponentially as the input dimension increases. This curse of dimensionality can only be addressed, if the response exhibits some special structure that can be discovered and exploited. A wide range of physical responses exhibit a special structure known as an active subspace (AS). An AS is a linear manifold of the stochastic space characterized by maximal response variation. The idea is that one should first identify this low dimensional manifold, project the high-dimensional input onto it, and then link the projection to the output. If the dimensionality of the AS is low enough, then learning the link function is a much easier problem than the original problem of learning a high-dimensional function. The classic approach to discovering the AS requires gradient information, a fact that severely limits its applicability. Furthermore, and partly because of its reliance to gradients, it is not able to handle noisy observations. The latter is an essential trait if one wants to be able to propagate uncertainty through stochastic simulators, e.g., through molecular dynamics codes. In this work, we develop a probabilistic version of AS which is gradient-free and robust to observational noise. Our approach relies on a novel Gaussian process regression with built-in dimensionality reduction. In particular, the AS is represented as an orthogonal projection matrix that serves as yet another covariance function hyper-parameter to be estimated from the data. To train the model, we design a two-step maximum likelihood optimization procedure that ensures the orthogonality of the projection matrix by exploiting recent results on the Stiefel manifold, i.e., the manifold of matrices with orthogonal columns. The additional benefit of our probabilistic formulation, is that it allows us to select the dimensionality of the AS via the Bayesian information criterion. We validate our approach by showing that it can discover the right AS in synthetic examples without gradient information using both noiseless and noisy observations. We demonstrate that our method is able to discover the same AS as the classical approach in a challenging one-hundred-dimensional problem involving an elliptic stochastic partial differential equation with random conductivity. Finally, we use our approach to study the effect of geometric and material uncertainties in the propagation of solitary waves in a one dimensional granular system.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS1991

Multiphase complete exchange on a circuit switched hypercube

On a distributed memory parallel computer, the complete exchange (all-to-all personalized) communication pattern requires each of n processors to send a different block of data to each of the remaining n - 1 processors. This pattern is at the heart of many important algorithms, most notably the matrix transpose. For a circuit switched hypercube of dimension d(n = 2(sup d)), two algorithms for achieving complete exchange are known. These are (1) the Standard Exchange approach that employs d transmissions of size 2(sup d-1) blocks each and is useful for small block sizes, and (2) the Optimal Circuit Switched algorithm that employs 2(sup d) - 1 transmissions of 1 block each and is best for large block sizes. A unified multiphase algorithm is described that includes these two algorithms as special cases. The complete exchange on a hypercube of dimension d and block size m is achieved by carrying out k partial exchange on subcubes of dimension d(sub i) Sigma(sup k)(sub i=1) d(sub i) = d and effective block size m(sub i) = m2(sup d-di). When k = d and all d(sub i) = 1, this corresponds to algorithm (1) above. For the case of k = 1 and d(sub i) = d, this becomes the circuit switched algorithm (2). Changing the subcube dimensions d, varies the effective block size and permits a compromise between the data permutation and block transmission overhead of (1) and the startup overhead of (2). For a hypercube of dimension d, the number of possible combinations of subcubes is p(d), the number of partitions of the integer d. This is an exponential but very slowly growing function and it is feasible over these partitions to discover the best combination for a given message size. The approach was analyzed for, and implemented on, the Intel iPSC-860 circuit switched hypercube. Measurements show good agreement with predictions and demonstrate that the multiphase approach can substantially improve performance for block sizes in the 0 to 160 byte range. This range, which corresponds to 0 to 40 floating point numbers per processor, is commonly encountered in practical numeric applications. The multiphase technique is applicable to all circuit-switched hypercubes that use the common e-cube routing strategy.

Bokhari, Shahid H.↗
DOE OSTI2021

Sparse Control Synthesis for Uncertain Responsive Loads With Stochastic Stability Guarantees

In this report, recent studies have demonstrated the potential of flexible loads in providing frequency response services, predominantly due to their availability and cost-effectiveness. However, uncertainty and variability in various weather-related and end-use behavioral factors often impact the reliability of demand-side control performance. This work addresses this problem with the design of a demand-side control to achieve frequency response under load uncertainties. Our approach involves modeling the load uncertainties via stochastic processes that appear as both multiplicative and additive in the power system dynamics. Recently developed mean square exponential stability (MSES) results for continuous-time linear stochastic systems are applied to pose the control synthesis problem which results in an LMI-based optimization problem. Additional costs and constraints are added to the LMI-based controller synthesis to ensure MSES, improve closed-loop transient performance, maximize tolerable uncertainties, and promote sparsity in the controller. Additionally, the fundamental limitations between the tolerable uncertainties and control efforts while ensuring MSES are discussed. Further, the control synthesis problem for the case of the full-state measurement is generalized to the case of partial-state measurements. The proposed control synthesis is illustrated on an IEEE 39 bus system with rigorous studies to demonstrate the role of sparsity, closed-loop transient performance, tolerable uncertainties, and control efforts while ensuring MSES and achieving frequency response.

42 ENGINEERING↗
DOE OSTI2012

Study of CP Violation in Dalitz-Plot Analyses of B-Meson Decays to Three Kaons

The Standard Model (SM) explains CP violation in terms of the CKM matrix. The BABAR experiment was designed mainly to test the CKM model in B decays. B decays that proceed through b → s loop diagrams, of which B {yields} KKK decays are an example, are sensitive to new physics effects that could lead to deviations from the CKM predictions for CP violation. We present studies of CP violation in the decays B + → K + K - K + , B + → K S 0 K S 0 K + , and B 0 → K + K - K S 0 , using a Dalitz plot amplitude analysis. These studies are based on approximately 470 million B$\bar{B}$ decays collected by BABAR at the PEP-II collider at SLAC. We perform measurements of time-dependent CP violation in B 0 → K + K - K S 0 , including B 0 → ΦK S 0 . We measure a CP-violating phase β eff (ΦK S 0 ) = 0.36 ± 0.11 ± 0.04 rad., in agreement with the SM. This is the world's most precise measurement of this quantity. We also measure direct CP asymmetries in all three decay modes, including the direct CP asymmetry A CP (ΦK + ) = (12.8 ± 4.4 ± 1.3)%, which is 2.8 sigma away from zero. This measurement is in tension with the SM, which predicts an asymmetry of a few percent. We also study the resonant and nonresonant features in the B → KKK Dalitz plots. We find that the hypothetical scalar f X (1500) resonance, introduced by prior analyses to explain an unknown peak in the m KK spectrum, cannot adequately describe the data. We conclude instead that the f X (1500) can be explained as the sum of the f 0 (1500), f' 2 (1525), and f 0 (1710) resonances, removing the need for the hypothetical f X (1500). We also find that an exponential nonresonant model, used by previous analyses to describe the broad nonresonant feature seen in B → KKK decays, cannot fully model the data. We introduce a new nonresonant model that contains more free parameters, allows for phase motion, and contains both S-wave and P-wave components.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2015

Investigations of Beam Dynamics Issues at Current and Future Hadron Accelerators (Final Report)

There is a synergy between the fields of Beam Dynamics (BD) in modern particle accelerators and Applied Mathematics (AMa). We have formulated significant problems in BD and have developed and applied tools within the contexts of dynamical systems, topological methods, numerical analysis and scientific computing, probability and stochastic processes, and mathematical statistics. We summarize the three main areas of our AMa work since 2011. First, we continued our study of Vlasov-Maxwell systems. Previously, we developed a state of the art algorithm and code (VM3@A) to calculate coherent synchrotron radiation in single pass systems. In this cycle we carefully analyzed the major expense, namely the integral-over-history (IOH), and developed two approaches to speed up integration. The first strategy uses a representation of the Bessel function J0 in terms of exponentials. The second relies on “local sequences” developed recently for radiation boundary conditions, which are used to reduce computational domains. Although motivated by practicality, both strategies involve interesting and rather deep analysis and approximation theory. As an alternative to VM3@A, we are integrating Maxwell’s equations by a time-stepping method, bypass- ing the IOH, using a Discontinuous Galerkin (DG) method. DG is a generalization of Finite Element and Finite Volume methods. It is spectrally convergent, unlike the commonly used Finite Difference methods, and can handle complicated vacuum chamber geometries. We have applied this in several contexts and have obtained very nice results including an explanation of an experiment at the Canadian Light Source, where the geometry is quite complex. Second, we continued our study of spin dynamics in storage rings. There is much current and proposed activity where spin polarized beams are being used in testing the Standard Model and its modifications. Our work has focused on invariant spin fields (ISFs) and amplitude dependent spin tunes (ADSTs), which are essential for estimating beam polarization. Several algorithms have been developed since the 1980s for computing the ISF, among them the Heinemann- Hoffstaetter method of stroboscopic averaging (SA) which is implemented in the code SPRINT. SA, which computes the ISF by using spin tracking data, can find the ISF to computer precision if it exists and thus can give evidence for existence of the ISF. Central to our work is resolving the ISF conjecture, which says that, off orbital resonance, an ISF exists. Thus Heinemann developed, in his 2010 PhD thesis, a new framework which unifies and generalizes the notions of ISF and ADST by using bundle theory. This lead to a long paper which was a major collaborative effort during the recent cycle. In a nutshell, our bundle approach elegantly unifies the dynamics of spin-1/2 and spin-1 particles, e.g., protons and deuterons. In fact it is well known that these two kinds of dynamics are driven by the same spin transfer matrix and in our approach one sees the deeper reason for this: the spin transfer matrix carries the natural dynamics of a principal bundle whereas the difference between the spin-1/2 and spin-1 dynamics lies in their different geometrical situation, i.e., different underlying associated bundles. Thus one arrives at new results for polarized beams, among them the Invariant Reduction Theorem (IRT) and the Cross Section Theorem (CST). The IRT gives a necessary and sufficient condition for the ISF to exist. The SA technique revealed, 20 years ago, that each ISF can be viewed as a complex agglomerate of spin tracking data. The bundle approach goes one step further by using the IRT and the CST to glue spin tracking data into agglomerates which are candidates for ISFs. We gain insight because the “good” agglomerates, in the presence of an ISF, are very different from the “bad” ones. Finally we mention that the bundle approach has analogies to the approach used in geometrical Yang-Mills theory. Third, we studied X-Ray Free Electron Lasers (FELs), which are electron accelerators producing coherent undulator radiation over a wide range of frequencies from microwaves to x-rays. The photon beams produced in FEL undulators are used to study material samples in biology, material science etc. We developed a mathematical analysis, based on the 6D Lorentz system, of energetic electrons moving through a planar undulator excited by a Maxwell traveling wave field of wavelength λ. Our Method of Averaging perturbation analysis yields non-resonant and near-to-resonant normal form approximations as a function of λ, which we present in two averaging theorems. We prove the theorems in detail, error bounds and giving a tutorial on mathematically rigorous perturbation theory in a context where proofs are easily understood. To our knowledge the planar problem has not been analyzed with the generality here nor has the standard FEL pendulum system, which appears on resonance, been derived with error bounds. In addition to the domains of validity of the normal forms we obtain new insights that require further study, including a more general low gain theory. With a firm foundation in the non-collective case above we are studying the 3D collective case from start up from noise to high gain and saturation. We have formulated the noise start up as a problem of going from the microscopic Klimontovich-Maxwell to the macroscopic Vlasov-Maxwell with a Vlasov correction term. In the 1D setting, we seek an alternative to the phenomenological Vlasov approach which models shot noise by a perturbation on an initial “smooth” density. The 1D wave equation with a Klimontovich source is often the starting point for the 1D FEL high gain theory. We have a new representation of solutions which may lead to new insights.

43 PARTICLE ACCELERATORS↗
DOE OSTI2018

Validation Data Acquisition in HTTF during PCC Events

A validation experimental campaign was conducted in an Integral Effect Test (IET) facility of High Temperature Gas Reactors (HTGR), the High-Temperature Test Facility (HTTF) at Oregon State University (OSU). The HTTF simulates Depressurized and Pressurized Conduction Cooldowns (DCC and PCC). This campaign required the development of a new laser spectroscopic diagnostic to measure velocity in the challenging conditions of the HTTF: low speed (~1 m/s) gas flows at HTGR prototypical temperature and 1/10th pressure. This was a collaborative effort between co-PIs at The George Washington University (GW), Oregon State University (OSU), and NASA Langley Research Center. The main accomplishments of this project include the record for dynamic range for velocimetry, highest accuracy obtained with this technique, successful deployment in an IET leading to new validation matrix for CFD. These are detailed below and in manuscript appended to this executive summary. For this project, we introduced a new class of laser spectroscopic diagnostics to Thermal-Hydraulics to measure velocity of gases; furthermore, the diagnostic was demonstrated in-situ in an IET during DCC events. In such scenarios, particles used in mainstream techniques, like Particle Image Velocimetry (PIV) are not appropriate as they settle down too rapidly and also contaminate the experimental facility. Molecular tracers stay mixed within the working gas and can seed the flow in a technique called Molecular Tagging Velocimetry (MTV). In MTV a molecular tracer is photo-dissociated by a first (write) laser into radicals or molecules. The pattern created by the write laser is then interrogated with planar laser-induced fluorescence (PLIF), the read pulse(s), which are recorded with a camera. The pattern is probed and matched at two times (interval or probe time, dt), resulting in a time-of-flight velocimetry technique. This project demonstrated a new application range for MTV in gases. MTV has been extensively used for high-speed gas dynamics: molecules do not suffer from inertia associated with particles in regions of high accelerations (such as shock waves). For high-speed flows, the probe time is on the order of 0.1-10 µs. In time-of-flight techniques, the velocity is reconstructed from the recorded displacement, dx, between time interval dt: U = dx/dt. To first order, the resolution of the pattern matching is typically a (fixed) fraction of a pixel; therefore the resolution of the velocity measurement increases with dx and, for a given flow velocity, dt. For the low-speed flows of interest here, this requires long probe time, and therefore finding tracers that are stable for a long time. After an extensive literature survey we identified 6 tracers that would be suitable. We also worked with Spectra Physics to design a custom laser system that is flexible enough to probe these various tracers. The final instrument is made of 4 lasers that are integrated into a compact and movable cart to facilitate the deployment of the instrument off-site. One unique aspect of the laser system is the tunable dye laser (needed to do PLIF of the radicals). By being pumped by two Nd:YAG lasers, this laser is able to provide two pulses with any probe time (dt) for wavelengths between 200 and 900 nm with a linewidth of 3 pm. Accommodating the dual-pulse capability required customizing the pump lasers and the dye laser, making this system unique. Additionally, a custom UV-intensified CCD camera capable of frame-straddling and exposure time as short as 10 ns was acquired. More than two-third of the instrument cost (~$200k) was covered by a cost sharing from the PI’s startup funds. Of the 6 potential tracers identified, two were tested extensively (H2O and N2O) at high-temperature and elevated pressure at GW prior to conducting the tests at OSU. For both tracers, an Ar-F laser (193 nm, 10 ns, 10 mJ/pulse) photo-dissociates trace amounts (0.1 – 5% molar fraction) of the gases into OH and NO. OH is probed at 281.905 nm and NO at 226.186 nm (4-30mJ, 10 ns pulse). NO is very stable in inert environment and we could reconstruct velocities precisely with probe delay times as long at 40 ms, compared to 2 ms for OH, resulting in increased precision. Therefore, for the tests in the HTTF, N2O was selected as the final tracer and its optimal concentration was determined to be 0.5% molar. In the GW laboratory, we obtained a velocity measurement precision of 4 mm/s at conditions mimicking the HTTF tests. We also demonstrated the benefit of molecular tracers over particles and the flexibility of our instrument by continuously measuring velocity from 1000 to 0.1 m/s in a blow down test. This represents a dynamic range on velocity of 80 dB, which is unprecedented for velocimetry! Once the diagnostic was optimized, two two-week campaigns were conducted in the HTTF. This experimental campaign required numerous practical solutions to successfully and rapidly deploy this very sensitive table-top diagnostic in-situ. This was accomplished thanks to very extensive planning from all parties involved in the project. Due to constraints on availability of the HTTF, only adiabatic DCC tests were conducted. Specifically, the velocity was measured at the exit of the hot leg into the Reactor Cavity Simulation Tank (RCST). We were able to record the gas velocity associated with the air ingress for long times (up to one hour). Depending on the initial conditions (density of the gases in the Reactor Pressure Vessel and the RCST) the lock exchange lasted 30-60 s past the valve openings. Large vortices were identified within the shear layer at the interface of the cross flow. These vortices are an indication of shear-driven instability between the counter streams of different density. After the lock-exchange, a small residual flow was still present and its strength decayed exponentially. It finds its source in a buoyancy difference between the gases in both vessels due to the mixing and diffusion of the gases in the cross-over duct and the RPV lower plenum. We did not expect to capture this part of the transient, which would have been less significant (if not negligible) in the presence of a heated core; the precision of our measurement in the HTTF was 6 mm/s, which exceeded our expectations for such a complex test! Using commercial software (Fluent) we simulated the experiment with Computational Fluid Dynamic (CFD) using Reynolds Averaged Navier-Stokes (RANS) turbulence models. We were not able to accurately reproduce the shear layer instability, mixing, and diffusion captured in our experimental data, even with large 3D meshes (>10 M points). We attributed this to the inadequacy of the RANS models for the flow of interest: our experimental Reynolds number was too low and was outside the applicability range of the RANS models. Instead, we should have employed CFD with less modeling, such as Large Eddy (LES) or even Direct Numerical (DNS) simulations. Unfortunately, we did not have the computational resources to do so. Finally, it should be noted that this experimental technique holds great promises for measuring velocity in thermalhydraulics. MTV has an application domain (highlighted in the narrative) that is complimentary to more established (and simpler) techniques, such as PIV. It has attracted the attention of several thermalhydraulics researchers and sponsors to measure velocity in harsh environments. The spectroscopic nature of this technique also enables to expand it to measure temperature and concentration simultaneously to velocity. In lieu of a narrative, 4 detailed journal publications are appended to this executive summary to describe the various development and accomplishments during this project. They include 2 published manuscripts (Experiments in Fluids and Measurement Science and Technology), and two invited contributions following the NURETH-17 conference (Nuclear Science and Technology). Our current Nuclear Engineering and Design draft is not included at this time.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗