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Studies of Analytic Evolution of Two-Photon Generalized Distribution Amplitude in QCD
We extend our method of analytic ERBL evolution for the case of distribution amplitudes that have jumps at some points x = {zeta}{sub I} inside the support region 0 < x < 1. As an application of the method, we use it for evolution of the two-photon generalized distribution amplitude. Our approach has advantages over the standard method of expansion in Gegenbauer polynomials, which requires infinite number of terms in order to accuretly reproduce functions in the vicinity of singular points, and over the method of straightforward iteration of initial distribution with evolution kernel which produces logarithmically divergent terms at each iteration. In our method, the logarithmic singularities are summed from the start, which immediately produces a continuous curve, with only one or two iterations needed afterwards in order to get precise results.
Computer method for identification of boiler transfer functions
An iterative computer method is described for identifying boiler transfer functions using frequency response data. An objective penalized performance measure and a nonlinear minimization technique are used to cause the locus of points generated by a transfer function to resemble the locus of points obtained from frequency response measurements. Different transfer functions can be tried until a satisfactory empirical transfer function to the system is found. To illustrate the method, some examples and some results from a study of a set of data consisting of measurements of the inlet impedance of a single tube forced flow boiler with inserts are given.
Temporal resolution and motion artifacts in single-source and dual-source cardiac CT
Purpose: The temporal resolution of a given image in cardiac computed tomography (CT) has so far mostly been determined from the amount of CT data employed for the reconstruction of that image. The purpose of this paper is to examine the applicability of such measures to the newly introduced modality of dual-source CT as well as to methods aiming to provide improved temporal resolution by means of an advanced image reconstruction algorithm. Methods: To provide a solid base for the examinations described in this paper, an extensive review of temporal resolution in conventional single-source CT is given first. Two different measures for assessing temporal resolution with respect to the amount of data involved are introduced, namely, either taking the full width at half maximum of the respective data weighting function (FWHM-TR) or the total width of the weighting function (total TR) as a base of the assessment. Image reconstruction using both a direct fan-beam filtered backprojection with Parker weighting as well as using a parallel-beam rebinning step are considered. The theory of assessing temporal resolution by means of the data involved is then extended to dual-source CT. Finally, three different advanced iterative reconstruction methods that all use the same input data are compared with respect to the resulting motion artifact level. For brevity and simplicity, the examinations are limited to two-dimensional data acquisition and reconstruction. However, all results and conclusions presented in this paper are also directly applicable to both circular and helical cone-beam CT. Results: While the concept of total TR can directly be applied to dual-source CT, the definition of the FWHM of a weighting function needs to be slightly extended to be applicable to this modality. The three different advanced iterative reconstruction methods examined in this paper result in significantly different images with respect to their motion artifact level, despite exactly the same amount of data being used in the reconstruction process. Conclusions: The concept of assessing temporal resolution by means of the data employed for reconstruction can nicely be extended from single-source to dual-source CT. However, for advanced (possibly nonlinear iterative) reconstruction algorithms the examined approach fails to deliver accurate results. New methods and measures to assess the temporal resolution of CT images need to be developed to be able to accurately compare the performance of such algorithms.
Erratum: “A finite difference Davidson procedure to sidestep full ab initio hessian calculation: Application to characterization of stationary points and transition state searches”
Certain key contributions to the field were not presented in the Introduction section of our recent paper. In addition to the variational method described in Ref. 14 and iterative diagonalization methods in Refs. 15–17, we wish to direct the attention of the reader to other significant efforts that have utilized similar principles for various applications. Reiher and Neugebauer developed a mode-tracking algorithm using the Davidson method for calculating only the modes relevant for vibrational analysis. Similarly, a block-Davidson approach via finite differentiation of the gradient was implemented for calculating hessian-vector products for normal mode analysis. Deglmann and Furche developed a method for stationary point characterization with density functional theory where full solution of all coupled-perturbed Kohn-Sham equations, the bottleneck in hessian calculation, could be avoided through iterative calculation of hessian-vector products to determine only the lowest eigenvalues.
A New Method for Analyzing Near-Field Faraday Probe Data in Hall Thrusters
This paper presents a new method for analyzing near-field Faraday probe data obtained from Hall thrusters. Traditional methods spawned from far-field Faraday probe analysis rely on assumptions that are not applicable to near-field Faraday probe data. In particular, arbitrary choices for the point of origin and limits of integration have made interpretation of the results difficult. The new method, called iterative pathfinding, uses the evolution of the near-field plume with distance to provide feedback for determining the location of the point of origin. Although still susceptible to the choice of integration limits, this method presents a systematic approach to determining the origin point for calculating the divergence angle. The iterative pathfinding method is applied to near-field Faraday probe data taken in a previous study from the NASA-300M and NASA-457Mv2 Hall thrusters. Since these two thrusters use centrally mounted cathodes, the current density associated with the cathode plume is removed before applying iterative pathfinding. A procedure is presented for removing the cathode plume. The results of the analysis are compared to far-field probe analysis results. This paper ends with checks on the validity of the new method and discussions on the implications of the results.
A New Method for Analyzing Near-Field Faraday Probe Data in Hall Thrusters
This paper presents a new method for analyzing near-field Faraday probe data obtained from Hall thrusters. Traditional methods spawned from far-field Faraday probe analysis rely on assumptions that are not applicable to near-field Faraday probe data. In particular, arbitrary choices for the point of origin and limits of integration have made interpretation of the results difficult. The new method, called iterative pathfinding, uses the evolution of the near-field plume with distance to provide feedback for determining the location of the point of origin. Although still susceptible to the choice of integration limits, this method presents a systematic approach to determining the origin point for calculating the divergence angle. The iterative pathfinding method is applied to near-field Faraday probe data taken in a previous study from the NASA-300M and NASA-457Mv2 Hall thrusters. Since these two thrusters use centrally mounted cathodes the current density associated with the cathode plume is removed before applying iterative pathfinding. A procedure is presented for removing the cathode plume. The results of the analysis are compared to far-field probe analysis results. This paper ends with checks on the validity of the new method and discussions on the implications of the results.
Hierarchical optimization for neutron scattering problems
We present a scalable optimization method for neutron scattering problems that determines confidence regions of simulation parameters in lattice dynamics models used to fit neutron scattering data for crystalline solids. The method uses physics-based hierarchical dimension reduction in both the computational simulation domain and the parameter space. We demonstrate for silicon that after a few iterations the method converges to parameters values (interatomic force-constants) computed with density functional theory simulations.
Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems
Multigrid methods are very efficient iterative solvers for system of algebraic equations arising from finite element and finite difference discretization of elliptic boundary value problems. The main principle of multigrid methods is to complement the local exchange of information in point-wise iterative methods by a global one utilizing several related systems, called coarse levels, with a smaller number of variables. The coarse levels are often obtained as a hierarchy of discretizations with different characteristic meshsizes, but this requires that the discretization is controlled by the iterative method. To solve linear systems produced by existing finite element software, one needs to create an artificial hierarchy of coarse problems. The principal issue is then to obtain computational complexity and approximation properties similar to those for nested meshes, using only information in the matrix of the system and as little extra information as possible. Such algebraic multigrid method that uses the system matrix only was developed by Ruge. The prolongations were based on the matrix of the system by partial solution from given values at selected coarse points. The coarse grid points were selected so that each point would be interpolated to via so-called strong connections. Our approach is based on smoothed aggregation introduced recently by Vanek. First the set of nodes is decomposed into small mutually disjoint subsets. A tentative piecewise constant interpolation (in the discrete sense) is then defined on those subsets as piecewise constant for second order problems, and piecewise linear for fourth order problems. The prolongation operator is then obtained by smoothing the output of the tentative prolongation and coarse level operators are defined variationally.
Development of iterative techniques for the solution of unsteady compressible viscous flows
Efficient iterative solution methods are being developed for the numerical solution of two- and three-dimensional compressible Navier-Stokes equations. Iterative time marching methods have several advantages over classical multi-step explicit time marching schemes, and non-iterative implicit time marching schemes. Iterative schemes have better stability characteristics than non-iterative explicit and implicit schemes. Thus, the extra work required by iterative schemes can also be designed to perform efficiently on current and future generation scalable, missively parallel machines. An obvious candidate for iteratively solving the system of coupled nonlinear algebraic equations arising in CFD applications is the Newton method. Newton's method was implemented in existing finite difference and finite volume methods. Depending on the complexity of the problem, the number of Newton iterations needed per step to solve the discretized system of equations can, however, vary dramatically from a few to several hundred. Another popular approach based on the classical conjugate gradient method, known as the GMRES (Generalized Minimum Residual) algorithm is investigated. The GMRES algorithm was used in the past by a number of researchers for solving steady viscous and inviscid flow problems with considerable success. Here, the suitability of this algorithm is investigated for solving the system of nonlinear equations that arise in unsteady Navier-Stokes solvers at each time step. Unlike the Newton method which attempts to drive the error in the solution at each and every node down to zero, the GMRES algorithm only seeks to minimize the L2 norm of the error. In the GMRES algorithm the changes in the flow properties from one time step to the next are assumed to be the sum of a set of orthogonal vectors. By choosing the number of vectors to a reasonably small value N (between 5 and 20) the work required for advancing the solution from one time step to the next may be kept to (N+1) times that of a noniterative scheme. Many of the operations required by the GMRES algorithm such as matrix-vector multiplies, matrix additions and subtractions can all be vectorized and parallelized efficiently.
SU-E-T-295: Simultaneous Beam Sampling and Aperture Shape Optimization for Station Parameter Optimized Radiation Therapy (SPORT)
Purpose: Station Parameter Optimized Radiation Therapy (SPORT) was recently proposed to fully utilize the technical capability of emerging digital LINACs, in which the station parameters of a delivery system, (such as aperture shape and weight, couch position/angle, gantry/collimator angle) are optimized altogether. SPORT promises to deliver unprecedented radiation dose distributions efficiently, yet there does not exist any optimization algorithm to implement it. The purpose of this work is to propose an optimization algorithm to simultaneously optimize the beam sampling and aperture shapes. Methods: We build a mathematical model whose variables are beam angles (including non-coplanar and/or even nonisocentric beams) and aperture shapes. To solve the resulting large scale optimization problem, we devise an exact, convergent and fast optimization algorithm by integrating three advanced optimization techniques named column generation, gradient method, and pattern search. Column generation is used to find a good set of aperture shapes as an initial solution by adding apertures sequentially. Then we apply the gradient method to iteratively improve the current solution by reshaping the aperture shapes and updating the beam angles toward the gradient. Algorithm continues by pattern search method to explore the part of the search space that cannot be reached by the gradient method. Results: The proposed technique is applied to a series of patient cases and significantly improves the plan quality. In a head-and-neck case, for example, the left parotid gland mean-dose, brainstem max-dose, spinal cord max-dose, and mandible mean-dose are reduced by 10%, 7%, 24% and 12% respectively, compared to the conventional VMAT plan while maintaining the same PTV coverage. Conclusion: Combined use of column generation, gradient search and pattern search algorithms provide an effective way to optimize simultaneously the large collection of station parameters and significantly improves quality of resultant treatment plans as compared with conventional VMAT or IMRT treatments.
SU-E-T-217: Intrinsic Respiratory Gating in Small Animal CT
Purpose: Preclinical animal models of lung cancer can provide a controlled test-bed for testing dose escalation or function-based-treatment-planning studies. However, to extract lung function, i.e. ventilation, one needs to be able to image the lung at different phases of ventilation (in-hale / ex-hale). Most respiratory-gated imaging using micro-CT involves using an external ventilator and surgical intervention limiting the utility in longitudinal studies. A new intrinsic respiratory retrospective gating method was developed and tested in mice. Methods: A fixed region of interest (ROI) that covers the diaphragm was selected on all projection images to estimate the mean intensity (M). The mean intensity depends on the projection angle and diaphragm position. A 3-point moving average (A) of consecutive M values: Mpre, Mcurrent and Mpost, was calculated to be subtracted from Mcurrent. A fixed threshold was used to enable amplitude based sorting into 4 different phases of respiration. Images at full-inhale and end-exhale phases of respiration were reconstructed using the open source OSCaR. Lung volumes estimated at the 2 phases of respiration were validated against literature values. Results: Intrinsic retrospective gating was accomplished without the use of any external breathing waveform. While projection images were acquired at 360 different angles. Only 138 and 104 projections were used to reconstruct images at full-inhale and end-exhale. This often results in non-uniform under-sampled angular projections leading to some minor streaking artifacts. The calculated expiratory, inspiratory and tidal lung volumes correlated well with the values known from the literature. Conclusion: Our initial result demonstrates an intrinsic gating method that is suitable for flat panel cone beam small animal CT systems. Reduction in streaking artifacts can be accomplished by oversampling the data or using iterative reconstruction methods. This initial experience will enable freebreathing small animal micro-CT imaging to fuel longitudinal studies of lung function.
A fast semidirect method for computing transonic aerodynamic flows
A fast, semidirect, iterative computational method, previously introduced for finite-difference solution of subsonic and slightly supercritical flow over airfoils, is extended both to apply to strongly supercritical conditions and to include full second-order accuracy in computing inviscid flows over airfoils. The nonlinear small-disturbance equations are solved iteratively by a direct, linear, elliptic solver. General, fully conservative, type-dependent difference equations are formulated, including parabolic- and shock-point transition operators that provide consistency with the integral conservation laws. These equations specialize to either first-order or to fully second-order-accurate equations. Various free parameters are evaluated for rapid convergence of the first-order scheme. Resulting pressure distributions and computing times are compared with the improved Murman-Cole line-relaxation method.
A tightly-coupled domain-decomposition approach for highly nonlinear stochastic multiphysics systems
Multiphysics simulations often involve nonlinear components that are driven by internally generated or externally imposed random fluctuations. When used with a domain-decomposition (DD) algorithm, such components have to be coupled in a way that both accurately propagates the noise between the subdomains and lends itself to a stable and cost-effective temporal integration. We develop a conservative DD approach in which tight coupling is obtained by using a Jacobian-free Newton–Krylov (JfNK) method with a generalized minimum residual iterative linear solver. This strategy is tested on a coupled nonlinear diffusion system forced by a truncated Gaussian noise at the boundary. Enforcement of path-wise continuity of the state variable and its flux, as opposed to continuity in the mean, at interfaces between subdomains enables the DD algorithm to correctly propagate boundary fluctuations throughout the computational domain. Reliance on a single Newton iteration (explicit coupling), rather than on the fully converged JfNK (implicit) coupling, may increase the solution error by an order of magnitude. Increase in communication frequency between the DD components reduces the explicit coupling's error, but makes it less efficient than the implicit coupling at comparable error levels for all noise strengths considered. Finally, the DD algorithm with the implicit JfNK coupling resolves temporally-correlated fluctuations of the boundary noise when the correlation time of the latter exceeds some multiple of an appropriately defined characteristic diffusion time.
Atomic resolution coherent x-ray imaging with physics-based phase retrieval
Coherent x-ray imaging and scattering from accelerator based sources such as synchrotrons continue to impact biology, medicine, technology, and materials science. Many synchrotrons around the world are currently undergoing major upgrades to increase their available coherent x-ray flux by approximately two orders of magnitude. The improvement of synchrotrons may enable imaging of materials in operando at the atomic scale which may revolutionize battery and catalysis technologies. Current algorithms used for phase retrieval in coherent x-ray imaging are based on the projection onto sets method. These traditional iterative phase retrieval methods will become more computationally expensive as they push towards atomic resolution and may struggle to converge. Additionally, these methods do not incorporate physical information that may additionally constrain the solution. In this work, we present an algorithm which incorporates molecular dynamics into Bragg coherent diffraction imaging (BCDI). This algorithm, which we call PRAMMol (Phase Retrieval with Atomic Modeling and Molecular Dynamics) combines statistical techniques with molecular dynamics to solve the phase retrieval problem. We present several examples where our algorithm is applied to simulated coherent diffraction from 3D crystals and show convergence to the correct solution at the atomic scale.
Discontinuous Finite Element Quasidiffusion Methods
Here in this paper, two-level methods for solving transport problems in one-dimensional slab geometry based on the quasi-diffusion (QD) method are developed. A linear discontinuous finite element method (LDFEM) is derived for the spatial discretization of the low-order QD (LOQD) equations. It involves special interface conditions at the cell edges based on the idea of QD boundary conditions (BCs). We consider different kinds of QD BCs to formulate the necessary cell-interface conditions. We develop two-level methods with independent discretization of the high-order transport equation and LOQD equations, where the transport equation is discretized using the method of characteristics and the LDFEM is applied to the LOQD equations. We also formulate closures that lead to the discretization consistent with a LDFEM discretization of the transport equation. The proposed methods are studied by means of test problems formulated with the method of manufactured solutions. Numerical experiments are presented demonstrating the performance of the proposed methods. Lastly, we also show that the method with independent discretization has the asymptotic diffusion limit.
A TWO-MOMENT RADIATION HYDRODYNAMICS MODULE IN ATHENA USING A TIME-EXPLICIT GODUNOV METHOD
We describe a module for the Athena code that solves the gray equations of radiation hydrodynamics (RHD), based on the first two moments of the radiative transfer equation. We use a combination of explicit Godunov methods to advance the gas and radiation variables including the non-stiff source terms, and a local implicit method to integrate the stiff source terms. We adopt the M{sub 1} closure relation and include all leading source terms to O({beta}{tau}). We employ the reduced speed of light approximation (RSLA) with subcycling of the radiation variables in order to reduce computational costs. Our code is dimensionally unsplit in one, two, and three space dimensions and is parallelized using MPI. The streaming and diffusion limits are well described by the M{sub 1} closure model, and our implementation shows excellent behavior for a problem with a concentrated radiation source containing both regimes simultaneously. Our operator-split method is ideally suited for problems with a slowly varying radiation field and dynamical gas flows, in which the effect of the RSLA is minimal. We present an analysis of the dispersion relation of RHD linear waves highlighting the conditions of applicability for the RSLA. To demonstrate the accuracy of our method, we utilize a suite of radiation and RHD tests covering a broad range of regimes, including RHD waves, shocks, and equilibria, which show second-order convergence in most cases. As an application, we investigate radiation-driven ejection of a dusty, optically thick shell in the ISM. Finally, we compare the timing of our method with other well-known iterative schemes for the RHD equations. Our code implementation, Hyperion, is suitable for a wide variety of astrophysical applications and will be made freely available on the Web.
Spectral partitioning with blends of eigenvectors
We report many common methods for data analysis rely on linear algebra. We provide new results connecting data analysis error to numerical accuracy in the context of spectral graph partitioning. We provide pointwise convergence guarantees so that spectral blends (linear combinations of eigenvectors) can be employed to solve data analysis problems with confidence in their accuracy. We apply this theory to an accessible model problem, the ring of cliques, by deriving the relevant eigenpairs and finding necessary and sufficient solver tolerances. Analysis of the ring of cliques provides an upper bound on eigensolver tolerances for graph partitioning problems. These results bridge the gap between linear algebra based data analysis methods and the convergence theory of iterative approximation methods. These results explain how the combinatorial structure of a problem can be recovered much faster than numerically accurate solutions to the associated linear algebra problem.