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DOE OSTI2012

PHURBAS: AN ADAPTIVE, LAGRANGIAN, MESHLESS, MAGNETOHYDRODYNAMICS CODE. II. IMPLEMENTATION AND TESTS

We present an algorithm for simulating the equations of ideal magnetohydrodynamics and other systems of differential equations on an unstructured set of points represented by sample particles. The particles move with the fluid, so the time step is not limited by the Eulerian Courant-Friedrichs-Lewy condition. Full spatial adaptivity is required to ensure the particles fill the computational volume and gives the algorithm substantial flexibility and power. A target resolution is specified for each point in space, with particles being added and deleted as needed to meet this target. We have parallelized the code by adapting the framework provided by GADGET-2. A set of standard test problems, including 10{sup -6} amplitude linear magnetohydrodynamics waves, magnetized shock tubes, and Kelvin-Helmholtz instabilities is presented. Finally, we demonstrate good agreement with analytic predictions of linear growth rates for magnetorotational instability in a cylindrical geometry. This paper documents the Phurbas algorithm as implemented in Phurbas version 1.1.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2021

Explicit physics-informed neural networks for nonlinear closure: The case of transport in tissues

In upscaling methods, closures for nonlinear problems present a well-known challenge. While a number of theoretical methods have been proposed for handling such closures, nonlinearities still remain a significant obstacle for many problems. In this work, we use a combination of formal upscaling and data-driven machine learning for explicitly closing a nonlinear transport and reaction process in multiscale tissues. The classical effectiveness factor model is used to formulate the macroscale reaction kinetics. We train a multilayer perceptron network using training data generated by direct numerical simulations over microscale examples. Once trained, the network is used in an algorithm for numerically solving the upscaled (coarse-grained) differential equation describing mass transport and reaction in two example tissues. The network is described as being explicit in the sense that the network is trained using macroscale concentrations and gradients of concentration as components of the feature space rather than incorporating them as part of a constraint in the optimization process. Network training and solutions to the macroscale transport equations were computed for two different tissues. The two tissue types (brain and liver) exhibit markedly different geometrical complexity and spatial scale (cell size and sample size). The upscaled solutions for the average concentration are compared with numerical solutions derived from the microscale concentration fields by a posteriori averaging. There are three outcomes of this work of particular note. 1) Our overall approach results in an upscaled nonlinear PDE. The PDE is closed using a neural network, and our approach results in the definition of the classical effectiveness factor for effecting closure. 2) We identify particular source terms for the closure problem that are important for representing the structure of the closure. These source terms involve macroscale concentrations and their gradients. We adopt these source terms to use as explicit features in the learning algorithm. We find the trained networks that include the macroscale source terms generate models that are able to predict the correction factor with increased fidelity over those that do not. 3) We find that the trained network exhibits good generalizability, and it is able to predict the effectiveness factor with high fidelity for realistically-structured tissues despite the significantly different scale and geometrical complexity of the two example tissue types. This latter result emphasizes our purposeful connection between conventional averaging methods with the use of machine learning for closure; this contrasts with some machine learning methods for upscaling where the exact form of the macroscale equation remains unknown.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1999

A Parallel Compact Multi-Dimensional Numerical Algorithm with Aeroacoustics Applications

In this study we propose a novel method to parallelize high-order compact numerical algorithms for the solution of three-dimensional PDEs (Partial Differential Equations) in a space-time domain. For this numerical integration most of the computer time is spent in computation of spatial derivatives at each stage of the Runge-Kutta temporal update. The most efficient direct method to compute spatial derivatives on a serial computer is a version of Gaussian elimination for narrow linear banded systems known as the Thomas algorithm. In a straightforward pipelined implementation of the Thomas algorithm processors are idle due to the forward and backward recurrences of the Thomas algorithm. To utilize processors during this time, we propose to use them for either non-local data independent computations, solving lines in the next spatial direction, or local data-dependent computations by the Runge-Kutta method. To achieve this goal, control of processor communication and computations by a static schedule is adopted. Thus, our parallel code is driven by a communication and computation schedule instead of the usual "creative, programming" approach. The obtained parallelization speed-up of the novel algorithm is about twice as much as that for the standard pipelined algorithm and close to that for the explicit DRP algorithm.

Povitsky, Alex↗

NASA NTRS1979

Maximum likelihood identification of aircraft parameters with unsteady aerodynamic modelling

A simplified aerodynamic force model based on the physical principle of Prandtl's lifting line theory and trailing vortex concept has been developed to account for unsteady aerodynamic effects in aircraft dynamics. Longitudinal equations of motion have been modified to include these effects. The presence of convolution integrals in the modified equations of motion led to a frequency domain analysis utilizing Fourier transforms. This reduces the integro-differential equations to relatively simple algebraic equations, thereby reducing computation time significantly. A parameter extraction program based on the maximum likelihood estimation technique is developed in the frequency domain. The extraction algorithm contains a new scheme for obtaining sensitivity functions by using numerical differentiation. The paper concludes with examples using computer generated and real flight data

Keskar, D. A.↗

DOE OSTI2018

Modeling the diffusion of heat energy within composites of homogeneous materials using the uncertainty principle

The goal of this paper is to develop a highly accurate and efficient numerical method for the solution of a time-dependent partial differential equation with a piecewise constant coefficient, on a finite interval with periodic boundary conditions. The resulting algorithm can be used, for example, to model the diffusion of heat energy in one space dimension, in the case where the spatial domain represents a medium consisting of two homogeneous materials. The resulting model has, to our knowledge, not yet been solved in closed form through analytical methods, and is difficult to solve using existing numerical methods, thus suggesting an alternative approach. The approach presented in this paper is to represent the solution as a linear combination of wave functions that change frequencies at the interfaces between different materials. It is demonstrated through numerical experiments that using the Uncertainty Principle to construct a basis of such functions, in conjunction with a spectral method, a mathematical model for heat diffusion through different materials can be solved much more efficiently than with conventional time-stepping methods.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2012

Trajectory Control of Rendezvous with Maneuver Target Spacecraft

In this paper, a nonlinear trajectory control algorithm of rendezvous with maneuvering target spacecraft is presented. The disturbance forces on the chaser and target spacecraft and the thrust forces on the chaser spacecraft are considered in the analysis. The control algorithm developed in this paper uses the relative distance and relative velocity between the target and chaser spacecraft as the inputs. A general formula of reference relative trajectory of the chaser spacecraft to the target spacecraft is developed and applied to four different proximity maneuvers, which are in-track circling, cross-track circling, in-track spiral rendezvous and cross-track spiral rendezvous. The closed-loop differential equations of the proximity relative motion with the control algorithm are derived. It is proven in the paper that the tracking errors between the commanded relative trajectory and the actual relative trajectory are bounded within a constant region determined by the control gains. The prediction of the tracking errors is obtained. Design examples are provided to show the implementation of the control algorithm. The simulation results show that the actual relative trajectory tracks the commanded relative trajectory tightly. The predicted tracking errors match those calculated in the simulation results. The control algorithm developed in this paper can also be applied to interception of maneuver target spacecraft and relative trajectory control of spacecraft formation flying.

Zhou, Zhinqiang↗

NASA NTRS1995

Aerodynamic parameter estimation via Fourier modulating function techniques

Parameter estimation algorithms are developed in the frequency domain for systems modeled by input/output ordinary differential equations. The approach is based on Shinbrot's method of moment functionals utilizing Fourier based modulating functions. Assuming white measurement noises for linear multivariable system models, an adaptive weighted least squares algorithm is developed which approximates a maximum likelihood estimate and cannot be biased by unknown initial or boundary conditions in the data owing to a special property attending Shinbrot-type modulating functions. Application is made to perturbation equation modeling of the longitudinal and lateral dynamics of a high performance aircraft using flight-test data. Comparative studies are included which demonstrate potential advantages of the algorithm relative to some well established techniques for parameter identification. Deterministic least squares extensions of the approach are made to the frequency transfer function identification problem for linear systems and to the parameter identification problem for a class of nonlinear-time-varying differential system models.

Pearson, A. E.↗

DOE OSTI2016

SU-G-IeP1-01: A Novel MRI Post-Processing Algorithm for Visualization of the Prostate LDR Brachytherapy Seeds and Calcifications Based On B0 Field Inhomogeneity Correction and Hough Transform

Purpose: This study aims at developing an MRI-only workflow for post-implant dosimetry of the prostate LDR brachytherapy seeds. The specific goal here is to develop a post-processing algorithm to produce positive contrast for the seeds and prostatic calcifications and differentiate between them on MR images. Methods: An agar-based phantom incorporating four dummy seeds (I-125) and five calcifications of different sizes (from sheep cortical bone) was constructed. Seeds were placed arbitrarily in the coronal plane. The phantom was scanned with 3T Philips Achieva MR scanner using an 8-channel head coil array. Multi-echo turbo spin echo (ME-TSE) and multi-echo gradient recalled echo (ME-GRE) sequences were acquired. Due to minimal susceptibility artifacts around seeds, ME-GRE sequence (flip angle=15; TR/TE=20/2.3/2.3; resolution=0.7×0.7×2mm3) was further processed.The induced field inhomogeneity due to the presence of titaniumencapsulated seeds was corrected using a B0 field map. B0 map was calculated using the ME-GRE sequence by calculating the phase difference at two different echo times. Initially, the product of the first echo and B0 map was calculated. The features corresponding to the seeds were then extracted in three steps: 1) the edge pixels were isolated using “Prewitt” operator; 2) the Hough transform was employed to detect ellipses approximately matching the dimensions of the seeds and 3) at the position and orientation of the detected ellipses an ellipse was drawn on the B0-corrected image. Results: The proposed B0-correction process produced positive contrast for the seeds and calcifications. The Hough transform based on Prewitt edge operator successfully identified all the seeds according to their ellipsoidal shape and dimensions in the edge image. Conclusion: The proposed post-processing algorithm successfully visualized the seeds and calcifications with positive contrast and differentiates between them according to their shapes. Further assessments on more realistic phantoms and patient study are required to validate the outcome.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2017

Distributed Automatic Differentiation for Ptychography

Synchrotron radiation light source facilities are leading the way to ultrahigh resolution X-ray imaging. High resolution imaging is essential to understanding the fundamental structure and interaction of materials at the smallest length scale possible. Diffraction based methods achieve nanoscale imaging by replacing traditional objective lenses by pixelated area detectors and computational image reconstruction. Among these methods, ptychography is quickly becoming the standard for sub-30 nanometer imaging of extended samples, but at the expense of increasingly high data rates and volumes. This paper presents a new distributed algorithm for solving the ptychographic image reconstruction problem based on automatic differentiation. Input datasets are subdivided between multiple graphics processing units (GPUs); each subset of the problem is then solved either entirely independent of other subsets (asynchronously) or through sharing gradient information with other GPUs (synchronously). The algorithm was evaluated on simulated and real data acquired at the Advanced Photon Source, scaling up to 192 GPUs. The synchronous variant of our method outperformed an existing multi-GPU implementation in terms of accuracy while running at a comparable execution time.

distributed algorithms↗

NASA NTRS1975

Special Perturbations Using Back-Correction Methods of Numerical Integration

A new class of linear multistep methods for numerical integration of differential equations is reported that permits satellite computation solutions to be corrected at certain points in the past as the integration advances in time. Algorithms have been developed for the solution of both first- and second-order differential equations. The back correction method appears to be more efficient than classical methods when dominant and perturbing forces can be separated.

Feagin, T.↗

DOE OSTI2018

On Periodic Solutions of Autonomous Systems

We propose a method for the investigation of periodic solutions of autonomous dynamical systems described by ordinary differential equations with phase and integral restrictions. We formulate the general problem of periodic solutions as a boundary-value problem with restrictions. By introducing a fictitious control, we transform the boundary-value problem into a control problem for dynamical systems with phase and integral restrictions. The control problem is solved by reducing it to an integral Fredholm equation of the first kind. We establish necessary and sufficient conditions for the existence of periodic solutions and propose an algorithm for finding periodic solution according to the limit points of the minimizing sequences.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Fast Multigrid Reduction-in-Time for Advection via Modified Semi-Lagrangian Coarse-Grid Operators

Many iterative parallel-in-time algorithms have been shown to be highly efficient for diffusion-dominated partial differential equations (PDEs) but are inefficient or even divergent when applied to advection-dominated PDEs. We consider the application of the multigrid reduction-in-time (MGRIT) algorithm to linear advection PDEs. Here, the key to efficient time integration with this method is using a coarse-grid operator that provides a sufficiently accurate approximation to the so-called ideal coarse-grid operator. For certain classes of semi-Lagrangian discretizations, we present a novel semi-Lagrangian-based coarse-grid operator that leads to fast and scalable multilevel time integration of linear advection PDEs. The coarse-grid operator is composed of a semi-Lagrangian discretization followed by a correction term, with the correction designed so that the leading-order truncation error of the composite operator is approximately equal to that of the ideal coarse-grid operator. Parallel results show substantial speed-ups over sequential time integration for variable-wave-speed advection problems in one and two spatial dimensions, and using high-order discretizations up to order five. The proposed approach establishes the first practical method that provides small and scalable MGRIT iteration counts for advection problems.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2017

Computational singular perturbation analysis of stochastic chemical systems with stiffness

Computational singular perturbation (CSP) is a useful method for analysis, reduction, and time integration of stiff ordinary differential equation systems. It has found dominant utility, in particular, in chemical reaction systems with a large range of time scales at continuum and deterministic level. On the other hand, CSP is not directly applicable to chemical reaction systems at micro or meso-scale, where stochasticity plays an non-negligible role and thus has to be taken into account. In this work we develop a novel stochastic computational singular perturbation (SCSP) analysis and time integration framework, and associated algorithm, that can be used to not only construct accurately and efficiently the numerical solutions to stiff stochastic chemical reaction systems, but also analyze the dynamics of the reduced stochastic reaction systems. Furthermore, the algorithm is illustrated by an application to a benchmark stochastic differential equation model, and numerical experiments are carried out to demonstrate the effectiveness of the construction.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2016

SU-F-R-22: Malignancy Classification for Small Pulmonary Nodules with Radiomics and Logistic Regression

Purpose: We conducted a retrospective study of Radiomics research for classifying malignancy of small pulmonary nodules. A machine learning algorithm of logistic regression and open research platform of Radiomics, IBEX (Imaging Biomarker Explorer), were used to evaluate the classification accuracy. Methods: The training set included 100 CT image series from cancer patients with small pulmonary nodules where the average diameter is 1.10 cm. These patients registered at Chang Gung Memorial Hospital and received a CT-guided operation of lung cancer lobectomy. The specimens were classified by experienced pathologists with a B (benign) or M (malignant). CT images with slice thickness of 0.625 mm were acquired from a GE BrightSpeed 16 scanner. The study was formally approved by our institutional internal review board. Nodules were delineated and 374 feature parameters were extracted from IBEX. We first used the t-test and p-value criteria to study which feature can differentiate between group B and M. Then we implemented a logistic regression algorithm to perform nodule malignancy classification. 10-fold cross-validation and the receiver operating characteristic curve (ROC) were used to evaluate the classification accuracy. Finally hierarchical clustering analysis, Spearman rank correlation coefficient, and clustering heat map were used to further study correlation characteristics among different features. Results: 238 features were found differentiable between group B and M based on whether their statistical p-values were less than 0.05. A forward search algorithm was used to select an optimal combination of features for the best classification and 9 features were identified. Our study found the best accuracy of classifying malignancy was 0.79±0.01 with the 10-fold cross-validation. The area under the ROC curve was 0.81±0.02. Conclusion: Benign nodules may be treated as a malignant tumor in low-dose CT and patients may undergo unnecessary surgeries or treatments. Our study may help radiologists to differentiate nodule malignancy for low-dose CT.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2017

Quinoa, Version 0.2

Quinoa is a set of computational tools that enables research and numerical analysis in fluid dynamics. At this time it remains a test-bed to experiment with various algorithms using fully asynchronous runtime systems. Currently, Quinoa consists of the following tools: (1) Walker, a numerical integrator for systems of stochastic differential equations in time. It is a mathematical tool to analyze and design the behavior of stochastic differential equations. It allows the estimation of arbitrary coupled statistics and probability density functions and is currently used for the design of statistical moment approximations for multiple mixing materials in variable-density turbulence. (2) Inciter, an overdecomposition-aware finite element field solver for partial differential equations using 3D unstructured grids. Inciter is used to research asynchronous mesh-based algorithms and to experiment with coupling asynchronous to bulk-synchronous parallel code. Two planned new features of Inciter, compared to the previous release (LA-CC-16-015), to be implemented in 2017, are (a) a simple Navier-Stokes solver for ideal single-material compressible gases, and (b) solution-adaptive mesh refinement (AMR), which enables dynamically concentrating compute resources to regions with interesting physics. Using the NS-AMR problem we plan to explore how to scale such high-load-imbalance simulations, representative of large production multiphysics codes, to very large problems on very large computers using an asynchronous runtime system. (3) RNGTest, a test harness to subject random number generators to stringent statistical tests enabling quantitative ranking with respect to their quality and computational cost. (4) UnitTest, a unit test harness, running hundreds of tests per second, capable of testing serial, synchronous, and asynchronous functions. (5) MeshConv, a mesh file converter that can be used to convert 3D tetrahedron meshes from and to either of the following formats: Gmsh, http://www.geuz.org/gmsh, Netgen, http://sourceforge.net/apps/mediawiki/netgen-mesher, ExodusII, http://sourceforge.net/projects/exodusii, HyperMesh, http://www.altairhyperworks.com/product/HyperMesh.

Bakosi, Jozsef↗

DOE OSTI2014

Differential visualisation of a spectrally selective structure of strongly scattering objects

We describe a modification of the algorithm for the fast approximate solution of the diffuse optical tomography inverse problem. In this modification the amount of a priori (auxiliary) information necessary for the visualisation of the internal structure of the object is reduced by using a differential measurement scheme. The experiment is performed at two different wavelengths, and some a priori information, necessary to reconstruct only the spectrally selective component of the internal structure (the difference structure of the spatial distributions of the extinction coefficient at the wavelength employed), is replaced by the data of one of these measurements. (laser biophotonics)

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1994

Methodology for sensitivity analysis, approximate analysis, and design optimization in CFD for multidisciplinary applications

The straightforward automatic-differentiation and the hand-differentiated incremental iterative methods are interwoven to produce a hybrid scheme that captures some of the strengths of each strategy. With this compromise, discrete aerodynamic sensitivity derivatives are calculated with the efficient incremental iterative solution algorithm of the original flow code. Moreover, the principal advantage of automatic differentiation is retained (i.e., all complicated source code for the derivative calculations is constructed quickly with accuracy). The basic equations for second-order sensitivity derivatives are presented; four methods are compared. Each scheme requires that large systems are solved first for the first-order derivatives and, in all but one method, for the first-order adjoint variables. Of these latter three schemes, two require no solutions of large systems thereafter. For the other two for which additional systems are solved, the equations and solution procedures are analogous to those for the first order derivatives. From a practical viewpoint, implementation of the second-order methods is feasible only with software tools such as automatic differentiation, because of the extreme complexity and large number of terms. First- and second-order sensitivities are calculated accurately for two airfoil problems, including a turbulent flow example; both geometric-shape and flow-condition design variables are considered. Several methods are tested; results are compared on the basis of accuracy, computational time, and computer memory. For first-order derivatives, the hybrid incremental iterative scheme obtained with automatic differentiation is competitive with the best hand-differentiated method; for six independent variables, it is at least two to four times faster than central finite differences and requires only 60 percent more memory than the original code; the performance is expected to improve further in the future.

Taylor, Arthur C., III↗

NASA NTRS2006

The Latitudinal Excursion of Coronal Magnetic Field Lines in Response to Differential Rotation: MHD Simulations

Solar energetic particles, which are believed to originate from corotating interacting regions (CIRS) at low heliographic latitude, were observed by the Ulysses spacecraft even as it passed over the Sun's poles. One interpretation of this result is that high-latitude field lines intercepted by Ulysses connect to low-latitude CIRs at much larger heliocentric distances. The Fisk model explains the latitudinal excursion of magnetic field lines in the solar corona and heliosphere as the inevitable consequence of the interaction of a tilted dipole in a differentially rotating photosphere with rigidly rotating coronal holes. We use a time-dependent three-dimensional magnetohydrodynamic (MHD) algorithm to follow the evolution of a simple model of the solar corona in response to the differential rotation of the photospheric magnetic flux. We examine the changes of the coronal-hole boundaries, the redistribution of the line-of-sight magnetic field, and the precession of field lines in the corona. Our results confirm the basic idea of the Fisk model, that differential rotation leads to changes in the heliographic latitude of magnetic field lines. However, the latitudinal excursion of magnetic field lines in this simple "tilted dipole" model is too small to explain the Ulysses observations. Although coronal holes in our model rotate more rigidly than do photospheric features (in general agreement with observations), they do not rotate strictly rigidly as assumed by Fisk. This basic difference between our model and Fisk's will be explored in the future by considering more realistic magnetic flux distributions, as observed during Ulysses polar excursions.

Lionello, Roberto↗