Reducing memory cost of exact diagonalization using singular value decomposition
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Weinstein, Marvin↗
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Parallel domain decomposition method for finite element approximation of 3D steady state non‐Newtonian fluids
Summary We introduce a stabilized finite element method for the 3D non‐Newtonian Navier–Stokes equations and a parallel domain decomposition method for solving the sparse system of nonlinear equations arising from the discretization. Non‐Newtonian flow problems are, generally speaking, more challenging than Newtonian flows because the nonlinearities are not only in the convection term but also in the viscosity term, which depends on the shear rate. Many good iterative methods and preconditioning techniques that work well for the Newtonian flows do not work well for the non‐Newtonian flows. We employ a Galerkin/least squares finite element method, with stabilization parameters adjusted to count the non‐Newtonian effect, to discretize the equations, and the resulting highly nonlinear system of equations is solved by a Newton–Krylov–Schwarz algorithm. In this study, we apply the proposed method to some inelastic power‐law fluid flows through the eccentric annuli with inner cylinder rotation and investigate the robustness of the method with respect to some physical parameters, including the power‐law index and the Reynolds number ratios. We then report the superlinear speedup achieved by the domain decomposition algorithm on a computer with up to 512 processors. Copyright © 2015 John Wiley & Sons, Ltd.
Shiu, Wen‐Shin↗
Certified dual-corrected radiation patterns of phased antenna arrays by offline–online order reduction of finite-element models
This paper presents a fast numerical method for computing certified far-field patterns of phased antenna arrays over broad frequency bands as well as wide ranges of steering and look angles. The proposed scheme combines finite-element analysis, dual-corrected model-order reduction, and empirical interpolation. To assure the reliability of the results, improved a posteriori error bounds for the radiated power and directive gain are derived. Both the reduced-order model and the error-bounds algorithm feature offline–online decomposition. A real-world example is provided to demonstrate the efficiency and accuracy of the suggested approach.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
WE-EF-207-09: Single-Scan Dual-Energy CT Using Primary Modulation
Purpose: Compared with conventional CT, dual energy CT (DECT) provides better material differentiation but requires projection data with two different effective x-ray spectra. Current DECT scanners use either a two-scan setting or costly imaging components, which are not feasible or available on open-gantry cone-beam CT systems. We propose a hardware-based method which utilizes primary modulation to enable single-scan DECT on a conventional CT scanner. The CT imaging geometry of primary modulation is identical to that used in our previous method for scatter removal, making it possible for future combination with effective scatter correction on the same CT scanner. Methods: We insert an attenuation sheet with a spatially-varying pattern - primary modulator-between the x-ray source and the imaged object. During the CT scan, the modulator selectively hardens the x-ray beam at specific detector locations. Thus, the proposed method simultaneously acquires high and low energy data. High and low energy CT images are then reconstructed from projections with missing data via an iterative CT reconstruction algorithm with gradient weighting. Proof-of-concept studies are performed using a copper modulator on a cone-beam CT system. Results: Our preliminary results on the Catphan(c) 600 phantom indicate that the proposed method for single-scan DECT is able to successfully generate high-quality high and low energy CT images and distinguish different materials through basis material decomposition. By applying correction algorithms and using all of the acquired projection data, we can reconstruct a single CT image of comparable image quality to conventional CT images, i.e., without primary modulation. Conclusion: This work shows great promise in using a primary modulator to perform high-quality single-scan DECT imaging. Future studies will test method performance on anthropomorphic phantoms and perform quantitative analyses on image qualities and DECT decomposition accuracy. We will use simulations to optimize the modulator material and geometry parameters.
60 APPLIED LIFE SCIENCES↗
Algebraic multigrid domain and range decomposition (AMG-DD / AMG-RD)*
In modern large-scale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMG-DD) and algebraic multigrid range decomposition (AMG-RD), that replace traditional AMG V-cycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods in Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135--155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224--233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 1411--1443], they differ primarily in that they are purely algebraic: AMG-RD and AMG-DD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMG-RD and AMG-DD is their novel residual communication process that enables effective parallel computation on composite grids, avoiding the all-to-all communication costs of the geometric methods. The main purpose of this paper is to study the potential of these two algebraic methods as possible alternatives to existing AMG approaches for future parallel machines. As a result, this paper develops some theoretical properties of these methods and reports on serial numerical tests of their convergence properties over a spectrum of problem parameters.
97 MATHEMATICS AND COMPUTING↗
Parallel performance of algebraic multigrid domain decomposition
Algebraic multigrid (AMG) is a widely used scalable solver and preconditioner for large-scale linear systems resulting from the discretization of a wide class of elliptic PDEs. While AMG has optimal computational complexity, the cost of communication has become a significant bottleneck that limits its scalability as processor counts continue to grow on modern machines. This article examines the design, implementation, and parallel performance of a novel algorithm, algebraic multigrid domain decomposition (AMG-DD), designed specifically to limit communication. The goal of AMG-DD is to provide a low-communication alternative to standard AMG V-cycles by trading some additional computational overhead for a significant reduction in communication cost. Numerical results show that AMG-DD achieves superior accuracy per communication cost compared with AMG, and speedup over AMG is demonstrated on a large GPU cluster.
97 MATHEMATICS AND COMPUTING↗
SNoGloDe: A Structured Nonlinear Global Decomposition Solver
Large-scale optimization problems often require decomposition strategies and customized algorithms to achieve optimal solutions within a reasonable time. Building on the work of Cao and Zavala (2019) for solving nonlinear two-stage stochastic programs to global optimality, we implement and extend their approach. We generalize to optimization problems reformulated with a block-angular constraint structure (e.g., temporal decomposition). Our framework, written in Python using Pyomo, is highly customizable and enables parallel execution of the decomposition. SNoGloDe allows tailored branching strategies, lower bounding problems, and candidate generators to leverage problem-specific knowledge. To demonstrate effectiveness, we compare SNoGloDe’s performance with Gurobi on a temporally decomposed produced water case study.
algorithms↗
New method for computer analysis of complex intermetallic compounds and nanocluster model of the samson phase Cd{sub 3}Cu{sub 4}
A new method is proposed for the computer analysis of crystal structures of complex intermetallic compounds (with more than 1000 atoms per unit cell) using a developed algorithm of the complete decomposition of the 3D graph of the structure into nanocluster substructures. This method has been implemented in the TOPOS software package and approved successfully in an analysis of the complex Cu{sub 3}Cd{sub 4} structure (Samson phase). Cu{sub 3}Cd{sub 4} structure models were used to establish a structural relationship between nanoclusters in this intermetallic compound and nanoclusters in other complex crystal structures: ZrZn{sub 22}, Ru{sub 7}Mg{sub 44}, NaCd{sub 2}, and Mg{sub 2}Al{sub 3}.
36 MATERIALS SCIENCE↗
Distributed Augmentation, Hypersweeps, and Branch Decomposition of Contour Trees for Scientific Exploration
Contour trees describe the topology of level sets in scalar fields and are widely used in topological data analysis and visualization. A main challenge of utilizing contour trees for large-scale scientific data is their computation at scale using highperformance computing. To address this challenge, recent work has introduced distributed hierarchical contour trees for distributed computation and storage of contour trees. However, effective use of these distributed structures in analysis and visualization requires subsequent computation of geometric properties and branch decomposition to support contour extraction and exploration. In this work, we introduce distributed algorithms for augmentation, hypersweeps, and branch decomposition that enable parallel computation of geometric properties, and support the use of distributed contour trees as query structures for scientific exploration. Finally, we evaluate the parallel performance of these algorithms and apply them to identify and extract important contours for scientific visualization.
97 MATHEMATICS AND COMPUTING↗
Validation and Demonstration of Control System Functional Capabilities within the IES Plug-and-Play Simulation Environment
The concept of an integrated energy system (IES) is meant to combine different energy technologies in synergistic ways to achieve a more secure and economical energy supply. The RAVEN-based HYBRID framework is used to find the optimal installed capacity and the optimal economical dispatch of each component of the IES. The new RAVEN plugin for grid and capacity optimization (HERON) only addresses the limits that affect the production variables and the corresponding rates of variation (explicit constraints). However, other variables are subject to constraints, and the associated limits should be accounted for (implicit constraints). In particular, for the power dispatch problem, the optimization algorithm takes into account the limits on the electrical power output and the corresponding hourly power variations but does not consider other constraints on process variables whose response affects the service life of the IES. This report describes a scheme that allows accounting for implicit constraints without increasing the size of the optimization problem. To obtain a more accurate approximation of the nonlinear dynamic behavior, a parametric version of the dynamic mode decomposition with control (DMDc) algorithm was developed to derive the state-space representation matrices of the IES components at different scheduling parameter. Thanks to this approach, a more accurate approximation of the system response can be obtained, the limits imposed by thermal mechanical implicit constraints can be translated into power dispatch limits, and the feedbacks to HERON power dispatcher can be provided. To assess the developed methodology, a power dispatching test case composed of three power generating and storage units (Balance of Plant, Secondary Energy Source, Thermal Energy Storage) was developed. The power output of each one of the three units was optimized to meet the imposed time-dependent load demand trajectory and to maximize the IES profitability by meeting both the explicit and implicit constraints.
24 POWER TRANSMISSION AND DISTRIBUTION↗
A new approach to detailed structural decomposition from the splash and phat surveys: Kicked-up disk stars in the Andromeda galaxy?
We characterize the bulge, disk, and halo subcomponents in the Andromeda galaxy (M31) over the radial range 4 kpc < R {sub proj} < 225 kpc. The cospatial nature of these subcomponents renders them difficult to disentangle using surface brightness (SB) information alone, especially interior to ∼20 kpc. Our new decomposition technique combines information from the luminosity function (LF) of over 1.5 million bright (20 < m {sub 814W} < 22) stars from the Panchromatic Hubble Andromeda Treasury survey, radial velocities of over 5000 red giant branch stars in the same magnitude range from the Spectroscopic and Photometric Landscape of Andromeda's Stellar Halo survey, and integrated I-band SB profiles from various sources. We use an affine-invariant Markov chain Monte Carlo algorithm to fit an appropriate toy model to these three data sets. The bulge, disk, and halo SB profiles are modeled as a Sérsic, exponential, and cored power law, respectively, and the LFs are modeled as broken power laws. We present probability distributions for each of 32 parameters describing the SB profiles and LFs of the three subcomponents. We find that the number of stars with a disk-like LF is 5.2% ± 2.1% larger than the number with disk-like (dynamically cold) kinematics, suggesting that some stars born in the disk have been dynamically heated to the point that they are kinematically indistinguishable from halo members. This is the first kinematical evidence for a 'kicked-up disk' halo population in M31. The fraction of kicked-up disk stars is consistent with that found in simulations. We also find evidence for a radially varying disk LF, consistent with a negative metallicity gradient in the stellar disk.
79 ASTRONOMY AND ASTROPHYSICS↗
Graph decomposition techniques for solving combinatorial optimization problems with variational quantum algorithms
The quantum approximate optimization algorithm (QAOA) has the potential to approximately solve complex combinatorial optimization problems in polynomial time. However, current noisy quantum devices cannot solve large problems due to hardware constraints. In this work, we develop an algorithm that decomposes the QAOA input problem graph into a smaller problem and solves MaxCut using QAOA on the reduced graph. The algorithm requires a subroutine that can be classical or quantum—in this work, we implement the algorithm twice on each graph. One implementation uses the classical solver Gurobi in the subroutine and the other uses QAOA. We solve these reduced problems with QAOA. On average, the reduced problems require only approximately 1/10 of the number of vertices than the original MaxCut instances. Furthermore, the average approximation ratio of the original MaxCut problems is 0.75, while the approximation ratios of the decomposed graphs are on average of 0.96 for both Gurobi and QAOA. With this decomposition, we are able to measure optimal solutions for ten 100-vertex graphs by running single-layer QAOA circuits on the Quantinuum trapped-ion quantum computer H1-1, sampling each circuit only 500 times. This approach is best suited for sparse, particularly k-regular graphs, as k-regular graphs on n vertices can be decomposed into a graph with at most $\frac{nk}{k+1}$ vertices in polynomial time. Further reductions can be obtained with a potential trade-off in computational time. In conclusion, while this paper applies the decomposition method to the MaxCut problem, it can be applied to more general classes of combinatorial optimization problems.
97 MATHEMATICS AND COMPUTING↗
A Screen Space GPGPU Surface LIC Algorithm for Distributed Memory Data Parallel Sort Last Rendering Infrastructures
The surface line integral convolution(LIC) visualization technique produces dense visualization of vector fields on arbitrary surfaces. We present a screen space surface LIC algorithm for use in distributed memory data parallel sort last rendering infrastructures. The motivations for our work are to support analysis of datasets that are too large to fit in the main memory of a single computer and compatibility with prevalent parallel scientific visualization tools such as ParaView and VisIt. By working in screen space using OpenGL we can leverage the computational power of GPUs when they are available and run without them when they are not. We address efficiency and performance issues that arise from the transformation of data from physical to screen space by selecting an alternate screen space domain decomposition. We analyze the algorithm's scaling behavior with and without GPUs on two high performance computing systems using data from turbulent plasma simulations.
Loring, Burlen↗
SU-G-JeP4-03: Anomaly Detection of Respiratory Motion by Use of Singular Spectrum Analysis
Purpose: The implementation and realization of automatic anomaly detection of respiratory motion is a very important technique to prevent accidental damage during radiation therapy. Here, we propose an automatic anomaly detection method using singular value decomposition analysis. Methods: The anomaly detection procedure consists of four parts:1) measurement of normal respiratory motion data of a patient2) calculation of a trajectory matrix representing normal time-series feature3) real-time monitoring and calculation of a trajectory matrix of real-time data.4) calculation of an anomaly score from the similarity of the two feature matrices. Patient motion was observed by a marker-less tracking system using a depth camera. Results: Two types of motion e.g. cough and sudden stop of breathing were successfully detected in our real-time application. Conclusion: Automatic anomaly detection of respiratory motion using singular spectrum analysis was successful in the cough and sudden stop of breathing. The clinical use of this algorithm will be very hopeful. This work was supported by JSPS KAKENHI Grant Number 15K08703.
60 APPLIED LIFE SCIENCES↗
A general framework of noise suppression in material decomposition for dual-energy CT
Purpose: As a general problem of dual-energy CT (DECT), noise amplification in material decomposition severely reduces the signal-to-noise ratio on the decomposed images compared to that on the original CT images. In this work, the authors propose a general framework of noise suppression in material decomposition for DECT. The method is based on an iterative algorithm recently developed in their group for image-domain decomposition of DECT, with an extension to include nonlinear decomposition models. The generalized framework of iterative DECT decomposition enables beam-hardening correction with simultaneous noise suppression, which improves the clinical benefits of DECT. Methods: The authors propose to suppress noise on the decomposed images of DECT using convex optimization, which is formulated in the form of least-squares estimation with smoothness regularization. Based on the design principles of a best linear unbiased estimator, the authors include the inverse of the estimated variance–covariance matrix of the decomposed images as the penalty weight in the least-squares term. Analytical formulas are derived to compute the variance–covariance matrix for decomposed images with general-form numerical or analytical decomposition. As a demonstration, the authors implement the proposed algorithm on phantom data using an empirical polynomial function of decomposition measured on a calibration scan. The polynomial coefficients are determined from the projection data acquired on a wedge phantom, and the signal decomposition is performed in the projection domain. Results: On the Catphan{sup ®}600 phantom, the proposed noise suppression method reduces the average noise standard deviation of basis material images by one to two orders of magnitude, with a superior performance on spatial resolution as shown in comparisons of line-pair images and modulation transfer function measurements. On the synthesized monoenergetic CT images, the noise standard deviation is reduced by a factor of 2–3. By using nonlinear decomposition on projections, the authors’ method effectively suppresses the streaking artifacts of beam hardening and obtains more uniform images than their previous approach based on a linear model. Similar performance of noise suppression is observed in the results of an anthropomorphic head phantom and a pediatric chest phantom generated by the proposed method. With beam-hardening correction enabled by their approach, the image spatial nonuniformity on the head phantom is reduced from around 10% on the original CT images to 4.9% on the synthesized monoenergetic CT image. On the pediatric chest phantom, their method suppresses image noise standard deviation by a factor of around 7.5, and compared with linear decomposition, it reduces the estimation error of electron densities from 33.3% to 8.6%. Conclusions: The authors propose a general framework of noise suppression in material decomposition for DECT. Phantom studies have shown the proposed method improves the image uniformity and the accuracy of electron density measurements by effective beam-hardening correction and reduces noise level without noticeable resolution loss.
60 APPLIED LIFE SCIENCES↗
Online randomized interpolative decomposition with a posteriori error estimator for temporal PDE data reduction
Traditional low-rank approximation is a powerful tool for compressing large data matrices that arise in simulations of partial differential equations (PDEs), but suffers from high computational cost and requires several passes over the PDE data. The compressed data may also lack interpretability thus making it difficult to identify feature patterns from the original data. Here, to address these issues, we present an online randomized algorithm to compute the interpolative decomposition (ID) of large-scale data matrices in situ. Compared to previous randomized IDs that used the QR decomposition to determine the column basis, we adopt a streaming ridge leverage score-based column subset selection algorithm that dynamically selects proper basis columns from the data and thus avoids an extra pass over the data to compute the coefficient matrix of the ID. In particular, we adopt a single-pass error estimator based on the non-adaptive Hutch++ algorithm to provide real-time error approximation for determining the best coefficients. As a result, our approach only needs a single pass over the original data and thus is suitable for large and high-dimensional matrices stored outside of core memory or generated in PDE simulations. A strategy to improve the accuracy of the reconstructed data gradient, when desired, within the ID framework is also presented. We provide numerical experiments on turbulent channel flow and ignition simulations, and on the NSTX Gas Puff Image dataset, comparing our algorithm with the offline ID algorithm to demonstrate its utility in real-world applications.
Column subset selection↗
An efficient parallel simulation of unsteady blood flows in patient-specific pulmonary artery
Simulation of blood flows in the pulmonary artery provides some insight into certain diseases by examining the relationship between some continuum metrics, eg, the wall shear stress acting on the vascular endothelium, which responds to flow-induced mechanical forces by releasing vasodilators/constrictors. V. Kheyfets, in his previous work, studies numerically a patient-specific pulmonary circulation to show that decreasing wall shear stress is correlated with increasing pulmonary vascular impedance. We develop a scalable parallel algorithm based on domain decomposition methods to investigate an unsteady model with patient-specific pulsatile waveforms as the inlet boundary condition. The unsteady model offers tremendously more information about the dynamic behavior of the flow field, but computationally speaking, the simulation is a lot more expensive since a problem which is similar to the steady-state problem has to be solved many times, and therefore, the traditional sequential approach is not suitable anymore. We show computationally that simulations using the proposed parallel approach with up to 10 000 processor cores can be obtained with much reduced compute time. Finally, this makes the technology potentially usable for the routine study of the dynamic behavior of blood flows in the pulmonary artery, in particular, the changes of the blood flows and the wall shear stress in the spatial and temporal dimensions.
60 APPLIED LIFE SCIENCES↗
Benchmarking ADMM in Nonconvex NLPs
Here, we study connections between the alternating direction method of multipliers (ADMM), the classical method of multipliers (MM), and progressive hedging (PH). The connections are used to derive benchmark metrics and strategies to monitor and accelerate convergence and to help explain why ADMM and PH are capable of solving complex nonconvex NLPs. Specifically, we observe that ADMM is an inexact version of MM and approaches its performance when multiple coordination steps are performed. In addition, we use the observation that PH is a specialization of ADMM and borrow Lyapunov function and primal-dual feasibility metrics used in ADMM to explain why PH is capable of solving nonconvex NLPs. This analysis also highlights that specialized PH schemes can be derived to tackle a wider range of stochastic programs and even other problem classes. Our exposition is tutorial in nature and seeks to to motivate algorithmic improvements and new decomposition strategies.
42 ENGINEERING↗