Mechanical impedance analysis for lumped parameter multi-degree of freedom/multi- dimensional systems
Mechanical impedance effects on dynamic response of multi-degree of freedom/multi-dimensional linear systems
LINEAR SYSTEM↗
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Results for “linear system”
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Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients
We present a robust and scalable preconditioner for the solution of large-scale linear systems that arise from the discretization of elliptic PDEs amenable to rank compression. The preconditioner is based on hierarchical low-rank approximations and the cyclic reduction method. The setup and application phases of the preconditioner achieve log-linear complexity in memory footprint and number of operations, and numerical experiments exhibit good weak and strong scalability at large processor counts in a distributed memory environment. Numerical experiments with linear systems that feature symmetry and nonsymmetry, definiteness and indefiniteness, constant and variable coefficients demonstrate the preconditioner applicability and robustness. Furthermore, it is possible to control the number of iterations via the accuracy threshold of the hierarchical matrix approximations and their arithmetic operations, and the tuning of the admissibility condition parameter. Together, these parameters allow for optimization of the memory requirements and performance of the preconditioner.
97 MATHEMATICS AND COMPUTING↗
High-order matrix-free incompressible flow solvers with GPU acceleration and low-order refined preconditioners
In this work, we present a matrix-free flow solver for high-order finite element discretizations of the incompressible Navier-Stokes and Stokes equations with GPU acceleration. For high polynomial degrees, assembling the matrix for the linear systems resulting from the finite element discretization can be prohibitively expensive, both in terms of computational complexity and memory. For this reason, it is necessary to develop matrix-free operators and preconditioners, which can be used to efficiently solve these linear systems without access to the matrix entries themselves. The matrix-free operator evaluations utilize GPU-accelerated sum-factorization techniques to minimize memory movement and maximize throughput. The preconditioners developed in this work are based on a low-order refined methodology with parallel subspace corrections, as described for diffusion problems in [1]. The saddle-point Stokes system is solved using block-preconditioning techniques, which are robust in mesh size, polynomial degree, time step, and viscosity. For the incompressible Navier-Stokes equations, we make use of projection (fractional step) methods, which require Helmholtz and Poisson solves at each time step. The performance of our flow solvers is assessed on several benchmark problems in two and three spatial dimensions.
97 MATHEMATICS AND COMPUTING↗
NeuroFEM
SAND2025-00525O NeuroFEM is a software tool that demonstrates a neuromorphic algorithm for solving finite element problems. It sets up a 2D finite element problem for the Poisson equation on a disk, constructs synaptic matrices, and simulates neural dynamics to solve the resulting sparse linear system. The software illustrates how the algorithm converges to the solution and plots the results, showcasing a neuromorphic counterpart to traditional methods like Conjugate Gradient or GMRES. This tool is designed to highlight the potential of neuromorphic algorithms for solving sparse linear systems, which are prevalent in various computational applications. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.
SciDAC↗
Dynamic Mode Decomposition of Solids
Dynamic mode decomposition (DMD) is a method that has gained notoriety in the field of turbulent fluid flows as a method for decomposing the flow field into modes that could be further deconstructed to understand their influence on the overall dynamics of the system. Forays into solids and non-linear systems were considered, but not applied. In this work, DMD was applied for the first time to the heat-diffusion and reactive heat diffusion equations on a random particle pack of uniform solid spheres. A verification of a linear heat-diffusion test problem was successful, showing equality between the normal modes and Koopman modes obtained from DMD. Further application to a non-linear reactive system revealed stability limits of the underlying modes which are dependent on microstructure and chemical kinetics. This work will enable the development of reactive material models based on further analysis using DMD to quantify the statistical dependencies of transient response on microstructural characteristics.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
On the stability of linear discrete dynamic systems
Equilibria stability of linear discrete dynamic systems involving elastic, nonconservative, dissipative and gyroscopic forces, using Liapunov-type energy method
Walker, J. A.↗
The RAPT control method for nonlinear systems
The Rapid Advancement Preview Tracking (RAPT) control method uses both a nonlinear system model and a linearized system model to provide a form of feedback/preview control for nonlinear systems. It uses measurements of the states to calculate the current preview control signal from a local linearization of the state equations, and then iteratively calculates a perturbation correction for that control using the nonlinear equations. The corrected control signal is then applied to the system, driving the end-effector to the next position along a specified trajectory. Computer simulations of tracking control for two-link planar manipulators indicate that this algorithm may be fast enough for real-time control.
Long, T. W.↗
Iterative methods in GPU-resident linear solvers for nonlinear constrained optimization
Linear solvers are major computational bottlenecks in a wide range of decision support and optimization computations. The challenges become even more pronounced on heterogeneous hardware, where traditional sparse numerical linear algebra methods are often inefficient. For example, methods for solving ill-conditioned linear systems have relied on conditional branching, which degrades performance on hardware accelerators such as graphical processing units (GPUs). To improve the efficiency of solving ill-conditioned systems, our computational strategy separates computations that are efficient on GPUs from those that need to run on traditional central processing units (CPUs). Our strategy maximizes the reuse of expensive CPU computations. Iterative methods, which thus far have not been broadly used for ill-conditioned linear systems, play an important role in our approach. In particular, we extend ideas from Arioli et al., (2007) to implement iterative refinement using inexact LU factors and flexible generalized minimal residual (FGMRES), with the aim of efficient performance on GPUs. In conclusion, we focus on solutions that are effective within broader application contexts, and discuss how early performance tests could be improved to be more predictive of the performance in a realistic environment.
97 MATHEMATICS AND COMPUTING↗
Chapter 4 - Recent Advances in Identification of Differential Equations from Noisy Data: IDENT Review
Differential equations and numerical methods are extensively used to model various real-world phenomena in science and engineering. With modern developments, we aim to find the underlying differential equation from a single observation of time-dependent data. If we assume that the differential equation is a linear combination of various linear and nonlinear differential terms, then the identification problem can be formulated as solving a linear system. The goal then reduces to finding the optimal coefficient vector that best represents the time derivative of the given data. We review some recent works on the identification of differential equations. We find some common themes for the improved accuracy: (i) The formulation of linear system with proper denoising is important, (ii) how to utilize sparsity and model selection to find the correct coefficient support needs careful attention, and (iii) there are ways to improve the coefficient recovery. We present an overview and analysis of recent developments on the topic.
97 MATHEMATICS AND COMPUTING↗
A Data-driven approach to Determining the Fidelity in the Hardware-in-the-loop Systems using Subspace Identification Method
One of the major questions in any Hardware-in-the-loop (HiL) simulation is to understand the fidelity of the HiL simulation itself which is most often indicated qualitatively as either high or low instead of quantifying it. Being cognizant of the level of fidelity forms the crux to assess the validity and credibility of the HiL simulation. In this work, we address this issue by developing a systematic, data-driven approach to assess the fidelity of any HiL simulation, and more specifically, the fidelity of the interface between the simulator and hardware in an HiL simulation. Applying a subspace identification method, a linear system representing the interface is obtained from the time-series data captured between the hardware and simulator in the HiL simulation. Finally, the fidelity is defined based on this linear system properties. The proposed data-driven fidelity quantification framework is illustrated on the IEEE 123 node feeder system running on HYPERSIM and interacting with virtual protection relays.
97 MATHEMATICS AND COMPUTING↗
Domain decomposition in time for PDE-constrained optimization
Here, PDE-constrained optimization problems have a wide range of applications, but they lead to very large and ill-conditioned linear systems, especially if the problems are time dependent. In this paper we outline an approach for dealing with such problems by decomposing them in time and applying an additive Schwarz preconditioner in time, so that we can take advantage of parallel computers to deal with the very large linear systems. We then illustrate the performance of our method on a variety of problems.
97 MATHEMATICS AND COMPUTING↗
Performance of an Astrophysical Radiation Hydrodynamics Code under Scalable Vector Extension Optimization
We present results of a performance study of an astrophysical radiation hydrodynamics code, V2D, on the Arm-based A64FX processor developed by Fujitsu. The code solves sparse linear systems, a task for which the A64FX architecture should be well suited. Here, we performed the performance analysis study on Ookami, an Apollo 80 platform utilizing the A64FX processor. We explored several compilers and performance anal-ysis packages and found the code did not perform as expected under scalable vector extension optimization, suggesting that a “deeper dive” into analyzing the code is worthwhile. However, a simple driver program that exercised basic sparse linear algebra routines used by V2D did show significant speedup with the use of the scalable vector extension optimization. We present the initial results from the study which used V2D on a relatively simple test problem that emphasized the repeated solution of sparse linear systems.
79 ASTRONOMY AND ASTROPHYSICS↗
Random Phase Approximation Correlation Energy Using Real-Space Density Functional Perturbation Theory
We present a real-space method for computing the random phase approximation (RPA) correlation energy within Kohn–Sham density functional theory, leveraging the low-rank nature of the frequency-dependent density response operator. In particular, we employ a cubic-scaling formalism based on density functional perturbation theory that circumvents the calculation of the response function matrix, instead relying on the ability to compute its product with a vector through the solution of the associated Sternheimer linear systems. We develop a large-scale parallel implementation of this formalism using the subspace iteration method in conjunction with the spectral quadrature method while employing the Kronecker product-based method for the application of the Coulomb operator and the conjugate orthogonal conjugate gradient method for the solution of the linear systems. We demonstrate convergence with respect to key parameters and verify the method’s accuracy by comparing with plane-wave results. We show that the framework achieves good strong scaling to many thousands of processors, reducing the time to solution for a lithium hydride system with 128 electrons to around 150 s on 4608 processors.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
A non‐intrusive domain‐decomposition model reduction method for linear steady‐state partial differential equations with random coefficients
Abstract Domain decomposition methods have been proved to be an effective strategy to reduce the dimension of parametric partial differential equations (PDEs). However, existing domain decomposition methods for parametric PDEs are usually intrusive, which means domain decomposition based solvers need to be implemented from scratch for each target parametric PDE. To address this issue, we develop a new non‐intrusive domain‐decomposition model reduction method for linear steady‐state PDEs with random‐field coefficients. As a variant of our previous work by Mu and Zhang, the new method only needs access to the final linear system, that is, the global stiffness matrix and the right hand side, of a deterministic PDE solver, in order to build a domain‐decomposition‐based reduced model without intrusive implementation from scratch. The key idea is to remove the interface condition between sub‐domains and rely on the correlation between columns of the linear system to couple the sub‐domains. The non‐intrusive feature enables the applicability of the proposed method to a broader class of uncertainty quantification problems, where many legacy codes/solvers can be fully reused by our method. Two numerical examples including diffusion equations with random diffusivity and convection‐dominated transport with random velocity, are provided to demonstrate the effectiveness and efficiency of our method.
Zhang, Guannan↗
Recent results on output feedback problems
Given a real linear system sigma = (A, B, C) with m inputs, p outputs and degree n, the problem of generic pole placement by output feedback is studied, which is to compute the constant C(m,p) such that the inequality C(m,p) not less than n is necessary and sufficient for generically positioning the poles of the generic linear system by constant output feedback. A constant C prime (m,p) is determined, which gives a sufficient condition for generic pole placement and which, to the best of the author's knowledge, is at least as good an estimate of C(m,p) as any in the literature. Some results on the construction of solutions in case mp = n are announced, based on the degree formula of Brockett and Byrnes and the Galois theory. In particular, a question raised by Anderson, Bose, and Jury, on the existence of a rational procedure for computing the feedback law from the desired characteristic polynomial is answered.
Byrnes, C. I.↗
PYRANDA
Pyranda Is fortran 90 code which solves compact finite difference equations on a three dimensional fortran arrays, domain decomposed using the Message Passing Interface (MPI). The fortran source solves the linear system of equations which forms a pentadtagona1 matrix. Derivatives are used to construct hyperbolic partlal differential equation (PDE) operators, namely gradient, divergence, Laplacian, and curl operators. 5. Solution Method: The library provides a python Interface which facilitates problem setup and description. However, the core solver Is written in the fortran 90 language. The solver couples MPI communication and linear solves together so that a direct (non-iterative) solution to the linear system of equations can be computed with minimal communication. Domain decomposition is limited to a uniform data size but provides perfect load balancing. 6. Computer for which the code Is written:
Olson, BrittonJ.↗
An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, part 2
It is shown how the look-ahead Lanczos process (combined with a quasi-minimal residual QMR) approach) can be used to develop a robust black box solver for large sparse non-Hermitian linear systems. Details of an implementation of the resulting QMR algorithm are presented. It is demonstrated that the QMR method is closely related to the biconjugate gradient (BCG) algorithm; however, unlike BCG, the QMR algorithm has smooth convergence curves and good numerical properties. We report numerical experiments with our implementation of the look-ahead Lanczos algorithm, both for eigenvalue problem and linear systems. Also, program listings of FORTRAN implementations of the look-ahead algorithm and the QMR method are included.
Freund, Roland W.↗
On the scalability of the Albany/FELIX first-order Stokes approximation ice sheet solver for large-scale simulations of the Greenland and Antarctic ice sheets
We examine the scalability of the recently developed Albany/FELIX finite-element based code for the first-order Stokes momentum balance equations for ice flow. We focus our analysis on the performance of two possible preconditioners for the iterative solution of the sparse linear systems that arise from the discretization of the governing equations: (1) a preconditioner based on the incomplete LU (ILU) factorization, and (2) a recently-developed algebraic multigrid (AMG) preconditioner, constructed using the idea of semi-coarsening. A strong scalability study on a realistic, high resolution Greenland ice sheet problem reveals that, for a given number of processor cores, the AMG preconditioner results in faster linear solve times but the ILU preconditioner exhibits better scalability. A weak scalability study is performed on a realistic, moderate resolution Antarctic ice sheet problem, a substantial fraction of which contains floating ice shelves, making it fundamentally different from the Greenland ice sheet problem. Here, we show that as the problem size increases, the performance of the ILU preconditioner deteriorates whereas the AMG preconditioner maintains scalability. This is because the linear systems are extremely ill-conditioned in the presence of floating ice shelves, and the ill-conditioning has a greater negative effect on the ILU preconditioner than on the AMG preconditioner.
54 ENVIRONMENTAL SCIENCES↗