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At least 289 records · Page 16
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Implicit Multirate GARK Methods

This article considers multirate generalized-structure additively partitioned Runge–Kutta methods for solving stiff systems of ordinary differential equations with multiple time scales. These methods treat different partitions of the system with different timesteps for a more targeted and efficient solution compared to monolithic single rate approaches. With implicit methods used across all partitions, methods must find a balance between stability and the cost of solving nonlinear equations for the stages. In order to characterize this important trade-off, we explore multirate coupling strategies, problems for assessing linear stability, and techniques to efficiently implement Newton iterations for stage equations. Unlike much of the existing multirate stability analysis which is limited in scope to particular methods, we present general statements on stability and describe fundamental limitations for certain types of multirate schemes. New implicit multirate methods up to fourth order are derived, and their accuracy and efficiency properties are verified with numerical tests.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Error estimates of finite difference methods for the Dirac equation in the massless and nonrelativistic regime

We present four frequently used finite difference methods and establish the error bounds for the discretization of the Dirac equation in the massless and nonrelativistic regime, involving a small dimensionless parameter 0 < ε &NestedLessLess; 1 inversely proportional to the speed of light. In the massless and nonrelativistic regime, the solution exhibits rapid motion in space and is highly oscillatory in time. Specifically, the wavelength of the propagating waves in time is at O(ε), while in space, it is at O(1) with the wave speed at O(ε -1 ). We adopt one leap-frog, two semi-implicit, and one conservative Crank-Nicolson finite difference methods to numerically discretize the Dirac equation in one dimension and establish rigorously the error estimates which depend explicitly on the time step τ, mesh size h, and the small parameter ε. The error bounds indicate that, to obtain the “correct” numerical solution in the massless and nonrelativistic regime, i.e., 0 < ε &NestedLessLess; 1, all these finite difference methods share the same ε-scalability as time step τ = O(ε 3/2 ) and mesh size h = O(ε 1/2 ). A large number of numerical results are reported to verify the error estimates.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Sparse invariant domain preserving discontinuous Galerkin methods with subcell convex limiting

In this paper, we develop high-order nodal discontinuous Galerkin (DG) methods for hyperbolic conservation laws that satisfy invariant domain preserving properties using subcell flux corrections and convex limiting. These methods are based on a subcell flux corrected transport (FCT) methodology that involves blending a high-order target scheme with a robust, low-order invariant domain preserving method that is obtained using a graph viscosity technique. Furthermore, the new low-order discretizations are based on sparse stencils which do not increase with the polynomial degree of the high-order DG method. As a result, the accuracy of the low-order method does not degrade when used with high-order target methods. The method is applied to both scalar conservation laws, for which the discrete maximum principle is naturally enforced, and to systems of conservation laws such as the Euler equations, for which positivity of density and a minimum principle for specific entropy are enforced. Numerical results are presented on a number of benchmark test cases.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

A pseudospectral implicit particle-in-cell method with exact energy and charge conservation

The standard particle-in-cell (PIC) method employs explicit finite-difference (FD) methods (e.g. the leap-frog scheme) for both spatial and temporal integrations. Here, we employ a pseudospectral method for solving the Poisson equation and a fully implicit time integration to achieve exact energy conservation. The advantage of a pseudospectral field solver is its spectral accuracy in solving field solutions. Earlier studies of implicit time integration of PIC FD equations can enforce exact energy exchange between field and particles, resulting in exact energy-conserving schemes. Here, we prove that the exact energy conservation property can be carried over to the pseudospectral scheme. Simultaneously, we provide a solution to ensure a pseudospectral charge continuity equation. We demonstrate the new scheme in a 2D electrostatic PIC code. In conclusion, theoretical results are confirmed via numerical examples.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Integration of discrete-event dynamics and machining dynamics for machine tool: Modeling, analysis and algorithms

Machining dynamics research lays a solid foundation for machining operations by providing stable combinations of spindle speed and depth of cut. Furthermore, machine learning has been applied to predict tool life as a function of cutting speed. However, the existing research does not consider the discrete-event dynamics in machine shop, i.e., the machine tool needs to process a series of parts in queue under various practical production requirements. This paper addresses the integration of discrete-event dynamics and machining dynamics to achieve cost savings in machining. A learning-based cost function is first proposed for the studied integrated optimization problem of machine tool. The proposed cost function utilizes the predicted tool life under different stable cutting speeds for further optimizing speed selection of machine tool to deal with the discrete-event dynamics in machine shop. Then, according to the practical production requirements, effective mathematical optimization models are developed for the related integrated optimization problems with the consideration of cost, makespan and due date, respectively. Numerical results show the effectiveness of our proposed methods and also the potential to be used in practice.

Ma, Mason↗
DOE OSTI2020

Exponential Decay in the Sensitivity Analysis of Nonlinear Dynamic Programming

In this report, we study the sensitivity of discrete-time dynamic programs with nonlinear dynamics and objective to perturbations in the initial conditions and reference parameters. Under uniform controllability and boundedness assumptions for the problem data, we prove that the directional derivative of the optimal state and control at time $k$, $x^*_k$, and $u^*_k$, with respect to the reference signal at time $i$, $d_i$, will have exponential decay in terms of $|k-i|$ with a decay rate $\rho$ independent of the temporal horizon length. The key technical step is to prove that a version of the convexification approach proposed by Verschueren et al. can be applied to the KKT conditions and results in a convex quadratic program with uniformly bounded data. In turn, Riccati techniques can be further employed to obtain the sensitivity result, borne from the observation that the directional derivatives are solutions of quadratic programs with structure similar to the KKT conditions themselves. We validate our findings with numerical experiments on a small nonlinear, nonconvex, dynamic program.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Brochure on the 2024 ASCR Workshop on Energy-Efficient Computing for Science

Large-scale computing has enabled numerous scientific discoveries, including ground-breaking achievements facilitated by the US Department of Energy (DOE) supercomputers and advances in applied mathematics and computer science. While important advances were made in energy efficiency to enable exascale computing, continued efforts are needed to dramatically improve the energy efficiency of the next generation of high-performance computing (HPC) systems and, more broadly, AI data centers. Without substantial improvements in energy efficiency, the energy consumption associated with computing could become a limiting factor for future scientific discovery, national security, and technological advancement.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2021

Shadow Lagrangian dynamics for superfluidity

Motivated by a similar approach for Born-Oppenheimer molecular dynamics, this paper proposes an extended "shadow" Lagrangian density for quantum states of superfluids. The extended Lagrangian contains an additional field variable that is forced to follow the wave function of the quantum state through a rapidly oscillating extended harmonic oscillator. By considering the adiabatic limit for large frequencies of the harmonic oscillator, we can derive the two equations of motions, a Schrödinger-type equation for the quantum state and a wave equation for the extended field variable. The equations are coupled in a nonlinear way, but each equation individually is linear with respect to the variable that it defines. The computational advantage of this new system is that it can be easily discretized using linear time stepping methods, where we propose to use a Crank-Nicolson-type approach for the Schrödinger equation and an extended leapfrog scheme for the wave equation. Furthermore, the difference between the quantum state and the extended field variable defines a consistency error that should go to zero if the frequency tends to infinity. By coupling the time-step size in our discretization to the frequency of the harmonic oscillator we can extract an easily computable consistency error indicator that can be used to estimate the numerical error without additional costs. The findings are illustrated in numerical experiments.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Practical CO2—WAG Field Operational Designs Using Hybrid Numerical-Machine-Learning Approaches

Machine-learning technologies have exhibited robust competences in solving many petroleum engineering problems. The accurate predictivity and fast computational speed enable a large volume of time-consuming engineering processes such as history-matching and field development optimization. The Southwest Regional Partnership on Carbon Sequestration (SWP) project desires rigorous history-matching and multi-objective optimization processes, which fits the superiorities of the machine-learning approaches. Although the machine-learning proxy models are trained and validated before imposing to solve practical problems, the error margin would essentially introduce uncertainties to the results. In this paper, a hybrid numerical machine-learning workflow solving various optimization problems is presented. By coupling the expert machine-learning proxies with a global optimizer, the workflow successfully solves the history-matching and CO2 water alternative gas (WAG) design problem with low computational overheads. The history-matching work considers the heterogeneities of multiphase relative characteristics, and the CO2-WAG injection design takes multiple techno-economic objective functions into accounts. This work trained an expert response surface, a support vector machine, and a multi-layer neural network as proxy models to effectively learn the high-dimensional nonlinear data structure. The proposed workflow suggests revisiting the high-fidelity numerical simulator for validation purposes. The experience gained from this work would provide valuable guiding insights to similar CO2 enhanced oil recovery (EOR) projects.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2021

Caravel: A C++ framework for the computation of multi-loop amplitudes with numerical unitarity

We present the first public version of Caravel, a C++17 framework for the computation of multi-loop scattering amplitudes in quantum field theory, based on the numerical unitarity method. Caravel is composed of modules for the D-dimensional decomposition of integrands of scattering amplitudes into master and surface terms, the computation of tree-level amplitudes in floating point or finite-field arithmetic, the numerical computation of one- and two-loop amplitudes in QCD and Einstein gravity, and functional reconstruction tools. Here, we provide programs that showcase Caravel's main functionalities and allow to compute selected one- and two-loop amplitudes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Hardware acceleration for HPS algorithms in two and three dimensions

We provide a flexible, open-source framework for hardware acceleration, namely massively-parallel execution on general-purpose graphics processing units (GPUs), applied to the hierarchical Poincaré–Steklov (HPS) family of algorithms for building fast direct solvers for linear elliptic partial differential equations. To take full advantage of the power of hardware acceleration, we propose two variants of HPS algorithms to improve performance on two- and three-dimensional problems. In the two-dimensional setting, we introduce a novel recomputation strategy that minimizes costly data transfers to and from the GPU; in three dimensions, we modify and extend the adaptive discretization technique of Geldermans and Gillman [1] to greatly reduce peak memory usage. We provide an open-source implementation of these methods written in JAX, a high-level accelerated linear algebra package, which allows for the first integration of a high-order fast direct solver with automatic differentiation tools. We conclude with extensive numerical examples showing our methods are fast and accurate on two- and three-dimensional problems.

Fast direct solvers↗
DOE OSTI2024

Calculation of machine precision second order derivatives using dual-complex numbers

It is well known that both complex and dual numbers can be employed to obtain machine precision first-order derivatives; however, neither, on their own, can compute machine precision 2nd order derivatives. To address this limitation, it is demonstrated in this paper that combined dual-complex numbers can be used to compute machine precision 1st and 2nd order derivatives. The dual-complex approach is simpler than utilizing multicomplex or hyper-dual numbers as existing dual libraries can be used as is or easily augmented to accept complex numbers, and the complexity of developing, integrating, and deploying multicomplex or hyper-dual libraries is avoided. The efficacy of this approach is demonstrated for both univariate and multivariate functions. Finally, source code examples using the Python, Julia, and Mathematica languages are provided as supplemental material.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Generalization error guaranteed auto-encoder-based nonlinear model reduction for operator learning

Many physical processes in science and engineering are naturally represented by operators between infinite-dimensional function spaces. The problem of operator learning, in this context, seeks to extract these physical processes from empirical data, which is challenging due to the infinite or high dimensionality of data. An integral component in addressing this challenge is model reduction, which reduces both the data dimensionality and problem size. In this paper, we utilize low-dimensional nonlinear structures in model reduction by investigating Auto-Encoder-based Neural Network (AENet). AENet first learns the latent variables of the input data and then learns the transformation from these latent variables to corresponding output data. Our numerical experiments validate the ability of AENet to accurately learn the solution operator of nonlinear partial differential equations. Furthermore, we establish a mathematical and statistical estimation theory that analyzes the generalization error of AENet. Finally, our theoretical framework shows that the sample complexity of training AENet is intricately tied to the intrinsic dimension of the modeled process, while also demonstrating the robustness of AENet to noise.

Auto-encoder↗
DOE OSTI2022

Deep Learning-Assisted Real-Time Forward Modeling of Electromagnetic Logging in Complex Formations

Higher dimensional (i.e., 2-D and 3-D) modeling is indispensable to correctly evaluate the responses of electromagnetic logging tools in complex formation environments. However, limited by the high computational cost of rigorous modeling such as finite difference method and the finite element method, the real-time applications in the well logging industry primarily rely on the 1-D forward solver, which would result in erroneous formation evaluation for complex scenarios. As a result, aiming at realizing fast modeling for electromagnetic logging tools in complex formations, this paper proposes a general framework assisted by deep neural networks (DNNs). The framework consists of three modules: earth model classification, parameter extraction, and surrogate construction. Separate DNNs are trained and tested for different modules. The accuracy and efficiency of the DNN assisted fast modeling are validated by several experiments. Here, this study finds that the fast modeling assisted by DNNs is able to calculate the tool responses and reconstruct the subsurface formations in real-time.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

C3PU:Real-Time Predictive Charge Control Software for Battery Management Systems (Code for Charge Control and Predictive Unit) [SWR-21-48]

C3PU sets up a model predictive control formulation for the optimal charging control of battery packs and can be deployed for real-time operation on an embedded microprocessor. C3PU uses electrochemical and thermal models of battery packs to predict the charging trajectories of the battery pack over a time horizon under different operating conditions. It then selects the optimal charging trajectory such that a pre-defined objective is minimized, which in the present state of the code is to minimize the charging duration of the battery pack. The optimal charging trajectory is obtained by solving an underlying mathematical problem. However, C3PU is flexible to incorporate other charging objectives. The underlying battery models can be swapped as well, as long as they follow certain mathematical properties. Mathematically, optimal control problems (in this case, optimally controlling the charging current of the battery pack) are computationally expensive. C3PU implements advanced numerical techniques, namely pseudo-spectral optimization, to reduce the computational burden of the underlying problem to solve. This allows for the problem to be solved in real-time in an embedded system. Once the optimal charging trajectory is computed over a time horizon, C3PU can package such data and send it out via appropriate communication protocols.

Mishra, Partha↗
DOE OSTI2025

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↗
DOE OSTI2024

Hybrid programming-model strategies for GPU offloading of electronic structure calculation kernels

To address the challenge of performance portability and facilitate the implementation of electronic structure solvers, we developed the basic matrix library (BML) and Parallel, Rapid O(N), and Graph-based Recursive Electronic Structure Solver (PROGRESS) library. The BML implements linear algebra operations necessary for electronic structure kernels using a unified user interface for various matrix formats (dense and sparse) and architectures (CPUs and GPUs). Focusing on density functional theory and tight-binding models, PROGRESS implements several solvers for computing the single-particle density matrix and relies on BML. In this paper, we describe the general strategies used for these implementations on various computer architectures, using OpenMP target functionalities on GPUs, in conjunction with third-party libraries to handle performance critical numerical kernels. In this study, we demonstrate the portability of this approach and its performance in benchmark problems.

36 MATERIALS SCIENCE↗
DOE OSTI2020

An augmented Lagrangian filter method

Here, we introduce a filter mechanism to enforce convergence for augmented Lagrangian methods for nonlinear programming. In contrast to traditional augmented Lagrangian methods, our approach does not require the use of forcing sequences that drive the first-order error to zero. Instead, we employ a filter to drive the optimality measures to zero. Our algorithm is flexible in the sense that it allows for equality-constrained quadratic programming steps to accelerate local convergence. We also include a feasibility restoration phase that allows fast detection of infeasible problems. We provide a convergence proof that shows that our algorithm converges to first-order stationary points. We provide preliminary numerical results that demonstrate the effectiveness of our proposed method.

97 MATHEMATICS AND COMPUTING↗