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All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Fast Solution of Fully Implicit Runge--Kutta and Discontinuous Galerkin in Time for Numerical PDEs, Part I: the Linear Setting

Fully implicit Runge--Kutta (IRK) methods have many desirable properties as time integration schemes in terms of accuracy and stability, but high-order IRK methods are not commonly used in practice with numerical PDEs due to the difficulty of solving the stage equations. This paper introduces a theoretical and algorithmic preconditioning framework for solving the systems of equations that arise from IRK methods applied to linear numerical PDEs (without algebraic constraints). Additionally, this framework also naturally applies to discontinuous Galerkin discretizations in time. Under quite general assumptions on the spatial discretization that yield stable time integration, the preconditioned operator is proven to have condition number bounded by a small, order-one constant, independent of the spatial mesh and time-step size, and with only weak dependence on number of stages/polynomial order; for example, the preconditioned operator for 10th-order Gauss IRK has condition number less than two, independent of the spatial discretization and time step. The new method can be used with arbitrary existing preconditioners for backward Euler-type time-stepping schemes and is amenable to the use of three-term recursion Krylov methods when the underlying spatial discretization is symmetric. The new method is demonstrated to be effective on various high-order finite-difference and finite element discretizations of linear parabolic and hyperbolic problems, demonstrating fast, scalable solution of up to 10th-order accuracy. The new method consistently outperforms existing block preconditioning approaches, and in several cases, the new method can achieve 4th-order accuracy using Gauss integration with roughly half the number of preconditioner applications and wallclock time as required using standard diagonally IRK methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Xyce Parallel Electronic Simulator Users' Guide Version 6.6.

This manual describes the use of the Xyce Parallel Electronic Simulator. Xyce has been de- signed as a SPICE-compatible, high-performance analog circuit simulator, and has been written to support the simulation needs of the Sandia National Laboratories electrical designers. This development has focused on improving capability over the current state-of-the-art in the following areas: Capability to solve extremely large circuit problems by supporting large-scale parallel com- puting platforms (up to thousands of processors). This includes support for most popular parallel and serial computers. A differential-algebraic-equation (DAE) formulation, which better isolates the device model package from solver algorithms. This allows one to develop new types of analysis without requiring the implementation of analysis-specific device models. Device models that are specifically tailored to meet Sandia's needs, including some radiation- aware devices (for Sandia users only). Object-oriented code design and implementation using modern coding practices. Xyce is a parallel code in the most general sense of the phrase -- a message passing parallel implementation -- which allows it to run efficiently a wide range of computing platforms. These include serial, shared-memory and distributed-memory parallel platforms. Attention has been paid to the specific nature of circuit-simulation problems to ensure that optimal parallel efficiency is achieved as the number of processors grows. The information herein is subject to change without notice. Copyright c 2002-2016 Sandia Corporation. All rights reserved. Acknowledgements The BSIM Group at the University of California, Berkeley developed the BSIM3, BSIM4, BSIM6, BSIM-CMG and BSIM-SOI models. The BSIM3 is Copyright c 1999, Regents of the University of California. The BSIM4 is Copyright c 2006, Regents of the University of California. The BSIM6 is Copyright c 2015, Regents of the University of California. The BSIM-CMG is Copyright c 2012 and 2016, Regents of the University of California. The BSIM-SOI is Copyright c 1990, Regents of the University of California. All rights reserved. The Mextram model has been developed by NXP Semiconductors until 2007, Delft University of Technology from 2007 to 2014, and Auburn University since April 2015. Copyrights c of Mextram are with Delft University of Technology, NXP Semiconductors and Auburn University. The MIT VS Model Research Group developed the MIT Virtual Source (MVS) model. Copyright c 2013 Massachusetts Institute of Technology (MIT). The EKV3 MOSFET model was developed by the EKV Team of the Electronics Laboratory-TUC of the Technical University of Crete. Trademarks Xyce TM Electronic Simulator and Xyce TM are trademarks of Sandia Corporation. Orcad, Orcad Capture, PSpice and Probe are registered trademarks of Cadence Design Systems, Inc. Microsoft, Windows and Windows 7 are registered trademarks of Microsoft Corporation. Medici, DaVinci and Taurus are registered trademarks of Synopsys Corporation. Amtec and TecPlot are trademarks of Amtec Engineering, Inc. All other trademarks are property of their respective owners. Contacts World Wide Web http://xyce.sandia.gov https://info.sandia.gov/xyce (Sandia only) Email xyce@sandia.gov (outside Sandia) xyce-sandia@sandia.gov (Sandia only) Bug Reports (Sandia only) http://joseki-vm.sandia.gov/bugzilla http://morannon.sandia.gov/bugzilla

47 OTHER INSTRUMENTATION↗
DOE OSTI2025

Differential equations for cosmological correlators

Cosmological fluctuations retain a memory of the physics that generated them in their spatial correlations. The strength of correlations varies smoothly as a function of external kinematics, which is encoded in differential equations satisfied by cosmological correlation functions. In this work, we provide a broader perspective on the origin and structure of these differential equations. As a concrete example, we study conformally coupled scalar fields in a power-law cosmology. The wavefunction coefficients in this model have integral representations, with the integrands being the product of the corresponding flat-space results and “twist factors” that depend on the cosmological evolution. Similar twisted integrals arise for loop amplitudes in dimensional regularization, and their recent study has led to the discovery of rich mathematical structures and powerful new tools for computing multi-loop Feynman integrals in quantum field theory. The integrals of interest in cosmology are also part of a finite-dimensional basis of master integrals, which satisfy a system of first-order differential equations. We develop a formalism to derive these differential equations for arbitrary tree graphs. The results can be represented in graphical form by associating the singularities of the differential equations with a set of graph tubings. Upon differentiation, these tubings grow in a local and predictive fashion. In fact, a few remarkably simple rules allow us to predict — by hand — the equations for all tree graphs. While the rules of this “kinematic flow” are defined purely in terms of data on the boundary of the spacetime, they reflect the physics of bulk time evolution. We also study the analogous structures in tr ϕ 3 theory, and see some glimpses of hidden structure in the sum over planar graphs. This suggests that there is an autonomous combinatorial or geometric construction from which cosmological correlations, and the associated spacetime, emerge.

Cosmological models↗
DOE OSTI2018

A Weighted Least-Squares Transport Equation Compatible with Source Iteration and Voids

We present that second-order forms of the transport equation allow the use of continuous finite elements (CFEMs). This can be desired in multiphysics calculations where other physics require CFEM discretizations. Second-order transport operators are generally self-adjoint, yielding symmetric positive-definite (SPD) matrices, which allow the use of efficient linear algebra solvers with an enormous advantage in memory usage. Least-squares (LS) forms of the transport equation can circumvent the void problems of other second-order forms but are almost always nonconservative. Additionally, the standard LS form is not compatible with discrete ordinates method (SN) iterative solution techniques such as source iteration. A new form of the LS transport equation has recently been developed that is compatible with voids and standard SN iterative solution techniques. Performing nonlinear diffusion acceleration (NDA) using an independently differenced low-order equation enforces conservation for the whole system and makes this equation suitable for reactor physics calculations. In this context, “independent” means that both the transport and low-order solutions converge to the same scalar flux and current as the spatial mesh is refined, but for a given mesh, the solutions are not necessarily equal. In this paper we show that introducing a weight function into this LS equation improves issues with causality and can render our equation equal to the self-adjoint angular flux (SAAF) equation. Causality is a principle of the transport equation that states that information travels only downstream along characteristics. This principle can be violated numerically. We show how to limit the weight function in voids and demonstrate the effect of this limit on accuracy. Using the C5G7 benchmark, we compare our method to the SAAF formulation with a void treatment (SAAFτ) that is not self-adjoint and has a nonsymmetric coefficient matrix. Finallyl, we show that the weighted LS equation with NDA gives acceptable accuracy relative to the SAAFτ equation while maintaining a SPD system matrix.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Communication: Charge-population based dispersion interactions for molecules and materials

We introduce a system-independent method to derive effective atomic C{sub 6} coefficients and polarizabilities in molecules and materials purely from charge population analysis. This enables the use of dispersion-correction schemes in electronic structure calculations without recourse to electron-density partitioning schemes and expands their applicability to semi-empirical methods and tight-binding Hamiltonians. We show that the accuracy of our method is en par with established electron-density partitioning based approaches in describing intermolecular C{sub 6} coefficients as well as dispersion energies of weakly bound molecular dimers, organic crystals, and supramolecular complexes. We showcase the utility of our approach by incorporation of the recently developed many-body dispersion method [Tkatchenko et al., Phys. Rev. Lett. 108, 236402 (2012)] into the semi-empirical density functional tight-binding method and propose the latter as a viable technique to study hybrid organic-inorganic interfaces.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Theory of Trotter Error with Commutator Scaling

The Lie-Trotter formula, together with its higher-order generalizations, provides a simple approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic structure, our approach holds in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, nearly matching or even outperforming the best previous results. Our applications include: (i) a simulation of second-quantized plane-wave electronic structure, nearly matching the interaction-picture algorithm of Low and Wiebe; (ii) a simulation of $k$-local Hamiltonians almost with induced one-norm scaling, faster than the qubitization algorithm of Low and Chuang; (iii) a simulation of rapidly decaying power-law interactions, outperforming the Lieb-Robinson-based approach of Tran et al.; (iv) a hybrid simulation of clustered Hamiltonians, dramatically improving the result of Peng, Harrow, Ozols, and Wu; and (v) quantum Monte Carlo simulations of the transverse field Ising model and quantum ferromagnets, tightening previous analyses of Bravyi and Gosset. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound nearly matching a recent result of Tran et al. Our analysis reproduces known tight bounds for first- and second-order formulas. We further investigate the tightness of our bounds for higher-order formulas. For quantum simulation of a one-dimensional Heisenberg model with an even-odd ordering of terms, our result overestimates the complexity by only a factor of $5$. Our bound is also close to tight for power-law interactions and other orderings of terms. This suggests that our theory can accurately characterize Trotter error in terms of both the asymptotic scaling and the constant prefactor.

quantum computing, numerical analysis↗
DOE OSTI2017

Initial CEED API (CEED-MS10)

Key components of CEED software development involve fast finite element operator storage and evaluation, architecture optimizations, performant algorithms for all orders, global kernels for finite element operators, efficient use of the memory sub-system, optimal data locality and motion, enhanced scalability and parallelism, and fast tensor contractions. As ultimate goals of the project, CEED is exploring and identifying the best algorithms for the full range of discretizations and applying algorithmic and software development to support ECP applications’ needs. In this milestone, we initiated the development and implementation of a common CEED API consisting of two main components: low-level and high-level APIs. We accomplished the goal of having the low-level component ready for the release of CEED v1.0 in the next quarter, see CEED-MS13. In this report we also provide a highlight from CEED’s integration with the MARBL app and the corresponding Laghos miniapp. A main theme in this work has been the systematic transition of advanced algorithms, developed and tested in Laghos, to the large-scale multi-physics settings of the MARBL code. The low-level CEED API provides a set of FE kernels and components for writing new low-level kernels. Examples include: vector and sparse linear algebra, element matrix assembly over a batch of elements, partial assembly and action for efficient high-order operators like mass, diffusion, advection, etc. The main goal of the low-level API is to establish the basis for the high-level API. Also, identifying such low-level kernels and providing a reference implementation for them serves as the basis for collaboration with vendors and STs. Another goal for the low-level API is to make it easy for applications with their own discretization infrastructure, who do not want to switch directly to using the high-level CEED API, to still evaluate and use the core operations provided by the low-level API. One of the challenges with high-order methods is that a global sparse matrix is no longer a good representation of a high-order linear operator, both with respect to the FLOPs needed for its evaluation, as well as the memory transfer needed for a matvec. Thus, another important question we tried to address while developing the low-level API is: is it possible to abstract out in a natural and efficient way the different implementations and data structures that arise when dealing with various computational device types: CPUs, GPUs, etc. Fully answering this question is beyond the scope of this particular milestone, since that requires longer time as we develop and expand the devices we support. The future high-level API will be the frontend provided by CEED consisting of high-level abstractions such as meshes, finite element spaces and solutions, bilinear forms, solvers, etc. At this high-level, an important aspect of the API will be to provide a uniform abstraction layer for writing code in a portable manner. To achieve that, we plan to utilize the MPI programming model combined with a device abstraction layer (DAL) for finite element computations that we will develop. The DAL will serve as the bridge between the low- and the high-level APIs, allowing a variety of programming models to be used for the low-level kernel implementations. The software artifacts delivered as part of this milestone include an initial CEED API software implementation (libCEED) provided through the CEED website, http://ceed.exascaleproject.org and the CEED GitHub organization, http://github.com/ceed. In addition to details and results from the above R&D efforts, in this document we are also reporting on other project-wide activities performed in Q1 of FY18, including: the Nek5000 v17.0 MFEM v3.3.2 and PETSc v3.8 releases, collaborations with ECP/ST and SciDAC projects, the inaugural Nek5000 hackathon and various outreach activities.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Machine-Learning Assisted Identification of Battery Life Models

Predictive battery life models are commonly utilized to extrapolate degradation trends observed during accelerated aging tests for simulation of degradation in real-world applications. Thus, fitting accelerated aging data as accurately as possible and with low uncertainty is crucial for making believable projections of battery lifetime, but it is challenging to identify algebraic expressions that accurately fit multivariate degradation trends. A review of models published in literature reveal some common expressions for fitting calendar aging data, which is only dependent on temperature and state-of-charge, but no consistency across many models for fitting cycle aging data, indicating the need for a statistically rigorous data driven approach for developing empirical models. This talk will describe a machine-learning assisted method for identification of predictive battery life models utilizing bilevel optimization and symbolic regression. Bilevel optimization with cross-validation is used to statistically determine cell- and stress-dependent model parameters, while symbolic regression identifies both linear and multiplicative candidate expressions to predict stress-dependent degradation rates by selecting low-order subsets of features from a generated feature library. Because model expressions are identified empirically, it is crucial to ensure resulting models behave according to physical expectations, so the stability of models for interpolation or extrapolation is interrogated qualitatively through simulation and quantitatively through cross-validation and uncertainty quantification via bootstrap resampling. This model identification approach substantially improves upon models identified purely using expert judgement in terms of both accuracy and uncertainty. Model simulation and validation is then conducted by deriving a state-equation form of the predictive model, enabling simulation of battery aging under dynamic stresses. This enables validation of the predictive battery model on lab-based tests with varying conditions or on drive-cycle or application-cycle testing protocols. Parameter uncertainty can be carried forward into model simulation, giving lifetime estimates and confidence windows for cell- or system-level lifetime. The financial impact of battery model uncertainty can be estimated by incorporating uncertainty into a technoeconomic model.

battery↗
DOE OSTI2026

Optimal Transfer Operators in Algebraic Two-Level Methods for Nonsymmetric and Indefinite Problems

Consider an algebraic two-level method applied to the 𝑛-dimensional linear system 𝐴⁢𝒙 = 𝒃 using fine-space preconditioner (i.e., “relaxation” or “smoother”) 𝑀, with 𝑀 ≈ 𝐴, restriction and interpolation 𝑅 and 𝑃, and algebraic coarse-space operator 𝐴 𝑐 : = 𝑅 ∗ ⁢𝐴⁢𝑃. Then, what are the best possible transfer operators 𝑅 and 𝑃 of a given dimension 𝑛 𝑐 < 𝑛? Brannick et al. [12] showed that when 𝐴 and 𝑀 are Hermitian positive definite (HPD), the optimal interpolation is such that its range contains the 𝑛 𝑐 smallest generalized eigenvectors of the matrix pencil (𝐴, 𝑀). Recently, in Ali et al. [5] we generalized this framework to the non-HPD setting, by considering both right (interpolation) and left (restriction) generalized eigenvectors of (𝐴, 𝑀) and defining corresponding nonsymmetric transfer operators {𝑅#, 𝑃#}. Tight convergence bounds for {𝑅#, 𝑃#} are derived in spectral radius, as well as a proof of pseudo-optimality. Note, {𝑅#, 𝑃#} are typically complex valued, which is not practical for real-valued problems. Here, in this work, we build on [5], first characterizing all inner products in which the coarse-space correction defined by {𝑅#, 𝑃#} is orthogonal. We then develop tight two-level convergence bounds in these norms, and prove that the underlying transfer operators {𝑅#, 𝑃#} are genuinely optimal. As a special case, our theory both recovers and extends the HPD results from [12]. Finally, we show how to construct optimal, real-valued transfer operators in the case of that 𝐴 and 𝑀 are real valued, but are not HPD. Numerical examples arising from a discretized advection-reaction equation, wave-equation, and Stokes equations are used to verify and illustrate the theory.

97 MATHEMATICS AND COMPUTING↗