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DOE OSTI2023

Batched Sparse Linear Algebra (Final Report for Subcontract B648960)

This report finalizes design specifications for developing batched kernels for small tensor operations for unassembled matrix-free iterative solvers, batched solvers for partially assembled operators, and batched solvers with support for various sparse formats. The outcome of the project milestones is a set of interfaces to Batched Sparse LA solvers running on hardware accelerators for use in ECP Libraries and Applications. It is part of the development of sparse batched kernels, solvers/preconditioners as well as creating interoperability in xSDK libraries with sparse and dense batched functions to benefit ECP applications. The participants included representatives from ECP libraries (not limited to the xSDK project), applications, and vendors (AMD, Intel, and NVIDIA). Batched sparse linear algebra solvers form the new frontier for algorithmic development and performance engineering. Many applications (ECP and non-ECP alike) require simultaneous solutions of small linear systems of equations that are structurally sparse. To move towards high hardware utilization, it is important to provide these applications with appropriate interfaces to efficient batched sparse solvers running on modern hardware accelerators. We present interface designs in use by HPC software libraries supporting batched sparse linear algebra and the development of sparse batched kernel codes for solvers and preconditioners. We also address the potential interoperability opportunities to keep the software portable between the major hardware accelerators from AMD, Intel, and NVIDIA. The presented interface specifications includes batched band, sparse iterative, and sparse direct solvers. This report summarizes progress in Kokkos Kernels and the xSDK libraries MAGMA, Ginkgo, hypre, SUNDIALS, and SuperLU_dist.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Algebraic expressions for estimating the impact depths of a surface barrier over a homogeneous soil

Engineered surface barriers are used to isolate subsurface contaminants for effective long-term containment of municipal solid waste, other nonhazardous solid and liquid waste, hazardous and toxic wastes, and radioactive waste. The impact depths of a surface barrier are affected by the pre-barrier recharge rate and the properties of the soil beneath the barrier. In this paper, the pore-size-specific (PSS) water velocity is defined and an algebraic expression of PSS velocity is derived based on the stream tube concept and the Brooks and Corey hydraulic retention model. Algebraic expressions are developed to estimate drainage velocities and barrier impact depths after the emplacement of a surface barrier. Four impact depth terms are used to convey the protective effect: drainage front, average drainage, the location with 50% impact, and drainage tail. The drainage front depth is the deepest point at which the barrier has a detectable impact at a specific time (also called the near zero-impact depth). The average-impact depth is the depth at which average drainage occurs. At the 50% impact depth, the water flux rate is reduced by half because of the surface barrier. Lastly, the drainage tail depth (also called the full-impact depth) is the deepest depth at which the water conditions above it are in equilibrium with the barrier. The algebraic expressions show that the average-impact depth is no more than 1/3 of near zero-impact depth, while the 50% impact depth is slightly larger than 1/2 of the near zero-impact depth. The full-impact depth, depending on the final recharge rate from the surface barrier, is usually much smaller than the other impact depths. These differences lead to a very large transition zone beneath a surface barrier. Numerical simulations were conducted to replicate the same conditions. The numerical results corroborated the analytical models by predicting very similar water content profiles and near zero-, average-, 50%, and full-impact depths. The algebraic expressions provided in this paper are useful for quickly identifying sites where the depth of the existing contaminants could be beyond the protection of a surface barrier.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

The IDAES process modeling framework and model library—Flexibility for process simulation and optimization

Abstract Energy systems and manufacturing processes of the 21st century are becoming increasingly dynamic and interconnected, which require new capabilities to effectively model and optimize their design and operations. Such next generation computational tools must leverage state‐of‐the‐art techniques in optimization and be able to rapidly incorporate new advances. To address these requirements, we have developed the Institute for the Design of Advanced Energy Systems (IDAES) Integrated Platform, which builds on the strengths of both process simulators (model libraries) and algebraic modeling languages (advanced solvers). This paper specifically presents the IDAES Core Modeling Framework (IDAES‐CMF), along with a case study demonstrating the application of the framework to solve process optimization problems. Capabilities provided by this framework include a flexible, modifiable, open‐source platform for optimization of process flowsheets utilizing state‐of‐the‐art solvers and solution techniques, fully open and extensible libraries of dynamic unit operations models and thermophysical property models, and integrated support for superstructure‐based conceptual design and optimization under uncertainty.

42 ENGINEERING↗

DOE OSTI2025

Self-supervised physics-informed generative networks for phase retrieval from a single X-ray hologram

X-ray phase contrast imaging significantly improves the visualization of structures with weak or uniform absorption, broadening its applications across a wide range of scientific disciplines. Propagation-based phase contrast is particularly suitable for time- or dose-critical in vivo/in situ/operando (tomography) experiments because it requires only a single intensity measurement. However, the phase information of the wave field is lost during the measurement and must be recovered. Conventional algebraic and iterative methods often rely on specific approximations or boundary conditions that may not be met by many samples or experimental setups. In addition, they require manual tuning of reconstruction parameters by experts, making them less adaptable for complex or variable conditions. Here we present a self-learning approach for solving the inverse problem of phase retrieval in the near-field regime of Fresnel theory using a single intensity measurement (hologram). A physics-informed generative adversarial network is employed to reconstruct both the phase and absorbance of the unpropagated wave field in the sample plane from a single hologram. Unlike most state-of-the-art deep learning approaches for phase retrieval, our approach does not require paired, unpaired, or simulated training data. This significantly broadens the applicability of our approach, as acquiring or generating suitable training data remains a major challenge due to the wide variability in sample types and experimental configurations. The algorithm demonstrates robust and consistent performance across diverse imaging conditions and sample types, delivering quantitative, high-quality reconstructions for both simulated data and experimental datasets acquired at beamline P05 at PETRA III (DESY, Hamburg), operated by Helmholtz-Zentrum Hereon. Furthermore, it enables the simultaneous retrieval of both phase and absorption information.

36 MATERIALS SCIENCE↗

DOE OSTI2021

QLiG: Query Like a Graph For Subgraph Matching

A graph is a natural and flexible modeling approach to represent entities and relationships between them in real-world. A Knowledge Graphs (KG) is a specialized graph with formal and structured representation of facts, relationships, annotated with semantic descriptions. Subgraph matching is one of the fundamental graph problems to identify relationships, interactions and activities of interest within a large graph. A query specification is a collection of abstract components, operations, and constraints to express a pattern. The specification can be implemented in different ways based on underlying data model. Various graph query specifications have been developed over the years and have led to the development of different open-sourced and vendor-specific query languages. Such specification are modeled as an extension of relational algebra used to develop relational query languages such as SQL. Such relational concepts do not inherently support graph queries. There is a need to represent graph queries in terms on graph-based components to expedite query construction by non-database experts. We present a graph-based query approach QLiG (pronounced cleeg), to perform subgraph matching in Labeled Property Graph. We present the query specifications, salient features, and a use case to show functional examples.

Purohit, Sumit↗

DOE OSTI2020

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↗

DOE OSTI2021

Accelerated basis adaptation in homogeneous chaos spaces

Polynomial chaos expansions (PCE) provide an efficient approach to uncertainty quantification (UQ) and have been adapted to diverse applications across the spectrum of science and engineering. For situations involving large stochastic parameterizations, the curse of dimensionality renders PCE-based methods computationally prohibitive. A basis adaptation approach for PCE was proposed by Tipireddy and Ghanem (2014) which transforms the input random variables through an isometry such that sufficient probabilistic characterization of specific quantities of interest (QoI) is concentrated in an algebraic manifold embedded in the linear span of the dominant transformed variables. While quite versatile, that original version of the PCE basis adaptation exhibited slow convergence for a number of problems of practical significance. In the present paper, we propose two novel methods to accelerate the convergence of the original basis adaptation approach, thus expanding its reach while also providing insight into its performance. In the first method, information gained from a pilot PCE representation is used to correct the mean and Gaussian coefficients in the adapted space. By taking advantages of probabilistic information in higher dimensional adaptation gleaned from an initial adaptation, the second method updates the rotation matrix used to identify the dominant transformed variables. In this manner, the new rotation matrix concentrates even more probabilistic information in its first few dimensions. These two method can be combined to achieve even better performance, the combined method is referred to as sequentially optimized adaptation method. Here, the methods are demonstrated on an analytical test function and a model of a space structure with several sub-components and a non-smooth quantity of interest representing the maximum acceleration over time. Both methods achieve accelerated convergence of the basis adaptation approach with negligible additional costs.

Accelerated convergence↗

DOE OSTI2025

Scalable Implementation of Mean-Field and Correlation Methods Based on Lie-Algebraic Similarity Transformation of Spin Hamiltonians in the Jordan–Wigner Representation

Recent work has highlighted that the strong correlation inherent in spin Hamiltonians can be effectively reduced by mapping spins to Fermions via the Jordan−Wigner transformation (JW). The Hartree−Fock method is straightforward in the Fermionic domain and may provide a reasonable approximation to the ground state. Correlation with respect to the Fermionic mean field can be recovered based on Lie-algebraic similarity transformation (LAST) with two-body correlators. Specifically, a unitary LAST variant eliminates the dependence on site ordering, while a nonunitary LAST yields size-extensive correlation energies. Whereas the first recent demonstration of such methods was restricted to small spin systems, we present efficient implementations using analytical gradients for the optimization with respect to the mean-field reference and the LAST parameters, thereby enabling the treatment of larger clusters, including systems with local spins s > $\frac{1}{2}$.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

CSRI Summer Proceedings 2021

The Computer Science Research Institute (CSRI) brings university faculty and students to Sandia National Laboratories for focused collaborative research on Department of Energy (DOE) computer and computational science problems. The institute provides an opportunity for university researches to learn about problems in computer and computational science at DOE laboratories, and help transfer results of their research to programs at the labs. Some specific CSRI research interest areas are: scalable solvers, optimization, algebraic preconditioners, graph-based, discrete, and combinatorial algorithms, uncertainty estimation, validation and verification methods, mesh generation, dynamic load-balancing, virus and other malicious-code defense, visualization, scalable cluster computers, beyond Moore’s Law computing, exascale computing tools and application design, reduced order and multiscale modeling, parallel input/output, and theoretical computer science. The CSRI Summer Program is organized by CSRI and includes a weekly seminar series and the publication of a summer proceedings.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

CSRI Summer Proceedings 2021

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

On infinite tensor networks, complementary recovery and type II factors

We initiate a study of local operator algebras at the boundary of infinite tensor networks, using the mathematical theory of inductive limits. In particular, we consider tensor networks in which each layer acts as a quantum code with complementary recovery, a property that features prominently in the bulk-to-boundary maps intrinsic to holographic quantum error-correcting codes. In this case, we decompose the limiting Hilbert space and the algebras of observables in a way that keeps track of the entanglement in the network. As a specific example, we describe this inductive limit for the holographic Harlow-Pastawski-Preskill-Yoshida code model and relate its algebraic and error-correction features. We find that the local algebras in this model are given by the hyperfinite type II$_\infty$ factor. Next, we discuss other networks that build upon this framework and comment on a connection between type II factors and stabilizer circuits. We conclude with a discussion of multiscale entanglement renormalization ansatz networks in which complementary recovery is broken. We argue that this breaking possibly permits a limiting type III von Neumann algebra, making them more suitable ansätze for approximating subregions of quantum field theories.

holographic dualities↗

DOE OSTI2022

Providing performance portable numerics for Intel GPUs

Summary With discrete Intel GPUs entering the high‐performance computing landscape, there is an urgent need for production‐ready software stacks for these platforms. In this article, we report how we enable the Ginkgo math library to execute on Intel GPUs by developing a kernel backed based on the DPC++ programming environment. We discuss conceptual differences between the CUDA and DPC++ programming models and describe workflows for simplified code conversion. We evaluate the performance of basic and advanced sparse linear algebra routines available in Ginkgo's DPC++ backend in the hardware‐specific performance bounds and compare against routines providing the same functionality that ship with Intel's oneMKL vendor library.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Evaluation of data driven low-rank matrix factorization for accelerated solutions of the Vlasov equation

Low-rank methods have shown success in accelerating simulations of a collisionless plasma described by the Vlasov equation, but still rely on computationally costly linear algebra every time step. We propose a data-driven factorization method using artificial neural networks, specifically with convolutional layer architecture, that trains on existing simulation data. At inference time, the model outputs a low-rank decomposition of the distribution field of the charged particles, and we demonstrate that this step is faster than the standard linear algebra technique. Numerical experiments show that the method achieves comparable reconstruction accuracy for interpolation tasks, generalizing to unseen test data in a manner beyond just memorizing training data; patterns in factorization also inherently followed the same numerical trend as those within algebraic methods (e.g., truncated singular-value decomposition). However, when training on the first 70% of a time-series data and testing on the remaining 30%, the method fails to meaningfully extrapolate. Despite this limiting result, the technique may have benefits for simulations in a statistical steady-state or otherwise showing temporal stability. These results suggest that while the model offers a computationally efficient alternative for datasets with temporal stability, its current formulation is best suited for interpolation rather than for predicting future states in time-evolving systems. This study thus lays the groundwork for further refinement of neural network-based approaches to low-rank matrix factorization in high-dimensional plasma simulations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Multilevel-in-width training for deep neural network regression

A common challenge in regression is that for many problems, the degrees of freedom required for a high-quality solution also allows for overfitting. Regularization is a class of strategies that seek to restrict the range of possible solutions so as to discourage overfitting while still enabling good solutions, and different regularization strategies impose different types of restrictions. In this paper, we present a multilevel regularization strategy that constructs and trains a hierarchy of neural networks, each of which has layers that are wider versions of the previous network's layers. Here, we draw intuition and techniques from the field of Algebraic Multigrid (AMG), traditionally used for solving linear and nonlinear systems of equations, and specifically adapt the Full Approximation Scheme (FAS) for nonlinear systems of equations to the problem of deep learning. Training through V-cycles then encourage the neural networks to build a hierarchical understanding of the problem. We refer to this approach as multilevel-in-width to distinguish from prior multilevel works which hierarchically alter the depth of neural networks. The resulting approach is a highly flexible framework that can be applied to a variety of layer types, which we demonstrate with both fully connected and convolutional layers. We experimentally show with PDE regression problems that our multilevel training approach is an effective regularizer, improving the generalize performance of the neural networks studied.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

A phase transition model and temporal logic specifications for smart energy systems - revisited

In this paper, we revisit the method of a phase transition model for representing hybrid systems and temporal logic specifications (TLSs) for specifying desired behaviors of systems, and discuss their usefulness for smart energy systems. On the one hand, the phase transition model incorporates the continuous model of the relay device action with a particular structural form that allows for the construction of a single, global differential-algebraic equation for hybrid systems (thus smoothed hybrid systems). On the other hand, the TLS allows sophisticated descriptions of control specifications addressing both magnitude and time simultaneously, which has recently been applied to the control strategy of several types of continuous and hybrid systems. We provide high-level descriptions of each of the two techniques and present simulation results in the context of smart energy systems.

Park, Byungkwon↗

DOE OSTI2023

TAMM: Tensor algebra for many-body methods

Tensor algebra operations such as contractions in computational chemistry consume a significant fraction of the computing time on large-scale computing platforms. The widespread use of tensor contractions between large multi-dimensional tensors in describing electronic structure theory has motivated the development of multiple tensor algebra frameworks targeting heterogeneous computing platforms. In this paper, we present Tensor Algebra for Many-body Methods (TAMM), a framework for productive and performance-portable development of scalable computational chemistry methods. TAMM decouples the specification of the computation from the execution of these operations on available high-performance computing systems. With this design choice, the scientific application developers (domain scientists) can focus on the algorithmic requirements using the tensor algebra interface provided by TAMM, whereas high-performance computing developers can direct their attention to various optimizations on the underlying constructs, such as efficient data distribution, optimized scheduling algorithms, and efficient use of intra-node resources (e.g., graphics processing units). The modular structure of TAMM allows it to support different hardware architectures and incorporate new algorithmic advances. We describe the TAMM framework and our approach to the sustainable development of scalable ground- and excited-state electronic structure methods. We present case studies highlighting the ease of use, including the performance and productivity gains compared to other frameworks.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

Hydrodynamics of quantum spin liquids

Quantum spin liquids are topological states of matter that arise in frustrated quantum magnets at low temperatures. At low energies, such states exhibit emergent gauge fields and fractionalized quasiparticles and can also possess enhanced global symmetries compared to their parent microscopic Hamiltonians. Here we study the consequences of this emergent gauge and symmetry structure for the hydrodynamics of quantum spin liquids. Specifically, we analyze two cases, the U(1) spin liquid with a Fermi surface and the SU(4)-symmetric “algebraic” spin liquid. We show that the emergent degrees of freedom in the spin liquid phase lead to a variety of additional hydrodynamic modes compared to the high-temperature paramagnetic phase. We identify a hydrodynamic regime for the internal U(1) gauge field common to both states, characterized by slow diffusion of the internal transverse photon.

36 MATERIALS SCIENCE↗

DOE OSTI2025

Ensemble Simulation Techniques and Fast Randomized Algorithms

The major goals of the project were to develop and analyze new ensemble simulation techniques, including trajectory stratification and preconditioned MCMC techniques, as well as develop fast numerical linear algebra techniques closely related to ensemble simulation ideas. The trajectory stratification techniques involve simulating in parallel short trajectory fragments of a Markov process confined to a specific region of space‐time and then patching together the statistics gathered to assemble estimates of very general dynamical properties. We have also developed this approach for rare event simulation and extended the techniques to applications requiring a more general framework (such as electronic structure calculations). The preconditioned MCMC techniques involve simulating multiple Markov chains in parallel and then using information from the ensemble to speed the mixing of each individual chain. The fast randomized linear algebra methods are motivated by the diffusion Monte Carlo technique, but are applicable to finding the dominant eigenvalue of (almost) general matrices. For most non‐negative matrices, the schemes result in an error (compared to the power method) that is constant in the dimension of the problem. For more general matrices, we see a very clear sublinear cost trend in computational tests.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗