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DOE OSTI2018

Numerical approach based on fractional-order Lagrange polynomials for solving a class of fractional differential equations

In this study, we propose a new set of fractional functions based on the Lagrange polynomials to solve a class of fractional differential equations. Fractional differential equations are the best tools for modelling natural phenomenon that are elaborated by fractional calculus. Therefore, we need an accurate and efficient technique for solving them. The main purpose of this article is to generalize new functions based on Lagrange polynomials to the fractional calculus. At first, we present a new representation of Lagrange polynomials and in continue, we propose a new set of fractional-order functions which are called fractional-order Lagrange polynomials (FLPs). Besides, a general formulation for operational matrices of fractional integration and derivative of FLPs on arbitrary nodal points are extracted. These matrices are obtained using Laplace transform. The initial value problems is reduced to the system of algebraic equations using the operational matrix of fractional integration and collocation method. Also, we find the upper bound of error vector for the fractional integration operational matrix and we indicate convergence of approximations of FLPs. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

Brochure on Basic Research Needs for Scientific Machine Learning: Core Technologies for Artificial Intelligence [Slides]

Scientific Machine Learning (SciML) and Artificial Intelligence (AI) will have broad use and transformative effects across the Department of Energy. Accordingly, the January 2018 Basic Research Needs workshop identified six Priority Research Directions (PRDs). The first three PRDs describe foundational research themes that correspond to the need for domain-awareness (PRD #1), interpretability (PRD #2), and robustness (PRD #3). The other three PRDs describe capability research themes and correspond to the three major use cases for massive scientific data analysis (PRD #4), machine learning-enhanced modeling and simulation (PRD #5), and intelligent automation and decision-support for complex systems (PRD #6). The Priority Research Directions provide a sound basis for a coherent, long-term research and development strategy in SciML and AI. Over the last decade, DOE investments in applied mathematics have laid the groundwork for the type of basic research that will underpin key advances in the six PRDs. Such advances will build on the work from leading researchers in optimization, linear algebra, high-performance solvers and algorithms, multiscale modeling and simulation, complex systems research, uncertainty quantification, and the new basic research areas that will emerge from the pursuit of transformative technologies.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Block smoothers and generalized ideal interpolation in AMG (Final Report)

The Pennsylvania State University (“Subcontractor”) worked on developing new parallel algebraic multilevel methods suitable for solving PDEs. Specifically, work on the design of multigrid solvers for coupled systems of partial differential equations arising in numerical modeling of various applications was completed. A main emphasis was on the design of new ideal algebraic multigrid interpolation for problems such as Maxwell’s equations where block smoothers are needed and the standard form of ideal interpolation is not an effective choice.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Statistical and machine learning models for optimizing energy in parallel applications

Rising power costs and constraints are driving a growing focus on the energy efficiency of high performance computing systems. The unique characteristics of a particular system and workload and their effect on performance and energy efficiency are typically difficult for application users to assess and to control. Settings for optimum performance and energy efficiency can also diverge, so we need to identify trade-off options that guide a suitable balance between energy use and performance. We present statistical and machine learning models that only require a small number of runs to make accurate Pareto-optimal trade-off predictions using parameters that users can control. We study model training and validation using several parallel kernels and more complex workloads, including Algebraic Multigrid (AMG), Large-scale Atomic Molecular Massively Parallel Simulator, and Livermore Unstructured Lagrangian Explicit Shock Hydrodynamics. Here, we demonstrate that we can train the models using as few as 12 runs, with prediction error of less than 10%. Our AMG results identify trade-off options that provide up to 45% improvement in energy efficiency for around 10% performance loss. We reduce the sample measurement time required for AMG by 90%, from 13 h to 74 min.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Compressed basis GMRES on high-performance graphics processing units

Krylov methods provide a fast and highly parallel numerical tool for the iterative solution of many large-scale sparse linear systems. To a large extent, the performance of practical realizations of these methods is constrained by the communication bandwidth in current computer architectures, motivating the investigation of sophisticated techniques to avoid, reduce, and/or hide the message-passing costs (in distributed platforms) and the memory accesses (in all architectures). This article leverages Ginkgo’s memory accessor in order to integrate a communication-reduction strategy into the (Krylov) GMRES solver that decouples the storage format (i.e., the data representation in memory) of the orthogonal basis from the arithmetic precision that is employed during the operations with that basis. Given that the execution time of the GMRES solver is largely determined by the memory accesses, the cost of the datatype transforms can be mostly hidden, resulting in the acceleration of the iterative step via a decrease in the volume of bits being retrieved from memory. Together with the special properties of the orthonormal basis (whose elements are all bounded by 1), this paves the road toward the aggressive customization of the storage format, which includes some floating-point as well as fixed-point formats with mild impact on the convergence of the iterative process. We develop a high-performance implementation of the “compressed basis GMRES” solver in the Ginkgo sparse linear algebra library using a large set of test problems from the SuiteSparse Matrix Collection. We demonstrate robustness and performance advantages on a modern NVIDIA V100 graphics processing unit (GPU) of up to 50% over the standard GMRES solver that stores all data in IEEE double-precision.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

Hybrid Finite Element--Spectral Method for the Fractional Laplacian: Approximation Theory and Efficient Solver

Here, a numerical scheme is presented for approximating fractional order Poisson problems in two and three dimensions. The scheme is based on reformulating the original problem posed over $\Omega$ on the extruded domain $\mathcal{C}=\Omega\times[0,\infty)$ following. The resulting degenerate elliptic integer order PDE is then approximated using a hybrid FEM-spectral scheme. Finite elements are used in the direction parallel to the problem domain $\Omega$, and an appropriate spectral method is used in the extruded direction. The spectral part of the scheme requires that we approximate the true eigenvalues of the integer order Laplacian over $\Omega$. We derive an a priori error estimate which takes account of the error arising from using an approximation in place of the true eigenvalues. We further present a strategy for choosing approximations of the eigenvalues based on Weyl's law and finite element discretizations of the eigenvalue problem. The system of linear algebraic equations arising from the hybrid FEM-spectral scheme is decomposed into blocks which can be solved effectively using standard iterative solvers such as multigrid and conjugate gradient. Numerical examples in two and three dimensions suggest that the approach is quasi-optimal in terms of complexity.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Workshop Report on Basic Research Needs for Scientific Machine Learning: Core Technologies for Artificial Intelligence

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

Deploy threading in Nalu solver stack

The goal of the ExaWind project is to enable predictive simulations of wind farms composed of many MW-scale turbines situated in complex terrain. Predictive simulations will require computational fluid dynamics (CFD) simulations for which the mesh resolves the geometry of the turbines, and captures the rotation and large deflections of blades. Whereas such simulations for a single turbine are arguably petascale class, multi-turbine wind farm simulations will require exascale-class resources. The primary code in the ExaWind project is Nalu, which is an unstructured-grid solver for the acousticallyincompressible Navier-Stokes equations, and mass continuity is maintained through pressure projection. The model consists of the mass-continuity Poisson-type equation for pressure and a momentum equation for the velocity. For such modeling approaches, simulation times are dominated by linear-system setup and solution for the continuity and momentum systems. For the ExaWind challenge problem, the moving meshes greatly affect overall solver costs as re-initialization of matrices and re-computation of preconditioners is required at every time step In this Milestone, we examine the effect of threading on the solver stack performance against flat-MPI results obtained from previous milestones using Haswell performance data full-turbine simulations. Whereas the momentum equations are solved only with the Trilinos solvers, we investigate two algebraic-multigrid preconditioners for the continuity equations: Trilinos/Muelu and HYPRE/BoomerAMG. These two packages embody smoothed-aggregation and classical Ruge-Stiiben AMG methods, respectively. In our FY18 Q2 report, we described our efforts to improve setup and solve of the continuity equations under flat-MPI parallelism. While significant improvement was demonstrated in the solve phase, setup times remained larger than expected. Starting with the optimized settings described in the Q2 report, we explore here simulation performance where OpenMP threading is employed in the solver stack. For Trilinos, threading is acheived through the Kokkos abstraction where, whereas HYPRE/BoomerAMG employs straight OpenMP. We examined results for our mid-resolution baseline turbine simulation configuration (229M DOF). Simulations on 2048 Haswell cores explored the effect of decreasing the number of MPI ranks while increasing the number of threads. Both HYPRE and Trilinos exhibited similar overal solution times, and both showed dramatic increases in simulation time in the shift from MPI ranks to OpenMP threads. This increase is attributed to the large amount of work per MPI rank starting at the single-thread configuration. Decreasing MPI ranks, while increasing threads, may be increasing simulation time due to thread synchronization and start-up overhead contributing to the latency and serial time in the model. These result showed that an MPI+OpenMP parallel decomposition will be more effective as the amount per MPI rank computation per MPI rank decreases and the communication latency increases. This idea was demonstrated in a strong scaling study of our low-resolution baseline model (29M DOF) with the Trilinos-HYPRE configuration. While MPI-only results showed scaling improvement out to about 1536 cores, engaging threading carried scaling improvements out to 4128 cores — roughly 7000 DOF per core. This is an important result as improved strong scaling is needed for simulations to be executed over sufficiently long simulated durations (i.e., for many timesteps). In addition to threading work described above, the team examined solver-performance improvements by exploring communication-overhead in the HYPRE-GMRES implementation through a communicationoptimal- GMRE algorithm (CO-GMRES), and offloading compute-intensive solver actions to GPUs. To those ends, a HYPRE mini-app was allow us to easily test different solver approaches and HYPRE parameter settings without running the entire Nalu code. With GPU acceleration on the Summitdev supercomputer, a 20x speedup was achieved for the overall preconditioner and solver execution time for the mini-app. A study on Haswell processors showed that CO-GMRES provides benefits as one increases MPI ranks.

17 WIND ENERGY↗

DOE OSTI2018

Finite Combinatorial Generation of Metabelian T-Ideal

In this work, we develop our idea on the construction of a system of combinatorial generators in a T-ideal of a free associative algebra, which is a full analogy of a Gröbner–Shirshov basis in a polynomial ideal. We prove a theorem on multilinear monomials that enables us to establish the existence of a finite set of combinatorial generators in a metabelian T-ideal.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Strong-Weak Duality via Jordan-Wigner Transformation: Using Fermionic Methods for Strongly Correlated su(2) Spin Systems

The Jordan-Wigner transformation establishes a duality between $su(2)$ and fermionic algebras. We present qualitative arguments and numerical evidence that when mapping spins to fermions, the transformation makes strong correlation weaker, as demonstrated by the Hartree-Fock approximation to the transformed Hamiltonian. This result can be rationalized in terms of rank reduction of spin shift terms when transformed to fermions. Conversely, the mapping of fermions to qubits makes strong correlation stronger, complicating its solution when one uses qubit-based correlators. The presence of string operators poses challenges to the implementation of quantum chemistry methods on classical computers, but these can be dealt with using established techniques of low computational cost. Here, our proof of principle results for XXZ and J1-J2 Heisenberg (in 1D and 2D) indicate that the JW transformed fermionic Hamiltonian has reduced complexity in key regions of their phase diagrams, and provides a better starting point for addressing challenging spin problems.

74 ATOMIC AND MOLECULAR PHYSICS↗

DOE OSTI2021

Report on hypre performance on AMD GPUs

In this report, the performance of the algebraic multigrid solver used as a preconditioner for conjugate gradient is investigated on 2 nodes of Spock, an early access system at ORNL with 4 MI-100 GPUs and a 64-core Rome CPU per node. We compare GPU and CPU performance for three different diffusion problems using increasing problem sizes.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Scalable Approaches to Selecting Key Entities in Large Networked Infrastructure Systems

This work aims at bringing advances in discrete optimization algorithms to solving practical engineering problems at scale. Often times, in many engineering design problems, there is a need to select a small set of influential or representative elements from a large ground set of entities in an optimal fashion. Submodular optimization provides for a formal way to solve such problems. Common examples with infrastructure systems involve sensor placement and identification of key entities with certain objectives. However, scaling these approaches to large infrastructure systems can be challenging because of the high computational complexity of the overall framework that include the optimization algorithms as well as high-complexity compute-oracles that provide the necessary objective function values. In this work, we explore a well-studied and widely-applicable paradigm, namely leader-selection in a multi-agent networked setting in the context of scalable methodologies. We demonstrate novel frameworks that utilize variations of accelerated submodular optimization algorithms along with linear-algebraic methods that can help accelerate the oracle computations. We further explore this combination in conjunction with graph partitioning paradigms to take advantage of the accelerated algorithms in a distributed setting. Finally we demonstrate the key findings on a practical problem in an operational setting. For this, we leverage an example road network with approximately 18k nodes and 27k edges in a traffic control application, where we seek a limited number of k=200 key intersections. This problem can be solved in a serial setting in just under 5 hours providing more than 2 orders of magnitude speed-up over methods that do not consider acceleration techniques.

Visweswara Sathanur, Arun↗

DOE OSTI2018

Evaluation of the Index and Singular Points of Linear Differential-Algebraic Equations of Higher Order

We study linear systems of ordinary differential equations of higher order with an identically singular matrix at the higher derivative of the required vector-valued function in the domain of definition. We give definitions of the index and singular points for these systems, formulate the conditions of solvability, and deduce a formula for the general solution. The algorithms for finding the index and singular points are presented.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Path integrals, complex probabilities and the discrete Weyl representation

Abstract A discrete formulation of the real-time path integral as the expectation value of a functional of paths with respect to a complex probability on a sample space of discrete valued paths is explored. The formulation in terms of complex probabilities is motivated by a recent reinterpretation of the real-time path integral as the expectation value of a potential functional with respect to a complex probability distribution on cylinder sets of paths. The discrete formulation in this work is based on a discrete version of the Weyl algebra that can be applied to any observable with a finite number of outcomes. The origin of the complex probability in this work is the completeness relation. In the discrete formulation the complex probability exactly factors into products of conditional probabilities and exact unitarity is maintained at each level of approximation. The approximation of infinite dimensional quantum systems by discrete systems is discussed. The method is illustrated by applying it to scattering theory and quantum field theory. The implications of these applications for quantum computing is discussed.

Physics↗

DOE OSTI2018

Robust spatial memory maps encoded by networks with transient connections

The spiking activity of principal cells in mammalian hippocampus encodes an internalized neuronal representation of the ambient space-a cognitive map. Once learned, such a map enables the animal to navigate a given environment for a long period. However, the neuronal substrate that produces this map is transient: the synaptic connections in the hippocampus and in the downstream neuronal networks never cease to form and to deteriorate at a rapid rate. How can the brain maintain a robust, reliable representation of space using a network that constantly changes its architecture? We address this question using a computational framework that allows evaluating the effect produced by the decaying connections between simulated hippocampal neurons on the properties of the cognitive map. Using novel Algebraic Topology techniques, we demonstrate that emergence of stable cognitive maps produced by networks with transient architectures is a generic phenomenon. The model also points out that deterioration of the cognitive map caused by weakening or lost connections between neurons may be compensated by simulating the neuronal activity. Lastly, the model explicates the importance of the complementary learning systems for processing spatial information at different levels of spatiotemporal granularity.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2026

Selective Amnesia using Contrastive Subnet Erasure for Class Level Unlearning in Vision Models

We study concept-level forgetting in pretrained vision models: removing an entire semantic category so the system no longer recognizes that object in unseen images and contexts, rather than merely forgetting specific training examples. Prior work either applies blunt global projections or fine-tunes parameters, which can introduce collateral damage to unrelated features, add compute, and become unstable as forgetting strength increases. We introduce Contrastive Subnet Erasure (CSE), a training-free, encoder-centric edit that targets a compact set of channels most responsible for the class and attenuates them in a calibrated manner. The modification is algebraically folded into the subsequent layer, yielding no inference-time overhead and leaving task heads unchanged. To evaluate whether forgetting generalizes beyond the data used to specify the class, we introduce a cross dataset protocol in which the class is defined on a source dataset and performance is measured on a disjoint target dataset drawn from a different distribution with no shared images. This setup tests whether the model still fails to recognize the object when it looks different or appears in new scenes, and it helps avoid overfitting to patterns in the source dataset. Across CIFAR 10, CIFAR 100, and ImageNet under this protocol, CSE achieves stronger forgetting of the target class while better preserving non target utility than existing baselines in both single class and multi class settings. Overall, CSE provides a simple, stable, and deployment-ready mechanism for class-level unlearning in vision.

Kotevska, Olivera [ORNL] (ORCID:0000000316772243)↗

DOE OSTI2026

Neuralized fermionic tensor networks for quantum many-body systems

In this work, we describe a class of neuralized fermionic tensor network states (NN-fTNSs) that introduce nonlinearity into fermionic tensor networks through configuration-dependent neural network transformations of the local tensors. The construction uses the fTNS algebra to implement a natural fermionic sign structure and is compatible with standard tensor network algorithms but gains enhanced expressivity through the neural network parametrization. Using the 1D and 2D Fermi-Hubbard models as benchmarks, we demonstrate that NN-fTNSs achieve order of magnitude improvements in the ground-state energy compared to pure fTNSs with the same bond dimension and can be systematically improved through both the tensor network bond dimension and the neural network parametrization. Compared to existing fermionic neural quantum states based on Slater determinants and Pfaffians, NN-fTNSs offer a physically motivated alternative fermionic structure. Furthermore, compared to such states, NN-fTNSs naturally exhibit improved computational scaling and we demonstrate a construction that achieves linear scaling with the lattice size.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2018

Systems with Parameters, or Efficiently Solving Systems of Polynomial Equations: 33 Years Later. I

Consider a system of polynomial equations with parametric coefficients over an arbitrary ground field. We show that the variety of parameters can be represented as a union of strata. For values of parameters from each stratum, the solutions of the system are given by algebraic formulas depending only on this stratum. Each stratum is a quasiprojective algebraic variety with degree bounded from above by a subexponential function in the size of the input data. Also, the number of strata is subexponential in the size of the input data. Thus, here we avoid double exponential upper bounds on the degrees and solve a long-standing problem.

97 MATHEMATICS AND COMPUTING↗