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All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Simultaneous stoquasticity

Stoquastic Hamiltonians play a role in the computational complexity of the local Hamiltonian problem as well as the study of classical simulability. In particular, stoquastic Hamiltonians can be straightforwardly simulated using Monte Carlo techniques. We address the question of whether two or more Hamiltonians may be made simultaneously stoquastic via a unitary transformation. This question has important implications for the complexity of simulating quantum annealing where quantum advantage is related to the stoquasticity of the Hamiltonians involved in the anneal. We find that for almost all problems no such unitary exists and show that the problem of determining the existence of such a unitary is equivalent to identifying if there is a solution to a system of polynomial (in)equalities in the matrix elements of the initial and transformed Hamiltonians. Furthermore, solving such a system of equations is NP-hard. We highlight a geometric understanding of this problem in terms of a collection of generalized Bloch vectors.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Complexity Phase Diagram for Interacting and Long-Range Bosonic Hamiltonians

Here, we classify phases of a bosonic lattice model based on the computational complexity of classically simulating the system. We show that the system transitions from being classically simulable to classically hard to simulate as it evolves in time, extending previous results to include on-site number-conserving interactions and long-range hopping. Specifically, we construct a complexity phase diagram with easy and hard “phases” and derive analytic bounds on the location of the phase boundary with respect to the evolution time and the degree of locality. We find that the location of the phase transition is intimately related to upper bounds on the spread of quantum correlations and protocols to transfer quantum information. Remarkably, although the location of the transition point is unchanged by on-site interactions, the nature of the transition point does change. Specifically, we find that there are two kinds of transitions, sharp and coarse, broadly corresponding to interacting and noninteracting bosons, respectively. Our Letter motivates future studies of complexity in many-body systems and its interplay with the associated physical phenomena.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Spatially adaptive stochastic methods for fluid–structure interactions subject to thermal fluctuations in domains with complex geometries

We develop stochastic mixed finite element methods for spatially adaptive simulations of fluid–structure interactions when subject to thermal fluctuations. To account for thermal fluctuations, we introduce a discrete fluctuation–dissipation balance condition to develop compatible stochastic driving fields for our discretization. We perform analysis that shows our condition is sufficient to ensure results consistent with statistical mechanics. We show the Gibbs–Boltzmann distribution is invariant under the stochastic dynamics of the semi-discretization. To generate efficiently the required stochastic driving fields, we develop a Gibbs sampler based on iterative methods and multigrid to generate fields with O(N) computational complexity. Our stochastic methods provide an alternative to uniform discretizations on periodic domains that rely on Fast Fourier Transforms. To demonstrate in practice our stochastic computational methods, we investigate within channel geometries having internal obstacles and no-slip walls how the mobility/diffusivity of particles depends on location. Our methods extend the applicability of fluctuating hydrodynamic approaches by allowing for spatially adaptive resolution of the mechanics and for domains that have complex geometries relevant in many applications.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

On the hardness of learning ground state entanglement of geometrically local Hamiltonians

Characterizing the entanglement structure of ground states of local Hamiltonians is a fundamental problem in quantum information. In this work we study the computational complexity of this problem, given the Hamiltonian as input. Our main result is that to show it is cryptographically hard to determine if the ground state of a geometrically local, polynomially gapped Hamiltonian on qudits (d=O(1)) has near-area law vs near-volume law entanglement. This improves prior work of Bouland et al. (arXiv:2311.12017) showing this for non-geometrically local Hamiltonians. In particular we show this problem is roughly factoring-hard in 1D, and LWE-hard in 2D. Our proof works by constructing a novel form of public-key pseudo-entanglement which is highly space-efficient, and combining this with a modification of Gottesman and Irani's quantum Turing machine to Hamiltonian construction. Our work suggests that the problem of learning so-called "gapless" quantum phases of matter might be intractable.

Computational Complexity (cs.CC)↗
DOE OSTI2014

Simulation of dilute polymeric fluids in a three-dimensional contraction using a multiscale FENE model

The multiscale FENE model is applied to a 3D square-square contraction flow problem. For this purpose, the stochastic Brownian configuration field method (BCF) has been coupled with our fully parallelized three-dimensional Navier-Stokes solver NaSt3DGPF. The robustness of the BCF method enables the numerical simulation of high Deborah number flows for which most macroscopic methods suffer from stability issues. The results of our simulations are compared with that of experimental measurements from literature and show a very good agreement. In particular, flow phenomena such as a strong vortex enhancement, streamline divergence and a flow inversion for highly elastic flows are reproduced. Due to their computational complexity, our simulations require massively parallel computations. Using a domain decomposition approach with MPI, the implementation achieves excellent scale-up results for up to 128 processors.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2017

Multiscale high-order/low-order (HOLO) algorithms and applications

We review the state of the art in the formulation, implementation, and performance of so-called high-order/low-order (HOLO) algorithms for challenging multiscale problems. HOLO algorithms attempt to couple one or several high-complexity physical models (the high-order model, HO) with low-complexity ones (the low-order model, LO). The primary goal of HOLO algorithms is to achieve nonlinear convergence between HO and LO components while minimizing memory footprint and managing the computational complexity in a practical manner. Key to the HOLO approach is the use of the LO representations to address temporal stiffness, effectively accelerating the convergence of the HO/LO coupled system. The HOLO approach is broadly underpinned by the concept of nonlinear elimination, which enables segregation of the HO and LO components in ways that can effectively use heterogeneous architectures. The accuracy and efficiency benefits of HOLO algorithms are demonstrated with specific applications to radiation transport, gas dynamics, plasmas (both Eulerian and Lagrangian formulations), and ocean modeling. Across this broad application spectrum, HOLO algorithms achieve significant accuracy improvements at a fraction of the cost compared to conventional approaches. It follows that HOLO algorithms hold significant potential for high-fidelity system scale multiscale simulations leveraging exascale computing.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Complexity and near extremal charged black branes

Highlights: • Trying to explore a remedy to overcome the puzzle in holographic complexity by assuming a cutoff behind the horizon and the corresponding boundary term. • Holographic complexity growth obtained by the WDW patch on shell action is insensitive to the UV cutoff at the late time. Therefore one may conclude that either on shell action does not compute complexity, or the late time is not given by the physical charges, or in our computations we have missed something! • We find that one needs to carefully compute on shell action taking into account the behind the horizon and the corresponding boundary term. Doing so, one arrives at a consistent picture. • We check this consistency by evaluating the late time complexity growth for near horizon limit of the near extremal black hole using two different methods. We would have obtained non-consistent picture, if we had not considered the behind the horizon cutoff as well as the corresponding boundary term. We compute holographic complexity of charged black brane solutions in arbitrary dimensions for the near horizon limit of near extremal case using two different methods. The corresponding complexity may be obtained either by taking the limit from the complexity of the charged black brane, or by computing the complexity for near horizon limit of near extremal solution. One observes that these results coincide if one assumes to have a cutoff behind horizon whose value is fixed by UV cutoff and also taking into account a proper counterterm evaluated on this cutoff. We also consider the situation for Vaidya charged black branes too.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Learning of networked spreading models from noisy and incomplete data

Recent years have seen a lot of progress in algorithms for learning parameters of spreading dynamics from both full and partial data. Some of the remaining challenges include model selection under the scenarios of unknown network structure, noisy data, missing observations in time, as well as an efficient incorporation of prior information to minimize the number of samples required for an accurate learning. Here, in this work, we introduce a universal learning method based on a scalable dynamic message-passing technique that addresses these challenges often encountered in real data. The algorithm leverages available prior knowledge on the model and on the data, and reconstructs both network structure and parameters of a spreading model. We show that a linear computational complexity of the method with the key model parameters makes the algorithm scalable to large network instances.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

VAN-DAMME: GPU-accelerated and symmetry-assisted quantum optimal control of multi-qubit systems

We present an open-source software package, VAN-DAMME (Versatile Approaches to Numerically Design, Accelerate, and Manipulate Magnetic Excitations), for massively-parallelized quantum optimal control (QOC) calculations of multi-qubit systems. To enable large QOC calculations, the VAN-DAMME software package utilizes symmetry-based techniques with custom GPU-enhanced algorithms. This combined approach allows for the simultaneous computation of hundreds of matrix exponential propagators that efficiently leverage the intra-GPU parallelism found in high-performance GPUs. In addition, to maximize the computational efficiency of the VAN-DAMME code, we carried out several extensive tests on data layout, computational complexity, memory requirements, and performance. These extensive analyses allowed us to develop computationally efficient approaches for evaluating complex-valued matrix exponential propagators based on Padé approximants. To assess the computational performance of our GPU-accelerated VAN-DAMME code, we carried out QOC calculations of systems containing 10 - 15 qubits, which showed that our GPU implementation is 18.4× faster than the corresponding CPU implementation. Our GPU-accelerated enhancements allow efficient calculations of multi-qubit systems, which can be used for the efficient implementation of QOC applications across multiple domains.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

One-shot omnidirectional pressure integration through matrix inversion

In this work, we present a method to perform 2D and 3D omnidirectional pressure integration from velocity measurements with a single-iteration matrix inversion approach. This work builds upon our previous work, where the rotating parallel ray approach was extended to the limit of infinite rays by taking continuous projection integrals of the ray paths and recasting the problem as an iterative matrix inversion problem. This iterative matrix equation is now 'fast-forwarded' to the 'infinity' iteration, leading to a different matrix equation that can be solved in a single step, thereby presenting the same computational complexity as the Poisson equation. We observe computational speedups of ~10 6 when compared to brute-force omnidirectional integration methods, enabling the treatment of grids of ~10 9 points and potentially even larger in a desktop setup at the time of publication. Further examination of the boundary conditions of our one-shot method shows that omnidirectional pressure integration implements a boundary condition where the boundary points are treated as interior points to the extent that information is available. Finally, we show how the method can be extended from the regular grids typical of particle image velocimetry to the unstructured meshes characteristic of particle tracking velocimetry data.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

SPIKANs: separable physics-informed Kolmogorov–Arnold networks

Physics-Informed Neural Networks (PINNs) have emerged as a promising method for solving partial differential equations (PDEs) in scientific computing. While PINNs typically use multilayer perceptrons (MLPs) as their underlying architecture, recent advancements have explored alternative neural network structures. One such innovation is the Kolmogorov–Arnold Network (KAN), which has demonstrated benefits over traditional MLPs, including faster neural scaling and better interpretability. The application of KANs to physics-informed learning has led to the development of Physics-Informed KANs (PIKANs), enabling the use of KANs to solve PDEs. However, despite their advantages, KANs often suffer from slower training speeds, particularly in higher-dimensional problems where the number of collocation points grows exponentially with the dimensionality of the system. To address this challenge, we introduce Separable Physics-Informed Kolmogorov–Arnold Networks (SPIKANs). This novel architecture applies the principle of separation of variables to PIKANs, decomposing the problem such that each dimension is handled by an individual KAN. This approach drastically reduces the computational complexity of training without sacrificing accuracy, facilitating their application to higher-dimensional PDEs. Through a series of benchmark problems, we demonstrate the effectiveness of SPIKANs, showcasing their superior scalability and performance compared to PIKANs and highlighting their potential for solving complex, high-dimensional PDEs in scientific computing.

Kolmogorov-Arnold networks↗
DOE OSTI2023

Efficient Integration of Algebraic Constraint for Exponential Time Integration

This work is a continuation of a previous project where the high efficiency of exponential integration methods for DRMHD systems was demonstrated. Often algebraic constraints must also be enforced for DRMHD systems of interest. Straightforward application of exponential methods to DRMHD equations with constraints leads to prohibitively computationally expensive methods. In this work, we propose new exponential schemes that allow the constraints to be removed from evaluation of exponential matrix functions which drastically reduces computational complexity by eliminating the need to perform computations enforcing constraints in exponential calculations while still preserving a high order of accuracy and allowing for a large time step even when the problem is stiff. This idea is similar to the W-methods and is achieved by carefully designing a method with the desired order of accuracy, even with an incomplete Jacobian matrix used as an argument of exponential-like functions. The constraints are accounted for by including them in the evaluation of the right-hand-side forcing function of the spatially discretized system. We study performance of the new methods on test problems and outline future research directions that this work opens.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Adaptive Extremum Seeking Control Based LCL Filter Resonant Frequency Online Estimation

The LCL filter has been widely used in the grid-tied inverter systems. However, the resonance of the LCL filter can reduce the system stability margin and the control performance. Moreover, the grid impedance variations can lead to the drift of the resonant frequency, which can further worsen the system robustness. Thus, it is important to know the actual resonant frequency of the LCL filter. In this letter, an adaptive extremum seeking control (AESC) based estimation scheme is proposed to estimate the resonant frequency of the LCL filter online. Further by injecting a high-frequency (HF) signal into the inverter output voltage, the AESC scheme can identify the extremum of the LCL filter amplitude response, i.e., resonant peak. The amplitude of injection signal is adaptive based on the inverter HF response, which can address the tradeoff between the dynamic response and inverter output current quality. Most importantly, compare to other method, the proposed scheme has very low computational complexity, which minimizes the burden to the normal inverter controller operation. Stability analysis is given in this letter, and experimental studies are conducted to validate the effectiveness of the proposed scheme.

42 ENGINEERING↗
DOE OSTI2019

Big data analytics in smart grids: state-of-the-art, challenges, opportunities, and future directions

Big data has a potential to unlock novel groundbreaking opportunities in the power grid sector that enhances a multitude of technical, social, and economic gains. The currently untapped potential of applying the science of big data for better planning and operation of the power grid is a very challenging task and needs significant efforts all-around. As power grid technologies evolve in conjunction with measurement and communication technologies, this results in unprecedented amount of heterogeneous big data sets from diverse sources. In particular, computational complexity, data security, and operational integration of big data into utility decision frameworks are the key challenges to transform the heterogeneous large dataset into actionable outcomes. Moreover, due to the complex nature of power grids along with the need to balance power in real time, seamless integration of big data into utility operations is very critical. In this context, big data analytics combined with grid visualization can lead to better situational awareness and predictive decisions. This paper presents a comprehensive state-of-the-art review of big data analytics and its applications in power grids, and also identifies challenges and opportunities from utility, industry, and research perspectives. The paper analyzes research gaps and presents insights on future research directions to integrate big data analytics into electric utility decision framework. Detailed information for utilities looking to apply big data analytics and details insights on how utilities can enhance revenue streams and bring disruptive innovation in the industry are discussed. More importantly, general guidelines for utilities to make the right investment in the adoption of big data analytics by unveiling interdependencies among critical infrastructures and operations are also provided.

13 HYDRO ENERGY↗
DOE OSTI2025

Accelerating resonant spectroscopy simulations using multishifted biconjugate gradient

Resonant spectroscopies, which involve intermediate states with finite lifetimes, provide important insights into collective excitations in quantum materials that are otherwise inaccessible. However, theoretical understanding in this area is often limited by the numerical challenges of solving Kramers-Heisenberg-type response functions for large-scale systems. To address this, we introduce a multishifted biconjugate gradient algorithm that exploits the shared structure of Krylov subspaces across spectra with varying incident energies, effectively reducing the computational complexity to that of linear spectroscopies. Both mathematical proofs and numerical benchmarks confirm that this algorithm substantially accelerates spectral simulations, achieving constant complexity independent of the number of incident energies, while ensuring accuracy and stability. This development provides a scalable, versatile framework for simulating advanced spectroscopies in quantum materials.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2016

Complexity, action, and black holes

In an earlier paper "Complexity Equals Action" we conjectured that the quantum computational complexity of a holographic state is given by the classical action of a region in the bulk (the `Wheeler-DeWitt' patch). We provide calculations for the results quoted in that paper, explain how it fits into a broader (tensor) network of ideas, and elaborate on the hypothesis that black holes are the fastest computers in nature.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

Real-Time Interactive 4D-STEM Phase-Contrast Imaging From Electron Event Representation Data: Less computation with the right representation

The arrival of direct electron detectors (DED) with high frame-rates in the field of scanning transmission electron microscopy has enabled many experimental techniques that require collection of a full diffraction pattern at each scan position, a field which is subsumed under the name four dimensional-scanning transmission electron microscopy (4D-STEM). DED frame rates approaching 100 kHz require data transmission rates and data storage capabilities that exceed commonly available computing infrastructure. Current commercial DEDs allow the user to make compromises in pixel bit depth, detector binning or windowing to reduce the per-frame file size and allow higher frame rates. This change in detector specifications requires decisions to be made before data acquisition that may reduce or lose information that could have been advantageous during data analysis. The 4D Camera, a DED with 87 kHz frame-rate developed at Lawrence Berkeley National Laboratory, reduces the raw data to a linear-index encoded electron event representation (EER). Here we show with experimental data from the 4D Camera that linear-index encoded EER and its direct use in 4D-STEM phase contrast imaging methods enables real-time, interactive phase-contrast from large-area 4D-STEM datasets. Furthermore, we detail the computational complexity advantages of the EER and the necessary computational steps to achieve real-time interactive ptychography and center-of-mass differential phase contrast using commonly available hardware accelerators.

4D-STEM↗
DOE OSTI2018

Adiabatic Quantum Computation Applied to Deep Learning Networks

Training deep learning networks is a difficult task due to computational complexity, and this is traditionally handled by simplifying network topology to enable parallel computation on graphical processing units (GPUs). However, the emergence of quantum devices allows reconsideration of complex topologies. We illustrate a particular network topology that can be trained to classify MNIST data (an image dataset of handwritten digits) and neutrino detection data using a restricted form of adiabatic quantum computation known as quantum annealing performed by a D-Wave processor. We provide a brief description of the hardware and how it solves Ising models, how we translate our data into the corresponding Ising models, and how we use available expanded topology options to explore potential performance improvements. Although we focus on the application of quantum annealing in this article, the work discussed here is just one of three approaches we explored as part of a larger project that considers alternative means for training deep learning networks. The other approaches involve using a high performance computing (HPC) environment to automatically find network topologies with good performance and using neuromorphic computing to find a low-power solution for training deep learning networks. Our results show that our quantum approach can find good network parameters in a reasonable time despite increased network topology complexity; that HPC can find good parameters for traditional, simplified network topologies; and that neuromorphic computers can use low power memristive hardware to represent complex topologies and parameters derived from other architecture choices.

97 MATHEMATICS AND COMPUTING↗