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NASA NTRS2012

Computational Ghost Imaging for Remote Sensing

This work relates to the generic problem of remote active imaging; that is, a source illuminates a target of interest and a receiver collects the scattered light off the target to obtain an image. Conventional imaging systems consist of an imaging lens and a high-resolution detector array [e.g., a CCD (charge coupled device) array] to register the image. However, conventional imaging systems for remote sensing require high-quality optics and need to support large detector arrays and associated electronics. This results in suboptimal size, weight, and power consumption. Computational ghost imaging (CGI) is a computational alternative to this traditional imaging concept that has a very simple receiver structure. In CGI, the transmitter illuminates the target with a modulated light source. A single-pixel (bucket) detector collects the scattered light. Then, via computation (i.e., postprocessing), the receiver can reconstruct the image using the knowledge of the modulation that was projected onto the target by the transmitter. This way, one can construct a very simple receiver that, in principle, requires no lens to image a target. Ghost imaging is a transverse imaging modality that has been receiving much attention owing to a rich interconnection of novel physical characteristics and novel signal processing algorithms suitable for active computational imaging. The original ghost imaging experiments consisted of two correlated optical beams traversing distinct paths and impinging on two spatially-separated photodetectors: one beam interacts with the target and then illuminates on a single-pixel (bucket) detector that provides no spatial resolution, whereas the other beam traverses an independent path and impinges on a high-resolution camera without any interaction with the target. The term ghost imaging was coined soon after the initial experiments were reported, to emphasize the fact that by cross-correlating two photocurrents, one generates an image of the target. In CGI, the measurement obtained from the reference arm (with the high-resolution detector) is replaced by a computational derivation of the measurement-plane intensity profile of the reference-arm beam. The algorithms applied to computational ghost imaging have diversified beyond simple correlation measurements, and now include modern reconstruction algorithms based on compressive sensing.

Erkmen, Baris I.↗

NASA NTRS2012

An Examination of Unsteady Airloads on a UH-60A Rotor: Computation Versus Measurement

An unsteady Reynolds-averaged Navier-Stokes solver for unstructured grids is used to simulate the flow over a UH-60A rotor. Traditionally, the computed pressure and shear stresses are integrated on the computational mesh at selected radial stations and compared to measured airloads. However, the corresponding integration of experimental data uses only the pressure contribution, and the set of integration points (pressure taps) is modest compared to the computational mesh resolution. This paper examines the difference between the traditional integration of computed airloads and an integration consistent with that used for the experimental data. In addition, a comparison of chordwise pressure distributions between computation and measurement is made. Examination of this unsteady pressure data provides new opportunities to understand differences between computation and flight measurement.

Biedron, Robert T.↗

NASA NTRS2012

Flexible Launch Vehicle Stability Analysis Using Steady and Unsteady Computational Fluid Dynamics

Launch vehicles frequently experience a reduced stability margin through the transonic Mach number range. This reduced stability margin can be caused by the aerodynamic undamping one of the lower-frequency flexible or rigid body modes. Analysis of the behavior of a flexible vehicle is routinely performed with quasi-steady aerodynamic line loads derived from steady rigid aerodynamics. However, a quasi-steady aeroelastic stability analysis can be unconservative at the critical Mach numbers, where experiment or unsteady computational aeroelastic analysis show a reduced or even negative aerodynamic damping.Amethod of enhancing the quasi-steady aeroelastic stability analysis of a launch vehicle with unsteady aerodynamics is developed that uses unsteady computational fluid dynamics to compute the response of selected lower-frequency modes. The response is contained in a time history of the vehicle line loads. A proper orthogonal decomposition of the unsteady aerodynamic line-load response is used to reduce the scale of data volume and system identification is used to derive the aerodynamic stiffness, damping, and mass matrices. The results are compared with the damping and frequency computed from unsteady computational aeroelasticity and from a quasi-steady analysis. The results show that incorporating unsteady aerodynamics in this way brings the enhanced quasi-steady aeroelastic stability analysis into close agreement with the unsteady computational aeroelastic results.

Bartels, Robert E.↗

NASA NTRS2014

HRP's Healthcare Spin-Offs Through Computational Modeling and Simulation Practice Methodologies

Spaceflight missions expose astronauts to novel operational and environmental conditions that pose health risks that are currently not well understood, and perhaps unanticipated. Furthermore, given the limited number of humans that have flown in long duration missions and beyond low Earth-orbit, the amount of research and clinical data necessary to predict and mitigate these health and performance risks are limited. Consequently, NASA's Human Research Program (HRP) conducts research and develops advanced methods and tools to predict, assess, and mitigate potential hazards to the health of astronauts. In this light, NASA has explored the possibility of leveraging computational modeling since the 1970s as a means to elucidate the physiologic risks of spaceflight and develop countermeasures. Since that time, substantial progress has been realized in this arena through a number of HRP funded activates such as the Digital Astronaut Project (DAP) and the Integrated Medical Model (IMM). Much of this success can be attributed to HRP's endeavor to establish rigorous verification, validation, and credibility (VV&C) processes that ensure computational models and simulations (M&S) are sufficiently credible to address issues within their intended scope. This presentation summarizes HRP's activities in credibility of modeling and simulation, in particular through its outreach to the community of modeling and simulation practitioners. METHODS: The HRP requires all M&S that can have moderate to high impact on crew health or mission success must be vetted in accordance to NASA Standard for Models and Simulations, NASA-STD-7009 (7009) [5]. As this standard mostly focuses on engineering systems, the IMM and DAP have invested substantial efforts to adapt the processes established in this standard for their application to biological M&S, which is more prevalent in human health and performance (HHP) and space biomedical research and operations [6,7]. These methods have also generated substantial interest by the broader medical community though institutions like the National Institutes of Health (NIH) and the Food and Drug Administration (FDA) to develop similar standards and guidelines applicable to the larger medical operations and research community. DISCUSSION: Similar to NASA, many leading government agencies, health institutions and medical product developers around the world are recognizing the potential of computational M&S to support clinical research and decision making. In this light, substantial investments are being made in computational medicine and notable discoveries are being realized [8]. However, there is a lack of broadly applicable practice guidance for the development and implementation of M&S in clinical care and research in a manner that instills confidence among medical practitioners and biological researchers [9,10]. In this presentation, we will give an overview on how HRP is working with the NIH's Interagency Modeling and Analysis Group (IMAG), the FDA and the American Society of Mechanical Engineers (ASME) to leverage NASA's biomedical VV&C processes to establish a new regulatory standard for Verification and Validation in Computational Modeling of Medical Devices, and Guidelines for Credible Practice of Computational Modeling and Simulation in Healthcare.

Mulugeta, Lealem↗

NASA NTRS2014

Computational Aerodynamic Simulations of a 1215 ft/sec Tip Speed Transonic Fan System Model for Acoustic Methods Assessment and Development

Computational Aerodynamic simulations of a 1215 ft/sec tip speed transonic fan system were performed at five different operating points on the fan operating line, in order to provide detailed internal flow field information for use with fan acoustic prediction methods presently being developed, assessed and validated. The fan system is a sub-scale, low-noise research fan/nacelle model that has undergone extensive experimental testing in the 9- by 15-foot Low Speed Wind Tunnel at the NASA Glenn Research Center. Details of the fan geometry, the computational fluid dynamics methods, the computational grids, and various computational parameters relevant to the numerical simulations are discussed. Flow field results for three of the five operating points simulated are presented in order to provide a representative look at the computed solutions. Each of the five fan aerodynamic simulations involved the entire fan system, which for this model did not include a split flow path with core and bypass ducts. As a result, it was only necessary to adjust fan rotational speed in order to set the fan operating point, leading to operating points that lie on a fan operating line and making mass flow rate a fully dependent parameter. The resulting mass flow rates are in good agreement with measurement values. Computed blade row flow fields at all fan operating points are, in general, aerodynamically healthy. Rotor blade and fan exit guide vane flow characteristics are good, including incidence and deviation angles, chordwise static pressure distributions, blade surface boundary layers, secondary flow structures, and blade wakes. Examination of the flow fields at all operating conditions reveals no excessive boundary layer separations or related secondary-flow problems.

Tweedt, Daniel L.↗

NASA NTRS2015

2nd Generation QUATARA Flight Computer Project

Single core flight computer boards have been designed, developed, and tested (DD&T) to be flown in small satellites for the last few years. In this project, a prototype flight computer will be designed as a distributed multi-core system containing four microprocessors running code in parallel. This flight computer will be capable of performing multiple computationally intensive tasks such as processing digital and/or analog data, controlling actuator systems, managing cameras, operating robotic manipulators and transmitting/receiving from/to a ground station. In addition, this flight computer will be designed to be fault tolerant by creating both a robust physical hardware connection and by using a software voting scheme to determine the processor's performance. This voting scheme will leverage on the work done for the Space Launch System (SLS) flight software. The prototype flight computer will be constructed with Commercial Off-The-Shelf (COTS) components which are estimated to survive for two years in a low-Earth orbit.

Falker, Jay↗

NASA NTRS2018

Nozzle Plume/Shock Interaction Experimental and Computational Sonic Boom Analyses from the NASA Ames 9- by 7-Foot Supersonic Wind Tunnel

A wind tunnel test and a computational study were conducted to investigate the complex interactions between a supersonic nozzle plume and shock waves of differing strengths generated from various aft surfaces typical of supersonic aircraft. These analytically-defined aft surfaces were representative of horizontal tails of various sizes, and an aft deck. CFD simulations of many proposed model configurations allowed for assessments of the detailed flow interactions of components in close proximity to the nozzle, as well as assessments of the nozzle jet flow itself. The evaluation of the computational results for many candidate configurations guided the design of model components. The interactions of the waveforms from these surfaces with the jet exhaust plume can have significant adverse effects on the loudness of the sonic boom if the surfaces are not carefully integrated into an aircraft design. The greatest discrepancy in estimating sonic boom loudness for low-boom flight vehicles is currently in predicting the signatures from the aft part of an aircraft, including the interactions with the plume flow. The objectives of this test were to gain a better understanding of these interactions, and to provide a detailed experimental database from multiple sources for use as validation cases for CFD tool development. The subject test was run in the NASA Ames 9- by 7-Ft Supersonic Wind Tunnel in February 2016 at Mach numbers of 1.6 and 2.0, and was funded by the NASA Commercial Supersonics Technology (CST) Project. The nozzle flow was provided by high-pressure air (HPA) pumped through the model, and pressure signature data were acquired with the NASA 14-inch sonic boom pressure rail. The rail measured the locations of the shocks and expansions at various distances and off-track angles from the model. This enabled the impact of the nozzle plume/shock interactions on the near- and mid-field sonic boom pressure waveforms to be quantified. Schlieren images of the flow field around and behind the model were obtained with an RBOS (Retroreflective Background-Oriented Schlieren) technique to determine the origins of the shock and expansion waves, to identify the shape and boundaries of the plume, and to determine the changes in incoming and exiting waveforms within the plume. A total pressure rake was positioned closely behind the model nozzle in order to measure the total pressure profiles of the flow above, within, and below the nozzle exhaust. Model angles and positions in the tunnel were measured by photogrammetry using two cameras since the lack of a model force balance prevented the measurement of model deflections under load.Navier-Stokes computations using two different CFD codes were compared to the experimental sonic boom pressure signature data, and the rake total pressure data in the plume. A computational schlieren technique was used to compare the computed flow field with the RBOS images. The computational results were also used to complement the test data with flow field quantities that could not be measured, such as Mach number and pressure distributions to distinguish shock waves and expansion waves.

sonic boom↗

NASA NTRS2019

High-Performance Spaceflight Computing (HPSC) Middleware Overview

High Performance Spacecraft Computing (HPSC) is a joint project between the National Aeronautics and Space Administration (NASA) and Air Force Research Lab (AFRL) to develop a high-performance multi-core radiation hardened flight processor. HPSC offers a new flight computing architecture to meet the needs of NASA missions through 2030 and beyond. Providing on the order of 100X the computational capacity of current flight processors for the same amount of power, the multicore architecture of the HPSC processor, or "Chiplet" provides unprecedented flexibility in a flight computing system by enabling the operating point to be set dynamically, trading among needs for computational performance, energy management and fault tolerance. The HPSC Chiplet is being developed by Boeing under contract to NASA, and is expected to provide prototypes in 2021. The HPSC Chiplet prototypes will be delivered with an evaluation board, system emulators, comprehensive system software, and a software development kit. In addition to the vendor deliverables, the AFRL is funding the development of a flexible Middleware to be developed by NASA Jet Propulsion Laboratory and NASA Goddard Space Flight Center. The HPSC Middleware provides a suite of thirteen high level services to manage the compute, memory and I/O resources of this complex device.This presentation will provide an overview of the HPSC project, including a hardware overview, system software overview, Middleware overview, and mission use cases. The hardware overview will provide a look at the 8 core High Performance Processing Subsystem (HPPS), the Real Time Processing Subsystem (RTPS), the Chiplet Configuration Management Subsystem, on chip peripherals, and high speed I/O. The system software overview will introduce the boot loaders, operating systems, device drivers, and software development environment. The Middleware overview will provide insight into the high-level services that will be provided to help mission developers manage the many resources and configurations made possible with the Chiplet. Finally, the presentation will provide a brief look at the mission use cases that can be enabled with this next generation architecture.

middleware↗

NASA NTRS2019

Using Graph Coloring to Compute Total Derivatives More Efficiently in OpenMDAO

When they are applicable, gradient based optimization algorithms are the most efficient way to solve design optimization problems. Although gradient based methods are generally efficient, they can be made significantly more so through the usage of analytic techniques to compute the necessary total derivatives. The traditional forward (direct) and reverse (adjoint) analytic techniques have computational costs that scale linearly with the number of design variables and the number of constraints, respectively. In this work, we present an application of a graph coloring algorithm to the analytic techniques for computing total derivative Jacobians in order to achieve much better computational scaling than the pure analytic methods can provide alone. A detailed theoretical explanation of how coloring algorithms interact with analytic derivative methods is presented that illustrates specific types of sparsity patterns that must be present in total derivative Jacobians in order for this coloring technique to be effective. The new technique has been implemented as a feature in the OpenMDAO framework and the implementation is demonstrated on two example problems. The performance on the example problems up to 50% reduction in compute cost for optimizations with bi-directional coloring compared to traditional constraint aggregation. Additionally, the results show how coloring technique alleviates some of the numerical difficulties that constraint aggregation can cause, leading to the ability to solve larger problems. It is expected that the new method will have wide applicability to multidisciplinary optimization problems, and that its availability in OpenMDAO will offer significant computational savings for users without the need for them to implement the coloring algorithm themselves.

Trajectory↗

NASA NTRS2019

High Performance Spacecraft Computing (HPSC) Middleware Update

High Performance Spacecraft Computing (HPSC) is a joint project between the National Aeronautics and Space Administration (NASA) and Air Force Research Lab (AFRL) to develop a high-performance multi-core radiation hardened flight processor. HPSC offers a new flight computing architecture to meet the needs of NASA missions through 2030 and beyond. Providing on the order of 100X the computational capacity of current flight processors for the same amount of power, the multicore architecture of the HPSC processor, or "Chiplet" provides unprecedented flexibility in a flight computing system by enabling the operating point to be set dynamically, trading among needs for computational performance, energy management and fault tolerance. The HPSC Chiplet is being developed by Boeing under contract to NASA, and is expected to provide prototypes, an evaluation board, system emulators, comprehensive system software, and a software development kit. In addition to the vendor deliverables, the AFRL is funding the development of a flexible Middleware to be developed by NASA Jet Propulsion Laboratory and NASA Goddard Space Flight Center. The HPSC Middleware provides a suite of thirteen high level services to manage the compute, memory and I/O resources of this complex device.This presentation will provide an HPSC project update, an overview of the latest HPSC System Software release, an overview of HPSC Middleware Release 2, and a preview of the third HPSC Middleware release. The presentation will begin with a project update that will provide a look at the high-level changes since the project was introduced at the Flight Software Workshop last year. Next, the presentation will provide an overview of the current suite of HPSC System Software which includes the vendor provided bootloaders, operating systems, emulator, and development tools. Next, the HPSC Middleware progress will be presented, which includes an overview of the features and capabilities of HPSC Middleware Release 2, followed by a look at the reference flight software applications which utilize the Middleware. Finally, the presentation will give a preview of the HPSC Middleware Release 3.

Cudmore, Alan↗

NASA NTRSDate not supplied

Sparse Linear Algebra Toolkit for Computational Aerodynamics

Finding solutions to sparse linear systems of equations is an essential step in Computational Engineering applications of interest to NASA. Linear systems of equations are composed and solved in almost every computational engineering application. The characteristics of linear systems vary greatly from one application to another. Accordingly, there are a wide variety of methods for the solution of linear systems of equations. The operations and methods prepared by the authors are focused on linear systems of interest to NASA, primarily those associated with Computational Fluid Dynamics (CFD), Aeroelasticity, and Aeroacoustics. The Sparse Linear Algebra Toolkit (SLAT) is a coordinated collection of software featuring operations, methods, and data structures that are useful when solving sparse linear systems of equations on modern computer architectures. The implemented operations and methods are designed and tuned for parallelism in shared memory, in distributed memory, and across the hybrid combination of distributed-shared memory. The toolkit includes novel methods and implementations for modern architectures and facilitates development of new approaches for meeting NASA’s evolving computational engineering challenges using evolving computer architectures that are not available in vendor libraries. In this paper, significant features and interfaces within SLAT are presented and verified for simulations performed with NASA’s CFD solver, FUN3D. The runtime and scaling performance of the Generalized Minimum Residual (GMRES) method implemented in SLAT is analyzed for the linear subproblems within the solution of turbulent Navier-Stokes equations employed in the simulation of high-lift configurations. Prior to this work, the SPARSKIT GMRES implementation was the only Krylov subspace method available within FUN3D. A strong scaling study shows the SLAT GMRES implementation facilitates accurate Reynolds-averaged Navier-Stokes CFD solutions between 15% and 56% faster than the SPARSKIT GMRES implementation.

Stephen L Wood↗

NASA NTRSDate not supplied

Improving Computational Efficiency of Prognostics Algorithms in Resource-Constrained Settings

The field of prognostics and health management provides quantitative methods for monitoring and predicting the health of physical systems. Prognostics algorithms are useful in that they can be employed to assess the current state of a system, propagate the system state throughout time, and predict potential anomalies or failures that may occur. However, effective prognosis can be challenging to achieve in resource-constrained settings due to computational limitations and high computational latency, leading to obsolete predictions. Thus, computationally efficient and accurate algorithms are necessary for some prognostics applications. In this work, we implement three new algorithmic approaches to prediction (sampling methods, variable prediction time step, variable prediction sample size) with the goal of improving computational efficiency while minimizing decrease in model accuracy. To quantitatively analyze our results, we examine a use-case of degradation of a Lithium-ion battery. Notably, through this work it was found that none of the sampling approaches had a significant impact on computational efficiency or model accuracy in predicting EOD of the battery. However, our results show that prediction accuracy is highly dependent on the time step used, and that an appropriate time step can optimize both model accuracy and simulation efficiency. Finally, implementing a variable sample size also affected prediction, and our results show that tuning both the magnitude and timing of the sample size adjustment in an application-specific manner may prove useful in some applications. Taken together, our findings highlight the challenge of performing prognostics in resource-constrained settings, and illustrate the potential of developing new prediction algorithms to improve computational efficiency of prognosis.

Prognostics↗

NASA NTRSDate not supplied

NASA’s Vision for Spaceflight Computing

Future NASA mission applications demand onboard computing performance, power efficiency, and flexibility not available from current products. To address these needs, NASA’s High Performance Spaceflight Computing (HPSC) project is developing a radiation hardened, general purpose multi-core processor. Key HPSC objectives include natural space radiation hardness, fault tolerance, computation performance and extensibility, and power scalability. This presentation will first highlight how advances in spaceflight computing are key to NASA’s envisioned future for advanced avionics. Descriptions will then be provided for NASA mission applications and use cases that demand advanced spaceflight computing. The presentation will then provide an overview of NASA’s HPSC project and how it will address the computational demands of future missions.

Wesley Powell↗

DOE OSTI2024

Empowering Scientific Innovation Through An Integrated Research Infrastructure: The Role of the Advanced Computing Ecosystem

As the landscape of computational science evolves, the Department of Energy (DOE) is reimagining the roles of its large-scale computing facilities to meet emerging research challenges. The Integrated Research Infrastructure (IRI) program aims to transform how experiments are designed, conducted, and shared, with significant impacts on all stakeholders. In response, the Oak Ridge Leadership Computing Facility (OLCF) has established the Advanced Computing Ecosystem (ACE), a strategic framework to prepare its hardware, software, and experimental capabilities for the IRI era. ACE focuses on integrating novel compute environments, orchestrating advanced workflows, and developing foundational technologies, ensuring a seamless transition to IRI while accelerating scientific discovery. This paper outlines ACE's role in advancing OLCF's mission and its impact on the future of computational science.

Widener, Patrick↗

DOE OSTI2024

Approaches for the Simulation of Coupled Processes in Evolving Fractured Porous Media Enabled by Exascale Computing

Models have historically represented fractured porous media with continuum descriptions that characterize the media using bulk parameters. The impact of small-scale features is not captured in these models, although they may be controlling the performance of subsurface applications. Pore-scale models can simulate processes in small-scale features by representing the pore space geometry explicitly but are computationally expensive for large domains. The alternative multiscale approach entails the combination of pore-scale and continuum-scale descriptions in a single framework. We use Chombo-Crunch, a computational capability that discretizes complex geometries with an adaptive, embedded boundary method to contrast these two approaches. Chombo-Crunch takes advantage of recent computational performance and memory bandwidth improvements resulting from the emergence of exascale computing resources. These combined improvements enable the efficient simulation of reactive transport in fractured media with a high degree of fidelity and the ability to capture the control small-scale processes exert on the overall medium evolution.

42 ENGINEERING↗

DOE OSTI2024

Integrating quantum computing resources into scientific HPC ecosystems

Quantum Computing (QC) offers significant potential to enhance scientific discovery in fields such as quantum chemistry, optimization, and artificial intelligence. Yet QC faces challenges due to the noisy intermediate-scale quantum era’s inherent external noise issues. Here, this paper discusses the integration of QC as a computational accelerator within classical scientific high-performance computing (HPC) systems. By leveraging a broad spectrum of simulators and hardware technologies, we propose a hardware-agnostic framework for augmenting classical HPC with QC capabilities. Drawing on the HPC expertise of the Oak Ridge National Laboratory (ORNL) and the HPC lifecycle management of the Department of Energy (DOE), our approach focuses on the strategic incorporation of QC capabilities and acceleration into existing scientific HPC workflows. This includes detailed analyses, benchmarks, and code optimization driven by the needs of the DOE and ORNL missions. Our comprehensive framework integrates hardware, software, workflows, and user interfaces to foster a synergistic environment for quantum and classical computing research. This paper outlines plans to unlock new computational possibilities, driving forward scientific inquiry and innovation in a wide array of research domains.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2026

Scalable quantum computational science: A perspective from block-encodings and polynomial transformations

Significant developments made in quantum hardware and error correction recently have been driving quantum computing toward practical utility. However, gaps remain between abstract quantum algorithmic development and practical applications in computational sciences. In this perspective article, we propose several properties that scalable quantum computational science methods should possess. We further discuss how block-encodings and polynomial transformations can potentially serve as a unified framework with the desired properties. Recent advancements on these topics are presented, including the construction and assembly of block-encodings, and various generalizations of quantum signal processing (QSP) algorithms to perform polynomial transformations. The scalability of QSP methods on parallel and distributed quantum architectures is also highlighted. Promising applications in simulation and observable estimation in chemistry, physics, and optimization problems are presented. We hope this perspective serves as a gentle introduction to state-of-the-art quantum algorithms for the computational science community and inspires future development of scalable quantum computational science methodologies that bridge theory and practice.

Bayesian inference↗

DOE OSTI2026

Analog and symbolic computation through the Koopman framework

We develop a Koopman operator framework for studying the computational structure of dynamical systems. Specifically, we show that the resolvent of the Koopman operator provides a natural abstraction of halting, yielding a ‘Koopman halting problem’ that is recursively enumerable in general. For symbolic systems, such as those defined on Cantor space, this operator formulation captures reachability between clopen sets, while for equicontinuous systems we prove that the Koopman halting problem is decidable. Our framework demonstrates that absorbing (halting) states in coarse-grained finite automata correspond to Koopman eigenfunctions with eigenvalue one, while cycles in the transition graph impose spectral constraints associated with periodic dynamics. These results provide a unifying perspective on computation in symbolic and analog systems, showing how computational universality is reflected in operator spectra, invariant subspaces, and algebraic structures. Beyond symbolic dynamics, this operator-theoretic lens opens pathways to analyze the computational properties of a broader class of dynamical systems, including polynomial and analog models, and suggests that computational hardness may admit dynamical signatures in terms of Koopman spectral structure.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗