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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 253 records · Page 14
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DOE OSTI2024

A Comprehensive Northern Hemisphere Particle Microphysics Data Set From the Precipitation Imaging Package

Microphysical observations of precipitating particles are critical data sources for numerical weather prediction models and remote sensing retrieval algorithms. However, obtaining coherent data sets of particle microphysics is challenging as they are often unindexed, distributed across disparate institutions, and have not undergone a uniform quality control process. This work introduces a unified, comprehensive Northern Hemisphere particle microphysical data set from the National Aeronautics and Space Administration precipitation imaging package (PIP), accessible in a standardized data format and stored in a centralized, public repository. Data is collected from 10 measurement sites spanning 34° latitude (37°N–71°N) over 10 years (2014–2023), which comprise a set of 1,070,000 precipitating minutes. The provided data set includes measurements of a suite of microphysical attributes for both rain and snow, including distributions of particle size, vertical velocity, and effective density, along with higher-order products including an approximation of volume-weighted equivalent particle densities, liquid equivalent snowfall, and rainfall rate estimates. The data underwent a rigorous standardization and quality assurance process to filter out erroneous observations to produce a self-describing, scalable, and achievable data set. Case study analyses demonstrate the capabilities of the data set in identifying physical processes like precipitation phase-changes at high temporal resolution. Bulk precipitation characteristics from a multi-site intercomparison also highlight distinct microphysical properties unique to each location. This curated PIP data set is a robust database of high-quality particle microphysical observations for constraining future precipitation retrieval algorithms, and offers new insights toward better understanding regional and seasonal differences in bulk precipitation characteristics.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Weighted Composition Operators for Learning Nonlinear Dynamics

Operator theoretic methods in dynamical system have been dominated by the use of Koopman operators and their continuous time counterparts, such as Koopman Generators and Liouville Operators. The advantage gained from their use primarily stems from the ability to extract subspaces and eigenfunctions within a space of observables that are invariant with respect to the Koopman operator over that space. When this occurs, a dynamic mode decomposition of the systems state provides a linear model for the dynamical system. Not all Koopman operators have eigenfunctions that may be exploited in this manner. However, the framework can still be leveraged for approximations using other operators. In this setting, we present a different operator for the study of dynamical systems, the weighted composition operator. These operators are compact for a wide range of dynamics and spaces, and through their interactions with occupation kernels and vector valued kernels, they admit an estimation of the underlying dynamics. Here, this manuscript presents a new algorithm for the data driven study of dynamical systems from data, and also provides two numerical experiments where convergence is achieved as a proof of concept.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Quantum Circuits for the Preparation of Spin Eigenfunctions on Quantum Computers

The application of quantum algorithms to the study of many-particle quantum systems requires the ability to prepare wave functions that are relevant in the behavior of the system under study. Hamiltonian symmetries are important instruments used to classify relevant many-particle wave functions and to improve the efficiency of numerical simulations. In this work, quantum circuits for the exact and approximate preparation of total spin eigenfunctions on quantum computers are presented. Two different strategies are discussed and compared: exact recursive construction of total spin eigenfunctions based on the addition theorem of angular momentum, and heuristic approximation of total spin eigenfunctions based on the variational optimization of a suitable cost function. The construction of these quantum circuits is illustrated in detail, and the preparation of total spin eigenfunctions is demonstrated on IBM quantum devices, focusing on three- and five-spin systems on graphs with triangle connectivity.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Improving the five-point bootstrap

We present a new algorithm for the numerical evaluation of five-point conformal blocks in d-dimensions, greatly improving the efficiency of their computation. To do this we use an appropriate ansatz for the blocks as a series expansion in radial coordinates, derive a set of recursion relations for the unknown coefficients in the ansatz, and evaluate the series using a Padé approximant to accelerate its convergence. We then study the 〈σσϵσσ〉 correlator in the 3d critical Ising model by truncating the operator product expansion (OPE) and only including operators with conformal dimension below a cutoff ∆ ⩽ ∆cutoff. We approximate the contributions of the operators above the cutoff by the corresponding contributions in a suitable disconnected five-point correlator. Using this approach, we compute a number of OPE coefficients with greater accuracy than previous methods.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Solving the homogeneous Bethe-Salpeter equation with a quantum annealer

The homogeneous Bethe-Salpeter equation (hBSE), describing a bound system in a genuinely relativistic quantum-field theory framework, was solved for the first time by using a D-Wave quantum annealer. After applying standard techniques of discretization, the hBSE, in ladder approximation, can be formally transformed in a generalized eigenvalue problem (GEVP), with two square matrices: one symmetric and the other nonsymmetric. The latter matrix poses the challenge of obtaining a suitable formal approach for investigating the GEVP by means of a quantum annealer, i.e., to recast it as a quadratic unconstrained binary optimization problem. A broad numerical analysis of the proposed algorithms, applied to matrices of dimension up to 64, was carried out by using both the simulated-annealing package and the D-Wave . The numerical results very nicely compare with those obtained with standard classical algorithms, and also show interesting scalability features. Published by the American Physical Society 2024

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Frozen Freedom: Unleashing Grocery Store Demand Flexibility: Preprint

Grocery stores consumed approximately 3% of total electricity used by commercial buildings in the U.S. in 2018 (EIA 2018), representing a unique end-use load profile characterized by the critical use of refrigerated display cases. Exploring demand response (DR) scenarios in grocery stores presents an opportunity to enhance the efficiency and sustainability of surrounding communities. In addition, recent studies demonstrate that implementing control algorithms considering demand flexibility strategies can lead to load and peak reductions in standalone refrigerated display cases. Because small business grocery stores operate on thin margins, the energy bill cost savings DR might provide could make a positive difference toward continued operations. Still, uncertainty remains about the extent of demand flexibility potential controls could provide when coupling refrigeration with whole building operation. To enhance economic viability and grid stability, it is essential to quantify the load flexibility capability of grocery stores. Advanced controls can optimize energy consumption by responding to load shedding, shifting, and DR events, as well as daily Time-of-Use (TOU) rates without compromising food safety. Using both quantitative data and interviews with community-based organizations, we developed a full-size store model and two small store models with controlled refrigerated cases, HVAC, and lighting systems based on actual grocery store properties. Through simulations, we have assessed load flexibility strategies with varied DR events. The results highlight potential for energy and peak reduction with advanced or basic controls. However, interviews and data indicate that more support is needed to make DR strategies consistently accessible to small grocery stores.

demand flexibility↗
DOE OSTI2026

Latent Twins

Over the past decade, scientific machine learning has transformed the development of mathematical and computational frameworks for analyzing, modeling, and predicting complex systems. From inverse problems to numerical partial differential equations (PDEs), dynamical systems, and model reduction, these advances have pushed the boundaries of what can be simulated. Yet they have often progressed in parallel, with representation learning and algorithmic solution methods evolving largely as separate pipelines. With Latent Twins, we propose a unifying mathematical framework that creates a hidden surrogate in latent space for the underlying equations. Whereas digital twins mirror physical systems in the digital world, Latent Twins mirror mathematical systems in a learned latent space governed by operators. Through this lens, classical modeling, inversion, model reduction, and operator approximation all emerge as special cases of a single principle. We establish the fundamental approximation properties of Latent Twins for both ordinary differential equations (ODEs) and PDEs and demonstrate the framework across three representative settings: (i) canonical ODEs, capturing diverse dynamical regimes; (ii) a PDE benchmark using the shallow-water equations, contrasting Latent Twin simulations with deep operator network and forecasts with a four-dimensional variational method baseline; and (iii) a challenging real-data geopotential reanalysis dataset, reconstructing and forecasting from sparse, noisy observations. Latent Twins provide a compact, interpretable surrogate for solution operators that evaluate across arbitrary time gaps in a single-shot, while remaining compatible with scientific pipelines such as assimilation, control, and uncertainty quantification. Looking forward, this framework offers scalable, theory-grounded surrogates that bridge data-driven representation learning and classical scientific modeling across disciplines.

Latent Twins↗
DOE OSTI2026

CONUS-wide Projected Flood Frequency and Uncertainty Estimates, Version 1.0

This dataset presents a large-ensemble of CONUS-wide projected flood frequency and uncertainty estimates across ~2.7 million NHDPlusV2 river reaches over the CONUS. The framework producing this dataset leverages a multi-model, uncertainty-aware modeling framework that allows evaluating shifts in flood frequences at the stream reach level across the CONUS. CONUS-wide ensemble streamflow projections generated from hydrologic simulations driven by downscaled and bias-corrected Coupled Model Intercomparison Project Phase 6 (CMIP6) outputs are used to derive these flood frequency and uncertainty estimates over the period 1980 - 2099. A spatially consistent regional L-moment algorithm is applied across clusters defined by the US Hydrologic Unit Code Subregions (HUC4s and HUC8s) and NHDPlusV2 stream orders to estimate flood frequencies. The dataset also includes at-site based flood estimates that allow for the comparison between local and regional approach-based estimates, assess projected changes, and characterize their uncertainties. For more reliable estimation of rare flood frequencies such as 500 and 1000-year return periods, super-ensemble based estimates are also included in the dataset. This dataset is derived to support the "Impact-Informed Dam Safety Risk Assessment for Securing Hydropower Assests" project for the US Department of Energy (DOE) Hydropower and Hydrokinetic Office (H2O). For further details, refer to Kao et al. (2022), Ghimire et al. (2023), Ghimire et al. (2025), and Hosking and Wallis (1997).

Ghimire, Ganesh [ORNL] (ORCID:0000000242843941)↗
DOE OSTI2024

Commissioning of the Mu2e tracker DAQ, planning for the Vertical Slice Test and pre-pattern recognition studies

The primary objective of the Mu2e experiment at Fermilab is to search for the neutrino-less coherent $\mu \rightarrow e$ conversion in the field of an aluminum nucleus ($\mu^- \text{Al} \rightarrow e^- \text{Al}$). The signature of this process is a monochromatic Conversion Electron (CE) with an energy of approximately 104.97 MeV \cite{bartoszek2015mu2e}. Within the Standard Model (SM), the branching ratio for this process, including neutrino masses and oscillation, is expected to be less than $\mathcal{O}(10^{-50})$. This value is far beyond current experimental capabilities. However, models of physics beyond the SM predict much higher relative rates, approaching an observable level. The SINDRUM II experiment set an upper limit on muon conversion at $7 \times 10^{-13}$ (90\% CL) on Au target \cite{SINDRUMII:2006dvw}, and the Mu2e collaboration aims to improve this limit by four orders of magnitude. Observing this process would provide a clear evidence of physics beyond the Standard Model. A brief discussion of the theoretical and experimental aspects is provided in Chapter \ref{intr}. Mu2e adopts a sophisticated experimental setup to achieve its goals, further described in Chapter \ref{mu2echapter}. The central part of the Mu2e detector is the tracker, that consists of 18 tracking stations. The tracker must provide excellent momentum resolution, approximately 1 MeV/c, to distinguish the monochromatic CE signal from the background. To minimize the energy losses, a straw tube tracker will be used \cite{bobbb}. Chapter \ref{chaptertrk} provides an overview of the straw tracker design and its working principles. This Thesis presents a comprehensive study of the Mu2e tracker, covering complementary aspects from initial commissioning to optimization and first steps of the calibration processes. My work at Fermilab has been focused on the complete Data Acquisition (DAQ) testing from both hardware and software perspectives. I was involved in the commissioning of the Mu2e DAQ system and the Vertical Slice Test (VST) of the tracker. The VST encompasses the entire testing chain, from the straws to the readout, and to processed data on disk. I was also focused on the offline analysis, especially on pre-pattern recognition studies, to explore the best methods for identifying $\delta$-electrons during the data taking. Chapter \ref{commissioning} details the commissioning of the tracker DAQ system, emphasizing the importance of understanding of the readout process before the data acquisition. This includes validating the readout logic and firmware through Monte Carlo simulations to confirm functionality and buffering, monitoring the quality of the data from the tracker preamplifiers and front-end electronics, and assessing overall DAQ performance to ensure reliability during future calibration and data-taking. Chapter \ref{planning} discusses the initial steps towards the tracker calibration. The ultimate goal is to perform a time calibration of the first assembled station of the tracker using cosmic muons, aiming for a longitudinal hit position resolution better than 4 cm. This involves determining the signal propagation times and channel-to-channel delays. I performed a Monte Carlo study to determine the impact of the station orientation on the quality of the calibration, in particular on the cosmic track reconstruction, focusing on potential biases that could arise. These studies provide essential insights into the operation, optimization, and calibration of the Mu2e tracker system. Given the high data volume expected during Mu2e operations, estimated at approximately 7 PBytes per year, optimizing memory usage and minimizing CPU consumption are critical. A significant challenge lies in effectively flagging $\delta$-electron hits, which are the primary source of hits in the tracker, without compromising the efficiency of CE hit detection and track reconstruction. A detailed study of pre-pattern recognition and a thorough comparison of two $\delta$-electron flagging algorithms is provided in Chapter \ref{delta}. In Chapter \ref{conclusions}, the findings are concisely summarized, offering a comprehensive synthesis of the research and emphasizing the key insights derived from this study.

43 PARTICLE ACCELERATORS↗
DOE OSTI2025

Surrogate optimization of variational quantum circuits

Variational quantum eigensolvers are touted as a near-term algorithm capable of impacting many applications. However, the potential has not yet been realized, with few claims of quantum advantage and high resource estimates, especially due to the need for optimization in the presence of noise. Finding algorithms and methods to improve convergence is important to accelerate the capabilities of near-term hardware for VQE or more broad applications of hybrid methods in which optimization is required. To this goal, we look to use modern approaches developed in circuit simulations and stochastic classical optimization, which can be combined to form a surrogate optimization approach to quantum circuits. Using an approximate (classical CPU/GPU) state vector simulator as a surrogate model, we efficiently calculate an approximate Hessian, passed as an input for a quantum processing unit or exact circuit simulator. This method will lend itself well to parallelization across quantum processing units. We demonstrate the capabilities of such an approach with and without sampling noise and a proof-of-principle demonstration on a quantum processing unit utilizing 40 qubits.

Gustafson, Erik J. [RIACS, Mtn. View] (ORCID:00000↗
DOE OSTI2025

Classical eikonal from Magnus expansion

In a classical scattering problem, the classical eikonal is defined as the generator of the canonical transformation that maps in-states to out-states. It can be regarded as the classical limit of the log of the quantum S-matrix. In a classical analog of the Born approximation in quantum mechanics, the classical eikonal admits an expansion in oriented tree graphs, where oriented edges denote retarded/advanced worldline propagators. The Magnus expansion, which takes the log of a time-ordered exponential integral, offers an efficient method to compute the coefficients of the tree graphs to all orders. We exploit a Hopf algebra structure behind the Magnus expansion to develop a fast algorithm which can compute the tree coefficients up to the 12th order (over half a million trees) in less than an hour. In a relativistic setting, our methods can be applied to the post-Minkowskian (PM) expansion for gravitational binaries in the worldline formalism. We demonstrate the methods by computing the 3PM eikonal and find agreement with previous results based on amplitude methods. Importantly, the Magnus expansion yields a finite eikonal, while the naïve eikonal based on the time-symmetric propagator is infrared-divergent from 3PM on.

Black Holes↗
DOE OSTI2025

Emergent hydrodynamic mode on SU(2) plaquette chains and quantum simulation

We search for emergent hydrodynamic modes in real-time Hamiltonian dynamics of 2+1-dimensional SU(2) lattice gauge theory on a quasi-one-dimensional plaquette chain, by numerically computing symmetric correlation functions of energy densities on lattice sizes of about 20 with the local Hilbert space truncated at 𝑗 max = $\frac{1}{2}$. Because of the Umklapp processes, we only find a mode for energy diffusion. The symmetric correlator exhibits transport peak near zero frequency with a width approximately proportional to momentum squared at small momentum, when the system is fully quantum ergodic, as indicated by the eigenenergy level statistics. This transport peak leads to a power-law 𝑡 −$\frac{1}{2}$ decay of the symmetric correlator at late time, also known as the long-time tail, as well as diffusionlike spreading in position space. We also introduce a quantum algorithm for computing the symmetric correlator on a quantum computer and find it gives results consistent with exact diagonalization when tested on the IBM emulator. Finally we discuss the future prospect of searching for the sound modes.

Hamiltonian systems↗
DOE OSTI2024

CIRCLES: Congestion Impacts Reduction via CAV-in-the-loop Lagrangian Energy Smoothing

The energy efficiency of today’s vehicular mobility relies on the un-integrated combination of i) control via static assets (traffic lights, metering, variable speed limits, etc.); and ii) onboard vehicle automation (adaptive cruise control (ACC), ecodriving, etc.). These two families of control were not co-designed and are not engineered to work in coordination. Recent studies have shown i) limitations of controls, and even ii) negative impacts of ACC. This project focused on the technology development, implementation and prototyping, and validation of Mobile Traffic Control (MTC). MTC can be viewed as an extension of classical traffic control (in which static infrastructure actuates traffic flow). In the MTC paradigm, automated vehicles actuate the entire flow via their behavior, offering enhanced possibilities to optimize the energy footprint of traffic, if designed correctly. We set out to demonstrate for the first time that considerably reduced fuel consumption of all vehicles in traffic can be achieved via distributed control of a small proportion of CAVs. Compared to baseline vehicular technologies, our work offers a significant design departure: control algorithms for the CAVs consider the impact one vehicle can have on overall traffic, improving resulting overall fuel consumption. We focus on using a few vehicles (as traffic controllers via CAV technology) to improve the energy efficiency of traffic flow to further optimize energy efficiency. A live-traffic demonstration in November 2022 featured the deployment of 100 specially-equipped CAVs, constituting an approximate local penetration rate upwards of 2%.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2024

Quantum simulations of hadron dynamics in the Schwinger model using 112 qubits

Hadron wave packets are prepared and time evolved in the Schwinger model using 112 qubits of IBM’s 133-qubit Heron quantum computer ibm_torino. The initialization of the hadron wave packet is performed in two steps. First, the vacuum is prepared across the whole lattice using the recently developed SC-ADAPT-VQE algorithm and workflow. SC-ADAPT-VQE is then extended to the preparation of localized states, and used to establish a hadron wave packet on top of the vacuum. This is done by adaptively constructing low-depth circuits that maximize the overlap with an adiabatically prepared hadron wave packet. Due to the localized nature of the wavepacket, these circuits can be determined on a sequence of small lattices using classical computers, and then robustly scaled to prepare wave packets on large lattices for simulations using quantum computers. Time evolution is implemented with a second-order Trotterization. To reduce both the required qubit connectivity and circuit depth, an approximate quasilocal interaction is introduced. This approximation is made possible by the emergence of confinement at long distances, and converges exponentially with increasing distance of the interactions. Using multiple error-mitigation strategies, up to 14 Trotter steps of time evolution are performed, employing 13,858 two-qubit gates (with a CNOT depth of 370). The propagation of hadrons is clearly identified, with results that compare favorably with Matrix Product State simulations. Finally, prospects for a near-term quantum advantage in simulations of hadron scattering are discussed.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

OpenSn: A massively parallel, open-source simulation environment for discrete ordinates radiation transport

OpenSn is an open-source, massively parallel deterministic radiation transport code for solving the discrete-ordinates ( S N ) form of the Boltzmann transport equation on unstructured, arbitrary polyhedral meshes. It supports high-fidelity simulations involving steady-state, eigenvalue, and adjoint problems for neutral particles (e.g., neutrons, photons, multi-particles), using the multigroup approximation in energy. OpenSn combines angular discretization via discrete ordinates with a discontinuous Galerkin finite element method (DGFEM) in space, enabling accurate resolution of transport physics on arbitrary polyhedral cells, included locally refined spatial grids. It includes multiple angular quadrature types, including locally refined angular quadratures. Written in modern C++ with a Python API, OpenSn runs efficiently on platforms ranging from laptops to supercomputers. The transport sweep algorithm is implemented using a task-based, directed-acyclic-graph (DAG) approach for each angle and supports asynchronous parallelism across thousands of MPI ranks. Group-set aggregation improves compute intensity, and synthetic acceleration techniques (e.g., diffusion synthetic acceleration, second-moment method) enhance solver convergence. OpenSn has been verified on reactor physics problems and demonstrated excellent weak and strong scaling performance on more than 32,768 processes, making it a versatile and robust platform for large-scale transport simulations in complex geometries.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

Distribution of centrality measures on undirected random networks via the cavity method

The Katz centrality of a node in a complex network is a measure of the node’s importance as far as the flow of information across the network is concerned. For ensembles of locally tree-like undirected random graphs, this observable is a random variable. Its full probability distribution is of interest but difficult to handle analytically because of its “global” character and its definition in terms of a matrix inverse. Leveraging a fast Gaussian Belief Propagation-Cavity algorithm to solve linear systems on tree-like structures, we show that i) the Katz centrality of a single instance can be computed recursively in a very fast way, and ii) the probability P ( K ) that a random node in the ensemble of undirected random graphs has centrality K satisfies a set of recursive distributional equations, which can be analytically characterized and efficiently solved using a population dynamics algorithm. We test our solution on ensembles of Erdős-Rényi and Scale Free networks in the locally tree-like regime, with excellent agreement. The analytical distribution of centrality for the configuration model conditioned on the degree of each node can be employed as a benchmark to identify nodes of empirical networks with over- and underexpressed centrality relative to a null baseline. We also provide an approximate formula based on a rank- 1 projection that works well if the network is not too sparse, and we argue that an extension of our method could be efficiently extended to tackle analytical distributions of other centrality measures such as PageRank for directed networks in a transparent and user-friendly way.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Enhancing ACPF Analysis: Integrating Newton-Raphson Method with Gradient Descent and Computational Graphs

This paper presents a new method for enhancing Alternating Current Power Flow (ACPF) analysis. The method integrates the Newton-Raphson (NR) method with Enhanced-Gradient Descent (GD) and computational graphs. The integration of renewable energy sources in power systems introduces variability and unpredictability, and this method addresses these challenges. It leverages the robustness of NR for accurate approximations and the flexibility of GD for handling variable conditions, all without requiring Jacobian matrix inversion. Furthermore, computational graphs provide a structured and visual framework that simplifies and systematizes the application of these methods. The goal of this fusion is to overcome the limitations of traditional ACPF methods and improve the resilience, adaptability, and efficiency of modern power grid analyses. We validate the effectiveness of our advanced algorithm through comprehensive testing on established IEEE benchmark systems. Furthermore, our findings demonstrate that our approach not only speeds up the convergence process but also ensures consistent performance across diverse system states, representing a significant advancement in power flow computation.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2026

A simple fourth order propagator based on the Magnus expansion in the Liouville space: Application to a Λ-system and assessment of the rotating wave approximation

A simple fourth-order propagator [Ture and Jang, J. Phys. Chem. A 128, 2871 (2024)] based on the Magnus expansion is extended to the Liouville space for both closed-system and Lindbladian open-system quantum dynamics. For both dynamics, commutator free versions of fourth-order propagators are provided as well. These propagators are then applied to the dynamics of a driven Λ-system, where Lindblad terms represent the effect of a photonic bath. For both dynamics, the accuracy of the rotating wave approximation (RWA) for the matter–radiation interaction is assessed. We confirmed reasonable performance of RWA for weak and resonant fields. However, small errors appear for moderate fields and substantial errors can be found for strong fields where coherent population trapping can still be expected. We also found that the presence of bath for open-system quantum dynamics consistently reduces the errors of the RWA. These results provide quantitative information on how the RWA breaks down beyond weak field or for non-resonant cases. Major results are benchmarked against results of our sixth-order ME-based propagator. Finally, we also provide numerical comparison of our algorithms with other fourth-order algorithms for the Λ-system. These confirm reasonable performance of our simple propagators and the improvement gained through commutator-free expressions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗