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Results for “Riemannian optimization”

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DOE OSTI2024

Riemannian Optimization Applied to AC Optimal Power Flow: Preprint

The nonlinear, nonconvex AC optimal power flow problem is of growing importance as the nature of the power grid evolves. This problem can be difficult to solve for interior point methods. However, the advent of optimization algorithms over smooth Riemannian manifolds presents an alternative approach. The nonlinear, nonconvex constraints in the AC power flow problem form an embedded submanifold of Euclidean space. In this paper, the authors explore the performance of Riemannian optimization algorithms for the ACOPF problem where the optimization is performed directly on the AC power flow manifold. They demonstrate that these are viable computational alternatives to interior point methods. This is done by using Julia and the packages PowerModels.jl and Manopt.jl.

manifold optimization↗

DOE OSTI2024

Riemannian Optimization Applied to AC Optimal Power Flow

The nonlinear, nonconvex AC optimal power flow problem is of growing importance as the nature of the power grid evolves. This problem can be difficult to solve for interior point methods. However, the advent of optimization algorithms over smooth Riemannian manifolds presents an alternative approach. The nonlinear, nonconvex constraints in the AC power flow problem form an embedded submanifold of Euclidean space. In this paper, the authors explore the performance of Riemannian optimization algorithms for the ACOPF problem where the optimization is performed directly on the AC power flow manifold. This is done by using the Julia programming language and the Julia packages PowerModels.jl and Manopt.jl.

AC optimal power flow↗

DOE OSTI2024

Exploiting Power Flow Manifold to Solve AC Optimal Power Flow

AC optimal power flow has proven difficult to solve with interior point methods on GPUs. This is largely due to challenging linear algebra problems that current state of the art massively parallel linear solvers struggle with. However, the advent of Riemannian optimization techniques and the fact that the power flow equations form a smooth manifold present an alternative approach. In this talk, we present the basics of Riemannian optimization techniques in which optimization is done directly on a manifold. Then we present computational results showing that Riemannian techniques are capable of producing solutions of comparable quality as interior point methods.

AC optimal power flow↗

DOE OSTI2026

Locally purified maximally mixed states at scale: Entanglement pruning and symmetries

Locally Purified Density Operators (LPDOs) are state-of-the-art tensor network ansatze candidates that efficiently represent mixed quantum states at scale. However, given their non-uniqueness, their representational complexity is generally sub-optimal in practical computations. Here, in this work we perform a comprehensive numerical and analytical analysis and resolve this issue in the experimentally relevant limit where noise depolarizes the density operator into a maximally mixed state. To resolve the sub-optimality issue, we analyze two numerical tools, one analytic method, and detail the relations between them. The numerical tools used are fidelity-preserving truncations and isometric gauge transformations leveraging Riemannian optimizations over entropic objective functions. In addition, by invoking the injectivity and symmetry constraints of the maximally mixed LPDO, we also present analytical closed-form expressions for the disentangler and discuss their relation to numerical optimizers. Further, away from the maximally mixed state, our simulations highlight how the truncation threshold smoothly interpolate, as a function of depolarization, between established matrix product results and our new results. Our work shows how, by minimizing the resources required to represent key states of practical interest in experiment, the efficiency of tensor network algorithms can be substantially increased. This paves the path for uncovering tensor network’s fundamental scalability limits and latent potential in representing the wide locus of mixed quantum states that are accessible on near-term quantum devices.

Gangapuram, Amit Jamadagni [Oak Ridge National Lab↗

DOE OSTI2021

Geometry optimization speedup through a geodesic approach to internal coordinates

We present a new geodesic-based method for geometry optimization in a basis set of redundant internal coordinates. Overall, our method updates the molecular geometry by following the geodesic generated by a displacement vector on the internal coordinate manifold, which dramatically reduces the number of steps required to converge to a minimum. Our method can be implemented in any existing optimization code, requiring only implementation of derivatives of the Wilson B-matrix and the ability to numerically solve an ordinary differential equation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

Absence of Barren Plateaus and Scaling of Gradients in the Energy Optimization of Isometric Tensor Network States

Abstract Vanishing gradients can pose substantial obstacles for high-dimensional optimization problems. Here we consider energy minimization problems for quantum many-body systems with extensive Hamiltonians and finite-range interactions, which can be studied on classical computers or in the form of variational quantum eigensolvers on quantum computers. Barren plateaus correspond to scenarios where the average amplitude of the energy gradient decreases exponentially with increasing system size. This occurs, for example, for quantum neural networks and for brickwall quantum circuits when the depth increases polynomially in the system size. Here we prove that the variational optimization problems for matrix product states, tree tensor networks, and the multiscale entanglement renormalization ansatz are free of barren plateaus. The derived scaling properties for the gradient variance provide an analytical guarantee for the trainability of randomly initialized tensor network states (TNS) and motivate certain initialization schemes. In a suitable representation, unitary tensors that parametrize the TNS are sampled according to the uniform Haar measure. We employ a Riemannian formulation of the gradient based optimizations which simplifies the analytical evaluation.

Barthel, Thomas↗

DOE OSTI2025

Power Flow Geometry and Approximation

Here, the power flow equations are important in numerous power systems problems of practical interest which consider alternating current power flow (ACPF) physics. Perhaps the most well studied being the alternating current optimal power flow problem (ACOPF), seeking to optimize the operation of an electric power system. Due to their non-linearity, problems which include the power flow equations are typically challenging, particularly in optimization. Interestingly, the set of solutions to the power flow equations forms a smooth manifold. As a result, differential geometry can be used to describe and analyze this set of equations. This approach has proven effective in several engineering applications (e.g., solving ACOPF and analyzing the solution space boundary). Central to the success of this approach is an understanding of the power flow manifold's geometry. In this work, we develop the geometric and topological properties of this manifold using concepts from differential geometry. After demonstrating the convenience of this manifold's representation as a function's graph, computational methods are emphasized: we develop retractions, error bounds for linear approximation, and formulas for evaluating the Riemannian metric (including associated objects such as geodesics and the curvature tensor). Scalar curvature and the second fundamental form play a new role in quantifying the quality of linear approximations, like the popular direct current approximation. All functions are implemented in Julia and available in an online repository. Proofs are included for completeness.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2025

Conflict Detection in Open RAN with Recurrent Neural Networks Using Geometric Manifolds

Allowing third-party applications on Radio Access Network (RAN) Intelligent Controllers (RICs) within the OpenRAN (O-RAN) framework introduces conflicting interactions that are often difficult to detect in advance. These conflicts, occurring between third-party applications in the Near RealTime RIC (Near-RT RIC), known as xApps, can lead to performance degradation and instability in O-RAN if not identified early. Existing conflict detection and mitigation solutions in the literature assume that the conflicts are known beforehand, which is not always accurate due to the complex and often hidden relationships between control parameters and Key Performance Indicators (KPIs). In this paper, we propose a novel Recurrent Neural Network (RNN) to detect both known and unknown conflicts in O-RAN xApps as specified in the O-RAN standards. We model the xApps, control parameters, and KPIs with nodes and edges to create graph structures and use the hidden nonEuclidean geometric properties of the Riemannian manifold to train the RNN model. The performance of this proposed model is validated using evaluation metrics and compared with benchmarks. Results demonstrate that the proposed RNN model, leveraging Riemannian geometric properties, can achieve 100% of the F1-score provided by an optimal solution in just 20 iterations.

5G↗

DOE OSTI2025

Conflict Detection in Open RAN with Recurrent Neural Networks Using Geometric Manifolds

5G↗

DOE OSTI2024

SPECTER: efficient evaluation of the spectral EMD

The Energy Mover’s Distance (EMD) has seen use in collider physics as a metric between events and as a geometric method of defining infrared and collinear safe observables. Recently, the Spectral Energy Mover’s Distance (SEMD) has been proposed as a more analytically tractable alternative to the EMD. In this work, we obtain a closed-form expression for the Riemannian-like p = 2 SEMD metric between events, eliminating the need to numerically solve an optimal transport problem. Additionally, we show how the SEMD can be used to define event and jet shape observables by minimizing the distance between events and parameterized energy flows (similar to the EMD), and we obtain closed-form expressions for several of these observables. We also present the Specter framework, an efficient and highly parallelized implementation of the SEMD metric and SEMD-derived shape observables as an analogue of the previously-introduced Shaper for EMD-based computations. We demonstrate that computing the SEMD with Specter can be up to a thousand times faster than computing the EMD with standard optimal transport libraries.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2023

Covariance Shaping Over Riemannian Manifolds for Massive MIMO Communication

Acquiring accurate instantaneous channel state information (CSI) is a challenging aspect of massive multi-input multi-output (MIMO) communication. Utilizing statistical information, such as channel covariance matrix, to design statistical beamforming vectors is robust when compared to instantaneous CSI. In this paper, we propose a novel MIMO covariance shaping scheme over Riemannian manifolds. It serves as an effective statistical beamforming solution to a number of close proximity user equipment (UE) that are undergoing substantial channel correlation. Proposed algorithm exploits the Hermitian positive definite nature of covariance matrices lying over Riemannian manifold. We introduce Wasserstein distance function as a Riemannian metric to measure distances between channel covariance matrices. Furthermore, K-means clustering technique is utilized to effectively identify the optimal shape of effective optimal covariance matrices. Our findings suggest that maximizing the geodesic distance between covariance matrices ultimately leads to a corresponding increase in the network throughput, as determined by the beamforming vector used to shape the covariance matrices. Simulation results validate that the proposed solution converges faster than Euclidean-based state-of-the-art, while maintaining the same computational complexity. Finally, the sum rate performance asymptotically achieves full capacity for two-UE case and more than 96% of the upper bound exhaustive search benchmark for multi-UE scenario.

42 ENGINEERING↗

DOE OSTI2023

Certifiably Correct Range-Aided SLAM

We present the first algorithm capable of efficiently computing certifiably optimal solutions to range-aided simultaneous localization and mapping (RA-SLAM) problems. Robotic navigation systems are increasingly incorporating point-to-point ranging sensors, leading state estimation which takes the form of RA-SLAM. However, the RA-SLAM problem is more difficult to solve than traditional pose-graph SLAM; ranging sensor models introduce additional non-convexity, unlike pose-pose or pose-landmark measurements, a single range measurement does not uniquely determine the relative transform between the involved sensors, and RA-SLAM inference is highly sensitive to initial estimates. Our approach relaxes the RA-SLAM problem to a semidefinite program (SDP), which we show how to solve efficiently using the Riemannian staircase methodology. The solution of this SDP provides a high-quality initialization for our original RA-SLAM problem, which is subsequently refined via local optimization, as well as a lower-bound on the RA-SLAM problem's optimal value. Our algorithm, named certifiably correct RA-SLAM (CORA), applies to problems comprised of arbitrary pose-pose, pose-landmark, and ranging measurements. Evaluation on simulated and real-world marine examples shows that our algorithm frequently produces certifiably optimal RA-SLAM solutions; moreover, even suboptimal estimates are typically within 1-2\% of the optimal value.

Papalia, Alan↗

DOE OSTI2022

Fast Characterization of Inducible Regions of Atrial Fibrillation Models With Multi-Fidelity Gaussian Process Classification

Computational models of atrial fibrillation have successfully been used to predict optimal ablation sites. A critical step to assess the effect of an ablation pattern is to pace the model from different, potentially random, locations to determine whether arrhythmias can be induced in the atria. In this work, we propose to use multi-fidelity Gaussian process classification on Riemannian manifolds to efficiently determine the regions in the atria where arrhythmias are inducible. We build a probabilistic classifier that operates directly on the atrial surface. We take advantage of lower resolution models to explore the atrial surface and combine seamlessly with high-resolution models to identify regions of inducibility. We test our methodology in 9 different cases, with different levels of fibrosis and ablation treatments, totalling 1,800 high resolution and 900 low resolution simulations of atrial fibrillation. When trained with 40 samples, our multi-fidelity classifier that combines low and high resolution models, shows a balanced accuracy that is, on average, 5.7% higher than a nearest neighbor classifier. We hope that this new technique will allow faster and more precise clinical applications of computational models for atrial fibrillation. All data and code accompanying this manuscript will be made publicly available at: https://github.com/fsahli/AtrialMFclass.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2023

Covariant bit threads

We derive several new reformulations of the Hubeny-Rangamani-Takayanagi covariant holographic entanglement entropy formula. These include: (1) a minimax formula, which involves finding a maximal-area achronal surface on a timelike hypersurface homologous to D(A) (the boundary causal domain of the region A whose entropy we are calculating) and minimizing over the hypersurface; (2) a max V-flow formula, in which we maximize the flux through D(A) of a divergenceless bulk 1-form V subject to an upper bound on its norm that is non-local in time; and (3) a min U-flow formula, in which we minimize the flux over a bulk Cauchy slice of a divergenceless timelike 1-form U subject to a lower bound on its norm that is non-local in space. The two flow formulas define convex programs and are related to each other by Lagrange duality. For each program, the optimal configurations dynamically find the HRT surface and the entanglement wedges of A and its complement. The V-flow formula is the covariant version of the Freedman-Headrick bit thread reformulation of the Ryu-Takayanagi formula. We also introduce a measure-theoretic concept of a “thread distribution”, and explain how Riemannian flows, V-flows, and U-flows can be expressed in terms of thread distributions.

79 ASTRONOMY AND ASTROPHYSICS↗