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All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Matrix-free preconditioning for high-order H (curl) discretizations

The greater arithmetic intensity of high-order finite element discretizations makes them attractive for implementation on next-generation hardware, but assembly of high-order finite element operators as matrices is prohibitively expensive. As a result, the development of general algebraic solvers for such operators has been an open research challenge. Fast matrix-free application of high-order operators has received significant attention in the literature in the context of Poisson-type problems, but preconditioners and solvers for inverting more general operators are not very well-developed. In this paper, we consider the problem of preconditioning a definite Maxwell operator at high polynomial order without assembling a matrix. We show that given efficient preconditioners for high-order H 1 finite element problems on the same mesh, efficient H(curl) preconditioners can be constructed in an auxiliary space framework. We demonstrate the resulting preconditioners in a practical setting with tensor-product basis functions on an unstructured mesh of quadrilaterals. Overall, our approach uses a sparsified H 1 solver constructed on a low-order mesh of the nodal points of the underlying high-order space, and we show that the resulting H(curl) preconditioner is effective at very high polynomial orders for two-dimensional model problems with complicated geometry, varying piecewise constant coefficients, and curved elements. The resulting preconditioner scales with nearly optimal O(p d+1 ) floating point operation count and optimal O(p d ) memory transfer requirements, outperforming existing Maxwell preconditioners in the high-order regime.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Universal bounds on CFT Distance Conjecture

For any unitary conformal field theory in two dimensions with the central charge c, we prove that, if there is a nontrivial primary operator whose conformal dimension ∆ vanishes in some limit on the conformal manifold, the Zamolodchikov distance t to the limit is infinite, the approach to this limit is exponential ∆ = exp(−αt + O(1)), and the decay rate obeys the universal bounds c−1/2 ≤ α ≤ 1. In the limit, we also find that an infinite tower of primary operators emerges without a gap above the vacuum and that the conformal field theory becomes locally a tensor product of a sigma-model in the large radius limit and a compact theory. As a corollary, we establish a part of the Distance Conjecture about gravitational theories in three-dimensional anti-de Sitter space. In particular, our bounds on α indicate that the emergence of exponentially light states is inevitable as the moduli field corresponding to t rolls beyond the Planck scale along the steepest path and that this phenomenon can begin already at the curvature scale of the bulk geometry. We also comment on implications of our bounds for gravity in asymptotically flat spacetime by taking the flat space limit and compare with the Sharpened Distance Conjecture.

AdS-CFT Correspondence↗
DOE OSTI2025

Three-Receiver Quantum Broadcast Channels: Classical Communication with Quantum Non-unique Decoding

In network communication, it is common in broadcasting scenarios for there to exist a hierarchy among receivers based on information they decode due, for example, to different physical conditions or premium subscriptions. This hierarchy may result in varied information quality, such as higher-quality video for certain receivers. This is modeled mathematically as a degraded message set, indicating a hierarchy between messages to be decoded by different receivers, where the default quality corresponds to a common message intended for all receivers, a higher quality is represented by a message for a smaller subset of receivers, and so forth. We extend these considerations to quantum communication, exploring three-receiver quantum broadcast channels with two- and three-degraded message sets. Our technical tool involves employing quantum non-unique decoding, a technique we develop by utilizing the simultaneous pinching method. Here, we construct one-shot codes for various scenarios and find achievable rate regions relying on various quantum Rényi mutual information error exponents. Our investigation includes a comprehensive study of pinching across tensor product spaces, presenting our findings as the asymptotic counterpart to our one-shot codes. By employing the non-unique decoding, we also establish a simpler proof to Marton’s inner bound for two-receiver quantum broadcast channels without the need for more involved techniques. Additionally, we derive no-go results and demonstrate their tightness in special cases.

Salek, Farzin [Technical University of Munich (Ger↗
DOE OSTI2021

Scaled boundary cubature scheme for numerical integration over planar regions with affine and curved boundaries

This paper introduces the scaled boundary cubature (SBC) scheme for accurate and efficient integration of functions over polygons and two-dimensional regions bounded by parametric curves. Over two-dimensional domains, the SBC method reduces integration over a region bounded by m curves to integration over m regions (referred to as curved triangular regions), where each region is bounded by two line segments and a curve. With proper (counterclockwise) orientation of the boundary curves, the scheme is applicable to convex and nonconvex domains. Additionally, for star-convex domains, a tensor-product cubature rule with positive weights and integration points in the interior of the domain is obtained. If the integrand is homogeneous, we show that this new method reduces to the homogeneous numerical integration scheme; however, the SBC scheme is more versatile since it is equally applicable to both homogeneous and non-homogeneous functions. This paper also introduces several methods for smoothing integrands with point singularities and near-singularities. When these methods are used, highly efficient integration of weakly singular functions is realized. Finally, the SBC method is applied to a number of benchmark problems, which reveal its broad applicability and superior performance (in terms of time to generate a rule and accuracy per cubature point) when compared to existing methods for integration.

42 ENGINEERING↗
DOE OSTI2025

General field evaluation in high-order meshes on GPUs

Robust and scalable function evaluation at any arbitrary point in the finite/spectral element mesh is required for querying the partial differential equation solution at points of interest, comparison of solution between different meshes, and Lagrangian particle tracking. This is a challenging problem, particularly for high-order unstructured meshes partitioned in parallel with MPI, as it requires identifying the element that overlaps a given point and computing the corresponding reference space coordinates. Here, we present a robust and efficient technique for general field evaluation in large-scale high-order meshes with quadrilaterals and hexahedra. In the proposed method, a combination of globally partitioned and processor-local maps are used to first determine a list of candidate MPI ranks, and then locally candidate elements that could contain a given point. Next, element-wise bounding boxes further reduce the list of candidate elements. Finally, Newton’s method with trust region is used to determine the overlapping element and corresponding reference space coordinates. Since GPU-based architectures have become popular for accelerating computational analyses using meshes with tensor-product elements, specialized kernels have been developed to utilize the proposed methodology on GPUs. The method is also extended to enable general field evaluation on surface meshes. The paper concludes by demonstrating the use of the proposed method in various applications ranging from mesh-to-mesh transfer during r-adaptivity to Lagrangian particle tracking.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A higher-order finite-element implementation of the nonlinear Fokker–Planck collision operator for charged particle collisions in a low density plasma

Collisions between particles in a low density plasma are described by the Fokker–Planck collision operator. In applications, this nonlinear integro-differential operator is often approximated by linearised or ad-hoc model operators due to computational cost and complexity. In this work, we present an implementation of the nonlinear Fokker–Planck collision operator written in terms of Rosenbluth potentials in the Rosenbluth–MacDonald–Judd (RMJ) form. The Rosenbluth potentials may be obtained either by direct integration or by solving partial differential equations (PDEs) similar to Poisson's equation: we optimise for performance and scalability by using sparse matrices to solve the relevant PDEs. We represent the distribution function using a tensor-product continuous-Galerkin finite-element representation and we derive and describe the implementation of the weak form of the collision operator. We present tests demonstrating a successful implementation using an explicit time integrator and we comment on the speed and accuracy of the operator. Finally, we speculate on the potential for applications in the current and next generation of kinetic plasma models.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Flow-driven spectral chaos (FSC) method for simulating long-time dynamics of arbitrary-order non-linear stochastic dynamical systems

Uncertainty quantification techniques such as the time-dependent generalized polynomial chaos (TD-gPC) use an adaptive orthogonal basis to better represent the stochastic part of the solution space (aka random function space) in time. However, because the random function space is constructed using tensor products, TD-gPC-based methods are known to suffer from the curse of dimensionality. Here, we introduce a new numerical method called the flow-driven spectral chaos (FSC) which overcomes this curse of dimensionality at the random-function-space level. The proposed method is not only computationally more efficient than existing TD-gPC-based methods but is also far more accurate. The FSC method uses the concept of enriched stochastic flow maps to track the evolution of a finite-dimensional random function space efficiently in time. To transfer the probability information from one random function space to another, two approaches are developed and studied herein. In the first approach, the probability information is transferred in the mean-square sense, whereas in the second approach the transfer is done exactly using a new theorem that was developed for this purpose. The FSC method can quantify uncertainties with high fidelity, especially for the long-time response of stochastic dynamical systems governed by ODEs of arbitrary order. Six representative numerical examples, including a nonlinear problem (the Van-der-Pol oscillator), are presented to demonstrate the performance of the FSC method and corroborate the claims of its superior numerical properties. Finally, a parametric, high-dimensional stochastic problem is used to demonstrate that when the FSC method is used in conjunction with Monte Carlo integration, the curse of dimensionality can be overcome altogether.

(nonlinear) stochastic dynamical systems↗
DOE OSTI2026

Definitive Assessment of the Accuracy, Variationality, and Convergence of Relativistic Coupled Cluster and Density Matrix Renormalization Group in 100-Orbital Space

Accuracy, variationality, and convergence underpin the reliability of modern electronic structure methods, yet definitive benchmarks in the relativistic regime remain elusive due to the absence of numerically exact full configuration interaction (CI) references. Recent algorithmic advances in the CI framework, enabled by the small-tensor-product (STP) decomposition approach, have dramatically extended the tractable size of the configuration space, making numerically exact CI calculations feasible in large active spaces previously beyond reach. In this paper, we employ the recently developed STP-CI framework to perform large-scale numerically exact CI calculations and directly benchmark relativistic coupled cluster and density matrix renormalization group methods. Definitive benchmarking of approximate relativistic electronic structure methods is ensured through the application of the gap theorem, which provides rigorous error bounds on the CI reference and establishes a controlled standard for assessing accuracy, variationality, and convergence.

Chemical calculations↗
DOE OSTI2025

Numerically exact configuration interaction at quadrillion-determinant scale

The combinatorial growth of configuration interaction (CI) has long limited this formally exact quantum chemistry method to only the smallest molecules. Here, we report a numerically exact CI calculation exceeding one quadrillion (10 15 ) determinants, made possible by a lossless categorical compression strategy within the small-tensor-product distributed active space (STP-DAS) framework. This approach overcomes the traditional memory bottlenecks of CI by a numerically exact compression of the wavefunction representation and reformulating the most computationally demanding matrix–vector operations. Using this method, we performed a fully relativistic CI calculation of the ground state of HBrTe with over 10 15 complex-valued determinants in just 34.5 h on 1000 computing nodes—the largest CI calculation ever reported. We further achieved fast computation for systems with hundreds of billions of determinants on only a few compute nodes. Extensive benchmarks confirm that the method retains full numerical exactness while cutting memory and computational cost by orders of magnitude. Compared to previous state-of-the-art CI calculations, this work achieves a 1000 times increase in CI space, a 10 6 -fold increase in floating-point operations performed, and a 10 6 -fold improvement in computational speed.

Computational chemistry↗
DOE OSTI2023

Symmetry-projected cluster mean-field theory applied to spin systems

We introduce S z spin-projection based on cluster mean-field theory and apply it to the ground state of strongly correlated spin systems. In cluster mean-fields, the ground state wavefunction is written as a factorized tensor product of optimized cluster states. In previous work, we have focused on unrestricted cluster mean-field, where each cluster is S z symmetry adapted. We here remove this restriction by introducing a generalized cluster mean-field (GcMF) theory, where each cluster is allowed to access all S z sectors, breaking S z symmetry. In addition, a projection scheme is used to restore global S z , which gives rise to the S z spin-projected generalized cluster mean-field (S z GcMF). Both of these extensions contribute to accounting for inter-cluster correlations. We benchmark these methods on the 1D, quasi-2D, and 2D J 1 – J 2 and XXZ Heisenberg models. Furthermore, our results indicate that the new methods (GcMF and S z GcMF) provide a qualitative and semi-quantitative description of the Heisenberg lattices in the regimes considered, suggesting them as useful references for further inter-cluster correlations, which are discussed in this work.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

The polarized-signal density matrix: A practical way to recover molecular frame information from isotropic samples

We present a novel approach to model ultrafast time-dependent nonlinear optical polarization sensitive signals emitted from randomly oriented molecules. By projecting the laboratory-frame analyzer polarization axis into the molecular-frame and linking that axis with the density matrix through a tensor product, we demonstrate an approach to find a specific molecular orientation that yields a good approximation to simulated four-wave mixing signals produced by the same model but with averaging over molecular orientation.

Thurston, Richard L↗
DOE OSTI2022

Characterizing Tradeoffs in Memory, Accuracy, and Speed for Chemistry Tabulation Techniques

Chemistry tabulation is a common approach in practical simulations of turbulent combustion at engineering scales. Linear interpolants have traditionally been used for accessing precomputed multidimensional tables but suffer from large memory requirements and discontinuous derivatives. Higher-degree interpolants address some of these restrictions but are similarly limited to relatively low-dimensional tabulation. Artificial neural networks (ANNs) can be used to overcome these limitations but cannot guarantee the same accuracy as interpolants and introduce challenges in reproducibility and reliable training. These challenges are enhanced as the physics complexity to be represented within the tabulation increases. Here, we assess the efficiency, accuracy, and memory requirements of Lagrange polynomials, tensor product B-splines, and ANNs as tabulation strategies. We analyze results in the context of nonadiabatic flamelet modeling where higher dimension counts are necessary. While ANNs do not require structuring of data, providing benefits for complex physics representation, interpolation approaches often rely on some structuring of the table. Interpolation using structured table inputs that are not directly related to the variables transported in a simulation can incur additional query costs. This is demonstrated in the present implementation of heat losses. We show that ANNs, despite being difficult to train and reproduce, can be advantageous for high-dimensional, unstructured datasets relevant to nonadiabatic flamelet models. Furthermore we demonstrate that Lagrange polynomials show significant speedup for similar accuracy compared to B-splines.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Discovering Active Subspaces for High-Dimensional Computer Models

Dimension reduction techniques have long been an important topic in statistics, and active subspaces (AS) have received much attention this past decade in the computer experiments literature. The most common approach towards estimating the AS is to use Monte Carlo with numerical gradient evaluation. While sensible in some settings, this approach has obvious drawbacks. Recent research has demonstrated that active subspace calculations can be obtained in closed form, conditional on a Gaussian process (GP) surrogate, which can be limiting in high-dimensional settings for computational reasons. In this paper, we produce the relevant calculations for a more general case when the model of interest is a linear combination of tensor products. These general equations can be applied to the GP, recovering previous results as a special case, or applied to the models constructed by other regression techniques including multivariate adaptive regression splines (MARS). Furthermore, using a MARS surrogate has many advantages including improved scaling, better estimation of active subspaces in high dimensions and the ability to handle a large number of prior distributions in closed form. In one real-world example, we obtain the active subspace of a radiation-transport code with 240 inputs and 9,372 model runs in under half an hour.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Duality defect of the monster CFT

Abstract We show that the fermionization of the Monster CFT with respect to Z 2 A is the tensor product of a free fermion and the Baby Monster CFT. The chiral fermion parity of the free fermion implies that the Monster CFT is self-dual under the Z 2 A orbifold, i.e. it enjoys the Kramers–Wannier duality. The Kramers–Wannier duality defect extends the Monster group to a larger category of topological defect lines that contains an Ising subcategory. We introduce the defect McKay – Thompson series defined as the Monster partition function twisted by the duality defect, and find that the coefficients can be decomposed into the dimensions of the (projective) irreducible representations of the Baby Monster group. We further prove that the defect McKay–Thompson series is invariant under the genus-zero congruence subgroup 16 D 0 of P S L ( 2 , Z ) .

Physics↗
DOE OSTI2021

Fermionic systems for quantum information people

The operator algebra of fermionic modes is isomorphic to that of qubits, the difference between them is twofold: the embedding of subalgebras corresponding to mode subsets and multiqubit subsystems on the one hand, and the parity superselection in the fermionic case on the other. We discuss these two fundamental differences extensively, and illustrate these through the Jordan–Wigner representation in a coherent, self-contained, pedagogical way, from the point of view of quantum information theory. Our perspective leads us to develop useful new tools for the treatment of fermionic systems, such as the fermionic (quasi-)tensor product, fermionic canonical embedding, fermionic partial trace, fermionic products of maps and fermionic embeddings of maps. We formulate these by direct, easily applicable formulas, without mode permutations, for arbitrary partitionings of the modes. It is also shown that fermionic reduced states can be calculated by the fermionic partial trace, containing the proper phase factors. We also consider variants of the notions of fermionic mode correlation and entanglement, which can be endowed with the usual, local operation based motivation, if the parity superselection rule is imposed. We also elucidate some other fundamental points, related to joint map extensions, which make the parity superselection inevitable in the description of fermionic systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

Generalized Hall conductivities in local commuting projector models: Generalized symmetries and protected surface modes

Hall conductivities are important characterizations of phases of matter. It is known that nonzero Hall conductivities are difficult to realize in local commuting projector lattice models due to no-go theorems in (2+1)⁢D. In this work we construct local commuting projector models in (2+1)⁢D and (3+1)⁢D with nonzero generalized Hall conductivities for ordinary and higher-form continuous symmetries on tensor product Hilbert space of finite local dimension. The model is given by a standard ℤ 𝑁 toric code, but the symmetries do not admit expression in terms of on-site charge operators. The symmetry do not have local charges or currents on the lattice in the absence of boundaries, but there is still a notion of Hall conductivities that coincide with the continuum field theories. We construct protected gapless boundaries of the lattice models using modified Villain formalism. The generalized Hall conductivities are computed by surface currents as well as bulk flux insertion and many-body Chern number.

Anomalies↗
DOE OSTI2025

Covariant Quantum Error-Correcting Codes with Metrological Entanglement Advantage

Here, we show that a subset of the basis for the irreducible representations of a tensor-product SU(2) rotation forms a covariant approximate quantum error-correcting code with transversal U(1) logical gates. Generalizing previous work on “thermodynamic codes” to general local spin and different irreducible representations using only properties of the angular momentum algebra, we obtain bounds on the code inaccuracy under generic noise on any known 𝑑 sites, under independent and identically distributed noise, and under heralded 𝑑-local erasures. We demonstrate that this family of codes protects a probe state with quantum Fisher information surpassing the standard quantum limit when the sensing parameter couples to the generator of the U(1) logical gate.

quantum error correction↗
DOE OSTI2026

Spontaneously Broken Noninvertible Symmetries in Transverse-Field Ising Qudit Chains

Recent developments have revealed that symmetries need not form a group, but instead can be noninvertible. Here we use analytical arguments and numerical evidence to illuminate how spontaneous symmetry breaking of a noninvertible symmetry is similar yet distinct from ordinary, invertible, symmetry breaking. We consider one-dimensional chains of group-valued qudits, whose local Hilbert space is spanned by elements of a finite group 𝐺 (reducing to ordinary qubits when 𝐺=ℤ 2 ). We construct Ising-type transverse-field Hamiltonians with Rep⁡(𝐺) symmetry whose generators multiply according to the tensor product of irreducible representations (irreps) of the group 𝐺 . For non-Abelian 𝐺 , the symmetry is noninvertible. In the symmetry broken phase there is one ground state per irrep on a closed chain. The symmetry breaking can be detected by local order parameters but, unlike the invertible case, different ground states have distinct entanglement patterns. We show that for each irrep of dimension greater than one the corresponding ground state exhibits string order, entanglement spectrum degeneracies, and has gapless edge modes on an open chain—features usually associated with symmetry-protected topological order. Consequently, domain wall excitations behave as one-dimensional non-Abelian anyons with nontrivial internal Hilbert spaces and fusion rules. Our Letter identifies properties of noninvertible symmetry breaking that existing quantum hardware can probe.

1-dimensional spin chains↗