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Results for “polynomial relaxation”

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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15 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Weighted relaxation for multigrid reduction in time

Current trends in computer architectures now mean that faster computation speed must come primarily from increased concurrency, not faster clock speeds, which are stagnating. Thus, this situation creates bottlenecks for serial algorithms, including the well-known bottleneck for sequential time-integration, where each individual time-value (i.e., time-step) is computed sequentially. One approach to alleviate this and achieve parallelism in time is with multigrid. Here, in this work, we consider multigrid-reduction-in-time (MGRIT), a multilevel method applied to the time dimension that computes multiple time-steps in parallel. Like all multigrid methods, MGRIT relies on the complementary relationship between relaxation on a fine-grid and a correction from the coarse grid to solve the problem. All current MGRIT implementations are based on unweighted-Jacobi relaxation; here we introduce the concept of weighted relaxation to MGRIT. We derive new convergence bounds for weighted relaxation, and use this analysis to guide the selection of relaxation weights. Numerical results then demonstrate that by choosing appropriate non-unitary relaxation weights, one can achieve faster convergence rates and lower iteration counts for MGRIT when compared with unweighted relaxation. In most cases, weighted relaxation yields a 10%–20% saving in iterations, which is significant when using large high-performance computers. For A-stable integration schemes, results also illustrate that under-relaxation can restore convergence in some cases where unweighted relaxation is not convergent.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

McCormick envelopes in mixed-integer PDE-constrained optimization

McCormick envelopes are a standard tool for deriving convex relaxations of optimization problems that involve polynomial terms. Such McCormick relaxations provide lower bounds, for example, in branch-and-bound procedures for mixed-integer nonlinear programs but have not gained much attention in PDE-constrained optimization so far. This lack of attention may be due to the distributed nature of such problems, which on the one hand leads to infinitely many linear constraints (generally state constraints that may be difficult to handle) in addition to the state equation for a pointwise formulation of the McCormick envelopes and renders bound-tightening procedures that successively improve the resulting convex relaxations computationally intractable. We analyze McCormick envelopes for a model problem class that is governed by a semilinear PDE involving a bilinearity and integrality constraints. We approximate the nonlinearity and in turn the McCormick envelopes by averaging the involved terms over the cells of a partition of the computational domain on which the PDE is defined. This yields convex relaxations that underestimate the original problem up to an a priori error estimate that depends on the mesh size of the discretization. These approximate McCormick relaxations can be improved by means of an optimization-based bound-tightening procedure. We show that their minimizers converge to minimizers to a limit problem with a pointwise formulation of the McCormick envelopes when driving the mesh size to zero. We provide a computational example, for which we certify all of our imposed assumptions. The results point to both the potential of the methodology and the gaps in the research that need to be closed. Our methodology provides a framework first for obtaining pointwise underestimators for nonconvexities and second for approximating them with finitely many linear inequalities in an infinite-dimensional setting.

Approximations and Expansions↗
DOE OSTI2022

Relaxation mechanisms in a disordered system with Poisson-level statistics

Here we discuss the interplay between many-body localization and spin symmetry. To this end, we study the time evolution of several observables in the anisotropic t- j model. Like the Hubbard chain, the studied model contains charge and spin degrees of freedom, yet it has smaller Hilbert space and thus allows for numerical studies of larger systems. We compare the field disorder that breaks the $\mathbb{Z}_2$ spin symmetry and a potential disorder that preserves the latter symmetry. In the former case, sufficiently strong disorder leads to localization of all studied observables, at least for the studied system sizes. However, in the case of symmetry-preserving disorder, we observe that odd operators under the $\mathbb{Z}_2$ spin transformation relax towards the equilibrium value at relatively short timescales that grow only polynomially with the disorder strength. On the other hand, the dynamics of even operators and the level statistics within each symmetry sector are consistent with localization. Our results indicate that localization exists within each symmetry sector for symmetry-preserving disorder. Odd operators' apparent relaxation is due to their time evolution between distinct symmetry sectors.

36 MATERIALS SCIENCE↗
DOE OSTI2025

An eigenvalue-based method for computing the relaxed pressure in compressible multiphase flow with N phases

The modeling of compressible multiphase flows is a decades-old area of study with many applications across various fields. Many of these application areas use stiff pressure relaxation. This process involves the solution of a nonlinear system with N + 1 equations and N + 1 unknowns, where N is the number of phases. The resolution of this system with general equations of state (EOSs) is difficult. Furthermore, nonlinear systems can admit multiple solutions, and current solution methods do not address this possibility. Very recently, a thermodynamic relaxation method was introduced, which effectively maps a relatively simple predictor equation of state onto a more complex target equation of state. In this context, the target EOSs are the chosen EOSs for the thermodynamic model. Furthermore, this thermodynamic relaxation has the benefit of simplifying the stiff pressure relaxation system of equations. In this article, we show this system reduces to a polynomial of degree N, which can be recast as an eigenvalue problem through the use of the associated companion matrix. We show that although this eigenvalue method is generally less efficient than Newton–Raphson iteration, it does not suffer from convergence issues and finds all N roots of the polynomial. Hence, the method provides a fail-safe for root-finding iterative methods and a way to address the issue of multiple solutions to the nonlinear system of equations in stiff pressure relaxation.

Eigenvalue algorithm↗
DOE OSTI2024

Efficient estimation of the modified Gromov–Hausdorff distance between unweighted graphs

Abstract Gromov–Hausdorff distances measure shape difference between the objects representable as compact metric spaces, e.g. point clouds, manifolds, or graphs. Computing any Gromov–Hausdorff distance is equivalent to solving an NP-hard optimization problem, deeming the notion impractical for applications. In this paper we propose a polynomial algorithm for estimating the so-called modified Gromov–Hausdorff (mGH) distance, a relaxation of the standard Gromov–Hausdorff (GH) distance with similar topological properties. We implement the algorithm for the case of compact metric spaces induced by unweighted graphs as part of Python library , and demonstrate its performance on real-world and synthetic networks. The algorithm finds the mGH distances exactly on most graphs with the scale-free property. We use the computed mGH distances to successfully detect outliers in real-world social and computer networks.

Oles, Vladyslav (ORCID:0000000188727463)↗
DOE OSTI2021

Ensemble Learning Based Convex Approximation of Three-Phase Power Flow

Though the convex optimization has been widely used in power systems, it still cannot guarantee to yield a tight (accurate) solution to some problems. To mitigate this issue, this paper proposes an ensemble learning based convex approximation for alternating current (AC) power flow equations that differs from the existing convex relaxations. The proposed approach is based on three-phase quadratic power flow equations in rectangular coordinates. To develop this data-driven convex approximation of power flows, the polynomial regression (PR) is first deployed as a basic learner to fit convex relationships between the independent and dependent variables. Then, ensemble learning algorithms such as gradient boosting (GB) and bagging are introduced to combine learners to boost model performance. Based on the learned convex approximation of power flow, optimal power flow (OPF) is formulated as a convex quadratic programming problem. The simulation results on IEEE standard cases of both balanced and unbalanced systems show that, in the context of solving OPF, the proposed data-driven convex approximation outperforms the conventional semi-definite programming (SDP) relaxation in both accuracy and computational efficiency, especially in the cases that the conventional SDP relaxation fails

Convex approximation↗
DOE OSTI2022

Deterministic and Monte Carlo Nuclear Data Adjustment Methods [Slides]

For the Bayesian Monte Carlo methodology, a need to understand convergence of the posterior moments as a function of the number of parameter realizations is required. In high-dimensional systems, it can be very costly to sample entire parameter space and perform functional evaluation for every realization. Bayesian Monte Carlo allows one to relax the GLLS approximations of model linearity and prior/posterior PDF shape. The Bayesian Stochastic Collocation Method is a deterministic approach to “sample” the parameter space. It allows one to relax the GLLS approximations of model linearity and posterior PDF shape. Higher-order posterior moments (i.e., skewness, kurtosis, etc.) can be studied through polynomial expansion. Tensor product quadrature scales poorly and can use sparse grid quadrature methods.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2023

Advanced Ab Initio Methods for Nuclear Structure (Final Report)

Over the past decade, there has been enormous progress in the description of nuclear structure from first principles, using interactions from Chiral Effective Field Theory that are rooted in Quantum Chromodynamics, the fundamental theory of strong interactions, and many-body methods that solve the Schrödinger equation with systematically improvable approximations. Amongst those are the family of In-Medium Similarity Renormalization Group (IMSRG) framework developed by the PI and his co-workers. While these methods scale polynomially in the size N of the single-particle basis, computational efforts still grows dramatically as we increase the degrees of freedom for the nucleons by relaxing symmetries or introducing continuum couplings (see below), or as we push to improved truncations to provide precise inputs for experimental efforts, in particular in fundamental symmetry searches. In order to address this growing computational cost, one focus area of this award was the exploration of compression and factorization methods. The key to success or failure is the presence of low-rank structures within the matrix elements of NN and 3N interactions or the IMSRG evolution operator, and a means to reformulate the method that will let us exploit them.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2022

Electrochemical Modeling and Experimental Verification of Lithiation Gradients in Oxide Cathodes of Lithium-Ion Cells

Lithiated nickel-cobalt-manganese oxides, such as NCM523, are used in the positive electrode (cathode) of Li-ion cells. Using operando X-ray diffraction profilometry, lithiation gradients in the cathode matrix can be observed and quantified by expansion into Legendre polynomials with time-dependent weights. These weights (referred to as gradients) increase in magnitude when electric current flows through the cell, decrease during potentiostatic hold and finally relax to zero when the current is interrupted during open circuit rest. Both physics-based electrochemical models and operando X-ray experiments suggest that the time constants for gradient growth and abatement are primarily determined by ionic diffusion in the oxide particles, which in turn depends on their lithium content. In contrast, the magnitude of gradients depends mainly on the applied current. The X-ray profilometry provides a way of directly probing the formation and disappearance of Li gradients across the cathode during fast cycling, which can help to diagnose the effects of material degradation in the cells.

25 ENERGY STORAGE↗
DOE OSTI2022

Calorimetric study of skutterudite (CoAs2.92) and heazlewoodite (Ni3S2)

Abstract Nickel and cobalt arsenides, sulfarsenides, and sulfides occur in many hydrothermal ore deposits, but their thermodynamic properties are not well known, in some cases not known at all. In this work, we determined a full set of thermodynamic properties for heazlewoodite and skutterudite. Both phases were synthesized in evacuated silica tubes at elevated temperatures, and electron microprobe analyses gave their compositions as Ni3S2 and CoAs2.92, respectively. Enthalpies of formation were measured by high-temperature oxide-melt solution calorimetry. The reference phases were pure elements, thus eliminating any systematic errors related to such phases. The enthalpies of formation at T = 298.15 K and P = 105 Pa are –216.0 ± 8.4(2σ) and –88.2 ± 6.1 kJ·mol−1 for Ni3S2 and CoAs2.92, respectively. Entropies were calculated from low-temperature heat capacity (CP) data from relaxation (PPMS) calorimetry and are 133.8 ± 1.6 and 106.4 ± 1.3 J·mol–1·K–1, respectively. The calculated Gibbs free energies of formation are –210.0 ± 8.4 and –79.9 ± 6.2 kJ·mol−1 for Ni3S2 and CoAs2.92, respectively. The PPMS CP data, together with a set of differential scanning calorimetry measurements, were used to derive CP polynomials up to 700 K with the Kieffer model based on previously published frequencies of acoustic and optic modes. Equilibrium constants for selected reactions with an aqueous phase were calculated up to 700 K. Geochemical modeling in these systems, however, should await until more reliable data for other phases from the system Co-Ni-As-S are available.

Geochemistry & Geophysics↗
DOE OSTI2021

The impacts of convex piecewise linear cost formulations on AC optimal power flow

Despite strong connections through shared application areas, research efforts on power market optimization (e.g., unit commitment) and power network optimization (e.g., optimal power flow) remain largely independent. A notable illustration of this is the treatment of power generation cost functions, where nonlinear network optimization has largely used polynomial representations and market optimization has adopted piecewise linear encodings. This work combines state-of-the-art results from both lines of research to understand the best mathematical formulations of the nonlinear AC optimal power flow problem with piecewise linear generation cost functions. An extensive numerical analysis of non-convex models, linear approximations, and convex relaxations across fifty-four realistic test cases illustrates that nonlinear optimization methods are surprisingly sensitive to the mathematical formulation of piecewise linear functions. The results indicate that a poor formulation choice can slow down algorithm performance by a factor of ten, increasing the runtime from seconds to minutes. Furthermore, these results provide valuable insights into the best formulations of nonlinear optimal power flow problems with piecewise linear cost functions, an important step towards building a new generation of energy markets that incorporate the nonlinear AC power flow model.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2024

Accurate numerical simulations of open quantum systems using spectral tensor trains

Decoherence between qubits is a major bottleneck in quantum computations. Decoherence results from intrinsic quantum and thermal fluctuations as well as noise in the external fields that perform the measurement and preparation processes. With prescribed colored noise spectra for intrinsic and extrinsic noise, we present a numerical method, Quantum Accelerated Stochastic Propagator Evaluation (Q-ASPEN), to solve the time-dependent noise-averaged reduced density matrix in the presence of intrinsic and extrinsic noise. Q-ASPEN is arbitrarily accurate and can be applied to provide estimates for the resources needed to error-correct quantum computations. We employ spectral tensor trains, which combine the advantages of tensor networks and pseudospectral methods, as a variational ansatz to the quantum relaxation problem and optimize the ansatz using methods typically used to train neural networks. Here, the spectral tensor trains in Q-ASPEN make accurate calculations with tens of quantum levels feasible. We present benchmarks for Q-ASPEN on the spin-boson model in the presence of intrinsic noise and on a quantum chain of up to 32 sites in the presence of extrinsic noise. In our benchmark, the memory cost of Q-ASPEN scales as a low-order polynomial in the size of the system once the number of system states surpasses the number of basis functions used in the spectral expansion.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

A Generative Model for Realistic Galaxy Cluster X-Ray Morphologies

Abstract The X-ray morphologies of clusters of galaxies display significant variations, reflecting their dynamical histories and the nonlinear dependence of X-ray emissivity on the density of the intracluster gas. Qualitative and quantitative assessments of X-ray morphology have long been considered a proxy for determining whether clusters are dynamically active or “relaxed.” Conversely, the use of circularly or elliptically symmetric models for cluster emission can be complicated by the variety of complex features realized in nature, spanning scales from megaparsecs down to the resolution limit of current X-ray observatories. In this work, we use mock X-ray images from simulated clusters from The Three Hundred project to define a basis set of cluster image features. We take advantage of the clusters’ approximate self-similarity to minimize the differences between images before encoding the remaining diversity through a distribution of high-order polynomial coefficients. Principal component analysis then provides an orthogonal basis for this distribution, corresponding to natural perturbations from an average model. This representation allows novel, realistically complex X-ray cluster images to be easily generated, and we provide code to do so. The approach provides a simple way to generate training data for cluster image analysis algorithms and could be straightforwardly adapted to generate clusters displaying specific types of features or selected by physical characteristics available in the original simulations.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Exponentially Reduced Circuit Depths Using Trotter Error Mitigation

Product formulas are a popular class of digital quantum simulation algorithms due to their conceptual simplicity, low overhead, and performance, which often exceeds theoretical expectations. Recently, Richardson extrapolation and polynomial interpolation have been proposed to mitigate the Trotter error incurred by the use of these formulas. This work provides a rigorous, general analysis of these techniques for computing time-evolved observables, simplifying the interpolation algorithm in the process, and shows that extrapolation generically improves the performance of product formulas for this task. We demonstrate that, to achieve error 𝜖 in a simulation of time 𝑇 using a 𝑝 ⁢th-order product formula with extrapolation, circuit depths of 𝑂⁡(𝑇 1+1/𝑝 ⁢polylog (1/𝜖)) are sufficient—an exponential improvement in the precision over product formulas alone. Furthermore, we prove that these algorithms achieve commutator scaling, and improve the 𝑇 complexity for the interpolation algorithm. By relaxing the requirement of performing exact Chebyshev interpolation, our simplified algorithm eliminates the need for fractional implementations of Trotter steps, reducing computational overhead. Finally, we show these techniques can be combined with the classical shadows method to estimate many time-evolved local observables. Taken together, our findings provide the strongest evidence yet for the utility of Trotter error-mitigation techniques in algorithmic applications.

quantum algorithms & computation↗
DOE OSTI2024

Beyond Single-Reference Fixed-Node Approximation in Ab Initio Diffusion Monte Carlo Using Antisymmetrized Geminal Power Applied to Systems with Hundreds of Electrons

Diffusion Monte Carlo (DMC) is an exact technique to project out the ground state (GS) of a Hamiltonian. Since the GS is always bosonic, in Fermionic systems, the projection needs to be carried out while imposing antisymmetric constraints, which is a nondeterministic polynomial hard problem. In practice, therefore, the application of DMC on electronic structure problems is made by employing the fixed-node (FN) approximation, consisting of performing DMC with the constraint of having a fixed, predefined nodal surface. How do we get the nodal surface? The typical approach, applied in systems having up to hundreds or even thousands of electrons, is to obtain the nodal surface from a preliminary mean-field approach (typically, a density functional theory calculation) used to obtain a single Slater determinant. This is known as single reference. In this paper, we propose a new approach, applicable to systems as large as the C 60 fullerene, which improves the nodes by going beyond the single reference. In practice, we employ an implicitly multireference ansatz (antisymmetrized geminal power wave function constraint with molecular orbitals), initialized on the preliminary mean-field approach, which is relaxed by optimizing a few parameters of the wave function determining the nodal surface by minimizing the FN-DMC energy. We highlight the improvements of the proposed approach over the standard single-reference method on several examples and, where feasible, the computational gain over the standard multireference ansatz, which makes the methods applicable to large systems. We also show that physical properties relying on relative energies, such as binding energies, are affordable and reliable within the proposed scheme.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗