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Results for “stochastic eigenvalue problem”

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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14 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2019

Low-Rank Solution Methods for Stochastic Eigenvalue Problems

In this report, we study efficient solution methods for stochastic eigenvalue problems arising from discretization of self-adjoint partial differential equations with random data, where the underlying operators depend linearly on the random parameters. With the stochastic Galerkin approach, the solutions are represented as generalized polynomial chaos expansions. When these solutions can be approximated well by low-rank objects, we introduce a low-rank variant of the inverse subspace iteration algorithm for computing one or several minimal eigenvalues and corresponding eigenvectors of parameter-dependent matrices. In the algorithm, the iterates are approximated by low-rank matrices, which leads to significant cost savings. Furthermore, the algorithm is tested on two benchmark problems, a stochastic diffusion problem with some poorly separated eigenvalues, and an operator derived from a discrete stochastic Stokes problem whose minimal eigenvalue is related to the inf-sup stability constant. Numerical experiments show that the low-rank algorithm produces accurate solutions compared to the Monte Carlo method, and it uses much less computational time than the original algorithm without low-rank approximation.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Stochastic gradient descent for optimization for nuclear systems

The use of gradient descent methods for optimizing k-eigenvalue nuclear systems has been shown to be useful in the past, but the use of k-eigenvalue gradients have proved computationally challenging due to their stochastic nature. ADAM is a gradient descent method that accounts for gradients with a stochastic nature. This analysis uses challenge problems constructed to verify if ADAM is a suitable tool to optimize k-eigenvalue nuclear systems. ADAM is able to successfully optimize nuclear systems using the gradients of k-eigenvalue problems despite their stochastic nature and uncertainty. Furthermore, it is clearly demonstrated that low-compute time, high-variance estimates of the gradient lead to better performance in the optimization challenge problems tested here.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2020

Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks

One of the open problems in scientific computing is the long-time integration of nonlinear stochastic partial differential equations (SPDEs), especially with arbitrary initial data. We address this problem by taking advantage of recent advances in scientific machine learning and the spectral dynamically orthogonal (DO) and borthogonal (BO) methods for representing stochastic processes. The recently introduced DO/BO methods reduce the SPDE to solving a system of deterministic PDEs and a system of stochastic ordinary differential equations. Specifically, we propose two new physics-informed neural networks (PINNs) for solving time-dependent SPDEs, namely the neural network (NN)-DO/BO methods. The proposed methods incorporate the DO/BO constraints into the loss function (along with the modal decomposition of the SPDE) with an implicit form instead of generating explicit expressions for the temporal derivatives of the DO/BO modes. Hence, the NN-DO/BO methods can overcome some of the drawbacks of the original DO/BO methods. For example, we do not need the assumption that the covariance matrix of the random coefficients is invertible as in the original DO method, and we can remove the assumption of no eigenvalue crossing as in the original BO method. Moreover, the NN-DO/BO methods can be used to solve time-dependent stochastic inverse problems with the same formulation and same computational complexity as for forward problems. Furthermore, we demonstrate the capability of the proposed methods via several numerical examples, namely: (1) A linear stochastic advection equation with deterministic initial condition: we obtain good results with the proposed methods, while the original DO/BO methods cannot be applied directly in this case. (2) Long-time integration of the stochastic Burgers' equation: we show the good performance of NN-DO/BO methods, especially the effectiveness of the NN-BO approach for such problems with many eigenvalue crossings during the whole time evolution, while the original BO method fails. (3) Nonlinear reaction diffusion equation: we consider both the forward problem and the inverse problems, including very noisy initial point values, to investigate the flexibility of the NN-DO/BO methods in handling inverse and mixed type problems. Taken together, these simulation results demonstrate that the NN-DO/BO methods can be employed to effectively quantify uncertainty propagation in a wide range of physical problems, but future work should address the efficiency issue of PINNs for forward problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Finite-volume pionless effective field theory for few-nucleon systems with differentiable programming

Finite-volume pionless effective field theory provides an efficient framework for the extrapolation of nuclear spectra and matrix elements calculated at finite volume in lattice QCD to infinite volume, and to nuclei with larger atomic number. In this work, it is demonstrated how this framework may be implemented via a set of correlated Gaussian wave functions optimized using differentiable programming and via solution of a generalized eigenvalue problem. This approach is shown to be significantly more efficient than a stochastic implementation of the variational method based on the same form of correlated Gaussian wave functions, yielding comparably accurate representations of the ground-state wave functions with an order of magnitude fewer terms. The efficiency of representation allows such calculations to be extended to larger systems than in previous work. Further, the method is demonstrated through calculations of the binding energies of nuclei with atomic number A ϵ {2,3,4} in finite volume, matched to lattice QCD calculations at quark masses corresponding to m π = 806 MeV, and infinite-volume effective field theory calculations of A ϵ {2,3,4,5,6} systems based on this matching.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2021

Solving the $k$-Sparse Eigenvalue Problem with Reinforcement Learning

We examine the possibility of using a reinforcement learning (RL) algorithm to solve large-scale eigenvalue problems in which the desired the eigenvector can be approximated by a sparse vector with at most k nonzero elements, where k is relatively small compare to the dimension of the matrix to be partially diagonalized. Here, this type of problem arises in applications in which the desired eigenvector exhibits localization properties and in large-scale eigenvalue computations in which the amount of computational resource is limited. When the positions of these nonzero elements can be determined, we can obtain the k-sparse approximation to the original problem by computing eigenvalues of a k × k submatrix extracted from k rows and columns of the original matrix. We review a previously developed greedy algorithm for incrementally probing the positions of the nonzero elements in a k-sparse approximate eigenvector and show that the greedy algorithm can be improved by using an RL method to refine the selection of k rows and columns of the original matrix. We describe how to represent states, actions, rewards and policies in an RL algorithm designed to solve the k-sparse eigenvalue problem and demonstrate the effectiveness of the RL algorithm on two examples originating from quantum many-body physics.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Stochastic noise can be helpful for variational quantum algorithms

Saddle points constitute a crucial challenge for first-order gradient descent algorithms. In notions of classical machine learning, they are avoided, for example, by means of stochastic gradient descent methods. In this work, we provide evidence that the saddle-points problem can be naturally avoided in variational quantum algorithms by exploiting the presence of stochasticity. We prove convergence guarantees and present practical examples in numerical simulations and on quantum hardware. We argue that the natural stochasticity of variational algorithms can be beneficial for avoiding strict saddle points, i.e., those saddle points with at least one negative Hessian eigenvalue. This insight that some levels of shot noise could help is expected to add a new perspective to notions of near-term variational quantum algorithms. Published by the American Physical Society 2025

Liu, Junyu↗
DOE OSTI2022

Computational Aspects of Single-Molecule Kinetics for Coupled Catalytic Cycles: A Spectral Analysis

Catalysis from single active sites is analyzed using methods developed from single molecule kinetics. Using a stochastic Markov state description, the observable properties of general catalytic networks of reactions are expressed using an eigenvalue decomposition of the transition matrix for the Markov process. By the use of a sensitivity analysis, the necessary eigenvalues and eigenvectors are related to the energies of controlling barriers and wells located along the reaction routes. A generalization of the energetic span theory allows the eigenvalues to be computed from several activation energies corresponding to distinct barrier-well pairings. The formalism is demonstrated accurately for model problems for a physically realistic mechanism for an alkene hydrogenation reaction on a single atom catalyst.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

On the Stochastic Stability of Deep Markov Models

Deep Markov models (DMM) are generative models which are scalable and expressive generalization of Markov models for representation, learning, and inference problems. DMMs using deep neural networks to parametrize the transition of Markov probability distributions have recently been shown to provide more expressiveness in modeling sequential data and dynamical system responses. However, the fundamental stochastic stability guarantees of such models have not been thoroughly investigated. In this paper, we present a rigorous analytical method to prove the necessary and sufficient conditions of DMM's stochastic stability. This task is achieved by spectral analysis of the efficiently computed Jacobians of probabilistic maps modeled by deep neural networks. We make theoretical connections between the eigenvalues of neural network's weights and the different activation function types used on the stability and overall dynamic behavior of DMMs with Gaussian distributions. We empirically substantiate our theoretical results on stochastic stability and eigenvalue spectra via several numerical experiments. Formal stability guarantees of DMMs can substantially improve their robustness and trustworthiness, necessary for reliable use in safety-critical real-world applications.

Drgona, Jan↗
DOE OSTI2022

A Note on the Probability of Initiation Problem

The purpose of this note is to draw attention to Pál’s derivation of the Pál Bell equation which is used by LLNL and LANL to model Probability of Initiation (POI) problems. Both laboratories credit the equation to Bell and Lee. Although the equation formulated by Bell and Lee and the equation derived by Pál appear to be the same, the derivation by Bell and Lee is flawed. Scattering, which was absent in Bell’s original formulation of a POI problem, was later introduced by Bell and Lee by simply replacing a fission term with a scattering term. Such a replacement, however, leads to an inaccurate average number v̅ of neutrons that is produced by a fission event. By approaching a POI problem from probabilistic point of view, Pál formulates the problem as a branching process that determines v̅ correctly. There are six reasons for writing this report. The first is to clarify the failing in Bell and Lee’s derivation. The second is to establish the origin of the equation used by the laboratories. The third is to elucidate the fixed point technique formulated by Mokhtar-Kharroubi for solving the Pál-Bell equation. The fourth is to derive the α eigenvalue that is associated with a time dependent Pál-Bell equation. The fifth is to show in Appendix A that the non-linear fission operator of the Pál-Bell equation is descended from the operator of the Galton-Watson process of stochastic theory. The sixth is to describe in Appendix B a second order positivity preserving technique that accelerates the convergence of the first order fixed point method developed by Mokhtar-Kharroubi.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2023

Real-Time Krylov Theory for Quantum Computing Algorithms

Quantum computers provide new avenues to access ground and excited state properties of systems otherwise difficult to simulate on classical hardware. New approaches using subspaces generated by real-time evolution have shown efficiency in extracting eigenstate information, but the full capabilities of such approaches are still not understood. In recent work, we developed the variational quantum phase estimation (VQPE) method, a compact and efficient real-time algorithm to extract eigenvalues on quantum hardware. Here we build on that work by theoretically and numerically exploring a generalized Krylov scheme where the Krylov subspace is constructed through a parametrized real-time evolution, which applies to the VQPE algorithm as well as others. We establish an error bound that justifies the fast convergence of our spectral approximation. We also derive how the overlap with high energy eigenstates becomes suppressed from real-time subspace diagonalization and we visualize the process that shows the signature phase cancellations at specific eigenenergies. We investigate various algorithm implementations and consider performance when stochasticity is added to the target Hamiltonian in the form of spectral statistics. To demonstrate the practicality of such real-time evolution, we discuss its application to fundamental problems in quantum computation such as electronic structure predictions for strongly correlated systems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Negative fluxes and cell-miss errors in the random ray method

The random ray method is a recently developed stochastic method for solving neutral particle transport problems based on the method of characteristics. Perhaps surprisingly for a characteristics-based method using flat sources, we note that the random ray method can produce negative fluxes which may be numerically troublesome in several situations. These occur most severely in fixed source problems where the source is in a region with a small cross section. Additionally, we briefly discuss another source of bias which can occur in similar situations, namely a ray missing a mesh with a strong source and small cross section, resulting in the entirety of the source being unphysically deposited locally. This paper describes the mechanism by which negative fluxes may occur and several different methods to mitigate their effects. These fixes are tested on an eigenvalue problem, a ‘fusion-like’ shielding problem, and a shielding problem featuring an adjoint calculation. Even when extremely coarse random ray quadratures are used such that 20%–30% of cells are missed during a given iteration, use of the preferred fix technique ensures local flux tally errors remain trivial (below 1%). The preferred fix is now the default option in SCONE and OpenMC.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

Rotational symmetry relation for efficient response function generation in the coarse mesh transport method COMET

The coarse mesh transport code COMET is a continuous energy hybrid stochastic-deterministic neutronics solver with high fidelity and formidable computation speed in solving reactor core problems. Its method is based on the incident flux expansion theory. In this work, we take advantage of the local geometric symmetry in many reactor cores lattices (e.g., fuel lattices and reflector blocks) to develop relations among the flux response expansion coefficients for symmetric surfaces to further improve the computational efficiency of the COMET response function generation tool (method). This is done by a rigorous derivation of the transformation matrices for the angular and spatial expansion moments resulting from a rotation of a coarse mesh by an arbitrary angle. The relations for the response coefficients for the symmetric surfaces can be then written as the Kronecker product of those transformation matrices. The method is implemented into COMET and tested on two advanced high temperature reactor (AHTR) full-length single assembly benchmark problems. The COMET results using the response function library based on the symmetry relations were compared to those using the library directly generated by continuous energy Monte Carlo for all surfaces. It was found that the eigenvalues and stripe-wise fission densities using the two libraries are in statistical agreement as expected. This indicates that the new method maintains the high fidelity of the original COMET method while improving the computational efficiency in the response function generation by 270% to 400%, depending on the local geometric symmetry. This method also reduces the size of the response function library by the same magnitude (270% to 400%). (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2022

Improved Cross Section Generation Capability of Griffin in FY22

The Griffin code is a Multiphysics Object-Oriented Simulation Environment (MOOSE) based reactor multiphysics analysis application jointly developed by Idaho National Laboratory and Argonne National Laboratory. The code includes a variety of deterministic steady-state transport solvers for fixed source, k-eigenvalue, adjoint, and subcritical multiplication as well as transient solvers for spatial dynamics with the improved quasi-static method. Griffin uses cross section data in the ISOXML format generated from external deterministic or Monte Carlo cross section generation codes. In recent years, the MC 2 -3 modules have been added to Griffin for fast reactor cross section generation, and the self-shielding application programming interface (SSAPI) was implemented in the ISOXML module for thermal reactor cross section generation. The on-the-fly slowing down method and double-heterogeneity treatment have been implemented to SSAPI and verified against particulate fuel-bearing graphite-moderated thermal reactor problems with high accuracy. This year, work has been focused on improving the cross section generation capability of ISOXML and streamlining the cross section generation procedures. In addition, the form function data were added to ISOXML in order to support the pin power reconstruction capability that was newly implemented in Griffin in this fiscal year. To facilitate the cross section generation using MC 2 -3 and SSAPI in Griffin, the cross section generation workflows have been set up for both fast and thermal spectrum reactors. The MOOSE action system tool was devised for fast spectrum problems, and the MOOSE stochastic tool was adopted to the branch calculation procedure for thermal spectrum problems. Meanwhile, to ensure the accuracy of group-constants, the thermal up-scattering kernel calculator accounting for resonance scattering was implemented in ISOXML, demonstrating the accurate computation of a Doppler-broadened scattering kernel of any Legendre order within a reasonable timescale. Other aspects of ISOXML, such as deletion solver and data, documentation, ISOXML file management, and the interface for Mixture, were improved as well.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗