Search NASA⌕ Search

SEARCH · Search NASA

Results for “Elliptic problems”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Image segmentation for analyzing galaxy-galaxy strong lensing systems

The goal of this Letter is to develop a machine learning model to analyze the main gravitational lens and detect dark substructure (subhalos) within simulated images of strongly lensed galaxies. Using the technique of image segmentation, we turn the task of identifying subhalos into a classification problem, where we label each pixel in an image as coming from the main lens, a subhalo within a binned mass range, or neither. Our network is only trained on images with a single smooth lens and either zero or one subhalo near the Einstein ring. On an independent test set with lenses with large ellipticities, quadrupole and octopole moments, and for source apparent magnitudes between 17–25, the area of the main lens is recovered accurately. On average, only 1.3% of the true area is missed and 1.2% of the true area is added to another part of the lens. In addition, subhalos as light as 10 8.5 M ⊙ can be detected if they lie in bright pixels along the Einstein ring. Furthermore, the model is able to generalize to new contexts it has not been trained on, such as locating multiple subhalos with varying masses or more than one large smooth lens.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2026

Integrated Simulation of Weld Residual Stress Evolution and Crack Propagation Using XFEM

Nuclear power plant components operate in environments that promote multiple degradation mecha- nisms, several of which involve crack initiation and growth. An ongoing effort in the U.S. Department of Energy’s Nuclear Energy Advanced Modeling and Simulation (NEAMS) program is developing a general capability within the Multiphysics Object Oriented Simulation Environment (MOOSE) framework for simulating three-dimensional crack growth under a range of driving conditions, including fatigue, stress corrosion cracking (SCC), brittle fracture, and stress-relaxation cracking. This report demonstrates an end-to-end workflow that uses this capability to model weld-residual-stress-driven SCC in the J-groove weld of a pressurized-water reactor control rod drive mechanism penetration in the vessel head. The workflow consists of a thermomechanical welding simulation with temperature-dependent plasticity, followed by cooldown to ambient conditions, and a restart of the simulation using the MOOSE extended finite element method (XFEM) module to propagate a three-dimensional crack through the residual stress field. New welding capabilities were developed to properly initialize newly activated elements in the weld region, and robustness improvements were made to the mesh-based algorithm for defining cutting planes in the 3D XFEM algorithm, allowing it to handle complex crack fronts and stress fields. Together these advances allowed the simulated SCC crack to grow from an initial elliptical flaw in the weld, across the weld, through the tube wall, and almost to the triple point (where the weld, tube, and reactor pressure vessel head intersect) over roughly 36 years of simulated service. These results demonstrate a workflow that can be extended to fully three-dimensional welding simulations and more complex crack interaction problems.

42 - ENGINEERING↗
DOE OSTI2024

Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes

In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and its partial derivatives, a result of the PDE constraint and for which fast algorithms are scarce. The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute ϵ-approximate inverse Cholesky factors of the kernel matrices with complexity O(N log d (N/ϵ)) in space and O(N log 2d (N/ϵ)) in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. We numerically illustrate our algorithm’s near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Ampère equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs and kernel methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Large-Time Behavior of Solutions to Vlasov-Poisson-Fokker-Planck Equations: From Evanescent Collisions to Diffusive Limit

The present contribution investigates the dynamics generated by the two-dimensional Vlasov-Poisson-Fokker-Planck equation for charged particles in a steady inhomogeneous background of opposite charges. We provide global in time estimates that are uniform with respect to initial data taken in a bounded set of a weighted L{sup 2} space, and where dependencies on the mean-free path τ and the Debye length δ are made explicit. In our analysis the mean free path covers the full range of possible values: from the regime of evanescent collisions τ→∞ to the strongly collisional regime τ→0. As a counterpart, the largeness of the Debye length, that enforces a weakly nonlinear regime, is used to close our nonlinear estimates. Accordingly we pay a special attention to relax as much as possible the τ-dependent constraint on δ ensuring exponential decay with explicit τ-dependent rates towards the stationary solution. In the strongly collisional limit τ→0, we also examine all possible asymptotic regimes selected by a choice of observation time scale. Here also, our emphasis is on strong convergence, uniformity with respect to time and to initial data in bounded sets of a L{sup 2} space. Our proofs rely on a detailed study of the nonlinear elliptic equation defining stationary solutions and a careful tracking and optimization of parameter dependencies of hypocoercive/hypoelliptic estimates.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

A unified hp-HDG framework for Friedrichs' PDE systems

Here, this work proposes a unified hp-adaptivity framework for hybridized discontinuous Galerkin (HDG) method for a large class of partial differential equations (PDEs) of Friedrichs' type. In particular, we present unified hp-HDG formulations for abstract one-field and two-field structures and prove their well-posedness. In order to handle non-conforming interfaces we simply take advantage of HDG built-in mortar structures. With split-type mortars and the approximation space of trace, a numerical flux can be derived via Godunov approach and be naturally employed without any additional treatment. As a consequence, the proposed formulations are parameter-free. We perform several numerical experiments for time-independent and linear PDEs including elliptic, hyperbolic, and mixed-type to verify the proposed unified hp-formulations and demonstrate the effectiveness of hp-adaptation. Two adaptivity criteria are considered: one is based on a simple and fast error indicator, while the other is rigorous but more expensive using an adjoint-based error estimate. The numerical results show that these two approaches are comparable in terms of convergence rate even for problems with strong gradients, discontinuities, and singularities.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Building a quantum computing architecture using 3D superconducting cavities

Quantum computers promise advantages over classical machines for solving certain complex problems, but building processors that truly deliver this advantage remains a central challenge, particularly due to limited coherence times. Three-dimensional superconducting radio-frequency (SRF) cavities offer an attractive platform due to their exceptionally long lifetimes. However, since these harmonic systems require nonlinear elements, such as transmons, for control, additional losses are often introduced. In this talk, I will present a multimode quantum system based on an elliptical SRF cavity hosting two cavity modes weakly coupled to an ancillary transmon circuit. This architecture is carefully engineered to preserve coherence while enabling efficient control. By optimizing the design to mitigate transmon-induced decoherence, we realize single-photon lifetimes of 20.6 ms and 15.6 ms in the two modes, with pure dephasing times exceeding 40 ms. Using sideband interactions and error-resilient protocols, such as measurement-based correction and post-selection, we demonstrate high-fidelity state control, including preparation of Fock states up to N=20 with fidelities above 95% (to our knowledge, the highest reported to date), as well as high-fidelity two-mode entanglement. These results highlight 3D SRF cavities as a robust foundation for qudit-based quantum information processing, harnessing the large Hilbert space of cavity modes. I will conclude by outlining strategies to further enhance coherence in both cavities and ancilla qubits, and discuss pathways toward scaling this architecture into a larger quantum computing platform.

Roy, Tanay [Fermilab]↗
DOE OSTI2022

The High Fraction of Thin Disk Galaxies Continues to Challenge ΛCDM Cosmology

Any viable cosmological framework has to match the observed proportion of early- and late-type galaxies. In this contribution, we focus on the distribution of galaxy morphological types in the standard model of cosmology (Lambda cold dark matter, ΛCDM). Using the latest state-of-the-art cosmological ΛCDM simulations known as Illustris, IllustrisTNG, and EAGLE, we calculate the intrinsic and sky-projected aspect ratio distribution of the stars in subhalos with stellar mass M * > 10 10 M ⊙ at redshift z = 0. There is a significant deficit of intrinsically thin disk galaxies, which however comprise most of the locally observed galaxy population. Consequently, the sky-projected aspect ratio distribution produced by these ΛCDM simulations disagrees with the Galaxy And Mass Assembly (GAMA) survey and Sloan Digital Sky Survey at ≥12.52σ (TNG50-1) and ≥14.82σ (EAGLE50) confidence. The deficit of intrinsically thin galaxies could be due to a much less hierarchical merger-driven build-up of observed galaxies than is given by the ΛCDM framework. It might also arise from the implemented sub-grid models, or from the limited resolution of the above-mentioned hydrodynamical simulations. We estimate that an 8 5 times better mass resolution realization than TNG50-1 would reduce the tension with GAMA to the 5.58σ level. Finally, we show that galaxies with fewer major mergers have a somewhat thinner aspect ratio distribution. Given also the high expected frequency of minor mergers in ΛCDM, the problem may be due to minor mergers. In this case, the angular momentum problem could be alleviated in Milgromian dynamics because of a reduced merger frequency arising from the absence of dynamical friction between extended dark matter halos.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2019

A cloud computing approach to massively-parallel distributed stochastic optimization with application to geomechanical reservoir characterization

Many search strategies used to optimize parameters during hydrologic model calibration are inherently limited in that simulations must be performed sequentially in order for the algorithm to converge. As a result, the overall time required to reach a solution increases linearly with the number of steps necessary. In contrast, sampling strategies that attempt to approximate the posterior distribution of the parameters do not require the model simulations to be run in sequence and therefore invite the use of a massively parallel approach across heterogeneous computing environments. We apply a cloud computing methodology to this problem where a head node running on Amazon Web Services continuously produces candidate models that are distributed to arbitrary compute environments to run the simulation, in our case Clemson’s Palmetto cluster computer. Thus the time required to obtain a sufficient number of samples to produce an adequate approximation of the posterior distribution of the parameters is limited by the number of available compute sites. Monte Carlo sampling of the posterior space would be duplicative and inefficient, thus we combine Latin hypercube methods with genetic algorithms to ensure that no large regions of the parameter space are left unexplored, and that high probability regions are investigated thoroughly. We independently assess the fit to different types of data sets without the need to construct a single objective function. Sets of Pareto optimal models are particularly valuable when the quality and information content of data is poorly known or conceptual model errors exist in our simulation. We illustrate our approach for a problem where in situ strain data are used to calibrate the parameters of an oil reservoir in the Avant oil field in Oklahoma. We use well injection and shut-in to stress a deep (500m) confined formation, and observe the geomechanical response using borehole strainmeters installed in the shallow subsurface (30m). These data were then used to calibrate the poroelastic properties of the reservoir and caprock along with the geometry of an elliptic channel associated with the formation. Using our approach we are able to identify issues with model error as well as substantial tradeoffs that occur within the parameters.

inversion, stochastic, geomechanics, poroelastic, ↗
DOE OSTI2017

Quantifying Environmental and Line-of-sight Effects in Models of Strong Gravitational Lens Systems

Matter near a gravitational lens galaxy or projected along the line of sight (LOS) can affect strong lensing observables by more than contemporary measurement errors. We simulate lens fields with realistic three-dimensional mass configurations (self-consistently including voids), and then fit mock lensing observables with increasingly complex lens models to quantify biases and uncertainties associated with different ways of treating the lens environment (ENV) and LOS. We identify the combination of mass, projected offset, and redshift that determines the importance of a perturbing galaxy for lensing. Foreground structures have a stronger effect on the lens potential than background structures, due to nonlinear effects in the foreground and downweighting in the background. There is dramatic variation in the net strength of ENV/LOS effects across different lens fields; modeling fields individually yields stronger priors for H {sub 0} than ray tracing through N-body simulations. Models that ignore mass outside the lens yield poor fits and biased results. Adding external shear can account for tidal stretching from galaxies at redshifts z⩾z{sub lens}, but it requires corrections for external convergence and cannot reproduce nonlinear effects from foreground galaxies. Using the tidal approximation is reasonable for most perturbers as long as nonlinear redshift effects are included. Even then, the scatter in H {sub 0} is limited by the lens profile degeneracy. Asymmetric image configurations produced by highly elliptical lens galaxies are less sensitive to the lens profile degeneracy, so they offer appealing targets for precision lensing analyses in future surveys like LSST and Euclid.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2017

Summer Proceedings 2016: The Center for Computing Research at Sandia National Laboratories

Solving sparse linear systems from the discretization of elliptic partial differential equations (PDEs) is an important building block in many engineering applications. Sparse direct solvers can solve general linear systems, but are usually slower and use much more memory than effective iterative solvers. To overcome these two disadvantages, a hierarchical solver (LoRaSp) based on H2-matrices was introduced in [22]. Here, we have developed a parallel version of the algorithm in LoRaSp to solve large sparse matrices on distributed memory machines. On a single processor, the factorization time of our parallel solver scales almost linearly with the problem size for three-dimensional problems, as opposed to the quadratic scalability of many existing sparse direct solvers. Moreover, our solver leads to almost constant numbers of iterations, when used as a preconditioner for Poisson problems. On more than one processor, our algorithm has significant speedups compared to sequential runs. With this parallel algorithm, we are able to solve large problems much faster than many existing packages as demonstrated by the numerical experiments.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Preconditioned steepest descent methods for some nonlinear elliptic equations involving p-Laplacian terms

We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulations for some important physical application problems – including thin film epitaxy with slope selection and the square phase field crystal model – are carried out to verify the efficiency of the scheme.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2020

Generalized Grad–Shafranov equation for non-axisymmetric MHD equilibria

The structure of static MHD equilibria that admit continuous families of Euclidean symmetries is well understood. Such field configurations are governed by the classical Grad–Shafranov equation, which is a single elliptic partial differential equation in two space dimensions. By revealing a hidden symmetry, here we show that in fact all smooth solutions of the equilibrium equations with non-vanishing pressure gradients away from the magnetic axis satisfy a generalization of the Grad–Shafranov equation. In contrast to solutions of the classical Grad–Shafranov equation, solutions of the generalized equation are not automatically equilibria, but instead only satisfy force balance averaged over the one-parameter hidden symmetry. We then explain how the generalized Grad–Shafranov equation can be used to reformulate the problem of finding exact three-dimensional smooth solutions of the equilibrium equations as finding an optimal volume-preserving symmetry.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2018

On the convergence and convergence order of finite volume gradient schemes for oblique derivative boundary value problems

This work is an improvement of the previous note (Bradji in: Fuhrmann et al., Finite volumes for complex applications VII–methods, theoretical aspects. Proceedings of the FVCA 7, Berlin, 2014) which dealt with the convergence analysis of a finite volume scheme for the Poisson’s equation with a linear oblique derivative boundary condition. The formulation of the finite volume scheme given in Bradji (in: Fuhrmann et al., Finite volumes for complex applications VII–methods, theoretical aspects. Proceedings of the FVCA 7, Berlin, 2014) involves the discrete gradient introduced recently in Eymard et al. (IMA J Numer Anal 30(4):1009–1043, 2010). In this paper, we consider the convergence analysis of finite volume schemes involving the discrete gradient of Eymard et al. (IMA J Numer Anal 30(4):1009–1043, 2010) for elliptic and parabolic equations with linear oblique derivative boundary conditions. Linear oblique derivative boundary conditions arise for instance in the study of the motion of water in a canal, cf. Lesnic (Commun Numer Methods Eng 23(12):1071–1080, 2007). We derive error estimates in several norms which allow us to get error estimates for the approximations of the exact solutions and its first derivatives. In particular, we provide an error estimate between the gradient of the exact solutions and the discrete gradient of the approximate solutions. Convergence of the family of finite volume approximate solutions towards the exact solution under weak regularity assumption is also investigated. In the case of parabolic equations with oblique derivative boundary conditions, we develop a new discrete a priori estimate result. The proof of this result is based on the use of a discrete mean Poincaré–Wirtinger inequality. Thanks to the stated a priori estimate and to a comparison with an appropriately chosen auxiliary finite volume scheme, we derive the convergence results. This work can be viewed as a continuation of the previous work (Bradji and Gallouët in Int J Finite Vol 3(2):1–35, 2006) where a convergence analysis for a finite volume scheme, based on the admissible mesh of Eymard et al. (In: Ciarlet and Lions, Handbook of numerical analysis, North-Holland, Amsterdam, 2000), for the Poisson’s equation with a linear oblique derivative boundary conditions is given. The obtained convergence results do not require any relation between the mesh sizes of the spatial and time discretizations. Some numerical tests are presented for both elliptic and parabolic equations. In particular, we present three methods to compute the discrete solution.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Field Emission Analysis in SRF Cavities for PIP-II Using GEANT4

Field emission (FE) remains a significant hurdle for achieving optimal performance and reliability in superconducting radiofrequency (SRF) cavities used in accelerator cryomodules. A thorough understanding of the generation and propagation of FE-induced radiation is therefore essential to mitigate this problem. The absence of standardized measurement protocols further complicates the comparison of radiation data across different testing phases and facilities. This highlights the need for a precise quantitative method to diagnose and analyze FE-induced radiation. Such efforts could prove beneficial for improving cavity preparation and cleanroom assembly techniques during the prototype and production stages of Fermilab's Proton Improvement Plan-II (PIP-II) project. This study presents the initial steps of detailed Geant4 simulations aimed at analyzing FE-induced radiation in the low-beta 650 MHz 5-cell elliptical (LB650) cavity. Our goal is to combine these results with radiation diagnostics to enhance diagnostic accuracy and optimize detector positioning. This integrated approach ultimately aims to improve the preparation, assembly, and testing procedures for PIP-II SRF cavities, ensuring the delivery of FE-free cryomodules.

Wijethunga, Sajini A.K. [Fermilab] (ORCID:00000002↗
DOE OSTI2025

Field Emission Analysis in SRF Cavities for PIP-II Using GEANT4

Field emission (FE) remains a significant hurdle for achieving optimal performance and reliability in super-conducting radiofrequency (SRF) cavities used in accelerator cryomodules. A thorough understanding of the generation and propagation of FE-induced radiation is therefore essential to mitigate this problem. The absence of standardized measurement protocols further complicates the comparison of radiation data across different testing phases and facilities. This highlights the need for a precise quantitative method to diagnose and analyze FE-induced radiation. Such efforts could prove beneficial for improving cavity preparation and cleanroom assembly techniques during the prototype and production stages of Fermilab's Proton Improvement Plan-II (PIP-II) project. This study presents the initial steps of detailed Geant4 simulations aimed at analyzing FE-induced radiation in the low-beta 650 MHz 5-cell elliptical (LB650) cavity. Our goal is to combine these results with radiation diagnostics to enhance diagnostic accuracy and optimize detector positioning. This integrated approach ultimately aims to improve the preparation, assembly, and testing procedures for PIP-II SRF cavities, ensuring the delivery of FE-free cryomodules.

Wijethunga, S. A.K. [Fermilab] (ORCID:000000028765↗
DOE OSTI2018

Quickest Detection for Changes in Maximal kNN Coherence of Random Matrices

This paper addresses the problem of quickest detection of a change in the maximal coherence between columns of a n p random matrix based on a sequence of matrix observations having a single unknown change point. The random matrix is assumed to have identically distributed rows and the maximal coherence is defined as the largest of the ( 2 p) correlation coefficients associated with any row. Likewise the κ nearest neighbor (kNN) coherence is defined as the κ-th largest of these correlation coefficients. The forms of the preand postchange distributions of the observed matrices are assumed to belong to the family of elliptically contoured densities with sparse dispersion matrices but are otherwise unknown. A nonparametric stopping rule is proposed that is based on the maximal kNN sample coherence between columns of each observed random matrix. This is a summary statistic that is related to a test of existence of a hub vertex in a sample correlation graph having degree at least κ. Performance bounds on the delay and false alarm performance of the proposed stopping rule are obtained in the purely high-dimensional regime where p → ∞ and n is fixed. When the prechange dispersion matrix is diagonal it is shown that, among all functions of the proposed summary statistic, the proposed stopping rule is asymptotically optimal under a minimax quickest change detection model as the stopping threshold approaches infinity. Furthermore, the theory developed also applies to sequential hypothesis testing and fixed sample size tests.

96 KNOWLEDGE MANAGEMENT AND PRESERVATION↗
DOE OSTI2018

Qualitative Properties of Solutions to Elliptic Equations with Nonpower Nonlinearities in ℝ{sub n}

We consider a class of anisotropic elliptic equations with nonpower nonlinearities. We establish the existence of a solution in the local Sobolev–Orlicz spaces without conditions on the data growth at infinity. We find sufficient uniqueness conditions, obtain estimates characterizing the behavior of the solution at infinity, and prove the continuous dependence of the solution on the right-hand side of the equation.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Explicit Weierstrass traveling wave solutions and bifurcation analysis for dissipative Zakharov–Kuznetsov modified equal width equation

We consider the dissipative Zakharov–Kuznetsov modified equal width (ZK-MEW) equation and discuss the effect of the dissipation on the existence and nature of traveling wave solutions of the equation. We use Lyapunov function and dynamical system theory in order to show that when viscosity term is added to the ZK-MEW equation, yet there exists still bounded traveling wave solutions in certain regions. Subsequently, we obtain general solution of the ZK-MEW equation in the presence and absence of viscosity in terms of Weirstrass ℘ functions and Jacobi elliptic functions. We also derive a new form of kink-type solution by exploring a factorization method based on functional transformation and the Abel’s first order nonlinear equation. Finally, we use the phase plane analysis and examine the stability of the viscous waves.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗