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At least 343 records · Page 19
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Fast Solution of Fully Implicit Runge--Kutta and Discontinuous Galerkin in Time for Numerical PDEs, Part II: Nonlinearities and DAEs

Fully implicit Runge--Kutta (IRK) methods have many desirable accuracy and stability properties as time integration schemes, but high-order IRK methods are not commonly used in practice with large-scale numerical PDEs because of the difficulty of solving the stage equations. This paper introduces a theoretical and algorithmic framework for solving the nonlinear equations that arise from IRK methods (and discontinuous Galerkin discretizations in time) applied to nonlinear numerical PDEs, including PDEs with algebraic constraints. Several new linearizations of the nonlinear IRK equations are developed, offering faster and more robust convergence than the often-considered simplified Newton, as well as an effective preconditioner for the true Jacobian if exact Newton iterations are desired. Inverting these linearizations requires solving a set of block 2 x 2 systems. Under quite general assumptions, it is proven that the preconditioned 2 x 2 operator's condition number is bounded by a small constant close to one, independent of the spatial discretization, spatial mesh, and time step, and with only weak dependence on the number of stages or integration accuracy. Moreover, the new method is built using the same preconditioners needed for backward Euler-type time stepping schemes, so can be readily added to existing codes. The new methods are applied to several challenging fluid flow problems, including the compressible Euler and Navier--Stokes equations, and the vorticity-streamfunction formulation of the incompressible Euler and Navier--Stokes equations. Up to 10th-order accuracy is demonstrated using Gauss IRK, while in all cases fourth-order Gauss IRK requires roughly half the number of preconditioner applications as required by standard Singly diagonally implicit Runge--Kutta methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Convergence Analysis of the Alternating Anderson–Picard Method for Nonlinear Fixed-Point Problems

Anderson acceleration (AA) has been widely used to solve nonlinear fixed-point problems due to its rapid convergence. This work focuses on a variant of AA in which multiple Picard iterations are performed between each AA step, referred to as the Alternating Anderson–Picard (AAP) method. Furthermore, despite introducing more “slow” Picard iterations, this method has been shown to be efficient and even more robust in both linear and nonlinear cases. However, there is a lack of theoretical analysis for AAP in the nonlinear case. In this paper, we address this gap by establishing the equivalence between AAP and a multisecant-GMRES method that uses GMRES to solve a multisecant linear system at each iteration. From this perspective, we show that AAP “converges” to the Newton-GMRES method. Specifically, as the residual approaches zero, the multisecant matrix, the approximate Jacobian inverse, the search direction, and the optimization gain of AAP converge to their counterparts in the Newton-GMRES method. These connections provide insights for analyzing the asymptotic convergence properties of AAP. Consequently, we show that AAP is locally 𝑞-linear convergent and provide an upper bound for the convergence factor of AAP. To validate the theoretical results, numerical examples are provided.

Anderson acceleration↗
DOE OSTI2026

Robust Optimal Experimental Design of Infinite-Dimensional Bayesian Nonlinear Inverse Problems

Abstract. We consider robust optimal experimental design (ROED) for nonlinear Bayesian inverse problems governed by partial differential equations (PDEs). An optimal design is one that maximizes some utility quantifying the quality of the solution of an inverse problem. However, the optimal design is dependent on elements of the inverse problem such as the simulation model, the prior, or the measurement error model. ROED aims to produce an optimal design that is aware of the additional uncertainties encoded in the inverse problem and remains optimal even after variations in them. We follow a worst-case scenario approach to develop a new framework for robust optimal design of nonlinear Bayesian inverse problems. The proposed framework (a) is scalable and designed for infinite-dimensional Bayesian nonlinear inverse problems constrained by PDEs; (b) develops efficient approximations of the utility, namely the expected information gain; (c) employs eigenvalue sensitivity techniques to develop analytical forms and efficient evaluation methods of the gradient of the utility with respect to the uncertainties against which we wish to be robust; and (d) employs a probabilistic optimization paradigm that properly defines and efficiently solves the resulting combinatorial max-min optimization problem. The effectiveness of the proposed approach is illustrated for optimal sensor placement problem in an inverse problem governed by an elliptic PDE.

Chowdhary, Abhijit↗
DOE OSTI2020

Mixed higher-order anisotropic flow and nonlinear response coefficients of charged particles in $\mathrm {PbPb}$ collisions at $\sqrt{\smash [b]{s_{_{\mathrm {NN}}}}} = 2.76$ and 5.02$\,\text {TeV}$

Anisotropies in the initial energy density distribution of the quark-gluon plasma created in high energy heavy ion collisions lead to anisotropies in the azimuthal distributions of the final-state particles known as collective anisotropic flow. Fourier harmonic decomposition is used to quantify these anisotropies. The higher-order harmonics can be induced by the same order anisotropies (linear response) or by the combined influence of several lower order anisotropies (nonlinear response) in the initial state. The mixed higher-order anisotropic flow and nonlinear response coefficients of charged particles are measured as functions of transverse momentum and centrality in $\mathrm {PbPb}$ collisions at nucleon-nucleon center-of-mass energies $\sqrt{\smash [b]{s_{_{\mathrm {NN}}}}} = 2.76$ and 5.02$\,\text {TeV}$ with the CMS detector. The results are compared with viscous hydrodynamic calculations using several different initial conditions, as well as microscopic transport model calculations. None of the models provides a simultaneous description of the mixed higher-order flow harmonics and nonlinear response coefficients.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Real-time Nonlinear Model Predictive Control (NMPC) Strategies using Physics-Based Models for Advanced Lithium-ion Battery Management System (BMS)

Optimal operation of lithium-ion batteries requires robust battery models for advanced battery management systems (ABMS). A nonlinear model predictive control strategy is proposed that directly employs the pseudo-two-dimensional (P2D) model for making predictions. Using robust and efficient model simulation algorithms developed previously, the computational time of the nonlinear model predictive control algorithm is quantified, and the ability to use such models for nonlinear model predictive control for ABMS is established.

25 ENERGY STORAGE↗
DOE OSTI2020

Dynamic Response of a Semiactive Suspension System with Hysteretic Nonlinear Energy Sink Based on Random Excitation by means of Computer Simulation

This paper aims to investigate the property and behavior of the hysteretic nonlinear energy sink (HNES) coupled to a half vehicle system which is a nine-degree-of-freedom, nonlinear, and semiactive suspension system in order to improve the ride comfort and increase the stability in shock mitigation by using the computer simulation method. The HNES model is a semiactive suspension device, which comprises the famous Bouc–Wen (B-W) model employed to describe the force produced by both the purely hysteretic spring and linear elastic spring of potentially negative stiffness connected in parallel, for the half vehicle system. Nine nonlinear motion equations of the half vehicle system are derived in terms of the seven displacements and the two dimensionless hysteretic variables, which are integrated numerically by employing the direct time integration method for studying both the variables of vertical displacements, velocities, accelerations, chassis pitch angle, and the ride comfort and driver safety, respectively, based on the bump and random road inputs of the pseudoexcitation method as excitation signal. Simulation results show that, compared with the HNES model and the magnetorheological (MR) model coupled to the half vehicle system, the ride comfort and stability have been evidently improved. A successful validation process has been performed, which indicated that both the ride comfort and driver safety properties of the HNES model coupled to half vehicle significantly improved.

Chen, Hui↗
DOE OSTI2024

Domain-decomposition nonlinear manifold reduced order model

This software combines nonlinear-manifold reduced order models (NM-ROMs) with domain decomposition (DD) techniques. NM-ROMs, which utilize a shallow, sparse autoencoder trained with full order model (FOM) snapshot data, approximate the FOM state on a nonlinear manifold. These models offer advantages over linear-subspace ROMs (LS-ROMs) particularly in scenarios with slowly decaying Kolmogorov n-width. However, the training of NM-ROMs involves a number of parameters that scale with the size of the FOM, and storing high-dimensional FOM snapshots can significantly increase the cost of ROM training for extreme-scale problems. To mitigate these costs, the software employs DD to partition the FOM into smaller subdomains, computes NM-ROMs for each, and then integrates these to form a global NM-ROM. This strategy offers multiple benefits: it enables parallel training of subdomain NM-ROMs, reduces the number of parameters needed, decreases the dimensional requirements of subdomain FOM training data, and allows for customization to the unique characteristics of each FOM subdomain. The use of a shallow, sparse autoencoder architecture in each subdomain NM-ROM facilitates the application of hyper-reduction (HR), simplifying the nonlinear complexities and enhancing computational speed. This software marks the inaugural application of NM-ROM combined with HR to a DD problem. It features an algebraic DD reformulation of the FOM, training of NM-ROMs with HR for each subdomain, and employs a sequential quadratic programming (SQP) solver for the evaluation of the coupled global NMROM. The effectiveness of the DD NM-ROM with HR is numerically demonstrated on the 2D steady-state Burgers' equation, showing an order of magnitude improvement in accuracy over the DD LS-ROM with HR.

Diaz, AlejandroN↗
DOE OSTI2021

Full Vorticity Budget of the Arabian Sea from a 0.1° Ocean Model: Sverdrup Dynamics, Rossby Waves, and Nonlinear Eddy Effects

Abstract The Arabian Sea, influenced by the Indian monsoon, has many unique features, including its basin-scale seasonally reversing surface circulation and the Great Whirl, a seasonal anticyclonic system appearing during the southwest monsoon close to the western boundary. To establish a comprehensive dynamical picture of the Arabian Sea, we utilize numerical model output and design a full vorticity budget that includes a fully decomposed nonlinear term. The ocean general circulation model has 0.1° resolution and is mesoscale eddy-resolving in the region. In the western boundary current system, we highlight the role of nonlinear eddies in the life cycle of the Great Whirl. The nonlinear eddy term is of leading-order importance in this feature’s vorticity balance. Specifically, it contributes to the Great Whirl’s persistence in boreal fall after the weakening of the southwesterly winds. In the open ocean, Sverdrup dynamics and annual Rossby waves are found to dominate the vorticity balance; the latter is considered as a key factor in the formation of the Great Whirl and the seasonal reversal of the western boundary current. In addition, we discuss different forms of vertically integrated vorticity equations in the model and argue that the bottom pressure torque term can be interpreted analogously as friction in the western boundary and vortex stretching in the open ocean.

58 GEOSCIENCES↗
DOE OSTI2025

Nonlinear Ensemble Filtering with Diffusion Models: Application to the Surface Quasigeostrophic Dynamics

The intersection between classical data assimilation methods and novel machine learning techniques has attracted significant interest in recent years. Here, we explore another promising solution in which diffusion models are used to formulate a robust nonlinear ensemble filter for sequential data assimilation. Unlike standard machine learning methods, the proposed ensemble score filter (EnSF) is completely training free and can efficiently generate a set of analysis ensemble members. Here, in this study, we apply the EnSF to a surface quasigeostrophic model and compare its performance against the popular local ensemble transform Kalman filter (LETKF), which makes Gaussian assumptions in the analysis step. Numerical tests demonstrate that EnSF maintains stable performance in the absence of localization and for a variety of experimental settings. We find that while LETKF maintains optimal performance in the case of linear observations of the entire state and a perfect model, EnSF shows improvements over LETKF when nonlinear observations are assimilated and the system is subject to unexpected model errors. A spectral decomposition of the analysis results in this nonlinear observation regime shows that the largest improvements over LETKF occur at large scales (small wavenumbers), where LETKF lacks sufficient ensemble spread. Overall, this initial application of EnSF to a geophysical model of intermediate complexity motivates further development of the algorithm for more realistic problems.

Artificial intelligence↗
DOE OSTI2021

Alignment tolerance analysis for divided-pulse nonlinear compression

We present an analytic model that describes the output pulse after Kerr-based spectral broadening with divided-pulse nonlinear compression that includes errors in unequal pulse division, birefringent plate retardance, and thermal drift. The model shows that alignment tolerances become impractically tight at high levels of nonlinearity and that the angle of incidence on the birefringent plates can be utilized as a compensator to loosen those tolerances. We present experimental verification of the model, which is expected to be a fast and flexible tool to design future divided-pulse nonlinear compression systems.

47 OTHER INSTRUMENTATION↗
DOE OSTI2024

Nonlinear propagation of chirped laser pulses through a dispersive and turbulent atmosphere

The evolution of ultrashort laser pulses in dispersive, turbulent, nonlinear, and dissipative media is discussed in connection with nonlinear self-focusing collapse and the onset of laser filamentation. In quiescent air, a laser pulse propagating with a peak power greater than a critical power for self-focusing will undergo a catastrophic, transverse collapse until the intensity is large enough for photoionization. At this point, self-focusing is arrested and balanced by plasma refraction, forming a laser filament. By applying an appropriate chirp, the dispersive properties of the medium can be used to enhance this process and control its onset, and to counter dissipative effects such as molecular absorption and atmospheric scattering. This paper presents an analysis of the effect of atmospheric turbulence on the propagation of nonlinear pulses with dispersion compensation (chirp). Furthermore, the analytical results are compared with wave optics simulations and found to be in reasonable agreement as long as the pulse maintains a near-Gaussian spatiotemporal profile.

47 OTHER INSTRUMENTATION↗
DOE OSTI2026

Nonlinear post-compression to sub-20 fs using a single-stage multipass cell filled with ambient air

Noble gas-filled multipass cells have proven to be very effective in compressing high-energy, hundreds-of-femtoseconds pulses down to tens of femtoseconds. Molecular gases can be an attractive alternative to noble gases since they provide additional Raman nonlinearities that can be much stronger than the electronic Kerr nonlinearity and can therefore enable more spectral broadening and shorter compressed pulses. Air at atmospheric pressure offers molecular gases in their simplest format, with both easy access and no costs for implementing a gas chamber. Here we demonstrate a single-stage, air-filled multipass cell that spectrally broadens 145- μ J, 185-fs pulses with >90% throughput, with a small fraction of the output compressed to sub-20 fs using a prism-pair compressor. The multipass cell uses standard broadband mirrors without dispersion engineering and ambient air as the nonlinear medium, making it the simplest and most cost-effective solution for generating few-cycle femtosecond pulses.

47 OTHER INSTRUMENTATION↗
DOE OSTI2024

Unidirectional amplification in the frozen mode regime enabled by a nonlinear defect

A stationary inflection point (SIP) is a spectral singularity of the Bloch dispersion relation ω(k) of a periodic structure where the first and the second derivatives of ω with respect to k vanish. An SIP is associated with a third-order exceptional point degeneracy in the spectrum of the unit-cell transfer matrix, where there is a collapse of one propagating and two evanescent Bloch modes. At the SIP frequency, the incident wave can be efficiently converted into the frozen mode with greatly enhanced amplitude and vanishing group velocity. This can be very attractive for applications, including light amplification. Due to its non-resonant nature, the frozen mode regime (FMR) has fundamental advantages over common cavity resonances. Furthermore, we propose, a novel, to the best of our knowledge, scheme for FMR-based unidirectional amplifiers by leveraging a tailored amplification/attenuation mechanism and a single nonlinear defect. The defect breaks the directional symmetry of the periodic structure and enables nonlinearity-related unidirectional amplification/attenuation in the vicinity of the SIP frequency. We demonstrate the robustness of the amplification mechanism to local impurities and parasitic nonlinearity.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Fundamental mode excitation via Joule–Thomson light expansion in nonlinear optical lattices

Under linear conditions, power injected from a single waveguide into a multi-core fiber array results in multimode propagation, progressively diminishing the spatial coherence of light. In this work, we introduce a comprehensive approach to mitigate this coherence loss by means of a nonlinear thermodynamic Joule–Thomson expansion. By leveraging the tools of optical thermodynamics, we demonstrate that as light undergoes a sudden transition from a small to a larger nonlinear optical array, it can abruptly drop its optical temperature to near-zero values. During this cooling process, light irreversibly flows into the system's fundamental mode with very high efficiency, synchronizing all elements of the lattice with the input port. We show that this nonlinear effect is highly predictable even in systems of arbitrary geometry and shape and can be controlled precisely by the initial conditions at the input of the array. In particular, for a single injection point, the reduction in optical temperature can be directly determined by the total power, irrespective of the input location.

Pyrialakos, Georgios G. (ORCID:0000000286129694)↗
DOE OSTI2025

Quantitative Nonlinear Optical Polarimetry with High Spatial Resolution

Nonlinear optical microscopy such as in the optical second-harmonic generation (SHG) modality has become a popular tool today for probing materials in the physical and biological sciences. While imaging and spectroscopy are widely used in the microscopy mode, nonlinear polarimetry, which can shed light on materials’ symmetry and microstructure, is relatively underdeveloped. This is partly because quantitative analytical modeling of the optical SHG response for anisotropic crystals and films largely assumes low-numerical aperture (NA) focusing of light, where the plane-wave approximation is sufficient. Tight focusing provides unique benefits in revealing out-of-plane polarization responses, which cannot be detected by near-plane-wave illumination at normal incidence. Here, we outline a method for quantitatively analyzing SHG polarimetry measurements obtained under high-NA focusing within a microscope geometry. Experiments and simulations of a variety of standard samples, from single crystals to thin films, are in good agreement, including measured and simulated spatial SHG maps of ferroelectric domains. A solution to the inverse problem is demonstrated, where the spatial distribution of an SHG tensor with unknown tensor coefficient magnitudes is determined by experimentally measured polarimetry. The ability to extract the out-of-plane component of the nonlinear polarization in normal incidence is demonstrated, which can be valuable for high-resolution polarimetry of 2D materials, thin films, heterostructures, and uniaxial crystals with a strong out-of-plane response.

36 MATERIALS SCIENCE↗
DOE OSTI2026

Electrically tunable nonlinear light generation in plasmonic tunnel junctions

Active and efficient control of nonlinear optical processes is essential for integrated photonics, with applications in signal processing, ultrafast switching, and quantum light manipulation. While nanophotonic structures are powerful for enhancing nonlinearities, achieving wide-range electrical tunability has remained a challenge. Plasmonic tunnel junctions offer a unique path to bridge this gap because they combine extreme optical field confinement with direct electrical control in a single nanoscale device. Here, we report the first, to our knowledge, demonstration of electrically tunable second-harmonic generation (SHG) in plasmonic tunnel junctions. Using ultra-stable epitaxial heterostructures, we achieve reproducible modulation of SHG with depths up to ∼500% and magnitudes above 1.3V −1 . We identify two mechanisms, electric-field-induced SHG (EFISH) and ion migration, that can either compete or cooperate depending on junction thickness and bias, enabling both broad tunability and ferroelectric-like hysteretic switching. These findings establish plasmonic tunnel junctions as a platform for electrically controlled nonlinear optics, with potential for nanoscale light sources, reconfigurable modulators, and neuromorphic photonic devices..

42 ENGINEERING↗
DOE OSTI2023

Improved measurements of nonlinear integrable optics at IOTA

Nonlinear integrable optics (NIO) are a promising novel approach at improving the stability of high intensity beams. Implementations of NIO based on specialized magnetic elements are being tested at the Integrable Optics Test Accelerator (IOTA) at Fermilab. One method of verifying proper implementation of these solutions is by measuring the analytic invariants predicted by theory. The initial measurements of nonlinear invariants were performed during IOTA run in 2019/20, however the covid-19 pandemic prevented the full-scale experimental program from being completed. Several important improvements were implemented in IOTA for the 2022/23 run, including the operation at higher beam energy of 150 MeV, improved optics control, and chromaticity correction. This report presents on the improved calibrations of the NIO for nonlinear invariant measurements.

43 PARTICLE ACCELERATORS↗
DOE OSTI2020

Nonlinear Tune-Shift Measurements in the Integrable Optics Test Accelerator

The first experimental run of Fermilab’s Integrable Optics Test Accelerator (IOTA) ring aimed at testing the concept of nonlinear integrable beam optics. In this report we present the preliminary results of the studies of a nonlinear focusing system with two invariants of motion realized with the special elliptic-potential magnet. The key measurement of this experiment was the horizontal and vertical betatron tune shift as a function of transverse amplitude. A vertical kicker strength was varied to change the betatron amplitude for several values of the nonlinear magnet strength. The turn-by-turn positions of the 100 MeV electron beam at twenty-one beam position monitors around the ring were captured and used for the analysis of phase-space trajectories.

43 PARTICLE ACCELERATORS↗