Optimal nonlinear dynamic sparse model selection and Bayesian parameter estimation for nonlinear systems
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Computer Science↗
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Linear and Nonlinear Solvers for Simulating Multiphase Flow within Large-Scale Engineered Subsurface Systems
Simulation of multiphase flow in the subsurface is well-known to be computationally challenging. While there have been many studies that have explored approaches to overcoming these challenges, they often utilize relatively simple case studies. In this paper, we focus on the unique numerical challenges posed by modeling large-scale engineered subsurface systems, characterized by discrete features embedded in a heterogeneous natural subsurface setting. The man-made features such as shafts, tunnels, and barriers often cause multiple challenges in modeling the domain for multiphase porous media flow. This flow scenario can have a wide range of applications such as nuclear waste repositories, enhanced recovery of a petroleum reservoir, geothermal engineering, and carbon sequestration. An example of these severe numerical challenges is the case of performance assessment (PA) for Waste Isolation Pilot Plant (WIPP), the only operating deep geological repository in the US, which simulates extreme material properties of bedded salt rock formation and extreme contrast due to open excavation next to the formation. The models have extremes not only of permeability and porosity but also of the constitutive models needed for multiphase flow; additionally, they have process models like salt creep closure reducing porosity over time, fracturing in clay and anhydrite interbeds of the bedded salt, gas generation from the waste materials, and unintentional human borehole intrusions in some scenarios. Numerical simulations require the solution of coupled systems of nonlinear PDEs; in our work, we use the open-source simulator PFLOTRAN which is based on Finite Volume discretization. The solution of the nonlinear equations requires use of the Newton-Raphson iteration at each time step, which entails the solution of the linearized Jacobian system at each iteration. The effects of all the processes (i.e., large number of unknowns, highly nonlinear constitutive relations, large contrasts in material properties in short distances) lead to an ill-conditioned Jacobian matrix that severely challenges traditional linear solver, i.e., stabilized biconjugate gradient with block Jacobi incomplete LU preconditioner (BCGS-ILU) leading to non-convergence for traditional Newton-Raphson nonlinear solver causing unacceptably long computation time for each model. This paper presents linear solvers such as constrained pressure residual (CPR) two-stage preconditioner with alternate-block-factorization (ABF) and quasi- implicit pressure and explicit saturation (QIMPES) decouplers and flexible generalized residual solver (FGMRES). The new general-purpose nonlinear solver, Newton trust-region dogleg Cauchy (NTRDC), is also introduced to resolve extreme nonlinearities in the models. We demonstrate the effectiveness of each method relative to the default BCGS-Newton solver. The two best cases had nearly 50 times speed-up and achieved completion of a simulation in 14 hours that never completed due to non-convergence with the default solver. We also investigate the strong scalability of each method and discuss some of the deficiencies found for Block Jacobi preconditioner using parallel domain decomposition, and node packing effects of modern processor architecture.
Preconditioner, Nonlinear, Porous media, Multiphas↗
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↗
Giant optical second- and third-order nonlinearities at a telecom wavelength
A material platform that excels in both optical second- and third-order nonlinearities at a telecom wavelength is theoretically and experimentally demonstrated. In this TiN-based coupled metallic quantum well structure, electronic subbands are engineered to support doubly resonant inter-subband transitions for an exceptionally high second-order nonlinearity and provide single-photon transitions for a remarkable third-order nonlinearity within the 1400–1600 nm bandwidth. The second-order susceptibility χ (2) reaches 2840 pm/V at 1440 nm, while the Kerr coefficient n 2 arrives at 2.8 × 10 −10 cm 2 /W at 1460 nm. The achievement of simultaneous strong second- and third-order nonlinearities in one material at a telecom wavelength creates opportunities for multi-functional advanced applications in the field of nonlinear optics.
Chen, Ching-Fu (ORCID:0000000262510722)↗
Nonlinear elastodynamic behavior of intact and fractured rock under in-situ stress and saturation conditions
Not provided.
Materials Science↗
Nonlinear Topological Photonics: Capturing Nonlinear Dynamics and Optical Thermodynamics
Combining multiple optical resonators or engineering dispersion of complex media has provided an effective method for demonstrating topological physics controlling photons in unprecedented ways such as unidirectional light propagation and spatially localized modes between an interface or on a corner. Further, adding nonlinear responses to those topological photonic systems has enabled achieving diverse phases of photons in both space and time, allowing for more functionalities in photonic devices that provide a new playground for studying dynamic features of nonlinear topological systems. However, most methods for describing nonlinear topological photonic systems rely on linear topological theories, making it challenging to accurately characterize the topology of nonlinear systems. Thus, substantial efforts have focused on rigorously describing nonlinear topological phases and developing effective tools to analyze nonlinear topological effects. Meanwhile, coupled multimode optical waveguides with nonlinear dynamic responses provide an excellent platform for the statistical description of photons, opening a new paradigm called “optical thermodynamics”. This review will introduce the basic concepts of nonlinear topological photonics and the recent development of theoretical approaches focusing on data-driven approaches for creating phase diagrams as well as the spectral localizer framework and the pseudospectrum method for understanding optical nonlinearities in topological systems. In addition, the new concept of optical thermodynamics will be introduced with some recent theoretical works.
Topological photonics↗
Transition of a network of nonlinear interactions into a regime of strong nonlinear fluctuations: A paradigm for the edge localized mode onset
We study how a network of nonlinear oscillators transits into a regime of strong nonlinear fluctuations when perturbed by a triad. In this regime, most of the potential energy contained in the waves is made available to the system through strong nonlinear fluctuations. This analysis is motivated by recent experimental observations [Dominski and Diallo, Plasma Phys. Control. Fusion 62, 095011 (2020)] where it was found that magnetic fluctuations trigger the onset of edge localized modes by suddenly exciting a network of nonlinear interactions. In our study, we consider the simplest system of many harmonic oscillators that are organized in a network of nonlinear triads. We model and simulate the sudden transition of this network of triads into a regime of strong nonlinear fluctuations—reminiscent of the onset of edge localized modes in tokamaks. This transition is triggered by the activation of a nonlinear perturbation. An abrupt rise of the system's disorder (an entropy-like quantity) is measured during the transition. This transition from weak to strong nonlinear fluctuations is even more abrupt when these fluctuations are chaotic, i.e., when the timescale of the nonlinear interaction is comparable to the timescale of the wave oscillations.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Statistical mechanics and pressure of composite multimoded weakly nonlinear optical systems
Statistical mechanics can provide a versatile theoretical framework for investigating the collective dynamics of weakly nonlinear-wave settings that can be utterly complex to describe otherwise. In optics, composite systems arise due to interactions between different frequencies and polarizations. The purpose of this work is to develop a thermodynamic theory that takes into account the synergistic action of multiple components. We find that the type of the nonlinearity involved can have important implications in the thermalization process and, hence, can lead to different thermal equilibrium conditions. Importantly, we derive closed-form expressions for the actual optomechanical pressure that is exerted on the system. In particular, the total optomechanical pressure is the sum of the partial pressures due to each component. Our results can be applied to a variety of weakly nonlinear optical settings such as multimode fibers, bulk waveguides, photonic lattices, and coupled microresonators. We present two specific examples, where two colors interact in a one-waveguide array with either a cubic or quadratic nonlinearity.
Efremidis, Nikolaos K. (ORCID:0000000298300268)↗
Nonlinear controller for nonlinear wave energy converters
The present invention is directed to a nonlinear controller for nonlinear wave energy converters (WECs). As an example of the invention, a nonlinear dynamic model is developed for a geometrically right-circular cylinder WEC design for the heave-only motion, or a single degree-of-freedom (DOF). The linear stiffness term is replaced by a nonlinear cubic hardening spring term to demonstrate the performance of a nonlinear WEC as compared to an optimized linear WEC. By exploiting the nonlinear physics in the nonlinear controller, equivalent power and energy capture, as well as simplified operational performance is observed for the nonlinear cubic hardening spring controller when compared to an optimized linear controller.
Wilson, David G.↗
Phenomenological Model of Nonlinear Dynamics and Deterministic Chaotic Gas Migration in Bentonite: Experimental Evidence and Diagnostic Parameters
Understanding gas migration in compacted clay materials, e.g., bentonite and claystone, is important for the design and performance assessment of an engineered barrier system of a radioactive waste repository system, as well as many practical applications. Existing field and laboratory data on gas migration processes in low-permeability clay materials demonstrate the complexity of flow and transport processes, including various types of instabilities, caused by nonlinear dynamics of coupled processes of liquid–gas exchange, dilation, fracturing, fracture healing, etc., which cannot be described by classical models of fluid dynamics in porous media. We here show that the complexity of gas migration processes can be explained using a phenomenological concept of nonlinear dynamics and deterministic chaos theory. To do so, we analyzed gas pressure and gas influx (i.e., input) and outflux (i.e., output), recorded during the gas injection experiment in the compact Mx80-D bentonite sample, and calculated a set of the diagnostic parameters of nonlinear dynamics and chaos, such a global embedding dimension, a correlation dimension, an information dimension, and a spectrum of Lyapunov exponents, as well as plotted 2D and 3D pseudo-phase-space strange attractors, based on the univariate influx and outflux time series data. These results indicate the presence of phenomena of low-dimensional deterministic chaotic behavior of gas migration in bentonite. In particular, during the onset of gas influx in the bentonite core, before the breakthrough, the development of gas flow pathways is characterized by the process of chaotic gas diffusion. After the breakthrough, with inlet-to-outlet movement of gas, the prevailing process is chaotic advection. During the final phase of the experiment, with no influx to the sample, the relaxation pattern of gas outflux is resumed back to a process of chaotic diffusion. The types of data analysis and a proposed phenomenological model can be used to establish the basic principles of experimental data-gathering, modeling predictions, and a research design.
36 MATERIALS SCIENCE↗
A Perturbative Solution for Nonlinear Stratified Upwelling over a Frictional Slope
Abstract A perturbative solution of simplified primitive equations for nonlinear weakly stratified upwelling over a frictional slope is found that resolves the vertical structure of velocity fields and can satisfy Ertel’s potential vorticity conservation in the stratified inviscid interior. The solution uses assumptions consistent with the model proposed by Lentz and Chapman, including a steady-state, constant cross-shore density gradient, no alongshore gradients, laterally inviscid, and consideration of cross-shore advection of alongshore momentum. The solution resolves the vertical structure of velocity fields (including subsurface maxima of compensational flow, not resolved by Lentz and Chapman) and can satisfy Ertel’s potential vorticity conservation in the stratified inviscid interior. The dynamics are similar to Lentz and Chapman; bottom stress balances alongshore wind stress in a homogeneous density ocean and is replaced by nonlinear cross-shore transport of alongshore momentum as the Burger number (S=αN/f, whereα,N, andfare the bottom slope, buoyancy frequency, Coriolis frequency, respectively) increases. When the solution uses the empirical relation between cross-shore and vertical density gradients proposed by Lentz and Chapman, vorticity conservation is not satisfied and the nonlinear momentum transport estimated by the solution linearly increases withS, asymptotically matching Lentz and Chapman forS< 1. When the solution conserves interior potential vorticity, the momentum transport is proportional toS 2 forS< 1 and is in better agreement with numerical simulations.
Oceanography↗
Nonlinear regenerative amplification using tunable intracavity cascaded quadratic nonlinearities
Conventional nonlinear regenerative amplification utilizes Kerr (i.e., self-focusing) nonlinearity to broaden the spectrum via self-phase modulation during the direct amplification of ultrashort pulses, which helps overcome spectral gain narrowing for high-energy, ultrashort-pulse generation. We introduce cascaded quadratic nonlinearities from a type-I second-harmonic generation crystal, which can be tuned in both magnitude and sign via the phase-mismatching condition, inside a regenerative amplifier to enable nonlinearity-tunable regenerative amplification. We demonstrate, both experimentally and numerically, a nonlinear Yb:YAG thin-disk regenerative amplifier with either positive or negative net nonlinearity that delivers 1-kHz, >0.4-mJ pulses as short as ∼200 fs and high-quality beams free of self-focusing. The laser’s average power can be scaled to >100 W when seeded and operated at a much higher repetition rate to benefit various applications.
47 OTHER INSTRUMENTATION↗
Experimental Limit on Nonlinear State-Dependent Terms in Quantum Theory
Linear time evolution is one of the fundamental postulates of quantum theory. Past theoretical attempts to introduce nonlinearity into quantum evolution have violated causality. However, a recent theory has introduced nonlinear state-dependent terms in quantum field theory, preserving causality [D. E. Kaplan and S. Rajendran, Phys. Rev. D 105, 055002 (2022)]. We report the results of an experiment that searches for such terms. Our approach, inspired by the Everett many-worlds interpretation of quantum theory, correlates a binary macroscopic classical voltage with the outcome of a projective measurement of a quantum bit, prepared in a coherent superposition state. Measurement results are recorded in a bit string, which is used to control a voltage switch. Presence of a nonzero voltage reading in cases of no applied voltage is the experimental signature of a nonlinear state-dependent shift of the electromagnetic field operator. We implement blinded measurement and data analysis with three control bit strings. Control of systematic effects is realized by producing one of the control bit strings with a classical random-bit generator. The other two bit strings are generated by measurements performed on a superconducting qubit in an IBM Quantum processor and on a N 15 nuclear spin in a nitrogen-vacancy center in diamond. Our measurements find no evidence for electromagnetic quantum state-dependent nonlinearity. We set a bound on the parameter that quantifies this nonlinearity | ε γ | < 4.7 × 10 - 11 , at 90% confidence level.
46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
Towards a Quantum Algorithm for the Incompressible Nonlinear Navier-Stokes Equations
In this work, we present novel concepts for quantum algorithms to solve transient, nonlinear partial differential equations (PDEs). The challenge lies in how to effectively represent, encode, process, and evolve the nonlinear system of PDEs on quantum computers. We will discuss the new techniques using the incompressible Navier-Stokes equations as an example, because it represents the fundamental nonlinear feature and yet removes certain complexity in physics, allowing us to focus on the design of quantum algorithms. Previous attempts solving nonlinear PDEs in quantum computation have often involved storing multiple copies of solutions or employing linearizations. Neither is practical due to exponential scaling with evolution time or insufficient solution accuracy. We propose a new framework based on matrix product states (MPSs) and matrix product operators (MPOs), in addition to the Krylov subspace methods. For example, the solution variables of the Navier-Stokes equations are represented by MPSs, and the linear and nonlinear terms are processed by MPOs. The time evolution of the operators is attained by a fast-forwarding algorithm using Krylov subspace methods. Furthermore, we discuss various techniques for efficient encoding of MPSs, measurement reduction for MPOs, and use of tensor operations to treat multi-variate, multi-physics characteristics of Navier-Stokes.
Gopalakrishnan Meena, Murali [ORNL] (ORCID:0000000↗
Gauge invariance and Ward identities in nonlinear response theory
We present a formal analysis of nonlinear response functions in terms of correlation functions in real- and imaginary-time domains. In particular, we show that causal nonlinear response functions, expressed in terms of nested commutators in real time, can be obtained from the analytic continuation of time-ordered response functions, which are more easily amenable to diagrammatic calculation. This generalizes the well-known result of linear response theory. We then use gauge invariance arguments to derive exact relations between second-order response functions in density and current channels. These identities, which are non-perturbative in the strength of inter-particle interactions, allow us to establish exact connections between nonlinear optics calculations done in different electromagnetic gauges.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Stability of trapped solutions of a nonlinear Schrödinger equation with a nonlocal nonlinear self-interaction potential
This work focuses on the study of the stability of trapped soliton-like solutions of a (1 + 1)-dimensional nonlinear Schrödinger equation (NLSE) in a nonlocal, nonlinear, self-interaction potential of the form [|Ψ(x,t)| 2 +|Ψ(-x,t)| 2 ] κ where κ is an arbitrary nonlinearity parameter. Although the system with κ = 1 (i.e. fully integrable case) was first reported by Yang (2018 Phys. Rev. E 98 042202), here in the present work, we extend this model to the one in which κ is arbitrary. This allows us to compare the stability properties of the now trapped solutions to previously found solutions of the more usual NLSE with κ ≠ 1 which are moving soliton solutions. We show that there is a simple, one-component, nonlocal Lagrangian and corresponding action governing the dynamics of the system. Using a collective coordinate method derived from the action as well as assuming the validity of Derrick's theorem, we find that these trapped solutions are stable for 0 < κ < 2 and unstable when κ > 2. At the critical value of κ, i.e. κ = 2, the solution can either collapse or blowup linearly in time when q 0 = 0, where q 0 is the center of the initial density ρ(x, t = 0) = ψ*ψ of the solution. For q 0 ≠ 0 the displaced solution collapses. When κ > 2 initial small displacements from the origin also lead to collapse of the wave function. This phenomenon is not seen in the usual NLSE.
collective coordinates↗
Nonlinear analog processing with anisotropic nonlinear films
Digital signal processing is the cornerstone of several modern-day technologies, yet in multiple applications it faces critical bottlenecks related to memory and speed constraints. Thanks to recent advances in metasurface design and fabrication, light-based analog computing has emerged as a viable option to partially replace or augment digital approaches. Several light-based analog computing functionalities have been demonstrated using patterned flat optical elements, with great opportunities for integration in compact nanophotonic systems. So far, however, the available operations have been restricted to the linear regime, limiting the impact of this technology to a compactification of Fourier optics systems. In this paper, we introduce nonlinear operations to the field of metasurface-based analog optical processing, demonstrating that nonlinear optical phenomena, combined with nonlocality in flat optics, can be leveraged to synthesize kernels beyond linear Fourier optics, paving the way to a broad range of new opportunities. As a practical demonstration, we report the experimental synthesis of a class of nonlinear operations that can be used to realize broadband, polarization-selective analog-domain edge detection.
analog image processing↗
Stability of exact solutions of the (2 + 1)-dimensional nonlinear Schrödinger equation with arbitrary nonlinearity parameter κ
In this work, we consider the nonlinear Schrödinger equation (NLSE) in 2+1 dimensions with arbitrary nonlinearity exponent κ in the presence of an external confining potential. Exact solutions to the system are constructed, and their stability as we increase the 'mass' (i.e., the L 2 norm) and the nonlinearity parameter κ is explored. Here we observe both theoretically and numerically that the presence of the confining potential leads to wider domains of stability over the parameter space compared to the unconfined case. Our analysis suggests the existence of a stable regime of solutions for all κ as long as their mass is less than a critical value M*(κ). Furthermore, we find that there are two different critical masses, one corresponding to width perturbations and the other one to translational perturbations. The results of Derrick's theorem are also obtained by studying the small amplitude regime of a four-parameter collective coordinate (4CC) approximation. A numerical stability analysis of the NLSE shows that the instability curve M*(κ) versus κ lies below the two curves found by Derrick's theorem and the 4CC approximation. In the absence of the external potential, κ = 1 demarcates the separation between the blowup regime and the stable regime. In this 4CC approximation, for κ < 1, when the mass is above the critical mass for the translational instability, quite complicated motions of the collective coordinates are possible. Energy conservation prevents the blowup of the solution as well as confines the center of the solution to a finite spatial domain. We call this regime the 'frustrated' blowup regime and give some illustrations. In an appendix, we show how to extend these results to arbitrary initial ground state solution data and arbitrary spatial dimension d.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗