Spectral Truncation Research Articles

Exponential integrator for stochastic strongly damped wave equation based on the Wong–Zakai approximation

We consider strong convergence of numerical approximations for a stochastic strongly damped wave equation driven by a class of additive space–time noises characterized by a parameter β∈(−1,2]. With the help of the Wong–Zakai (WZ) approximation to the noise, i.e., replacing the driven noise with its finite spectral expansion truncation, we obtain an approximate equation. Based on the consistency and high regularity of the approximate equation, we develop two exponential integrators for time-stepping discretization and use the spectral Galerkin method in space to develop full-discrete schemes. Performing error estimates in the strong sense, we show that the optimal strong order in time of the proposed WZ-approximation-based exponential Euler scheme is min{1,1+β/2−ϵ}, which is |β|/2 order higher than the order of min{1,1+β} in the existing works when β∈(−1,0). Moreover, we prove that the proposed WZ-approximation-based exponential trapezoidal scheme is of min{3/2,1+β/2−ϵ}-order strong convergence in time when β∈[−1/4,2], so it can break the first order barrier and obtain a higher accurate numerical solution. Numerical examples are performed to verify and develop the theoretical findings.

Spectral analysis and domain truncation for Maxwell's equations

We analyse how the spectrum of the anisotropic Maxwell system with bounded conductivity σ on a Lipschitz domain Ω is approximated by domain truncation. First we prove a new non-convex enclosure for the spectrum of the Maxwell system, with weak assumptions on the geometry of Ω and none on the behaviour of the coefficients at infinity. We also establish a simple criterion for non-accumulation of eigenvalues at iR as well as resolvent estimates. For asymptotically constant coefficients, we describe the essential spectrum and show that spectral pollution may occur only in the essential numerical range We(L∞)⊂R of the quadratic pencil L∞(ω)=μ∞−1curl2−ω2ε∞, acting on divergence-free vector fields. Further, every isolated spectral point of the Maxwell system lying outside We(L∞) and outside the part of the essential spectrum on iR is approximated by spectral points of the Maxwell system on the truncated domains. Our analysis is based on two new abstract results on the (limiting) essential spectrum of polynomial pencils and triangular block operator matrices, which are of general interest. We believe our strategy of proof could be used to establish domain truncation spectral exactness for more general classes of non-self-adjoint differential operators and systems with non-constant coefficients.

Simulación numérica de la cascada de energía en fluidos turbulentos

Transfer of energy from large to small scales in turbulent flows is described as a flux of energy from small wave numbers to large wave numbers in the spectral representation of the Navier-Stokes equation. The problem of resolving the relevant scales in the flow corresponds in the spectral representation to determining the spectral truncation at large wave numbers. The effective number of degrees of freedom in the flow depends on the Reynolds number and a numerical simulation of the Navier Stokes equation for high Reynolds numbers is impractical without some sort of reduction of the number of degrees of freedom. The shell models are constructed to obey the same conservation laws and symmetries as the Navier Stokes equation as an alternative. In this work we present the Sabra shell model for the study of the energy cascade of turbulence and we will show numerically that the energy dissipation is approximately -1/3 which is in agreement with the theory K41.

From absolute equilibrium to Kardar–Parisi–Zhang crossover: A short review of recent developments

I first recall the theoretical background relevant to spectral truncation: absolute equilibrium in helical flows and compressible effects. Thermalization phenomenology in Gross–Pitaevskii superflows and thermalization processes in classical systems are then briefly reviewed. The so-called ’tygers’ that appear in the truncated inviscid Burgers equation are demonstrated. The basic definitions that relate the Burgers equation to the Kardar–Parisi–Zhang system are recalled. Spectral truncation and conserved quantities are used to introduce the microcanonical and canonical stationary probabilities. The main results on the crossover from absolute equilibrium to Kardar–Parisi–Zhang scaling are finally given after a brief discussion of the relevant physical parameters. The present contribution is thus a short review of the publications, scientific developments and collaborations that went on during the last decade and led to the joint work (Cartes et al., 2022) that I presented at the XVIII Instabilities and Nonequilibrium Structures Workshop held (online) in December 2021 in Valparaiso (Chile), dedicated to the memory of the late Enrique Tirapegui.

Strong 1.5 order scheme for second-order stochastic differential equations without Levy area

In this work, a finite difference scheme combined with a truncated spectral expansion of white noise is presented for solving second-order stochastic differential equations (SDEs) with additive white noise. Consistency of the approximate equation, which is obtained by replacing the white noise with its spectral truncation, is considered in mean-square sense. Based on it, a full discrete scheme is constructed and the mean-square convergence order of numerical solution is investigated. Numerical examples show that the presented scheme is of 1.5-order convergence in mean-square sense, which verifies the theoretical findings and is a half order higher than converting the equation to a two-dimensional system with additive white noise and using the Euler-Maruyama scheme to solve the resulting problem. It is also illustrated that the piecewise version of the spectral approximation of white noise plus proper time discretization works well for long-time simulation.

Fourier ptychographic microscopy with untrained deep neural network priors.

We propose a physics-assisted deep neural network scheme in Fourier ptychographic microscopy (FPM) using untrained deep neural network priors (FPMUP) to achieve a high-resolution image reconstruction from multiple low-resolution images. Unlike the traditional training type of deep neural network that requires a large labelled dataset, this proposed scheme does not require training and instead outputs the high-resolution image by optimizing the parameters of neural networks to fit the experimentally measured low-resolution images. Besides the amplitude and phase of the sample function, another two parallel neural networks that generate the general pupil function and illumination intensity factors are incorporated into the carefully designed neural networks, which effectively improves the image quality and robustness when both the aberration and illumination intensity fluctuation are present in FPM. Reconstructions using simulated and experimental datasets are demonstrated, showing that the FPMUP scheme has better image quality than the traditional iterative algorithms, especially for the phase recovery, but at the expense of increasing computational cost. Most importantly, it is found that the FPMUP scheme can predict the Fourier spectrum of the sample outside synthetic aperture of FPM and thus eliminate the ringing effect of the recovered images due to the spectral truncation. Inspired by deep image prior in the field of image processing, we may impute the expansion of Fourier spectrums to the deep prior rooted in the architecture of the careful designed four parallel deep neural networks. We envisage that the resolution of FPM will be further enhanced if the Fourier spectrum of the sample outside the synthetic aperture of FPM is accurately predicted.

Water wave problem with inclined walls

We study free surface water waves in a 2-D symmetric triangular channel with sides that have a 45° slope. We develop models for small amplitude nonlinear waves, extending earlier studies that have considered the linearized problem. We show that a combination of heuristic small amplitude expansions and the equations of motion for free surface potential flow lead to a relatively simple system that we then use to study interactions of low frequency modes. The formalism relies on an explicit construction of the normal modes of the linear problem and a new way to represent the free surface. We argue that the construction can be applied to more general geometries. We also examine the structure and some dynamical features of spectral truncations for the lowest even modes.

Boundary stabilization and observation of a weak unstable heat equation in a general multi-dimensional domain

In this paper, we consider the exponential stabilization and observation of an unstable heat equation in a general multi-dimensional domain by combining the finite-dimensional spectral truncation technique and the recently developed dynamics compensation approach. In contrast to the unstable one-dimensional partial differential equation (PDE), such as the transport equation, wave equation and the heat equation, stabilization of unstable PDE in a general multi-dimensional domain by using the backstepping approach is still a challenging problem. We treat the stabilization and observation problems separately. A dynamical state feedback law is proposed firstly to stabilize the unstable heat equation exponentially and then a state observer is designed via a boundary measurement. Both the stability of the closed-loop system and the well-posedness of the observer are proved. Some of the theoretical results are validated by the numerical simulations.

Thermo-optically tunable spectral broadening in a nonlinear ultra-silicon-rich nitride Bragg grating

Spectral tunability methods used in optical communications and signal processing leveraging optical, electrical, and acousto-optic effects typically involve spectral truncation that results in energy loss. Here we demonstrate temperature tunable spectral broadening using a nonlinear ultra-silicon-rich nitride device consisting of a 3-mm-long cladding-modulated Bragg grating and a 7-mm-long nonlinear channel waveguide. By operating at frequencies close to the grating band edge, in an apodized Bragg grating, we access strong grating-induced dispersion while maintaining low losses and high transmissivity. We further exploit the redshift in the Bragg grating stopband due to the thermo-optic effect to achieve tunable dispersion, leading to varying degrees of soliton-effect compression and self-phase-modulation-induced spectral broadening. We observe an increase in the bandwidth of the output pulse spectrum from 69 to 106 nm as temperature decreases from 70°C to 25°C, in good agreement with simulated results using the generalized nonlinear Schrödinger equation. The demonstrated approach provides a new avenue to achieve on-chip laser spectral tuning without loss in pulse energy.

Accuracy of slender body theory in approximating force exerted by thin fiber on viscous fluid

AbstractWe consider the mapping properties of the integral operator arising in nonlocal slender body theory (SBT) for the model geometry of a straight, periodic filament. It is well known that the classical singular SBT integral operator suffers from high wavenumber instabilities, making it unsuitable for approximating the slender body inverse problem, where the fiber velocity is prescribed and the integral operator must be inverted to find the force density along the fiber. Regularizations of the integral operator must therefore be used instead. Here, we consider two regularization methods: spectral truncation and the ‐regularization of Tornberg and Shelley (2004). We compare the mapping properties of these approximations to the underlying partial differential equation (PDE) solution, which for the inverse problem is simply the Stokes Dirichlet problem with data constrained to be constant on cross sections. For the straight‐but‐periodic fiber with constant radius , we explicitly calculate the spectrum of the operator mapping fiber velocity to force for both the PDE and the approximations. We prove that the spectrum of the original SBT operator agrees closely with the PDE operator at low wavenumbers but differs at high frequencies, allowing us to define a truncated approximation with a wavenumber cutoff . For both the truncated and ‐regularized approximations, we obtain rigorous ‐based convergence to the PDE solution as : A fiber velocity with regularity gives convergence, while a fiber velocity with at least regularity yields convergence. Moreover, we determine the dependence of the ‐regularized error estimate on the regularization parameter .

On the Errors of Estimating Oceanic Eddy Kinetic Energy

AbstractWhen evaluating the kinetic energy of the eddy field, using a flow‐decomposition method to split the eddies from the mean‐flow is often necessary. However, the kinetic energies of the eddy and mean‐flow may not sum up to the total energy, because of the existence of a third term, the residual kinetic energy (RKE). This paper attempts to thoroughly assess the RKE from both theoretical and observational perspectives. It is shown that the instantaneous RKE can be quite large, therefore should not be ignored in the time‐varying energetics. The Reynolds decomposition, however, eliminate the time‐mean RKE. It is thus a good practice to study the energy budget in the sense of time‐mean. A running piece‐wise Reynolds decomposition is proposed to allow for choosing the scales of interest while retaining the feature of zero mean RKE. Another decomposition method, the temporal filtering using spectral truncation, are inspected in detail. Based on 26‐years realistic ocean observations, it is found that the ratio between the mean RKE and mean total energy (the Mean Residual Energy Error [MREE]) depends on the spectral shape, the cutoff period, and the available data length. Maximum MREE for cutoff period of 1 year is about 20%. Spatially, the MREE concentrates in zonal bands migrating poleward with increasing cutoff period. MREE from the temporal and spatial coarse‐graining filters with the boxcar and Gaussian kernels are also evaluated. The necessity to estimate the MREE before analyzing the energy budget is emphasized, lest the RKE takes up a non‐negligible portion.

Linear response based parameter estimation in the presence of model error

Recently, we proposed a method to estimate parameters of stochastic dynamics based on the linear response statistics. The method rests upon a nonlinear least-squares problem that takes into account the response properties that stem from the Fluctuation-Dissipation Theory. In this article, we address an important issue that arises in the presence of model error. In particular, when the equilibrium density function is high dimensional and non-Gaussian, and in some cases, is unknown, the linear response statistics are inaccessible. We show that this issue can be resolved by fitting the imperfect model to appropriate marginal linear response statistics that can be approximated using the available data and parametric or nonparametric models. The effectiveness of the parameter estimation approach is demonstrated in the context of molecular dynamical models (Langevin dynamics) with a non-uniform temperature profile, where the modeling error is due to coarse-graining, and a PDE (non-Langevin dynamics) that exhibits spatiotemporal chaos, where the model error arises from a severe spectral truncation. In these examples, we show how the imperfect models, the Langevin equation with parameters estimated using the proposed scheme, can predict the nonlinear response statistics of the underlying dynamics under admissible external disturbances.

Spectrally-truncated kernel ridge regression and its free lunch

Kernel ridge regression (KRR) is a well-known and popular nonparametric regression approach with many desirable properties, including minimax rate-optimality in estimating functions that belong to common reproducing kernel Hilbert spaces (RKHS). The approach, however, is computationally intensive for large data sets, due to the need to operate on a dense n×n kernel matrix, where n is the sample size. Recently, various approximation schemes for solving KRR have been considered, and some analyzed. Some approaches such as Nyström approximation and sketching have been shown to preserve the rate optimality of KRR. In this paper, we consider the simplest approximation, namely, spectrally truncating the kernel matrix to its largest r<n eigenvalues. We derive an exact expression for the maximum risk of this truncated KRR, over the unit ball of the RKHS. This result can be used to study the exact trade-off between the level of spectral truncation and the regularization parameter. We show that, as long as the RKHS is infinite-dimensional, there is a threshold on r, above which, the spectrally-truncated KRR surprisingly outperforms the full KRR in terms of the minimax risk, where the minimum is taken over the regularization parameter. This strengthens the existing results on approximation schemes, by showing that not only one does not lose in terms of the rates, truncation can in fact improve the performance, for all finite samples (above the threshold). Moreover, we show that the implicit regularization achieved by spectral truncation is not a substitute for Hilbert norm regularization. Both are needed to achieve the best performance.

FLUPS: A Fourier-Based Library of Unbounded Poisson Solvers

A Fourier-based Library of Unbounded Poisson Solvers (FLUPS) for 2D and 3D homogeneous distributed grids is presented. It is designed to handle every possible combination of periodic, symmetric, semi-unbounded and fully unbounded boundary conditions for the Poisson equation on rectangular domains with uniform resolution. FLUPS leverages a dedicated implementation of 3D Fourier transforms to solve the Poisson equation using Green's functions, in a fast and memory-efficient way. Several Green's functions are available, optionally with explicit regularization, spectral truncation, or using lattice Green's functions, and provide verified convergence orders from 2 to spectral-like. The algorithm depends on the FFTW library to perform 1D transforms, while Message Passing Interface (MPI) communications enable the required remapping of data in memory. For the latter operation, a first available implementation resorts to the standard all-to-all routines. A second implementation, featuring non-blocking and persistent point-to-point communications, is however shown to be more efficient in a majority of cases and especially while taking advantage of the shared memory parallelism with OpenMP. The scalability of the algorithm, aimed at massively parallel architectures, is demonstrated up to 73 720 cores. The results obtained with three different supercomputers show that the weak efficiency remains above 40\% and the strong efficiency above 30% when the number of cores is multiplied by 16, for typical problems. These figures are slightly better than those expected from a third party 3D Fast Fourier Transform (FFT) tool, with which a 20% longer execution time was also measured on average.

Gromov–Hausdorff convergence of state spaces for spectral truncations

We study the convergence aspects of the metric on spectral truncations of geometry. We find general conditions on sequences of operator system spectral triples that allows one to prove a result on Gromov–Hausdorff convergence of the corresponding state spaces when equipped with Connes’ distance formula. We exemplify this result for spectral truncations of the circle, Fourier series on the circle with a finite number of Fourier modes, and matrix algebras that converge to the sphere.

Functional lagged regression with sparse noisy observations

A functional (lagged) time series regression model involves the regression of scalar response time series on a time series of regressors that consists of a sequence of random functions. In practice, the underlying regressor curve time series are not always directly accessible, but are latent processes observed (sampled) only at discrete measurement locations. In this article, we consider the so‐called sparse observation scenario where only a relatively small number of measurement locations have been observed, possibly different for each curve. The measurements can be further contaminated by additive measurement error. A spectral approach to the estimation of the model dynamics is considered. The spectral density of the regressor time series and the cross‐spectral density between the regressors and response time series are estimated by kernel smoothing methods from the sparse observations. The impulse response regression coefficients of the lagged regression model are then estimated by means of ridge regression (Tikhonov regularization) or principal component analysis (PCA) regression (spectral truncation). The latent functional time series are then recovered by means of prediction, conditioning on all the observed data. The performance and implementation of our methods are illustrated by means of a simulation study and the analysis of meteorological data.

Spectral Truncations in Noncommutative Geometry and Operator Systems

In this paper we extend the traditional framework of noncommutative geometry in order to deal with spectral truncations of geometric spaces (i.e. imposing an ultraviolet cutoff in momentum space) and with tolerance relations which provide a coarse grain approximation of geometric spaces at a finite resolution. In our new approach the traditional role played by C^*-algebras is taken over by operator systems. As part of the techniques we treat C^*-envelopes, dual operator systems and stable equivalence. We define a propagation number for operator systems, which we show to be an invariant under stable equivalence and use to compare approximations of the same space. We illustrate our methods for concrete examples obtained by spectral truncations of the circle. These are operator systems of finite-dimensional Toeplitz matrices and their dual operator systems which are given by functions in the group algebra on the integers with support in a fixed interval. It turns out that the cones of positive elements and the pure state spaces for these operator systems possess a very rich structure which we analyze including for the algebraic geometry of the boundary of the positive cone and the metric aspect i.e. the distance on the state space associated to the Dirac operator. The main property of the spectral truncation is that it keeps the isometry group intact. In contrast, if one considers the other finite approximation provided by circulant matrices the isometry group becomes discrete, even though in this case the operator system is a C^*-algebra. We analyze this in the context of the finite Fourier transform on the cyclic group. The extension of noncommutative geometry to operator systems allows one to deal with metric spaces up to finite resolution by considering the relation d(x,y)< varepsilon between two points, or more generally a tolerance relation which naturally gives rise to an operator system.

Sparse representation approaches to parameter estimation of completely polarized wideband signals

Three sparse representation methods are proposed for estimating the direction-of-arrival (DOA) and polarization parameters of multiple completely polarized wideband signals. The array employed comprises a number of identically oriented vector antennas (VAs). The first two methods are based on spectral segmentation (SSE) and spectral sampling (SSA), both of which require no prior knowledge/estimation of the correlation functions (or power spectral densities) of signals. The third method is based on spectral truncation (STR) which can also be interpreted as the interpolation/reconstruction of signals’ common correlation coefficient. It is computationally simpler than SSE and SSA but requires preliminary estimation of the signals’ common correlation coefficient within a non-zero scalar multiple. Simulation results have validated the efficacy of the three methods.

Recovering the Eulerian energy spectrum from noisy Lagrangian tracers

We consider the time-sequential state estimation of a flow field given a stream of noisy measurements that are provided by instruments advected by the flow, known as Lagrangian tracers or drifters. Lagrangian drifters collect real-time data as they move through the velocity field and are an important data collection method for atmospheric and oceanic measurements. We quantify the recovery of the Eulerian energy spectra from observations of Lagrangian drifters. This is performed by utilizing special Lagrangian data assimilation algorithms, known as conditionally Gaussian nonlinear filters. We address the following questions: how much of the turbulent Eulerian energy spectra can be recovered from assimilation of Lagrangian trajectory data and how accurately are the various energetic scales recovered relative to the truth. These issues are primarily studied in the perfect model scenario, but we quantify recovery due to model error by reduced order models via spectral truncation of the forecast model. We demonstrate high recovery skill of the two-dimensional turbulent energy spectra for both an exact filter and an imperfect filter, based on extreme localization of the covariance matrix, which is vastly cheaper than the exact filter, for both an inverse cascade spectrum with slope k−5∕3 and a direct cascade spectrum with slope k−3. The dependence of the spectral energy recovery skill on the number of tracers and the spectral truncation grid size is also studied.

Tempered random attractors of a non-autonomous non-local fractional equation driven by multiplicative white noise

In this paper, we prove the existence and regularity of a tempered pullback attractor for a non-autonomous stochastic fractional equation involving an abstract non-local operator defined aswhere is the kernel of which satisfies the general fractional-type condition of order and is a ball in centered at x with radius ε. The asymptotical compactness of solutions in is demonstrated by the spectral decomposition and truncation technique, where is a Hilbert space with and As a special example, we obtain the existence and regularity of tempered pullback attractor of stochastic fractional reaction-diffusion equation.