Spectral Operators Research Articles

Marcinkiewicz-Type Spectral Multipliers on Hardy and Lebesgue Spaces on Product Spaces of Homogeneous Type

Let $X_1$ and $X_2$ be metric spaces equipped with doubling measures and let $L_1$ and $L_2$ be nonnegative self-adjoint second-order operators acting on $L^2(X_1)$ and $L^2(X_2)$ respectively. We study multivariable spectral multipliers $F(L_1, L_2)$ acting on the Cartesian product of $X_1$ and $X_2$. Under the assumptions of the finite propagation speed property and Plancherel or Stein--Tomas restriction type estimates on the operators $L_1$ and~$L_2$, we show that if a function~$F$ satisfies a Marcinkiewicz-type differential condition then the spectral multiplier operator $F(L_1, L_2)$ is bounded from appropriate Hardy spaces to Lebesgue spaces on the product space $X_1\times X_2$. We apply our results to the analysis of second-order elliptic operators in the product setting, specifically Riesz-transform-like operators and double Bochner--Riesz means.

Nonhomogeneous boundary conditions for the spectral fractional Laplacian

We present a construction of harmonic functions on bounded domains for the spectral fractional Laplacian operator and we classify them in terms of their divergent profile at the boundary. This is used to establish and solve boundary value problems associated with nonhomogeneous boundary conditions. We provide a weak-L1 theory to show how problems with measure data at the boundary and inside the domain are well-posed. We study linear and semilinear problems, performing a sub- and supersolution method. We finally show the existence of large solutions for some power-like nonlinearities.

A Spectral Element Method with Transparent Boundary Condition for Periodic Layered Media Scattering

We present a high-order spectral element method for solving layered media scattering problems featuring an operator that can be used to transparently enforce the far-field boundary condition. The incorporation of this Dirichlet-to-Neumann (DtN) map into the spectral element framework is a novel aspect of this work, and the resulting method can accommodate plane-wave radiation of arbitrary angle of incidence. In order to achieve this, the governing Helmholtz equations subject to quasi-periodic boundary conditions are rewritten in terms of periodic unknowns. We construct a spectral element operator to approximate the DtN map, thus ensuring nonreflecting outgoing waves on the artificial boundaries introduced to truncate the computational domain. We present an explicit formula that accurately computes the Fourier coefficients of the solution in the spectral element discretization space projected onto the boundary which is required by the DtN map. Our solutions are represented by the tensor product basis of one-dimensional Legendre---Lagrange interpolation polynomials based on the Gauss---Lobatto---Legendre grids. We study the scattered field in singly and doubly layered media with smooth and nonsmooth interfaces. We consider rectangular, triangular, and sawtooth interfaces that are accurately represented by the body-fitted quadrilateral elements. We use GMRES iteration to solve the resulting linear system, and we validate our results by demonstrating spectral convergence in comparison with exact solutions and the results of an alternative computational method.

‐triangular operators: spectral properties and important examples

We introduce a notion of ‐triangular operators in the Hilbert space using some basic ideas from triangular representation theory. Our notion generalizes the well‐known notion of the spectral operators so that many properties of the ‐triangular operators coincide with those of spectral operators. At the same time we show that wide classes of operators are ‐triangular.

Entropy Stable Staggered Grid Discontinuous Spectral Collocation Methods of any Order for the Compressible Navier--Stokes Equations

Staggered grid, entropy stable discontinuous spectral collocation operators of any order are developed for the compressible Euler and Navier--Stokes equations on unstructured hexahedral elements. This generalization of previous entropy stable spectral collocation work [M. H. Carpenter, T. C. Fisher, E. J. Nielsen, and S. H. Frankel, SIAM J. Sci. Comput., 36 (2014), pp. B835--B867, M. Parsani, M. H. Carpenter, and E. J. Nielsen, J. Comput. Phys., 292 (2015), pp. 88--113], extends the applicable set of points from tensor product, Legendre--Gauss--Lobatto (LGL), to a combination of tensor product Legendre--Gauss (LG) and LGL points. The new semidiscrete operators discretely conserve mass, momentum, energy, and satisfy a mathematical entropy inequality for the compressible Navier--Stokes equations in three spatial dimensions. They are valid for smooth as well as discontinuous flows. The staggered LG and conventional LGL point formulations are compared on several challenging test problems. The staggered LG ope...

Discrete Spectrum of 2 + 1-Dimensional Nonlinear Schrödinger Equation and Dynamics of Lumps

We consider a natural integrable generalization of nonlinear Schrödinger equation to2+1dimensions. By studying the associated spectral operator we discover a rich discrete spectrum associated with regular rationally decaying solutions, the lumps, which display interesting nontrivial dynamics and scattering. Particular interest is placed in the dynamical evolution of the associated pulses. For all cases under study we find that the relevant dynamics corresponds to acentral configurationof a certainN-body problem.

Localized Lp-estimates of eigenfunctions: A note on an article of Hezari and Rivière

We use a straightforward variation on a recent argument of Hezari and Rivière [8] to obtain localized Lp-estimates for all exponents larger than or equal to the critical exponent pc=2(n+1)n−1. We are able to do this directly by just using the Lp-bounds for spectral projection operators from our much earlier work [12]. The localized bounds we obtain here imply, for instance, that, for a density one sequence of eigenvalues on a manifold whose geodesic flow is ergodic, all of the Lp, 2<p≤∞, bounds of the corresponding eigenfunctions are relatively small compared to the general ones in [12], which are saturated on round spheres. The connection with quantum ergodicity was established for exponents 2<p<pc in the recent results of the author [13] and Blair and the author [3]; however, the article of Hezari and Rivière [8] was the first one to make this connection (in the case of negatively curved manifolds) for the critical exponent, pc. As is well known, and we indicate here, bounds for the critical exponent, pc, imply ones for all the other exponents 2<p≤∞. The localized estimates involve L2-norms over small geodesic balls Br of radius r, and we shall go over what happens for these in certain model cases on the sphere and on manifolds of nonpositive curvature. We shall also state a problem of determining when one can improve on the trivial O(r12) estimates for these L2(Br) bounds. If r=λ−1, one can improve on the trivial estimates if one has improved Lpc(M) bounds just by using Hölder's inequality; however, obtaining improved bounds for r≫λ−1 seems to be subtle.

Mini-Workshop: Eigenvalue Problems in Surface Superconductivity

The aim of the meeting is to discuss several classes of Schrödinger equations appearing within the Ginzburg-Landau theory of superconductivity. The related problems are discussed from several perspectives including semiclassical analysis, PDE in non-smooth domains, geometric spectral theory and operator theory, which should provide a new insight into various phenomena appearing in superconducting systems.

The gradient estimate of a Neumann eigenfunction on a compact manifold with boundary

Let eλ(x) be a Neumann eigenfunction with respect to the positive Laplacian Δ on a compact Riemannian manifold M with boundary such that Δeλ = λ2eλ in the interior of M and the normal derivative of eλ vanishes on the boundary of M. Let χλ be the unit band spectral projection operator associated with the Neumann Laplacian and f be a square integrable function on M. The authors show the following gradient estimate for χλ f as \(\lambda \geqslant 1:{\left\| {\nabla {\chi _\lambda }{\kern 1pt} \left. f \right\|} \right._\infty } \leqslant C\left( {\lambda \left\| {{\chi _\lambda }{{\left. f \right\|}_\infty } + \left. {{\lambda ^{ - 1}}} \right\|} \right.\Delta {\chi _\lambda }{{\left. f \right\|}_\infty }} \right)\), where C is a positive constant depending only on M. As a corollary, the authors obtain the gradient estimate of eλ: For every λ ≥ 1, it holds that \(\left\| {\nabla {{\left. {{e_\lambda }} \right\|}_\infty } \leqslant C\left. \lambda \right\|} \right.{\left. {{e_\lambda }} \right\|_\infty }\).

On index theory for non‐Fredholm operators: A (1 + 1)‐dimensional example

Using the general formalism of , a study of index theory for non‐Fredholm operators was initiated in . Natural examples arise from (1 + 1)‐dimensional differential operators using the model operator in of the type , where , and the family of self‐adjoint operators in studied here is explicitly given by urn:x-wiley:0025584X:media:mana201500065:mana201500065-math-0007Here has to be integrable on and tends to zero as and to 1 as (both functions are subject to additional hypotheses). In particular, , , has asymptotes (in the norm resolvent sense) urn:x-wiley:0025584X:media:mana201500065:mana201500065-math-0015 as , respectively. The interesting feature is that violates the relative trace class condition introduced in , Hypothesis 2.1 ]. A new approach adapted to differential operators of this kind is given here using an approximation technique. The approximants do fit the framework of enabling the following results to be obtained. Introducing , , we recall that the resolvent regularized Witten index of , denoted by , is defined by urn:x-wiley:0025584X:media:mana201500065:mana201500065-math-0023 whenever this limit exists. In the concrete example at hand, we prove urn:x-wiley:0025584X:media:mana201500065:mana201500065-math-0024Here denotes the spectral shift operator for the pair of self‐adjoint operators , and we employ the normalization, , .

Model of anisotropic nonlinearity in self-defocusing photorefractive media.

We develop a phenomenological model of anisotropy in self-defocusing photorefractive crystals. In addition to an independent term due to nonlinear susceptibility, we introduce a nonlinear, non-separable correction to the spectral diffraction operator. The model successfully describes the crossover between photovoltaic and photorefractive responses and the spatially dispersive shock wave behavior of a nonlinearly spreading Gaussian input beam. It should prove useful for characterizing internal charge dynamics in complex materials and for accurate image reconstruction through nonlinear media.

Towards quantized number theory: spectral operators and an asymmetric criterion for the Riemann hypothesis.

This research expository article not only contains a survey of earlier work but also contains a main new result, which we first describe. Given c≥0, the spectral operator [Formula: see text] can be thought of intuitively as the operator which sends the geometry onto the spectrum of a fractal string of dimension not exceeding c. Rigorously, it turns out to coincide with a suitable quantization of the Riemann zeta function ζ=ζ(s): a=ζ(∂), where ∂=∂(c) is the infinitesimal shift of the real line acting on the weighted Hilbert space [Formula: see text]. In this paper, we establish a new asymmetric criterion for the Riemann hypothesis (RH), expressed in terms of the invertibility of the spectral operator for all values of the dimension parameter [Formula: see text] (i.e. for all c in the left half of the critical interval (0,1)). This corresponds (conditionally) to a mathematical (and perhaps also, physical) 'phase transition' occurring in the midfractal case when [Formula: see text]. Both the universality and the non-universality of ζ=ζ(s) in the right (resp., left) critical strip [Formula: see text] (resp., [Formula: see text]) play a key role in this context. These new results are presented here. We also briefly discuss earlier joint work on the complex dimensions of fractal strings, and we survey earlier related work of the author with Maier and with Herichi, respectively, in which were established symmetric criteria for the RH, expressed, respectively, in terms of a family of natural inverse spectral problems for fractal strings of Minkowski dimension D∈(0,1), with [Formula: see text], and of the quasi-invertibility of the family of spectral operators [Formula: see text] (with [Formula: see text]).

Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport–diffusion equation

In this paper, we prove explicit lower bounds for the cost of fast boundary controls for a class of linear equations of parabolic or dispersive type involving the spectral fractional Laplace operator. We notably deduce the following striking result: in the case of the heat equation controlled on the boundary, Miller's conjecture formulated in Miller (2004) [16] is not verified. Moreover, we also give a new lower bound for the minimal time needed to ensure the uniform controllability of the one-dimensional convection–diffusion equation with negative speed controlled on the left boundary, proving that the conjecture formulated in Coron and Guerrero (2005) [2] concerning this problem is also not verified at least for negative speeds.The proof is based on complex analysis, and more precisely on a representation formula for entire functions of exponential type, and is quite related to the moment method.

Discrete symmetries in the heterotic-string landscape

We describe a new type of discrete symmetry that relates heterotic-string models. It is based on the spectral flow operator which normally acts within a general N = (2, 2) model and we use this operator to construct a map between N = (2, 0) models. The landscape of N = (2, 0) models is of particular interest among all heterotic-string models for two important reasons: Firstly, N =1 spacetime SUSY requires (2, 0) superconformal invariance and secondly, models with the well motivated by the Standard Model SO(10) unification structure are of this type. This idea was inspired by a new discrete symmetry in the space of fermionic ℤ2 × ℤ2 heterotic-string models that exchanges the spinors and vectors of the SO(10) GUT group, dubbed spinor-vector duality. We will describe how to generalize this to arbitrary internal rational Conformal Field Theories.

A Convex Formulation for Hyperspectral Image Superresolution via Subspace-Based Regularization

Hyperspectral remote sensing images (HSIs) usually have high spectral resolution and low spatial resolution. Conversely, multispectral images (MSIs) usually have low spectral and high spatial resolutions. The problem of inferring images which combine the high spectral and high spatial resolutions of HSIs and MSIs, respectively, is a data fusion problem that has been the focus of recent active research due to the increasing availability of HSIs and MSIs retrieved from the same geographical area. We formulate this problem as the minimization of a convex objective function containing two quadratic data-fitting terms and an edge-preserving regularizer. The data-fitting terms account for blur, different resolutions, and additive noise. The regularizer, a form of vector Total Variation, promotes piecewise-smooth solutions with discontinuities aligned across the hyperspectral bands. The downsampling operator accounting for the different spatial resolutions, the non-quadratic and non-smooth nature of the regularizer, and the very large size of the HSI to be estimated lead to a hard optimization problem. We deal with these difficulties by exploiting the fact that HSIs generally "live" in a low-dimensional subspace and by tailoring the Split Augmented Lagrangian Shrinkage Algorithm (SALSA), which is an instance of the Alternating Direction Method of Multipliers (ADMM), to this optimization problem, by means of a convenient variable splitting. The spatial blur and the spectral linear operators linked, respectively, with the HSI and MSI acquisition processes are also estimated, and we obtain an effective algorithm that outperforms the state-of-the-art, as illustrated in a series of experiments with simulated and real-life data.

Shadow detection improvement using spectral indices and morphological operators in high resolution images from urban areas

Abstract. While high-resolution remote sensing images have increased application possibilities for urban studies, the large number of shadow areas has created challenges to processing and extracting information from these images. Furthermore, shadows can reduce or omit information from the surface as well as degrading the visual quality of images. The pixels of shadows tend to have lower radiance response within the spectrum and are often confused with low reflectance targets. In this work, a shadow detection method was proposed using a morphological operator for dark pattern identification combined with spectral indices. The aims are to avoid misclassification in shadow identification through properties provided by them on color models and, therefore, to improve shadow detection accuracy. Experimental results were tested applying the panchromatic and multispectral band of WorldView-2 image from São Paulo city in Brazil, which is a complex urban environment composed by high objects like tall buildings causing large shadow areas. Black top-hat with area injunction was applied in PAN image and shadow identification performance has improved with index as Normalized Difference Vegetation Index (NDVI) and Normalized Saturation-Value Difference Index (NSDVI) ratio from HSV color space obtained from pansharpened multispectral WV-2 image. An increase in distinction between shadows and others objects was observed, which was tested for the completeness, correctness and quality measures computed, using a created manual shadow mask as reference. Therefore, this method can contribute to overcoming difficulties faced by other techniques that need shadow detection as a first necessary preprocessing step, like object recognition, image matching, 3D reconstruction, etc.

Investigations on several numerical methods for the non-local Allen–Cahn equation

This paper investigates some numerical methods for solving the non-local Allen–Cahn equation with a space–time dependent Lagrange multiplier. Several types of methods, including the Crank–Nicolson finite difference method, the finite difference operator splitting method, and the Fourier spectral operator splitting method are proposed respectively. Comparisons are made among these methods in terms of accuracy and computational efficiency for solving different types of problems. Numerical results show that the Fourier spectral operator splitting method is very efficient because of its spectral accuracy in space, especially for simulation of long time dynamics.

Entropy stable wall boundary conditions for the three-dimensional compressible Navier–Stokes equations

Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the three-dimensional compressible Navier–Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneous-approximation-term penalty technique. Although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, discontinuous Galerkin, and flux reconstruction/correction procedure via reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions.

On condition numbers of spectral operators in a hilbert space

We consider a linear unbounded operator \(A\) in a separable Hilbert space. with the following property: there is a normal operator \(D\) with a discrete spectrum, such \(\Vert A-D\Vert <\infty \). Besides, all the Eigen values of \(D\) are different. Under certain assumptions it is shown that \(A\) is similar to a normal operator and a sharp bound for the condition number is suggested. Applications of that bound to spectrum perturbations and operator functions are also discussed. As an illustrative example we consider a non-selfadjoint differential operator.

A Computable Characterization of the Extrinsic Mean of Reflection Shapes and Its Asymptotic Properties

The reflection shapes of configurations in ℜm with k landmarks consist of all the geometric information that is invariant under compositions of similarity and reflection transformations. By considering the corresponding Schoenberg embedding, we embed the reflection shape space into the Euclidean space of all (k - 1) by (k - 1) real symmetric matrices. In this paper, we provide a computable formula of the extrinsic mean of the reflection shapes in arbitrary dimensions. Moreover, the asymptotic analysis of the extrinsic mean of the reflection shapes is studied. By using the differentiability of spectral operators, we obtain a central limit theorem of the sample extrinsic mean of the reflection shapes. As a direct application, the two-example hypothesis test of the reflection shapes is also derived.