Spectral Projection Research Articles

Central limit theorem for smooth statistics of one‐dimensional free fermions

AbstractWe consider the determinantal point processes associated with the spectral projectors of a Schrödinger operator on , with a smooth confining potential. In the semiclassical limit, where the number of particles tends to infinity, we obtain a Szegő‐type central limit theorem for the fluctuations of smooth linear statistics. More precisely, the Laplace transform of any statistic converges without renormalisation to a Gaussian limit with a ‐type variance, which depends on the potential. In the one‐well (one‐cut) case, using the quantum action‐angle theorem and additional micro‐local tools, we reduce the problem to the asymptotics of Fredholm determinants of certain approximately Toeplitz operators. In the multi‐cut case, we show that for generic potentials, a similar result holds and the contributions of the different wells are independent in the limit.

Improved spectral projection estimates

We obtain new improved spectral projection estimates on manifolds of non-positive curvature, including sharp ones for relatively large spectral windows for general tori. Our results are stronger than those in an earlier work of the first and third authors (2019), and the arguments have been greatly simplified. We more directly make use of pointwise estimates that are implicit in the work of Bérard (1977) and avoid the use of weak-type spaces that were used in the previous works by the first and third authors (2017, 2019). We also simplify and strengthen the bilinear arguments by exploiting the use of microlocal Kakeya–Nikodym L^{2}\to L^{q_c} -estimates and avoiding the use of L^{2}\to L^{2} ones as in earlier results. This allows us to prove new results for manifolds of negative curvature and some new sharp estimates for tori. We also have new and improved techniques in two dimensions for general manifolds of non-positive curvature.

Approximation of Feedback Gains Using Spectral Projections: Application to the Oseen System

Approximation of Feedback Gains Using Spectral Projections: Application to the Oseen System

Reconstruction of blood flow velocity with deep learning information fusion from spectral ct projections and vessel geometry

In this work, we investigate a new deep learning reconstruction method of blood flow velocity within deformed vessels from contrast enhanced X-ray projections and vessel geometry. The principle of the method is to perform linear or nonlinear dimension reductions on the Radon projections and on the mesh of the vessel. These low dimensional projections are then fused to obtain the velocity field in the vessel. The accuracy of the reconstruction method is proved using various neural network architectures with realistic unsteady blood flows. The approach leverages the vessel geometry information and outperforms the simple PCA-net.

Superconvergent spectral projection and multi-projection methods for nonlinear Fredholm integral equations

In this article, we apply Galerkin and multi-Galerkin methods based on Kumar Sloan technique using Legendre polynomial basis functions for approximating the nonlinear Fredholm Hammerstein integral equations with the smooth kernels as well as the weakly singular algebraic and logarithmic kernels. We show that we are getting the improved superconvergence rates for Galerkin and multi-Galerkin methods based on Kumar Sloan technique without the need for the iterated Galerkin and the iterated multi-Galerkin methods. Infact without going to the iterated versions, we obtain the superconvergence rates equal to the convergence rates of iterated Galerkin and iterated multi-Galerkin methods. Theoretical results have been illustrated with numerical experiments.

Hybridized Brazilian–Bowein type spectral gradient projection method for constrained nonlinear equations

This paper proposes a hybridized Brazilian and Bowein derivative-free spectral gradient projection method for solving systems of convex-constrained nonlinear equations. The method avoids solving any subproblems in each iteration. Global convergence is established under appropriate assumptions on the functions involved. Additionally, numerical experiments are conducted to evaluate the algorithm’s performance, providing evidence of its efficiency compared to similar algorithms from the existing literature. The results demonstrate that the method outperforms some existing approaches in terms of the number of iterations, function evaluations, and time required to obtain a solution based on the examples considered.

Superconvergence results for hypersingular integral equation of first kind by Chebyshev spectral projection methods

In this article, we propose Chebyshev spectral projection methods to solve the hypersingular integral equation of first kind. The presence of strong singularity in Hadamard sense in the first part of the integral equation makes it challenging to get superconvergence results. To overcome this, we transform the first kind hypersingular integral equation into a second kind integral equation. This is achieved by defining a bounded inverse of the hypersingular integral operator in some suitable Hilbert space. Using iterated Chebyshev spectral Galerkin method on the equivalent second kind integral equation, we obtain improved convergence of O(N−2r), where N is the highest degree of Chebyshev polynomials employed in the approximation space and r is the smoothness of the solution. Further, using commutativity of projection operator and inverse of the hypersingular integral operator, we are able to obtain superconvergence of O(N−3r) and O(N−4r), by Chebyshev spectral multi-Galerkin method (CSMGM) and iterated CSMGM, respectively. Finally, numerical examples are presented to verify our theoretical results.

Integro-differential diffusion equations on graded Lie groups

We first study the existence, uniqueness and well-posedness of a general class of integro-differential diffusion equation on L p ( G ) ( 1 < p < + ∞, G is a graded Lie group). We show the explicit solution of the considered equation. The equation involves a nonlocal in time operator (with a general kernel) and a positive Rockland operator acting on G. Also, we provide L p ( G ) − L q ( G ) ( 1 < p ⩽ 2 ⩽ q < + ∞) norm estimates and time decay rate for the solutions. In fact, by using some contemporary results, one can translate the latter regularity problem to the study of boundedness of its propagator which strongly depends on the traces of the spectral projections of the Rockland operator. Moreover, in many cases, we can obtain time asymptotic decay for the solutions which depends intrinsically on the considered kernel. As a complement, we give some norm estimates for the solutions in terms of a homogeneous Sobolev space in L 2 ( G ) that involves the Rockland operator. We also give a counterpart of our results in the setting of compact Lie groups. Illustrative examples are also given.

Uncertainty quantification of acoustic metamaterial bandgaps with stochastic material properties and geometric defects

Acoustic metamaterials are a subject of increasing study and utility. Through designed combinations of geometries with material properties, acoustic metamaterials can be built to arbitrarily manipulate acoustic waves for various applications. Despite the theoretical advances in this field, however, acoustic metamaterials have seen limited penetration into industry and commercial use. This is largely due to the difficulty of manufacturing the intricate geometries that are integral to their function and the sensitivity of metamaterial designs to material batch variability and manufacturing defects. Capturing the effects of stochastic material properties and geometric defects requires empirical testing of manufactured samples, but this can quickly become prohibitively expensive with higher precision requirements or with an increasing number of input variables. This paper demonstrates how uncertainty quantification techniques, and more specifically the use of polynomial chaos expansions and spectral projections, can be used to greatly reduce sampling needs for characterizing acoustic metamaterial dispersion curves. With a novel method of encoding geometric defects in a 1D, interpretable, resolution-independent way, our uncertainty quantification approach allows for both stochastic material properties and geometric defects to be considered simultaneously. Two to three orders of magnitude sampling reductions down to ∼100 and ∼101 were achieved in 1D and 7D input space scenarios, respectively. Remarkably, this reduction in sampling was possible while preserving accurate output probability distributions of the metamaterial performance characteristics (bandgap size and location).

Label-free 3D molecular imaging of living tissues using Raman spectral projection tomography

The ability to image tissues in three dimensions (3D) with label-free molecular contrast at the mesoscale would be a valuable capability in biology and biomedicine. Here, we introduce Raman spectral projection tomography (RSPT) for volumetric molecular imaging with optical sub-millimeter spatial resolution. We have developed a RSPT imaging instrument capable of providing 3D molecular contrast in transparent and semi-transparent samples. We also created a computational pipeline for multivariate reconstruction to extract label-free spatial molecular information from Raman projection data. Using these tools, we demonstrate imaging and visualization of phantoms of various complex shapes with label-free molecular contrast. Finally, we apply RSPT as a tool for imaging of molecular gradients and extracellular matrix heterogeneities in fixed and living tissue-engineered constructs and explanted native cartilage tissues. We show that there exists a favorable balance wherein employing Raman spectroscopy, with its advantages in live cell imaging and label-free molecular contrast, outweighs the reduction in imaging resolution and blurring caused by diffuse photon propagation. Thus, RSPT imaging opens new possibilities for label-free molecular monitoring of tissues.

Investigation of the dimension of the spectral projection of a self-adjoint second order quasidifferential operator

Let λ1and λ2be real, λ1< λ2, functionsψ_( λi,t) be solutions to the second order quasidifferential equationsLψ_= λiP0ψ_,i= 1, 2, satisfying a homogeneous boundary condition at pointа. We express the number of eigenvalues of operatorL, belonging to the interval (λ1, λ2) (or the dimension of its spectral projection relative to the interval (λ1, λ2), in terms of the number of zeros of the Vronskian composed for the functionsψ_(λ1,t) andψ_(λ2,t).

Threshold approximations for the exponential of a factorized operator family with correctors taken into account

In a Hilbert space H \mathfrak H , consider a family of selfadjoint operators (a quadratic operator pencil) A ( t ) A(t) , t ∈ R t\in \mathbb {R} , of the form A ( t ) = X ( t ) ∗ X ( t ) A(t) = X(t)^* X(t) , where X ( t ) = X 0 + t X 1 X(t) = X_0 + t X_1 . It is assumed that the point λ 0 = 0 \lambda _0=0 is an isolated eigenvalue of finite multiplicity for the operator A ( 0 ) A(0) . Let F ( t ) F(t) be the spectral projection of the operator A ( t ) A(t) for the interval [ 0 , δ ] [0,\delta ] . Approximations for F ( t ) F(t) and A ( t ) F ( t ) A(t)F(t) for | t | ≤ t 0 |t| \leq t_0 (the so-called threshold approximations) are used to obtain approximations in the operator norm on H \mathfrak H for the operator exponential exp ⁡ ( − i τ A ( t ) ) \exp (-i \tau A(t)) , τ ∈ R \tau \in \mathbb {R} . The numbers δ \delta and t 0 t_0 are controlled explicitly. Next, the behavior for small ε > 0 \varepsilon >0 of the operator exp ⁡ ( − i ε − 2 τ A ( t ) ) \exp (-i \varepsilon ^{-2} \tau A(t)) multiplied by the “smoothing factor” ε s ( t 2 + ε 2 ) − s / 2 \varepsilon ^s (t^2 + \varepsilon ^2)^{-s/2} with a suitable s > 0 s>0 is studied. The obtained approximations are given in terms of the spectral characteristics of the operator A ( t ) A(t) near the lower edge of the spectrum. The results are aimed at application to homogenization of the Schrödinger-type equations with periodic rapidly oscillating coefficients.

Quantitative limit theorems and bootstrap approximations for empirical spectral projectors

Given finite i.i.d. samples in a Hilbert space with zero mean and trace-class covariance operator Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document}, the problem of recovering the spectral projectors of Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document} naturally arises in many applications. In this paper, we consider the problem of finding distributional approximations of the spectral projectors of the empirical covariance operator Σ^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\hat{\\Sigma }}$$\\end{document}, and offer a dimension-free framework where the complexity is characterized by the so-called relative rank of Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document}. In this setting, novel quantitative limit theorems and bootstrap approximations are presented subject to mild conditions in terms of moments and spectral decay. In many cases, these even improve upon existing results in a Gaussian setting.

Investigation of the Dimension of the Spectral Projection of a Self-Adjoint Second-Order Quasidifferential Operator

Investigation of the Dimension of the Spectral Projection of a Self-Adjoint Second-Order Quasidifferential Operator

Spectral Projections and Paley–Wiener Theorem for the Unit Ball in $$\mathbb {C}^{n}$$

Spectral Projections and Paley–Wiener Theorem for the Unit Ball in $$\mathbb {C}^{n}$$

Persistence of spectral projections for stochastic operators on large tensor products

Abstract It is proved that for families of stochastic operators on a countable tensor product, depending smoothly on parameters, any spectral projection persists smoothly, where smoothness is defined using norms based on ideas of Dobrushin. A rigorous perturbation theory for families of stochastic operators with spectral gap is thereby created. It is illustrated by deriving an effective slow two-state dynamics for a three-state probabilistic cellular automaton.

Partial Acquisition of Spectral Projections Accelerates Four-dimensional Spectral-spatial EPR Imaging for Mouse Tumor Models: A Feasibility Study.

Our study aimed to accelerate the acquisition of four-dimensional (4D) spectral-spatial electron paramagnetic resonance (EPR) imaging for mouse tumor models. This advancement in EPR imaging should reduce the acquisition time of spectroscopic mapping while reducing quality degradation for mouse tumor models. EPR spectra under magnetic field gradients, called spectral projections, were partially measured. Additional spectral projections were later computationally synthesized from the measured spectral projections. Four-dimensional spectral-spatial images were reconstructed from the post-processed spectral projections using the algebraic reconstruction technique (ART) and assessed in terms of their image qualities. We applied this approach to a sample solution and a mouse Hs766T xenograft model of human-derived pancreatic ductal adenocarcinoma cells to demonstrate the feasibility of our concept. The nitroxyl radical imaging agent 2H,15N-DCP was exogenously infused into the mouse xenograft model. The computation code of 4D spectral-spatial imaging was tested with numerically generated spectral projections. In the linewidth mapping of the sample solution, we achieved a relative standard uncertainty (standard deviation/| mean |) of 0.76 μT/45.38 μT = 0.017 on the peak-to-peak first-derivative EPR linewidth. The qualities of the linewidth maps and the effect of computational synthesis of spectral projections were examined. Finally, we obtained the three-dimensional linewidth map of 2H,15N-DCP in a Hs766T tumor-bearing leg in vivo. We achieved a 46.7% reduction in the acquisition time of 4D spectral-spatial EPR imaging without significantly degrading the image quality. A combination of ART and partial acquisition in three-dimensional raster magnetic field gradient settings in orthogonal coordinates is a novel approach. Our approach to 4D spectral-spatial EPR imaging can be applied to any subject, especially for samples with less variation in one direction.

An improved spectral conjugate gradient projection method for monotone nonlinear equations with application

An improved spectral conjugate gradient projection method for monotone nonlinear equations with application

The Chevalley--Jordan decomposition and spectral projections of complex matrices

In this paper, a novel and simple method for obtaining the Chevalley-Jordan decomposition and the spectral projections of matrices is presented. Our method is direct and elementary, it gives tractable and manageable formulas with minimum mathematical prerequisites. Moreover, knowing only some associated matrices of the matrix, we can simply provide the minimal polynomial of this matrix.

Universality for free fermions and the local Weyl law for semiclassical Schrödinger operators

We study local asymptotics for the spectral projector associated to a Schrödinger operator -\hbar^{2}\Delta+V on \mathbb{R}^{n} in the semiclassical limit as \hbar\to0 . We prove local uniform convergence of the rescaled integral kernel of this projector towards a universal model, inside the classically allowed region as well as on its boundary. This implies universality of microscopic fluctuations for the corresponding free fermions (determinantal) point processes, both in the bulk and around regular boundary points. Our results apply to a general class of smooth potentials in arbitrary dimension n\ge 1 . These results are complemented by studying both macroscopic and mesoscopic fluctuations of the point process. We obtain tail bounds for macroscopic linear statistics and, provided n\geq 2 , a central limit theorem for both macroscopic and mesoscopic linear statistics in the bulk.