Discrete Schrodinger Operators Research Articles

The Discrete Green's Function for Convective Heat Transfer—Part 2: Semi-Analytical Estimates of Boundary Layer Discrete Green's Function

Abstract This is the second paper in a set that defines the discrete Green's function (DGF). This paper focuses first on the turbulent boundary layer and presents two different methods to estimate the DGF. The long-element formulation defines the DGF with just two simple algebraic equations, but it is not quantitatively accurate for short element lengths. A short element correction is derived, but must be recalculated for each selection of flow parameters and element lengths. A similarity solution is derived that allows accurate estimates of the DGF diagonal elements for laminar boundary layers and for turbulent boundary layers discretized with short element lengths. To illustrate other methods to derive DGFs in more complex flows, a low-resolution DGF for laminar stagnation line boundary layers is determined using the skin-friction formulation combined with similarity solutions for two different thermal boundary conditions. Stagnation line flow is shown to be highly sensitive to the thermal boundary condition, and this can be analyzed effectively using the DGF.

The Discrete Green's Function for Convective Heat Transfer—Part 1: Definition and Physical Understanding

Abstract The discrete Green's function (DGF) is a superposition-based descriptor of the relationship between the surface temperature and the convective heat transfer from a surface. The surface is discretized into a finite number of elements and the DGF matrix elements relate the heat transfer out of any element i to the temperature rise on every element j of the surface. For a given flow situation, the DGF is insensitive to the thermal boundary condition so it allows direct calculation of the heat transfer for any temperature distribution and noniterative solution of conjugate heat transfer problems. The diagonal elements of the matrix are determined solely by the local velocity field while the off-diagonals are determined by the spread of the thermal wake downstream of a heated element. An analytical DGF for the laminar flat-plate boundary layers is included as an example.

The existence and location of eigenvalues of the one-particle discrete Schrodinger operator on a lattice

The existence and location of eigenvalues of the one-particle discrete Schrodinger operator on a lattice

Threshold Analysis of the Three Dimensional Lattice Schrödinger Operator with Non-Local Potential

We consider a family of the discrete Schrodinger operators $$H_{\lambda\mu}$$ , depending on parameters, in the $$3$$ -dimensional lattice, $$\mathbb{Z}^{3}$$ with a non-local potential constructed via the Dirac delta function and the shift operator. The existence of eigenvalues outside the essential spectrum and their dependence on the parameters of the operator are explicitly derived. The threshold eigenvalue is proven to be absorbed into the essential spectrum and it turns into an embedded eigenvalue at the left intercept of a particular parabola, and the threshold resonance at the other points of the parabola.

Wave scattering on lattice structures involving an array of cracks.

Scattering of waves as a result of a vertical array of equally spaced cracks on a square lattice is studied. The convenience of Floquet periodicity reduces the study to that of scattering of a specific wave-mode from a single crack in a waveguide. The discrete Green's function, for the waveguide, is used to obtain the semi-analytical solution for the scattering problem in the case of finite cracks whereas the limiting case of semi-infinite cracks is tackled by an application of the Wiener-Hopf technique. Reflectance and transmittance of such an array of cracks, in terms of incident wave parameters, is analysed. Potential applications include construction of tunable atomic-scale interfaces to control energy transmission at different frequencies.

A sketched finite element method for elliptic models

We consider a sketched implementation of the finite element method for elliptic partial differential equations on high-dimensional models. Motivated by applications in real-time simulation and prediction we propose an algorithm that involves projecting the finite element solution onto a low-dimensional subspace and sketching the reduced equations using randomised sampling. We show that a sampling distribution based on the leverage scores of a tall matrix associated with the discrete Laplacian operator, can achieve nearly optimal performance and a significant speedup. We derive an expression of the complexity of the algorithm in terms of the number of samples that are necessary to meet an error tolerance specification with high probability, and an upper bound for the distance between the sketched and the high-dimensional solutions. Our analysis shows that the projection not only reduces the dimension of the problem but also regularises the reduced system against sketching error. Our numerical simulations suggest speed improvements of two orders of magnitude in exchange for a small loss in the accuracy of the prediction.

Enumerating linear systems on graphs

The divisor theory of graphs views a finite connected graph G as a discrete version of a Riemann surface. Divisors on G are formal integral combinations of the vertices of G, and linear equivalence of divisors is determined by the discrete Laplacian operator for G. As in the case of Riemann surfaces, we are interested in the complete linear system |D| of a divisor D—the collection of nonnegative divisors linearly equivalent to D. Unlike the case of Riemann surfaces, the complete linear system of a divisor on a graph is always finite. We compute generating functions encoding the sizes of all complete linear systems on G and interpret our results in terms of polyhedra associated with divisors and in terms of the invariant theory of the (dual of the) Jacobian group of G. If G is a cycle graph, our results lead to a bijection between complete linear systems and binary necklaces. Our results also apply to a model in which the Laplacian is replaced by an invertible, integral M-matrix.

Virtual network embedding on massive substrate networks

AbstractTo alleviate the computational burden of previous virtual network embedding (VNE) approaches when the resource network scales up significantly, we propose an efficient node ranking strategy that considers both global and local topological characteristics of the substrate network in mapping virtual nodes to physical nodes. This method ranks the substrate network nodes in two stages. First, all nodes are ranked globally with respect to the stationary distribution of the entire network. Then, a connected subset of the ranked substrate nodes, forming the H‐admissible embedding subgraph, is extracted. Finally, the subgraph nodes are ranked according to a local node ranking vector derived from a random‐walking scheme. The local rank vector is resolved using discrete Green's function satisfying the Dirichelet boundary condition. The more accurate association of node demands and resources that our proposed method provides leads to both better acceptance ratio and lower computational overhead. These claims have been justified via theoretical and algorithmic presentation of our scheme and offer experimental results obtained through simulation, to confirm its execution efficiency and solution quality compared with a couple of previous VNE proposals.

A fast multi-resolution lattice Green's function method for elliptic difference equations

We propose a mesh refinement technique for solving elliptic difference equations on unbounded domains based on the fast lattice Green's function (FLGF) method. The FLGF method exploits the regularity of the Cartesian mesh and uses the fast multipole method in conjunction with fast Fourier transforms to yield linear complexity and decrease time-to-solution. We extend this method to a multi-resolution scheme and allow for locally refined Cartesian blocks embedded in the computational domain. Appropriately chosen interpolation and regularization operators retain consistency between the discrete Laplace operator and its inverse on the unbounded domain. Second-order accuracy and linear complexity are maintained, while significantly reducing the number of degrees of freedom and hence the computational cost.

Local ultraconvergence of linear and bilinear finite element method for second order elliptic problems

In this article, local ultraconvergence in gradient of the bilinear and linear finite element solutions to two-dimensional second order elliptic problems has been investigated. Special interpolation postprocessing algorithms of numerical solutions by Richardson extrapolation have been developed. The local symmetry theory and estimates of discrete Green functions are used in the analysis. Numerical experiments are provided to confirm the theoretical findings.

HLO: Half-kernel Laplacian operator for surface smoothing

This paper presents a simple yet effective and efficient method for feature-preserving surface smoothing. Through analyzing the differential property of surfaces, we show that the conventional discrete Laplacian operator with uniform weights is not applicable to feature points at which the surface is non-differentiable and the second order derivatives do not exist. To overcome this difficulty, we propose a Half-kernel Laplacian Operator (HLO) as an alternative to the conventional Laplacian. Given a vertex v, HLO first finds all pairs of its neighboring vertices and divides each pair into two subsets (called half windows); then computes the uniform Laplacians of all such subsets and subsequently projects the computed Laplacians to the full-window uniform Laplacian to alleviate flipping and degeneration. The half window with least regularization energy is then chosen for v. We develop an iterative approach to apply HLO for surface denoising. Our method is conceptually simple and easy to use because it has a single parameter, i.e., the number of iterations for updating vertices. We show that our method can preserve features better than the popular uniform Laplacian-based denoising and it significantly alleviates the shrinkage artifact. Extensive experimental results demonstrate that HLO is better than or comparable to state-of-the-art techniques both qualitatively and quantitatively and that it is particularly good at handling meshes with high noise. Also, it outperforms all other compared methods in terms of computational time. We make our executable program publicly available.

Superconvergence of the function value for pentahedral finite elements for an elliptic equation with varying coefficients

In this article, for an elliptic equation with varying coefficients, we first derive an interpolation fundamental estimate for the mathcal{P}_{2}(x,y)otimes mathcal{P}_{2}(z) pentahedral finite element over uniform partitions of the domain. Then combined with the estimate for the W^{2,1}-seminorm of the discrete Green function, superconvergence of the function value between the finite element approximation and the corresponding interpolant to the true solution is given.

Discrete Laplacian Operator and Its Applications in Signal Processing

Fractional calculus has increased in popularity in recent years, as the number of its applications in different fields has increased. Compared to the traditional operations in calculus (integration and differentiation) which are uniquely defined, the fractional-order operators have numerous definitions. Furthermore, a consensus on the most suitable definition for a given task is yet to be reached. Fractional operators are defined as continuous operators and their implementation requires a discretization step. In this article, we propose a discrete fractional Laplacian as a matrix operator. The proposed operator is real (non-complex) which makes it computationally efficient. The construction of the proposed fractional Laplacian utilizes the DCT transform avoiding the complexity associated with the discretization step which is typical in the constructions based on signal processing. We demonstrate the utility of the proposed operator on a number of data modeling and image processing tasks.

The fractional p-Laplacian emerging from homogenization of the random conductance model with degenerate ergodic weights and unbounded-range jumps

We study a general class of discrete p-Laplace operators in the random conductance model with long-range jumps and ergodic weights. Using a variational formulation of the problem, we show that under the assumption of bounded first moments and a suitable lower moment condition on the weights, the homogenized limit operator is a fractional p-Laplace operator. Under strengthened lower moment conditions, we can apply our insights also to the spectral homogenization of the discrete Laplace operator to the continuous fractional Laplace operator.

Spectral Enclosures for Non-self-adjoint Discrete Schrödinger Operators

We study location of eigenvalues of one-dimensional discrete Schrodinger operators with complex $$\ell ^{p}$$-potentials for $$1\le p\le \infty $$. In the case of $$\ell ^{1}$$-potentials, the derived bound is shown to be optimal. For $$p>1$$, two different spectral bounds are obtained. The method relies on the Birman–Schwinger principle and various techniques for estimations of the norm of the Birman–Schwinger operator.

Automatic spike correction using UNIFIT 2020

The improvement of the software UNIFIT 2020 from an analysis processing software for photoelectron spectroscopy (XPS) only to a powerful tool for XPS, Auger electron spectroscopy (AES), X‐ray absorption spectroscopy (XAS), and Raman spectroscopy requires new additional programme routines. Particularly, the implementation of the analysis of Raman spectra needs a well‐working automatic spike correction. The application of the modified discrete Laplace operator method allows for a perfect localization and correction of the spikes and finally a successful peak fit of the spectra. The theoretical basis is described. Test spectra allow for the evaluation of the presented method. A comparison of the original and spike‐corrected real measurements demonstrates the high quality of the method used.

Maximum error estimates of a MAC scheme for Stokes equations with Dirichlet boundary conditions

This paper presents a rigorous convergence analysis in ℓ2-norm and maximum norm for the Marker-And-Cell (MAC) scheme for solving the incompressible Stokes equations with Dirichlet boundary conditions in two dimensions. The numerical scheme is based on a finite-difference approximation on a staggered grid, where the computational boundary conditions are introduced by reflecting boundary conditions. This scheme is well known to numerically produce second order accuracy for velocity, pressure as well as the gradient of velocity. However, the reflecting boundary conditions lead to the fact that the truncation errors near the boundary is the order of O(1), which presents great difficulties in error analysis to obtain optimal order. In this work, by means of a discrete LBB condition and some auxiliary functions that satisfy discrete Stokes equations to a high-order accuracy, we first prove that the convergence is second order in the ℓ2-norm for velocity, pressure and the gradient of the velocity. After that, according to the property of discrete Green functions, we further provide rigorous analysis to show second-order accuracy in the maximum norm for both the velocity and its gradient. The extension of our results to three dimensional problems would be routine. Numerical results are also provided to demonstrate our theoretical analysis.

Construction of QuasiPeriodic Schrödinger Operators with Cantor Spectrum

It is well known that, for fixed Diophantine frequencies and generic small smooth or analytic quasiperiodic potentials, both continuous and discrete Schrodinger operators have Cantor spectrum. Although there have been several examples of Schrodinger operators with Cantor spectrum since Moser’s pioneering work (Bellissard, in: Luck, Moussa, Waldschmidt (eds) Number theory and physics (Les Houches, 1989), Springer, Berlin, 1990; Bellissard et al. in Phys Rev Lett 49:701–704, 1982; Damanik et al. in Ann Henri Poincare 15:1123–1144, 2014, J Spectr Theory 7:1101–1118, 2017; Moser in Comment Math Helv 56:198–224, 1981); however, so far there is no concrete quasiperiodic example in the continuous case, and there is no concrete quasiperiodic example in the discrete case besides the cosine-like potentials (Avila and Jitomirskaya in Ann Math 170:303–342, 2009; Puig in Commun Math Phys 244:297–309, 2004; Sinai in J Stat Phys 46:861–909; 1987; Wang and Zhang in Int Math Res Not 2017:2300–2336, 2017). In this paper, we present a strategy for explicitly constructing quasiperiodic Schrodinger operators with Cantor spectrum.

Semi-implicit methods for the dynamics of elastic sheets

Recent applications (e.g. active gels and self-assembly of elastic sheets) motivate the need to efficiently simulate the dynamics of thin elastic sheets. We present semi-implicit time stepping algorithms to improve the time step constraints that arise in explicit methods while avoiding much of the complexity of fully-implicit approaches. For a triangular lattice discretization with stretching and bending springs, our semi-implicit approach involves discrete Laplacian and biharmonic operators, and is stable for all time steps in the case of overdamped dynamics. For a more general finite-difference formulation that can allow for general elastic constants, we use the analogous approach on a square grid, and find that the largest stable time step is two to three orders of magnitude greater than for an explicit scheme. For a model problem with a radial traveling wave form of the reference metric, we find transitions from quasi-periodic to chaotic dynamics as the sheet thickness is reduced, wave amplitude is increased, and damping constant is reduced.

Ollivier Ricci curvature for general graph Laplacians: Heat equation, Laplacian comparison, non-explosion and diameter bounds

Discrete time random walks on a finite set naturally translate via a one-to-one correspondence to discrete Laplace operators. Typically, Ollivier curvature has been investigated via random walks. We first extend the definition of Ollivier curvature to general weighted graphs and then give a strikingly simple representation of Ollivier curvature using the graph Laplacian. Using the Laplacian as a generator of a continuous time Markov chain, we connect Ollivier curvature with the heat equation which is strongly related to continuous time random walks. In particular, we prove that a lower bound on the Ollivier curvature is equivalent to a certain Lipschitz decay of solutions to the heat equation. This is a discrete analogue to a celebrated Ricci curvature lower bound characterization by Renesse and Sturm. Our representation of Ollivier curvature via the Laplacian allows us to deduce a Laplacian comparison principle by which we prove non-explosion and improved diameter bounds.