Discrete Laplacian Research Articles

Polyharmonic fields and Liouville quantum gravity measures on tori of arbitrary dimension: From discrete to continuous

AbstractFor an arbitrary dimension , we study: the polyharmonic Gaussian field on the discrete torus , that is the random field whose law on given by where is the Lebesgue measure and is the discrete Laplacian; the associated discrete Liouville quantum gravity (LQG) measure associated with it, that is, the random measure on where is a regularity parameter. As , we prove convergence of the fields to the polyharmonic Gaussian field on the continuous torus , as well as convergence of the random measures to the LQG measure on , for all .

Discretization of Fractional Fully Nonlinear Equations by Powers of Discrete Laplacians

AbstractWe study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $$\sigma \in (0,2)$$ σ ∈ ( 0 , 2 ) since they involve fractional Laplace operators $$(-\Delta )^{\sigma /2}$$ ( - Δ ) σ / 2 . They arise e.g. in control and game theory as dynamic programming equations – HJB and Isaacs equation – and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of $$\sigma $$ σ . The accuracy of previous approximations of fractional fully nonlinear equations depend on $$\sigma $$ σ and are worse when $$\sigma $$ σ is close to 2. We show that the schemes are monotone, consistent, $$L^\infty $$ L ∞ -stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We also prove a second order error bound for smooth solutions and present many numerical examples.

Floquet isospectrality of the zero potential for discrete periodic Schrödinger operators

Let Γ=q1Z⊕q2Z⊕⋯⊕qdZ, with qj∈Z+ for each j ∈ {1, …, d}, and denote by Δ the discrete Laplacian on ℓ2Zd. Using Macaulay2, we first numerically find complex-valued Γ-periodic potentials V:Zd→C such that the operators Δ + V and Δ are Floquet isospectral. We then use combinatorial methods to validate these numerical solutions.

A connection between discrete and regularized Laplacian determinants on fractals

The spectral zeta function of the Laplacian on self-similar fractal sets has been previously studied and shown to meromorphically extend to the complex plane. In this work we establish that under certain conditions a relationship exists between the logarithm of the determinant of the discrete graph Laplacian on the sequence of graphs approximating the fractal and the regularized Laplacian determinant on the fractal itself which is defined via help of the spectral zeta function. We then at the end present some concrete examples of this phenomenon.

Littlewood–Paley–Stein square functions for the fractional discrete Laplacian on $$\mathbb {Z}$$

Littlewood–Paley–Stein square functions for the fractional discrete Laplacian on $$\mathbb {Z}$$

Precise error bounds for numerical approximations of fractional HJB equations

Abstract We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton–Jacobi–Bellman equations. We consider diffusion-corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations that are (formally) fractional order and second-order methods. It is well known in numerical analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: (i) strongly degenerate problems with Lipschitz solutions and (ii) weakly nondegenerate problems where we show that solutions have bounded fractional derivatives of order $\sigma \in (1,2)$. Our main results are optimal error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions. For strongly degenerate equations, these rates improve earlier results. For weakly nondegenerate problems of order greater than one, the results are new. Here we show improved rates compared to the strongly degenerate case, rates that are always better than $\mathcal{O}\big (h^{\frac{1}{2}}\big )$.

A MAC grid based FFT-AMIB solver for incompressible Stokes flows with interfaces and singular forces

In this paper, a second order accurate Augmented Matched Interface and Boundary (AMIB) method is introduced for solving the incompressible Stokes flows with interfaces and singular forces. Based on the MAC staggered grid for velocity and pressure, the zeroth order and first order jump conditions are repeatedly enforced in the AMIB scheme to generate fictitious values around the interface, which allow an accurate approximation of the Cartesian derivative jumps. By treating all jump quantities as auxiliary variables and coupling them with the corrected finite difference formulas, the Stokes equations can be discretized into an augmented system. The proposed FFT-AMIB solver does not need to take the pressure boundary condition into account. Moreover, in the Schur complement solution of the augmented system, the discrete Laplacian can be efficiently inverted by using the fast Fourier transform (FFT). Numerical results show that the FFT-AMIB solver is very efficient and can achieve second order accuracy for both velocity and pressure. The iteration number of Bi-CGSTAB in solving the resulting augmented system only weakly depends on the mesh size. When the interface is allowed to be moved under the surface tension, the proposed algorithm can maintain the divergence-free condition and area conservation very well.

Stiff-cut leap-frog scheme for fractional Laplacian diffusion equations

In this study, we investigate the numerical discretization of two-dimensional fractional Laplacian diffusion equations. After discretizing the space derivative, we split the discrete fractional Laplacian into two parts. One is the τ preconditioner multiplying a constant, solved using the leap-frog scheme, while the other part and the remaining nonlinear term are treated with an explicit scheme. This results in what is termed the stiff-cut leap-frog scheme, which is more efficient than the traditional leap-frog scheme. The main advantage of the proposed scheme is that the coefficient matrices of the generated linear systems belong to τ algebra, which can be diagonalized by fast discrete sine transform with the complexity of O(NlogN), where N is the freedom of unknowns. Hence, no iterative method is required at each time integration. Moreover, we theoretically prove that the proposed scheme is unconditionally stable and converges in second order. Ultimately, numerical experiments are conducted to demonstrate the precision and effectiveness of the stiff-cut leap-frog scheme.

Extracting Spatial Spectra Using Coarse‐Graining Based On Implicit Filters

AbstractScale analysis based on coarse‐graining has been proposed recently as an alternative to Fourier analysis. It requires interpolation to a regular mesh for data from unstructured‐mesh models. We propose an alternative coarse‐graining method which relies on implicit filters using powers of discrete Laplacians. This method can work on arbitrary (structured or unstructured) meshes and is applicable to the direct output of unstructured‐mesh models. Illustrations and detailed discussions are provided for discrete fields placed at vertices of triangular meshes. The case with placement on triangles is also briefly discussed.

Discrete Laplacian on Graphs, Dirichlet Problem, and Minimization of Energy


 
 
 This report is primarily interested in discrete Laplacian on graphs and Dirichlet prob- lem. We investigate two cases of graphs, their Dirichlet problems, and the method of finding solutions to such problems by minimization of their corresponding energy functionals. In the case of an infinite, locally-finite, undirected, simple graph, we find that the minimiza- tion of its energy functional yields a solution to its Dirichlet or boundary-value problem. In the case of finite, directed, locally-finite decision tree, we propose a type of such tree specified by a chosen vertex with no predecessors and by a chosen boundary set of vertices that have no successors. We then further consider a special case of such construction where all the paths of the tree terminate at the same level. We find that the minimization of such special case’s energy functional yields a solution to its Dirichlet problem only when some additional conditions on and close to the boundary (level of termination) are met.
 
 

Heat Kernel of Networks with Long-Range Interactions

The heat kernel associated with a discrete graph Laplacian is the basic solution to the heat diffusion equation of a strict graph or network. In addition, this kernel represents the heat transfer that occurs over time across the network edges. Its computation involves exponentiating the Laplacian eigensystem with respect to time. In this paper, we expand upon this concept by considering a novel network-theoretic approach developed in recent years, which involves defining the k-path Laplacian operator for networks. Prior studies have adopted the notion of integrating long-range interactions (LRI) in the transmission of “information” across the nodes and edges of the network. Various methods have been employed to consider long-range interactions. We explore here the incorporation of long-range interactions in network analysis through the use of Mellin and Laplace transforms applied to the k-path Laplacian matrix. The contribution of this paper is the computation of the heat kernel associated with thek-path Laplacian, called the generalized heat kernel (GHK), and its employment as the basis for extracting stable and useful novel versions of invariants for graph characterization. The results presented in this paper demonstrate that the use of LRI improves the results obtained with classical diffusion methods for networks characterization.

Lighter and faster simulations on domains with symmetries

A strategy to improve the performance and reduce the memory footprint of simulations on meshes with spatial reflection symmetries is presented in this work. By using an appropriate mirrored ordering of the unknowns, discrete partial differential operators are represented by matrices with a regular block structure that allows replacing the standard sparse matrix–vector product with a specialised version of the sparse matrix-matrix product, which has a significantly higher arithmetic intensity. Consequently, matrix multiplications are accelerated, whereas their memory footprint is reduced, making massive simulations more affordable. As an example of practical application, we consider the numerical simulation of turbulent incompressible flows using a low-dissipation discretisation on unstructured collocated grids. All the required matrices are classified into three sparsity patterns that correspond to the discrete Laplacian, gradient, and divergence operators. Therefore, the above-mentioned benefits of exploiting spatial reflection symmetries are tested for these three matrices on both CPU and GPU, showing up to 5.0x speed-ups and 8.0x memory savings. Finally, a roofline performance analysis of the symmetry-aware sparse matrix–vector product is presented.

Isospectrum of non-self-adjoint almost-periodic Schrödinger operators

For non-self-adjoint almost-periodic Schrödinger operators, a criterion is given to guarantee that they have both the same spectrum and the same Lyapunov exponents with the discrete free Laplacian. As a byproduct, we show that the Moser–Pöschel argument for opening gaps may not be valid for non-self-adjoint operators.

On (global) unique continuation properties of the fractional discrete Laplacian

We study various qualitative and quantitative (global) unique continuation properties for the fractional discrete Laplacian. We show that while the fractional Laplacian enjoys striking rigidity properties in the form of (global) unique continuation properties, the fractional discrete Laplacian does not enjoy these in general. While discretization thus counteracts the strong rigidity properties of the continuum fractional Laplacian, by discussing quantitative forms of unique continuation, we illustrate that these properties can be recovered if exponentially small (in the lattice size) correction terms are added. In particular, this allows us to deduce stability properties for a discrete, linear inverse problem for the fractional Laplacian. We complement these observations with a transference principle and the discussion of these properties on the discrete torus.

Li-Yau and Harnack inequalities via curvature-dimension conditions for discrete long-range jump operators including the fractional discrete Laplacian

Li-Yau and Harnack inequalities via curvature-dimension conditions for discrete long-range jump operators including the fractional discrete Laplacian

Fractional electrical impurity

We examine the localized mode and the transmission of plane waves across a capacitive impurity of strength Δ, in a 1D bi-inductive electrical transmission line where the usual discrete Laplacian is replaced by a fractional one characterized by a fractional exponent s. In the absence of the impurity, the plane wave dispersion is computed in closed form in terms of hypergeometric functions. It is observed that the bandwidth decreases steadily, as s decreases towards zero, reaching a minimum width at s = 0. The localized mode energy and spatial profiles are computed in closed form vía lattice Green functions. The profiles show a remnant of the staggered-unstaggered symmetry that is common in non-fractional chains. The width of the localized mode decreases with decreasing s, becoming completely localized at the impurity site at s = 0. The transmission coefficient of plane waves across the impurity is qualitatively similar to its non-fractional counterpart (s = 1), except at low s values ( s≪1 ). For a fixed exponent s, the transmission decreases with increasing Δ.

On the Fourier asymptotics of absolutely continuous measures with power-law singularities

We prove sharp estimates on the time-average behavior of the squared absolute value of the Fourier transform of some absolutely continuous measures that may have power-law singularities, in the sense that their Radon–Nikodym derivatives diverge with a power-law order. We also discuss an application to spectral measures of finite-rank perturbations of the discrete Laplacian.

Robustness of Flat Bands on the Perturbed Kagome and the Perturbed Super-Kagome Lattice

We study spectral properties of perturbed discrete Laplacians on two-dimensional Archimedean tilings. The perturbation manifests itself in the introduction of non-trivial edge weights. We focus on the two lattices on which the unperturbed Laplacian exhibits flat bands, namely the (3.6)2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(3.6)^2$$\\end{document} Kagome lattice and the (3.122)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(3.12^2)$$\\end{document} “Super-Kagome” lattice. We characterize all possible choices for edge weights which lead to flat bands. Furthermore, we discuss spectral consequences such as the emergence of new band gaps. Among our main findings is that flat bands are robust under physically reasonable assumptions on the perturbation, and we completely describe the perturbation-spectrum phase diagram. The two flat bands in the Super-Kagome lattice are shown to even exhibit an “all-or-nothing” phenomenon in the sense that there is no perturbation, which can destroy only one flat band while preserving the other.

Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some Applications

The fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulting numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by an integral operator with zero boundary condition. Second, because the solutions of fractional diffusion equations are usually singular near the boundary, we use a fractional interpolation function in the region near the boundary and use a classical interpolation function in other intervals. Then, we apply a finite difference scheme to the discrete fractional Laplacian operator and fractional diffusion equation with the above fractional interpolation function and classical interpolation function. Moreover, it is found that the differential matrix of the above scheme is a symmetric matrix and strictly row-wise diagonally dominant in special fractional interpolation functions. Third, we show a finite volume scheme for a discrete fractional diffusion equation by fractional interpolation function and classical interpolation function and analyze the properties of the differential matrix. Finally, the numerical experiments are given, and we verify the correctness of the theoretical results and the efficiency of the schemes.

Geometric triangulations and discrete Laplacians on manifolds: An update

This paper uses the technology of weighted triangulations to study discrete versions of the Laplacian on piecewise Euclidean manifolds. Given a collection of Euclidean simplices glued together along their boundary, a geometric structure on the Poincaré dual may be constructed by considering weights at the vertices. We show that this is equivalent to specifying sphere radii at vertices and generalized intersection angles at edges, or by specifying a certain way of dividing the edges. This geometric structure gives rise to a discrete Laplacian operator acting on functions on the vertices. We study these geometric structures in some detail, considering when dual volumes are nondegenerate, which corresponds to weighted Delaunay triangulations in dimension 2, and how one might find such nondegenerate weighted triangulations. Finally, we talk briefly about the possibilities of discrete Riemannian manifolds.