Discrete Schrodinger Operators Research Articles

MultiGrid Preconditioners for Mixed Finite Element Methods of the Vector Laplacian

Due to the indefiniteness and poor spectral properties, the discretized linear algebraic system of the vector Laplacian by mixed finite element methods is hard to solve. A block diagonal preconditioner has been developed and shown to be an effective preconditioner by Arnold et al. (Acta Numer 15:1–155, 2006). The purpose of this paper is to propose alternative and effective block diagonal and approximate block factorization preconditioners for solving these saddle point systems. A variable V-cycle multigrid method with the standard point-wise Gauss–Seidel smoother is proved to be a good preconditioner for the discrete vector Laplacian operator. The major benefit of our approach is that the point-wise Gauss–Seidel smoother is more algebraic and can be easily implemented as a black-box smoother. This multigrid solver will be further used to build preconditioners for the saddle point systems of the vector Laplacian. Furthermore it is shown that Maxwell’s equations with the divergent free constraint can be decoupled into one vector Laplacian and one scalar Laplacian equation.

On the number of eigenvalues of the discrete one-dimensional Dirac operator with a complex potential

In this paper we define a one-dimensional discrete Dirac operator on $${\mathbb {Z}}$$ . We study the eigenvalues of the Dirac operator with a complex potential. We obtain bounds on the total number of eigenvalues in the case where V decays exponentially at infinity. We also estimate the number of eigenvalues for the discrete Schrodinger operator with complex potential on $${\mathbb {Z}}$$ . That is we extend the result obtained by Hulko (Bull Math Sci, to appear) to the whole $${\mathbb {Z}}$$ .

Optimal Hardy inequalities for Schrödinger operators on graphs

For a given subcritical discrete Schrodinger operator H on a weighted infinite graph X, we construct a Hardy-weight w which is optimal in the following sense. The operator H − λw is subcritical in X for all λ 1. Our results rely on a criticality theory for Schrodinger operators on general weighted graphs.

Scaling Limit for a Long-Range Divisible Sandpile

We study the scaling limit of a divisible sandpile model associated to a truncated $\alpha$-stable random walk. We prove that the limiting distribution is related to an obstacle problem for a truncated fractional Laplacian. We also provide, as a fundamental tool, precise asymptotic expansions for the corresponding rescaled discrete Green's functions. In particular, the convergence rate of these Green's functions to its continuous counterpart is derived.

Простая аппроксимация дискретной функции Грина в частотной области при численном решении параболического уравнения

Простая аппроксимация дискретной функции Грина в частотной области при численном решении параболического уравнения

A Boundary Algebraic Formulation for Plane Strain Elastodynamic Scattering

Solving of elastodynamic problems arises in many scientific fields such as wave propagation in the ground, nondestructive testing, vibration design of buildings, or vibroacoustics in general. An integral formulation based on boundary algebraic equations is presented here. This formulation leads to a numerical method with a discretized boundary. An important advantage of the method over the standard boundary element method is that no contour (in two-dimensional problems) or surface (in three-dimensional problems) integral needs to be computed. This feature is helpful in order to obtain a discrete version of the combined field integral equations (designed to damp numerically the fictitious eigenfrequencies) without difficulties caused by the evaluation of hypersingular integrals. The key aspects are (i) the approach deals with discrete equations from the very beginning; (ii) discrete (instead of continuous) tensor Green's functions are considered (the methodology to evaluate them is demonstrated); (iii) the boundary must be described by means of a regular square grid. In order to overcome the drawback of this third condition the boundary integral is coupled, if needed, with a thin layer of finite elements. This improves the description of curved geometries and reduces numerical errors. The properties of the method are demonstrated by means of numerical examples: the scattering of waves by objects and holes in an unbounded elastic medium, and an interior elastic problem.

A class of renormalised meshless Laplacians for boundary value problems

A meshless approach to approximating spatial derivatives on scattered point arrangements is presented in this paper. Three various derivations of approximate discrete Laplace operator formulations are produced using the Taylor series expansion and renormalised least-squares correction of the first spatial derivatives. Numerical analyses are performed for the introduced Laplacian formulations, and their convergence rate and computational efficiency are examined. The tests are conducted on regular and highly irregular scattered point arrangements. The results are compared to those obtained by the smoothed particle hydrodynamics method and the finite differences method on a regular grid. Finally, the strong form of various Poisson and diffusion equations with Dirichlet or Robin boundary conditions are solved in two and three dimensions by making use of the introduced operators in order to examine their stability and accuracy for boundary value problems. The introduced Laplacian operators perform well for highly irregular point distribution and offer adequate accuracy for mesh and mesh-free numerical methods that require frequent movement of the grid or point cloud.

An Introduction to Laplacian Spectral Distances and Kernels: Theory, Computation, and Applications

Abstract In geometry processing and shape analysis, several applications have been addressed through the properties of the Laplacian spectral kernels and distances, such as commute-time, biharmonic, diffusion, and wave distances. Within this context, this book is intended to provide a common background on the definition and computation of the Laplacian spectral kernels and distances for geometry processing and shape analysis. To this end, we define a unified representation of the isotropic and anisotropic discrete Laplacian operator on surfaces and volumes; then, we introduce the associated differential equations, i.e., the harmonic equation, the Laplacian eigenproblem, and the heat equation. Filtering the Laplacian spectrum, we introduce the Laplacian spectral distances, which generalize the commute-time, biharmonic, diffusion, and wave distances, and their discretization in terms of the Laplacian spectrum. As main applications, we discuss the design of smooth functions and the Laplacian smoothing of noi...

Asymptotics of the determinant of the discrete Laplacian on the circle

We compute the coefficients of the terms with non negative powers of the asymptotic expansion of the determinant of the discrete Laplace operator of the de Rham complex of the circle associated to any orthogonal representation of the fundamental group.

Chung-Yau Invariants and Graphs with Symmetric Hitting Times

The Chung–Yau graph invariants were originated from Chung–Yau's work on discrete Green's function. We show how they could be used to derive new explicit formulas and estimates for hitting times of random walks. We also apply them to study graphs with symmetric hitting times.

Quantum ergodicity on graphs: From spectral to spatial delocalization

We prove a quantum-ergodicity theorem on large graphs, for eigenfunctions of Schrodinger operators in a very general setting. We consider a sequence of finite graphs endowed with discrete Schrodinger operators, assumed to have a local weak limit. We assume that our graphs have few short loops, in other words that the limit model is a random rooted tree endowed with a random discrete Schrodinger operator. We show that absolutely continuous spectrum for the infinite model, reinforced by a good control of the moments of the Green function, imply quantum ergodicity , a form of spatial delocalization for eigenfunctions of the finite graphs approximating the tree. This roughly says that the eigenfunctions become equidistributed in phase space. Our result applies in particular to graphs converging to the Anderson model on a regular tree, in the regime of extended states studied by Klein and Aizenman–Warzel.

On the spectral Hausdorff dimension of 1D discrete Schrödinger operators under power decaying perturbations

We show that spectral Hausdorff dimensional properties of discrete Schrodinger operators with (1) Sturmian potentials of bounded density and (2) a class of sparse potentials are preserved under suitable polynomial decaying perturbations, when the spectrum of these perturbed operators have some singular continuous component.

Exact discretization of fractional Laplacian

An exact discretization of fractional-order Laplacian for N-dimensional space is suggested. Particular solutions of fractional-order partial difference equations with the proposed discrete Laplace operators are suggested.

Spherical Model on a Cayley Tree: Large Deviations

We study the spherical model of a ferromagnet on a Cayley tree and show that in the case of empty boundary conditions a ferromagnetic phase transition takes place at the critical temperature \(T_\mathrm{c} =\frac{6\sqrt{2}}{5}J\), where J is the interaction strength. For any temperature the equilibrium magnetization, \(m_n\), tends to zero in the thermodynamic limit, and the true order parameter is the renormalized magnetization \(r_n=n^{3/2}m_n\), where n is the number of generations in the Cayley tree. Below \(T_\mathrm{c}\), the equilibrium values of the order parameter are given by \(\pm \rho ^*\), where $$\begin{aligned} \rho ^*=\frac{2\pi }{(\sqrt{2}-1)^2}\sqrt{1-\frac{T}{T_\mathrm{c}}}. \end{aligned}$$ One more notable temperature in the model is the penetration temperature $$\begin{aligned} T_\mathrm{p}=\frac{J}{W_\mathrm{Cayley}(3/2)}\left( 1-\frac{1}{\sqrt{2}}\left( \frac{h}{2J}\right) ^2\right) . \end{aligned}$$ Below \(T_\mathrm{p}\) the influence of homogeneous boundary field of magnitude h penetrates throughout the tree. The main new technical result of the paper is a complete set of orthonormal eigenvectors for the discrete Laplace operator on a Cayley tree.

Convergence of Discrete Green Functions with Neumann Boundary Conditions

In this article we prove convergence of Green functions with Neumann boundary conditions for the random walk to their continuous counterparts. Our methods rely on local central limit theorems for convergence of random walks on discretizations of smooth domains to Reflected Brownian motion.

Ultraconvergence of high order FEMs for elliptic problems with variable coefficients

In this paper, we investigate local ultraconvergence properties of the high-order finite element method (FEM) for second order elliptic problems with variable coefficients. Under suitable regularity and mesh conditions, we show that at an interior vertex, which is away from the boundary with a fixed distance, the gradient of the post-precessed kth $$(k\ge 2)$$(kź2) order finite element solution converges to the gradient of the exact solution with order $$\mathcal{O}(h^{k+2} (\mathrm{ln} h)^3)$$O(hk+2(lnh)3). The proof of this ultraconvergence property depends on a new interpolating operator, some new estimates for the discrete Green's function, a symmetry theory derived in [26], and the Richardson extrapolation technique in [20]. Numerical experiments are performed to demonstrate our theoretical findings.

A 3D+t Laplace operator for temporal mesh sequences

The Laplace operator plays a fundamental role in geometry processing. Several discrete versions have been proposed for 3D meshes and point clouds, among others. We define here a discrete Laplace operator for temporally coherent mesh sequences, which allows to process mesh animations in a simple yet efficient way. This operator is a discretisation of the Laplace–Beltrami operator using Discrete Exterior Calculus on CW complexes embedded in a four-dimensional space. A parameter is introduced to tune the influence of the motion with respect to the geometry. This enables straightforward generalisation of existing Laplacian static mesh processing works to mesh sequences. An application to spacetime editing is provided as example.

A simple and efficient finite difference method for the phase-field crystal equation on curved surfaces

We present a simple and efficient finite difference method for the phase-field crystal (PFC) equation on curved surfaces embedded in R3. We employ a narrow band neighborhood of a curved surface that is defined as a zero level set of a signed distance function. The PFC equation on the surface is extended to the three-dimensional narrow band domain. By using the closest point method and applying a pseudo-Neumann boundary condition, we can use the standard seven-point discrete Laplacian operator instead of the discrete Laplace–Beltrami operator on the surface. The PFC equation on the narrow band domain is discretized using an unconditionally stable scheme and the resulting implicit discrete system of equations is solved by using the Jacobi iterative method. Computational results are presented to demonstrate the efficiency and usefulness of the proposed method.

Multiple-scale theory of topology-driven patterns on directed networks

Dynamical processes on networks are currently being considered in different domains of cross-disciplinary interest. Reaction-diffusion systems hosted on directed graphs are in particular relevant for their widespread applications, from computer networks to traffic systems. Due to the peculiar spectrum of the discrete Laplacian operator, homogeneous fixed points can turn unstable, on a directed support, because of the topology of the network, a phenomenon which cannot be induced on undirected graphs. A linear analysis can be performed to single out the conditions that underly the instability. The complete characterization of the patterns, which are eventually attained beyond the linear regime of exponential growth, calls instead for a full nonlinear treatment. By performing a multiple time scale perturbative calculation, we here derive an effective equation for the nonlinear evolution of the amplitude of the most unstable mode, close to the threshold of criticality. This is a Stuart-Landau equation the complex coefficients of which appear to depend on the topological features of the embedding directed graph. The theory proves adequate versus simulations, as confirmed by operating with a paradigmatic reaction-diffusion model.

Deformable Model-Based Methods for Shape Control of a Haptic Jamming Surface.

Haptic Jamming, the approach of simultaneously controlling mechanical properties and surface deformation of a tactile display via particle jamming and pneumatics, shows promise as a tangible, shape-changing human-computer interface. Previous research introduced device design and described the force-displacement interactions for individual jamming cells. The work in this article analyzes the shape output capabilities of a multi-cell array. A spring-mass deformable body simulation combines models of the three actuation inputs of a Haptic Jamming surface: node pinning, chamber pressurization, and cell jamming. Surface measurements of a 12-cell prototype from a depth camera fit the mass and stiffness parameters to the device during pressurization tests and validate the accuracy of the model for various actuation sequences. The simulator is used to develop an algorithm that generates a sequence of actuation inputs for a Haptic Jamming array of any size in order to match a desired surface output shape. Data extracted from topographical maps and three-dimensional solid object models are used to evaluate the shape-matching algorithm and assess the utility of increasing array size and resolution. Results show that a discrete Laplace operator applied to the input is a suitable predictor of the correlation coefficient between the desired shape and the device output.