Discrete Schrodinger Operators Research Articles

Neural Laplacian Operator for 3D Point Clouds

The discrete Laplacian operator holds a crucial role in 3D geometry processing, yet it is still challenging to define it on point clouds. Previous works mainly focused on constructing a local triangulation around each point to approximate the underlying manifold for defining the Laplacian operator, which may not be robust or accurate. In contrast, we simply use the K- nearest neighbors (KNN) graph constructed from the input point cloud and learn the Laplacian operator on the KNN graph with graph neural networks (GNNs). However, the ground-truth Laplacian operator is defined on a manifold mesh with a different connectivity from the KNN graph and thus cannot be directly used for training. To train the GNN, we propose a novel training scheme by imitating the behavior of the ground-truth Laplacian operator on a set of probe functions so that the learned Laplacian operator behaves similarly to the ground-truth Laplacian operator. We train our network on a subset of ShapeNet and evaluate it across a variety of point clouds. Compared with previous methods, our method reduces the error by an order of magnitude and excels in handling sparse point clouds with thin structures or sharp features. Our method also demonstrates a strong generalization ability to unseen shapes. With our learned Laplacian operator, we further apply a series of Laplacian-based geometry processing algorithms directly to point clouds and achieve accurate results, enabling many exciting possibilities for geometry processing on point clouds. The code and trained models are available at https://github.com/IntelligentGeometry/NeLo.

Dynamics of a mistletoe-bird model on a weighted network.

Based on the patchy habitats of mistletoes and the mutualistic relationship between mistletoes and birds, we propose a mistletoe-bird model on a weighted network that is described by discrete Laplacian operators. Without considering mistletoes, the dynamics of a model of birds is investigated by a threshold parameter. Under the premise of the persistence of birds, the existence and uniqueness of solutions of a mistletoe-bird model are established, and the stability of solutions of the model is discussed by the ecological reproduction index , specifically, mistletoes go extinct when , and mistletoes coexist with birds when . Moreover, we show that network weights can induce changes of instantaneous dynamics of birds or mistletoes by the matrix perturbation method. By assuming that the weighted network is a river network and a star network, we simulate the extinction of mistletoes and the coexistence of mistletoes with birds, respectively.

Toward an efficient second‐order method for computing the surface gravitational potential on spherical‐polar meshes

AbstractAstrophysical accretion discs that carry a significant mass compared with their central object are subject to the effect of self‐gravity. In the context of circumstellar discs, this can, for instance, cause fragmentation of the disc gas, and—under suitable conditions—lead to the direct formation of gas‐giant planets. If one wants to study these phenomena, the disc's gravitational potential needs to be obtained by solving the Poisson equation. This requires to specify suitable boundary conditions. In the case of a spherical‐polar computational mesh, a standard multipole expansion for obtaining boundary values is not practicable. We hence compare two alternative methods for overcoming this limitation. The first method is based on a known Green's function expansion (termed “CCGF”) of the potential, while the second (termed “James' method”) uses a surface screening mass approach with a suitable discrete Green's function. We demonstrate second‐order convergence for both methods and test the weak scaling behavior when using thousands of computational cores. Overall, James' method is found superior owing to its favorable algorithmic complexity of compared with the scaling of the CCGF method.

Unconditional optimal [formula omitted]-norm error estimate and superconvergence analysis of a linearized nonconforming finite element variable-time-step BDF2 method for the nonlinear complex Ginzburg–Landau equation

In this paper, a linearized fully discrete scheme combining the variable-time-step two-step backward differentiation formula (VSBDF2) in time and the nonconforming finite element methods (FEMs) in space is constructed and analyzed for the nonlinear complex Ginzburg–Landau equation. A novel convergence analysis approach is proposed, which shows that the H1-norm error of this scheme can reach optimal order in space and sharp second-order convergence in time. The key ingredients of our analysis are temporal–spatial error splitting approach, a new positive definiteness property of discrete orthogonal convolution (DOC) kernels with respect to complex sequences, an energy projection operator and a discrete Laplace operator Δh related to the nonconforming element space, and a relationship between Δh and classical L2 projection operator in complex space. Furthermore, through a specific treatment of the error equation and the combination technique of interpolating operator and projection operator, we prove that the proposed scheme has the global superconvergence result of order O(τ2+h2) in H1-norm for the first time, which further improves the efficiency of numerical computation. Here τ and h are the maximum time and spatial step sizes, respectively. It should be pointed out that all these error results are rigorously established under mild assumptions of the time step, and get rid of the usual grid ratio constraints between temporal and spatial step sizes in linearized scheme. At last, numerical experiments are provided to verify the theoretical analysis.

A linearized L2-1[formula omitted] Galerkin FEM for Kirchhoff type quasilinear subdiffusion equation with memory

This article aims to construct a novel linearized Galerkin FEM based on Alikhanov’s L2-1σ scheme to solve a Kirchhoff type quasilinear subdiffusion equation with memory Dα. Near time t=0, the weak singularity of the solution to the equation Dα is taken into account by using the graded mesh in the time direction. The nonlocal behavior of Kirchhoff term causes huge computer storage thereby increasing computational cost. We reduce these costs by linearizing the Kirchhoff term. We derive a priori bounds and global rates of convergence for the developed numerical scheme. We prove that the convergence rate of the proposed numerical scheme is first-order in space and second-order in time in L∞(0,T;L2(Ω)). In addition, we prove the same convergence rate in L∞(0,T;H01(Ω)) by making use of a discrete Laplacian operator. A numerical experiment is presented to verify the theoretical results.

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.

Numerical analysis of finite element method for a stochastic active fluids model

In this paper, we first investigate the well-posedness and regularity of mild solution to a stochastic active fluids model driven by the additive noise. A fully-discrete scheme is proposed for solving the given model, which is based on the finite element method for spatial discretization and the backward Euler method for temporal discretization. By overcoming the difficulty of error analysis caused by the discrete Laplacian operator, we obtain the convergence results of the developed scheme. Finally, some numerical examples are provided to validate the theoretical results and we also simulate the motion states of the active fluids.

Pointwise superconvergence of block finite elements for the three-dimensional variable coefficient elliptic equation

<p>This study investigated the point-wise superconvergence of block finite elements for the variable coefficient elliptic equation in a regular family of rectangular partitions of the domain in three-dimensional space. Initially, the estimates for the three-dimensional discrete Greens function and discrete derivative Greens function were presented. Subsequently, employing an interpolation operator of projection type, two essential weak estimates were derived, which were crucial for superconvergence analysis. Ultimately, by combining the aforementioned estimates, we achieved superconvergence estimates for the derivatives and function values of the finite element approximation in the point-wise sense of the $ L^\infty $-norm. A numerical example illustrated the theoretical results.</p>

Асимптотика Сеге для системы Анжелеско

We discuss opportunity for extension the strong asymptotics (Szego asymptotics) for more wide class of the sequences of multiple orthogonal polynomials with respect to the system of weights supported by the non-intersecting intervals (Angelesco system). The interest for limiting properties of these polynomials, when their multi-index tends monotonically to in_nity (along of the edges of p-dimensional lattice) has appeared recently, in onnection with spectral problems for the discrete Schrodinger operator (multidimensional Jacobi matrix) on the graph-trees.

Contribution of Discrete and Continuum Wave Functions in the Ground State Polarizability of Hydrogen

We analytically derive a discrete radial Green function for a hydrogenic system in terms of associated Leguerre polynomials. We investigate the polarizability of the ground-state hydrogen using the so-called length gauge approximation of electrodynamics. The polarizability is the sum of the polarization matrix elements for the negative and the positive frequency values. We estimate the contribution of discrete wavefunctions to the static ground state polarizability. We found that the contribution of continuum states dominates over that of discrete states to the polarizability by an order of magnitude.

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.

Finite element analysis of time‐fractional integro‐differential equation of Kirchhoff type for non‐homogeneous materials

In this paper, we study a time‐fractional initial‐boundary value problem of Kirchhoff type involving memory term for non‐homogeneous materials. As a consequence of energy argument, we derive bound as well as bound on the solution of the considered problem by defining two new discrete Laplacian operators. Using these a priori bounds, existence and uniqueness of the weak solution to the considered problem are established. Further, we study semi discrete formulation of the problem by discretizing the space domain using a conforming finite element method (FEM) and keeping the time variable continuous. The semi discrete error analysis is carried out by modifying the standard Ritz‐Volterra projection operator in such a way that it reduces the complexities arising from the Kirchhoff type nonlinearity. Finally, we develop a new linearized L1 Galerkin FEM to obtain numerical solution of the problem under consideration. This method has a convergence rate of , where is the fractional derivative exponent and and are the discretization parameters in the space and time directions, respectively. This convergence rate is further improved to second order in the time direction by proposing a novel linearized L2‐1 Galerkin FEM. We conduct a numerical experiment to validate our theoretical claims.

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.

Fast construction of cluster interaction matrix for data-driven cluster-based reduced-order model and prediction of elastoplastic stress-strain curves and yield surface

The data-driven cluster-based reduced-order model (CROM) methods[47–50] are very effective for predicting the nonlinear effective properties of heterogeneous materials. The Cluster Interaction Matrix (CIM) in these methods can be regarded as a clustered discrete Green's function and is the critical part of CROM methods, which more and more scholars have realized. However, the explicit expression of Green's function is available only for homogeneous material. The CIM for heterogeneous materials needs to be calculated through numerical approaches, which face the challenge of computational efficiency. Therefore, this paper proposes an efficient method to construct the CIM of heterogeneous materials. By deriving the Green's function in Fourier space under the governing equation of FCA, the CIM of homogeneous materials that can provide the self-equilibrium stress field is calculated efficiently. Then, to avoid the effects of stress averaging on clusters, we augment the proposed cluster-element interaction matrix with the self-equilibrium stress basis vector of heterogeneous material without introducing additional computational cost. Based on the augmented cluster-element interaction matrix, the self-equilibrium stress space is improved, and the CIM of highly heterogeneous materials is calculated using the minimum complementary energy principle. The proposed algorithm significantly improves the offline computational accuracy of FCA, enabling efficient prediction of nonlinear effective properties and yield surface under complex load paths. Numerical examples demonstrate the efficiency and accuracy of the proposed method. In addition, we provide the relationship between the CIMs of a homogeneous material of FCA and SCA. Based on the proposed algorithm in this paper, it can be helpful for other CROM methods.

Exponential synchronization of 2D cellular neural networks with boundary feedback

In this work we propose a new mathematical model of 2D cellular neural networks (CNN) in terms of the lattice FitzHugh-Nagumo equations with boundary feedback. The model features discrete Laplacian operators and periodic boundary feedback instead of the interior-clamped or mean-field feedback. We first prove the globally dissipative dynamics of the solutions through a priori uniform estimates. In the main result we show that the 2D cellular neural networks are exponentially synchronized if the computable threshold condition is satisfied by the synaptic gap signals of pairwise boundary cells of the grid and the system parameters with a linear feedback coupling coefficient tunable in applications.

An FFT-Based MAC scheme for Stokes equations with periodic boundary conditions and its application to elasticity problems

This paper presents a fast Fourier transform (FFT) based finite difference MAC scheme to compute the Stokes equations with periodic boundary conditions. The numerical implementation of this solver is very convenient and only requires three simple steps. The remarkable superiorities of such a solver compared to classical iterative approaches are computational time-saving and memory storage saving. The application of the proposed solver to nearly incompressible linear elasticity problems is also introduced. In addition, following the similar spirit used in Dong et al. (2020) for Stokes equations with Dirichlet boundary conditions, a rigorous error analysis is provided for Stokes equations with periodic boundary conditions. The results include: a second-order convergence in the ℓ2-norm for velocity, pressure as well as the gradient of velocity; a second-order accuracy in maximum norm for both velocity and its gradient. Moreover, the properties of the discrete Green functions play an important part in maximum error estimation. Most notably, the major difficulties of the extension from Dirichlet boundary conditions to periodic boundary conditions are the discrete version of Green’s formulae, Poincaré inequality and LBB condition. A variety of two-dimensional and three-dimensional numerical examples are given to illustrate the efficiency of the proposed FFT solver and validate the theoretical results.

An improved pointwise error estimate of the LDG method for 1-d singularly perturbed reaction-diffusion problem

The local discontinuous Galerkin (LDG) method on a Shishkin mesh is studied for a one-dimensional singularly perturbed reaction-diffusion problem. Based on the discrete Green's function, we derive some improved pointwise error estimates in the regular and layer regions. For the LDG approximation to the solution itself, the convergence rate of the pointwise error in the regular region is sharp. For the LDG approximation to the derivative of the solution, the convergence rate of the pointwise error in the whole domain is optimal. We establish also optimal pointwise error estimates in the case that the regular component of the exact solution belongs to the finite element space. Numerical experiments are presented to validate the theoretical results.

Existence condition of an eigenvalue of the three particle Schr¨odinger operator on a lattice

We consider the three-particle discrete Schrodinger operator Hµ,γ(К), К ϵТ3 associated to a system of three particles (two particle are fermions with mass 1 and third one is an another particle with mass m = 1/y < 1) interacting through zero range pairwise potential µ> 0 on the three-dimensional lattice Z3. It is proved that for γϵ(1, γ0) (γ0≈4,7655) the operator Hµ,γ(π), π=(π,π,π), has no eigenvalue and has only unique eigenvalue with multiplicity three for γ>γ0 lying right of the essential spectrum for sufficiently large µ.

Gradient-descent-like scheme for the Allen–Cahn equation

The phase-field equations have many attractive characteristics. First, phase separation can be induced by the phase-field equations. It transforms from a single homogeneous mixture to two distinct phases in a nascent state. Second, the solution of the phase-field equations is bounded by a finite value. It is beneficial to ensure numerical stability. Third, the motion of the interface can be described by geometric features. It is helpful for expressing natural phenomena in mathematical terms. Fourth, the phase-field equations possess the energy dissipation law. This law is about degeneration and decay. It tells us in thermodynamics that all occurrences are irreversible processes. In this paper, we would like to investigate the numerical implementation of the Allen–Cahn (AC) equation, which is the classical one of the phase-field equations. In phase field modeling, the binary phase system is described using a continuous variable called the order parameter. The order parameter can be categorized into two forms: conserved, which represents the physical property such as concentration or mass, and non-conserved, which does not have the conserved physical property. We consider both the non-conservative and conservative AC equations. Our interest is more precisely to scrutinize the utilization of the discrete Laplacian operator in the AC equation by considering the conservative and non-conservative order parameter ϕ. Constructing linearly implicit methods for solving the AC equation, we formulate a gradient-descent-like scheme. Therefore, reinterpreting the implicit scheme for the AC equation, we propose a novel numerical scheme in which solutions are bounded by 1 for all t > 0. Together with the conservative Allen–Cahn equation, our proposed scheme is consistent when mass is conserved as well. From a numerical point of view, a linear, unconditionally energy stable splitting scheme is transformed into a gradient-descent-like scheme. Various numerical simulations are illustrated to demonstrate the validity of the proposed scheme. We also make distinctions between the proposed one and existing numerical schemes.

Second-Law Considerations in Monte Carlo Ray-Trace and Discrete Green’s Function Analysis of Coupled Radiation and Conduction Heat Transfer

Abstract A generic Monte Carlo ray-trace (MCRT) engine for computing radiation distribution factors (RDFs), working in tandem with an efficient finite volume formulation based on discrete Green’s functions (DGFs), has been used to solve a massive (2636 nodes) coupled radiation/conduction problem. Distribution factors computed using the MCRT method are known to produce unconditionally stable solutions to pure radiation problems even though the RDFs themselves do not conform exactly to the reciprocity principle. However, when these same RFDs are introduced into time-dependent finite volume conduction formulations based on DGFs, the resulting model is found to be fundamentally unstable due to inherent violations of the second law of thermodynamics. A novel approach to addressing this instability is presented and demonstrated for the case of a telescope typical of those employed to monitor the planetary energy budget from low Earth orbit.