Laplace-Beltrami Operator Research Articles

A semi-Lagrangian radial basis function partition of unity closest point method for advection-diffusion equations on surfaces

A semi-Lagrangian radial basis function partition of unity (RBF-PU) closest point method is designed for solving advection-diffusion equations on surfaces. This new meshfree method combines the semi-Lagrangian method with the RBF-PU closest point method. The semi-Lagrangian RBF-PU closest point method traces the departure point backwards along the velocity field based on patches at each time step. Therefore, it saves the computational cost compared with the semi-Lagrangian radial basis function finite difference (RBF-FD) closest point method. Our proposed RBF-PU closest point method for approximating the Laplace-Beltrami operator has two main advantages over the RBF-FD closest point method. Firstly, the RBF-PU closest point method to construct the local influence domain is not required to identify the size of the computational tube. Therefore, our method is more concise and easier to implement. Secondly, the RBF-PU closest point method not only improves the accuracy but also saves computational costs, and we confirm the advantage by testing differential accuracy. Numerical experiments verify the convergence and effectiveness of the semi-Lagrangian RBF-PU closest point method.

On periodic solution for the Boussinesq system on real hyperbolic manifolds

In this work, we study the existence, uniqueness and exponential stability of periodic mild solution of the Boussinesq system on the real hyperbolic manifold H d ( R ) ( d ⩾ 2 ). We consider a couple of Ebin-Marsden's Laplace and Laplace–Beltrami operators associated with the corresponding linear system. Our method is based on the dispersive and smoothing estimates of semigroups generated by Ebin-Marsden's Laplace and Laplace–Beltrami operators. First, we prove the existence and uniqueness of the bounded periodic mild solution for the linear system. Next, using the fixed point arguments, we pass from the linear system framework to the semilinear system framework to establish the existence of the periodic mild solution. Additionally, we prove the unconditional uniqueness of periodic mild solution for the Boussinesq system on hyperbolic manifolds. Finally, we establish the exponential stability of periodic mild solution.

Higher Cheeger ratios of features in Laplace-Beltrami eigenfunctions

This paper investigates links between the eigenvalues and eigenfunctions of the Laplace-Beltrami operator, and the higher Cheeger constants of smooth Riemannian manifolds, possibly weighted and/or with boundary. The higher Cheeger constants give a loose description of the major geometric features of a manifold. We give a constructive upper bound on the higher Cheeger constants, in terms of the eigenvalue of any eigenfunction with the corresponding number of nodal domains. Specifically, we show that for each such eigenfunction, a positive-measure collection of its superlevel sets have their Cheeger ratios bounded above in terms of the corresponding eigenvalue.Some manifolds have their major features entwined across several eigenfunctions, and no single eigenfunction contains all the major features. In this case, there may exist carefully chosen linear combinations of the eigenfunctions, each with large values on a single feature, and small values elsewhere. We can then apply a soft-thresholding operator to these linear combinations to obtain new functions, each supported on a single feature. We show that the Cheeger ratios of the level sets of these functions also give an upper bound on the Laplace-Beltrami eigenvalues. We extend these level set results to nonautonomous dynamical systems, and show that the dynamic Laplacian eigenfunctions reveal sets with small dynamic Cheeger ratios.

A unified structure-preserving parametric finite element method for anisotropic surface diffusion

We propose and analyze a unified structure-preserving parametric finite element method (SP-PFEM) for the anisotropic surface diffusion of curves in two dimensions ( d = 2 ) (d=2) and surfaces in three dimensions ( d = 3 ) (d=3) with an arbitrary anisotropic surface energy density γ ( n ) \gamma (\boldsymbol {n}) , where n ∈ S d − 1 \boldsymbol {n}\in \mathbb {S}^{d-1} represents the outward unit vector. By introducing a novel unified surface energy matrix G k ( n ) \boldsymbol {G}_k(\boldsymbol {n}) depending on γ ( n ) \gamma (\boldsymbol {n}) , the Cahn-Hoffman ξ \boldsymbol {\xi } -vector and a stabilizing function k ( n ) : S d − 1 → R k(\boldsymbol {n}):\ \mathbb {S}^{d-1}\to {\mathbb R} , we obtain a unified and conservative variational formulation for the anisotropic surface diffusion via different surface differential operators, including the surface gradient operator, the surface divergence operator and the surface Laplace-Beltrami operator. A SP-PFEM discretization is presented for the variational problem. In order to establish the unconditional energy stability of the proposed SP-PFEM under a very mild condition on γ ( n ) \gamma (\boldsymbol {n}) , we propose a new framework via local energy estimate for proving energy stability/structure-preserving properties of the parametric finite element method for the anisotropic surface diffusion. This framework sheds light on how to prove unconditional energy stability of other numerical methods for geometric partial differential equations. Extensive numerical results are reported to demonstrate the efficiency and accuracy as well as structure-preserving properties of the proposed SP-PFEM for the anisotropic surface diffusion with arbitrary anisotropic surface energy density γ ( n ) \gamma (\boldsymbol {n}) arising from different applications.

Learning on manifolds without manifold learning

Function approximation based on data drawn randomly from an unknown distribution is an important problem in machine learning. The manifold hypothesis assumes that the data is sampled from an unknown submanifold of a high dimensional Euclidean space. A great deal of research deals with obtaining information about this manifold, such as the eigendecomposition of the Laplace–Beltrami operator or coordinate charts, and using this information for function approximation. This two-step approach implies some extra errors in the approximation stemming from estimating the basic quantities of the data manifold in addition to the errors inherent in function approximation. In this paper, we project the unknown manifold as a submanifold of an ambient hypersphere and study the question of constructing a one-shot approximation using a specially designed sequence of localized spherical polynomial kernels on the hypersphere. Our approach does not require preprocessing of the data to obtain information about the manifold other than its dimension. We give optimal rates of approximation for relatively “rough” functions.

The logarithmic Dirichlet Laplacian on Ahlfors regular spaces

We introduce the logarithmic analogue of the Laplace-Beltrami operator on Ahlfors regular metric-measure spaces. This operator is intrinsically defined with spectral properties analogous to those of elliptic pseudo-differential operators on Riemannian manifolds. Specifically, its heat semigroup consists of compact operators which are trace-class after some critical point in time. Moreover, its domain is a Banach module over the Dini continuous functions and every Hölder continuous function is a smooth vector. Finally, the operator is compatible, in the sense of noncommutative geometry, with the action of a large class of non-isometric homeomorphisms.

The classical and quantum particle on a flag manifold

Abstract In the present paper we consider two related problems, i.e. the description of geodesics and the calculation of the spectrum of the Laplace–Beltrami operator on a flag manifold. We show that there exists a family of invariant metrics such that both problems can be solved simply and explicitly. In order to determine the spectrum of the Laplace–Beltrami operator, we construct natural, finite-dimensional approximations (of spin chain type) to the Hilbert space of functions on a flag manifold.

Spherical Logvinenko–Sereda–Kovrijkine type inequality and null-controllability of the heat equation on the sphere

It is shown that the restriction of a polynomial to a sphere satisfies a Logvinenko–Sereda–Kovrijkine type inequality (a specific type of uncertainty relation). This implies a spectral inequality for the Laplace–Beltrami operator, which, in turn, yields observability and null-controllability with explicit estimates on the control costs for the spherical heat equation that are sharp in the large and in the small time regime.

Non-Compact Inaudibility of Weak Symmetry and Commutativity via Generalized Heisenberg Groups

Two Riemannian manifolds are said to be isospectral if there exists a unitary operator which intertwines their Laplace-Beltrami operator. In this paper, we prove in the non-compact setting the inaudibility of the weak symmetry property and the commutative property using an isospectral pair of 23 dimensional generalized Heisenberg groups.

Clustering Molecules at a Large Scale: Integrating Spectral Geometry with Deep Learning.

This study conducts an in-depth analysis of clustering small molecules using spectral geometry and deep learning techniques. We applied a spectral geometric approach to convert molecular structures into triangulated meshes and used the Laplace-Beltrami operator to derive significant geometric features. By examining the eigenvectors of these operators, we captured the intrinsic geometric properties of the molecules, aiding their classification and clustering. The research utilized four deep learning methods: Deep Belief Network, Convolutional Autoencoder, Variational Autoencoder, and Adversarial Autoencoder, each paired with k-means clustering at different cluster sizes. Clustering quality was evaluated using the Calinski-Harabasz and Davies-Bouldin indices, Silhouette Score, and standard deviation. Nonparametric tests were used to assess the impact of topological descriptors on clustering outcomes. Our results show that the DBN + k-means combination is the most effective, particularly at lower cluster counts, demonstrating significant sensitivity to structural variations. This study highlights the potential of integrating spectral geometry with deep learning for precise and efficient molecular clustering.

An Improved Eigenvalue Estimate for Embedded Minimal Hypersurfaces in the Sphere

Abstract Suppose that $\Sigma ^{n}\subset \mathbb{S}^{n+1}$ is a closed embedded minimal hypersurface. We prove that the first non-zero eigenvalue $\lambda _{1}$ of the induced Laplace–Beltrami operator on $\Sigma $ satisfies $\lambda _{1} \geq \frac{n}{2}+ a_{n}(\Lambda ^{6} + b_{n})^{-1}$, where $a_{n}$ and $b_{n}$ are explicit dimensional constants and $\Lambda $ is an upper bound for the length of the second fundamental form of $\Sigma $. This provides the first explicitly computable improvement on Choi and Wang’s lower bound $\lambda _{1} \geq \frac{n}{2}$ without any further assumptions on $\Sigma $.

Right Conoids Demonstrating a Time-like Axis within Minkowski Four-Dimensional Space

In the four-dimensional Minkowski space, hypersurfaces classified as right conoids with a time-like axis are introduced and studied. The computation of matrices associated with the fundamental form, the Gauss map, and the shape operator specific to these hypersurfaces is included in our analysis. The intrinsic curvatures of these hypersurfaces are determined to provide a deeper understanding of their geometric properties. Additionally, the conditions required for these hypersurfaces to be minimal are established, and detailed calculations of the Laplace–Beltrami operator are performed. Illustrative examples are provided to enhance our comprehension of these concepts. Finally, the umbilical condition is examined to determine when these hypersurfaces become umbilic, and also the Willmore functional is explored.

The Laplace-Beltrami Operator on the Surface of the Ellipsoid

The Laplace-Beltrami Operator on the Surface of the Ellipsoid

Transparent boundary condition and its effectively local approximation for the Schrödinger equation on a rectangular computational domain

The transparent boundary condition for the free Schrödinger equation on a rectangular computational domain requires implementation of an operator of the form ∂t−i△Γ where △Γ is the Laplace-Beltrami operator. It is known that this operator is nonlocal in time as well as space which poses a significant challenge in developing an efficient numerical method of solution. The computational complexity of the existing methods scale with the number of time-steps which can be attributed to the nonlocal nature of the boundary operator. In this work, we report an effectively local approximation for the boundary operator such that the resulting complexity remains independent of number of time-steps. At the heart of this algorithm is a Padé approximant based rational approximation of certain fractional operators that handles corners of the domain adequately. For the spatial discretization, we use a Legendre-Galerkin spectral method with a new boundary adapted basis which ensures that the resulting linear system is banded. A compatible boundary-lifting procedure is also presented which accommodates the segments as well as the corners on the boundary. The proposed novel scheme can be implemented within the framework of any one-step time marching schemes. In particular, we demonstrate these ideas for two one-step methods, namely, the backward-differentiation formula of order 1 (BDF1) and the trapezoidal rule (TR). For the sake of comparison, we also present a convolution quadrature based scheme conforming to the one-step methods which is computationally expensive but serves as a golden standard. Finally, several numerical tests are presented to demonstrate the effectiveness of our novel method as well as to verify the order of convergence empirically.

Full asymptotic expansion for orbit-summable quadrant walks and discrete polyharmonic functions

Enumeration of walks with small steps in the quadrant has been a topic of great interest in combinatorics over the last few years. In this article, it is shown how to compute exact asymptotics of the number of such walks with fixed start- and endpoints for orbit-summable models with finite group, up to arbitrary precision. The resulting representation greatly resembles one conjectured for walks starting from the origin in 2020 by Chapon, Fusy and Raschel, differing only in terms appearing due to the periodicity of the model. We will see that the dependency on start- and endpoint is given by discrete polyharmonic functions, which are solutions of △nv=0 for a discretization △ of a Laplace–Beltrami operator. They can be decomposed into a sum of products of lower order polyharmonic functions of either the start- or the endpoint only, which leads to a partial extension of a theorem by Denisov and Wachtel.

Local invariants of noncommutative tori

The notion of a generic curved noncommutative torus is considered, which extends the notion of a conformally deformed noncommutative torus introduced by Connes and Tretkoff. For this manifold, an asymptotic expansion is established for the heat semigroup generated by the Laplace–Beltrami operator (in fact, for an arbitrary selfadjoint positive elliptic differential operator of order 2 2 ) and an algorithm is provided to compute the local invariants that arize as the coefficients in the expansion. This allows one to extend a series of previous results by several authors beyond the conformal case and/or for multidimensional tori.

The Möbius Addition and Generalized Laplace–Beltrami Operator in Octonionic Space

The Möbius Addition and Generalized Laplace–Beltrami Operator in Octonionic Space

Potentials on the conformally compactified Minkowski spacetime and their application to quark deconfinement

In this paper, we study a class of conformal metric deformations in the quasi-radial coordinate parametrizing the three-sphere in the conformally compactified Minkowski spacetime [Formula: see text]. Prior to reduction of the associated Laplace–Beltrami operators to a Schrödinger form, a corresponding class of exactly solvable potentials (each one containing a scalar and a gradient term) is found. In particular, the scalar piece of these potentials can be exactly or quasi-exactly solvable, and among them we find the finite range confining trigonometric potentials of Pöschl–Teller, Scarf and Rosen–Morse. As an application of the results developed in the paper, the large compactification radius limit of the interaction described by some of these potentials is studied, and this regime is shown to be relevant to a quantum mechanical quark deconfinement mechanism.

Simulating time-harmonic acoustic wave effects induced by periodic holes/inclusions on surfaces

This paper introduces a localized meshless method to analyze time-harmonic acoustic wave propagation on curved surfaces with periodic holes/inclusions. In particular, the generalized finite difference method is used as a localized meshless technique to discretize the surface gradient and Laplace-Beltrami operators defined extrinsically in the governing equations. An absorbing boundary condition is introduced to reduce reflections from boundaries and accurately simulate wave propagation on unclosed surfaces with periodic inclusions. Several benchmark examples demonstrate the efficiency and accuracy of the proposed method in simulating acoustic wave propagation on surfaces with diverse geometries, including complex shapes and periodic holes or inclusions.

Sharp Adams Type Inequalities for the Fractional Laplace–Beltrami Operator on Noncompact Symmetric Spaces

Sharp Adams Type Inequalities for the Fractional Laplace–Beltrami Operator on Noncompact Symmetric Spaces