Topic
Laplace operator
About: Laplace operator is a research topic. Over the lifetime, 13227 publications have been published within this topic receiving 239400 citations. The topic is also known as: Laplacian & Laplacian operator.
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Papers
Book•
19 Jun 1986
TL;DR: This paper presents the results of an analysis of the "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions and its applications to Mixed Approximation and Homogeneous Stokes Equations.
Abstract: I. Mathematical Foundation of the Stokes Problem.- 1. Generalities on Some Elliptic Boundary Value Problems.- 1.1. Basic Concepts on Sobolev Spaces.- 1.2. Abstract Elliptic Theory.- 1.3. Example 1: Dirichlet's Problem for the Laplace Operator.- 1.4. Example 2: Neumann's Problem for the Laplace Operator.- 1.5. Example 3: Dirichlet's Problem for the Biharmonic Operator.- 2. Function Spaces for the Stokes Problem.- 2.1. Preliminary Results.- 2.2. Some Properties of Spaces Related to the Divergence Operator.- 2.3. Some Properties of Spaces Related to the Curl Operator.- 3. A Decomposition of Vector Fields.- 3.1. Decomposition of Two-Dimensional Vector Fields.- 3.2. Application to the Regularity of Functions of H(div ?) ? H(curl ?).- 3.3. Decomposition of Three-Dimensional Vector Fields.- 3.4. The Imbedding of H(div ?) ? H0 (curl ?) into H1(?)3.- 3.5. The Imbedding of H0(div ?) ? H (curl ?) into H1(?)3.- 4. Analysis of an Abstract Variational Problem.- 4.1. A General Result.- 4.2. A Saddle-Point Approach.- 4.3. Approximation by Regularization or Penalty.- 4.4. Iterative Methods of Gradient Type.- 5. The Stokes Equations.- 5.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 5.2. The Stream Function Formulation of the Dirichlet Problem in Two Dimensions.- 5.3. The Three-Dimensional Case.- Appendix A. Results of Standard Finite Element Approximation.- A.l. Triangular Finite Elements.- A.2. Quadrilateral Finite Elements.- A.3. Interpolation of Discontinuous Functions.- II. Numerical Solution of the Stokes Problem in the Primitive Variables.- 1. General Approximation.- 1.1. An Abstract Approximation Result.- 1.2. Decoupling the Computation of uh and ?h.- 1.3. Application to the Homogeneous Stokes Problem.- 1.4. Checking the inf-sup Condition.- 2. Simplicial Finite Element Methods Using Discontinuous Pressures.- 2.1. A First Order Approximation on Triangular Elements.- 2.2. Higher-Order Approximation on Triangular Elements.- 2.3. The Three-Dimensional case: First and Higher-Order Schemes.- 3. Quadrilateral Finite Element Methods Using Discontinuous Pressures.- 3.1. A quadrilateral Finite Element of Order One.- 3.2. Higher-Order Quadrilateral Elements.- 3.3. An Example of Checkerboard Instability: the Q1 - P0 Element.- 3.4. Error Estimates for the Q1 - P0 Element.- 4. Continuous Approximation of the Pressure.- 4.1. A First Order Method: the "Mini" Finite Element.- 4.2. The "Hood-Taylor" Finite Element Method.- 4.3. The "Glowinski-Pironneau" Finite Element Method.- 4.4. Implementation of the Glowinski-Pironneau Scheme.- III. Incompressible Mixed Finite Element Methods for Solving the Stokes Problem.- 1. Mixed Approximation of an Abstract Problem.- 1.1. A Mixed Variational Problem.- 1.2. Abstract Mixed Approximation.- 2. The "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions.- 2.1. A Mixed Formulation.- 2.2. Mixed Approximation and Application to Finite Elements of Degree l.- 2.3. The Technique of Mesh-Dependent Norms.- 3. Further Topics on the "Stream Function-Vorticity-Pressure" Scheme.- 3.1. Refinement of the Error Analysis.- 3.2. Super Convergence Using Quadrilateral Finite Elements of Degree l.- 4. A "Stream Function-Gradient of Velocity Tensor" Method in Two Dimensions.- 4.1. The Hellan-Herrmann-Johnson Formulation.- 4.2. Approximation with Triangular Finite Elements of Degree l.- 4.3. Additional Results for the Hellan-Herrmann-Johnson Scheme.- 4.4. Discontinuous Approximation of the Pressure.- 5. A "Vector Potential-Vorticity" Scheme in Three Dimensions.- 5.1. A Mixed Formulation of the Three-Dimensional Stokes Problem.- 5.2. Mixed Approximation in H(curl ?).- 5.3. A Family of Conforming Finite Elements in H(curl ?).- 5.4. Error Analysis for Finite Elements of Degree l.- 5.5. Discontinuous Approximation of the Pressure.- IV. Theory and Approximation of the Navier-Stokes Problem.- 1. A Class of Nonlinear Problems.- s Problem for the Laplace Operator.- 1.5. Example 3: Dirichlet's Problem for the Biharmonic Operator.- 2. Function Spaces for the Stokes Problem.- 2.1. Preliminary Results.- 2.2. Some Properties of Spaces Related to the Divergence Operator.- 2.3. Some Properties of Spaces Related to the Curl Operator.- 3. A Decomposition of Vector Fields.- 3.1. Decomposition of Two-Dimensional Vector Fields.- 3.2. Application to the Regularity of Functions of H(div ?) ? H(curl ?).- 3.3. Decomposition of Three-Dimensional Vector Fields.- 3.4. The Imbedding of H(div ?) ? H0 (curl ?) into H1(?)3.- 3.5. The Imbedding of H0(div ?) ? H (curl ?) into H1(?)3.- 4. Analysis of an Abstract Variational Problem.- 4.1. A General Result.- 4.2. A Saddle-Point Approach.- 4.3. Approximation by Regularization or Penalty.- 4.4. Iterative Methods of Gradient Type.- 5. The Stokes Equations.- 5.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 5.2. The Stream Function Formulation of the Dirichlet Problem in Two Dimensions.- 5.3. The Three-Dimensional Case.- Appendix A. Results of Standard Finite Element Approximation.- A.l. Triangular Finite Elements.- A.2. Quadrilateral Finite Elements.- A.3. Interpolation of Discontinuous Functions.- II. Numerical Solution of the Stokes Problem in the Primitive Variables.- 1. General Approximation.- 1.1. An Abstract Approximation Result.- 1.2. Decoupling the Computation of uh and ?h.- 1.3. Application to the Homogeneous Stokes Problem.- 1.4. Checking the inf-sup Condition.- 2. Simplicial Finite Element Methods Using Discontinuous Pressures.- 2.1. A First Order Approximation on Triangular Elements.- 2.2. Higher-Order Approximation on Triangular Elements.- 2.3. The Three-Dimensional case: First and Higher-Order Schemes.- 3. Quadrilateral Finite Element Methods Using Discontinuous Pressures.- 3.1. A quadrilateral Finite Element of Order One.- 3.2. Higher-Order Quadrilateral Elements.- 3.3. An Example of Checkerboard Instability: the Q1 - P0 Element.- 3.4. Error Estimates for the Q1 - P0 Element.- 4. Continuous Approximation of the Pressure.- 4.1. A First Order Method: the "Mini" Finite Element.- 4.2. The "Hood-Taylor" Finite Element Method.- 4.3. The "Glowinski-Pironneau" Finite Element Method.- 4.4. Implementation of the Glowinski-Pironneau Scheme.- III. Incompressible Mixed Finite Element Methods for Solving the Stokes Problem.- 1. Mixed Approximation of an Abstract Problem.- 1.1. A Mixed Variational Problem.- 1.2. Abstract Mixed Approximation.- 2. The "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions.- 2.1. A Mixed Formulation.- 2.2. Mixed Approximation and Application to Finite Elements of Degree l.- 2.3. The Technique of Mesh-Dependent Norms.- 3. Further Topics on the "Stream Function-Vorticity-Pressure" Scheme.- 3.1. Refinement of the Error Analysis.- 3.2. Super Convergence Using Quadrilateral Finite Elements of Degree l.- 4. A "Stream Function-Gradient of Velocity Tensor" Method in Two Dimensions.- 4.1. The Hellan-Herrmann-Johnson Formulation.- 4.2. Approximation with Triangular Finite Elements of Degree l.- 4.3. Additional Results for the Hellan-Herrmann-Johnson Scheme.- 4.4. Discontinuous Approximation of the Pressure.- 5. A "Vector Potential-Vorticity" Scheme in Three Dimensions.- 5.1. A Mixed Formulation of the Three-Dimensional Stokes Problem.- 5.2. Mixed Approximation in H(curl ?).- 5.3. A Family of Conforming Finite Elements in H(curl ?).- 5.4. Error Analysis for Finite Elements of Degree l.- 5.5. Discontinuous Approximation of the Pressure.- IV. Theory and Approximation of the Navier-Stokes Problem.- 1. A Class of Nonlinear Problems.- 2. Theory of the Steady-State Navier-Stokes Equations.- 2.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 2.2. The Stream Function Formulation of the Homogeneous Problem..- 3. Approximation of Branches of Nonsingular Solutions.- 3.1. An Abstract Framework.- 3.2. Approximation of Branches of Nonsingular Solutions.- 3.3. Application to a Class of Nonlinear Problems.- 3.4. Non-Differentiable Approximation of Branches of Nonsingular Solutions.- 4. Numerical Analysis of Centered Finite Element Schemes.- 4.1. Formulation in Primitive Variables: Methods Using Discontinuous Pressures.- 4.2. Formulation in Primitive Variables: the Case of Continuous Pressures.- 4.3. Mixed Incompressible Methods: the "Stream Function-Vorticity" Formulation.- 4.4. Remarks on the "Stream Function-Gradient of Velocity Tensor" Scheme.- 5. Numerical Analysis of Upwind Schemes.- 5.1. Upwinding in the Stream Function-Vorticity Scheme.- 5.2. Error Analysis of the Upwind Scheme.- 5.3. Approximating the Pressure with the Upwind Scheme.- 6. Numerical Algorithms.- 2.11. General Methods of Descent and Application to Gradient Methods.- 2.12. Least-Squares and Gradient Methods to Solve the Navier-Stokes Equations.- 2.13. Newton's Method and the Continuation Method.- References.- Index of Mathematical Symbols.
6,135 citations
Proceedings Article•
3 Jan 2001TL;DR: The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering.
Abstract: Drawing on the correspondence between the graph Laplacian, the Laplace-Beltrami operator on a manifold, and the connections to the heat equation, we propose a geometrically motivated algorithm for constructing a representation for data sampled from a low dimensional manifold embedded in a higher dimensional space. The algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering. Several applications are considered.
5,349 citations
Proceedings Article•
9 Dec 2003TL;DR: These are linear projective maps that arise by solving a variational problem that optimally preserves the neighborhood structure of the data set by finding the optimal linear approximations to the eigenfunctions of the Laplace Beltrami operator on the manifold.
Abstract: Many problems in information processing involve some form of dimensionality reduction. In this paper, we introduce Locality Preserving Projections (LPP). These are linear projective maps that arise by solving a variational problem that optimally preserves the neighborhood structure of the data set. LPP should be seen as an alternative to Principal Component Analysis (PCA) – a classical linear technique that projects the data along the directions of maximal variance. When the high dimensional data lies on a low dimensional manifold embedded in the ambient space, the Locality Preserving Projections are obtained by finding the optimal linear approximations to the eigenfunctions of the Laplace Beltrami operator on the manifold. As a result, LPP shares many of the data representation properties of nonlinear techniques such as Laplacian Eigenmaps or Locally Linear Embedding. Yet LPP is linear and more crucially is defined everywhere in ambient space rather than just on the training data points. This is borne out by illustrative examples on some high dimensional data sets.
4,780 citations
TL;DR: In this article, the square root of the Laplacian (−△) 1/2 operator was obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition.
Abstract: The operator square root of the Laplacian (−△) 1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems.
3,157 citations
TL;DR: The Hessian-based locally linear embedding method for recovering the underlying parametrization of scattered data (mi) lying on a manifold M embedded in high-dimensional Euclidean space is described, where the isometric coordinates can be recovered up to a linear isometry.
Abstract: We describe a method for recovering the underlying parametrization of scattered data (m(i)) lying on a manifold M embedded in high-dimensional Euclidean space. The method, Hessian-based locally linear embedding, derives from a conceptual framework of local isometry in which the manifold M, viewed as a Riemannian submanifold of the ambient Euclidean Space R(n), is locally isometric to an open, connected subset Θ of Euclidean space R(d). Because Θ does not have to be convex, this framework is able to handle a significantly wider class of situations than the original ISOMAP algorithm. The theoretical framework revolves around a quadratic form H(f) = ∫(M)||H(f)(m)||²(F)dm defined on functions f: M--> R. Here Hf denotes the Hessian of f, and H(f) averages the Frobenius norm of the Hessian over M. To define the Hessian, we use orthogonal coordinates on the tangent planes of M. The key observation is that, if M truly is locally isometric to an open, connected subset of R(d), then H(f) has a (d + 1)-dimensional null space consisting of the constant functions and a d-dimensional space of functions spanned by the original isometric coordinates. Hence, the isometric coordinates can be recovered up to a linear isometry. Our method may be viewed as a modification of locally linear embedding and our theoretical framework as a modification of the Laplacian eigenmaps framework, where we substitute a quadratic form based on the Hessian in place of one based on the Laplacian.
2,029 citations
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