Spectral clustering methods use eigenvectors of a matrix, called Gaussian affinity matrix, in order to define a low-dimensional space in which data points can be clustered. This matrix is widely used and depends on a free parameter σ. It is usually interpreted as some discretization of the Heat Equation Green kernel. Combining tools from Partial Differential Equations and Finite Elements theory...

This paper develops and analyzes finite element Galerkin and spectral Galerkin methods for approximating viscosity solutions of the fully nonlinear Monge-Ampère equation det(D2u0) = f (> 0) based on the vanishing moment method which was developed by the authors in [17, 15]. In this approach, the Monge-Ampère equation is approximated by the fourth order quasilinear equation −ε∆2uε + det D2uε = f...

Spectral element methods (SEM) are superior to general finite element methods (FEM) in achieving high order accuracy through p-type refinement. Owing to orthogonal polynomials in both expansion and test functions, the discretization errors in SEM could be reduced exponentially to machine zero so that the spectral convergence rate can be achieved. Inherited the advantage of FEM, SEM can enhance ...

We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are considered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the origin...