1,150 research outputs found
Eigenvalues and eigenfunctions of the Laplacian via inverse iteration with shift
In this paper we present an iterative method, inspired by the inverse
iteration with shift technique of finite linear algebra, designed to find the
eigenvalues and eigenfunctions of the Laplacian with homogeneous Dirichlet
boundary condition for arbitrary bounded domains . This
method, which has a direct functional analysis approach, does not approximate
the eigenvalues of the Laplacian as those of a finite linear operator. It is
based on the uniform convergence away from nodal surfaces and can produce a
simple and fast algorithm for computing the eigenvalues with minimal
computational requirements, instead of using the ubiquitous Rayleigh quotient
of finite linear algebra. Also, an alternative expression for the Rayleigh
quotient in the associated infinite dimensional Sobolev space which avoids the
integration of gradients is introduced and shown to be more efficient. The
method can also be used in order to produce the spectral decomposition of any
given function .Comment: In this version the numerical tests in Section 6 were considerably
improved and the Section 5 entitled "Normalization at each step" was
introduced. Moreover, minor adjustments in the Section 1 (Introduction) and
in the Section 7 (Fi nal Comments) were made. Breno Loureiro Giacchini was
added as coautho
Computing the first eigenpair of the p-Laplacian via inverse iteration of sublinear supersolutions
We introduce an iterative method for computing the first eigenpair
for the -Laplacian operator with homogeneous Dirichlet
data as the limit of as , where
is the positive solution of the sublinear Lane-Emden equation
with same boundary data. The method is
shown to work for any smooth, bounded domain. Solutions to the Lane-Emden
problem are obtained through inverse iteration of a super-solution which is
derived from the solution to the torsional creep problem. Convergence of
to is in the -norm and the rate of convergence of
to is at least . Numerical evidence is
presented.Comment: Section 5 was rewritten. Jed Brown was added as autho
Quantitative Anderson localization of Schr\"odinger eigenstates under disorder potentials
This paper concerns spectral properties of linear Schr\"odinger operators
under oscillatory high-amplitude potentials on bounded domains. Depending on
the degree of disorder, we prove the existence of spectral gaps amongst the
lowermost eigenvalues and the emergence of exponentially localized states. We
quantify the rate of decay in terms of geometric parameters that characterize
the potential. The proofs are based on the convergence theory of iterative
solvers for eigenvalue problems and their optimal local preconditioning by
domain decomposition.Comment: accepted for publication in M3A
Approximate Inference in Continuous Determinantal Point Processes
Determinantal point processes (DPPs) are random point processes well-suited
for modeling repulsion. In machine learning, the focus of DPP-based models has
been on diverse subset selection from a discrete and finite base set. This
discrete setting admits an efficient sampling algorithm based on the
eigendecomposition of the defining kernel matrix. Recently, there has been
growing interest in using DPPs defined on continuous spaces. While the
discrete-DPP sampler extends formally to the continuous case, computationally,
the steps required are not tractable in general. In this paper, we present two
efficient DPP sampling schemes that apply to a wide range of kernel functions:
one based on low rank approximations via Nystrom and random Fourier feature
techniques and another based on Gibbs sampling. We demonstrate the utility of
continuous DPPs in repulsive mixture modeling and synthesizing human poses
spanning activity spaces
Steklov Spectral Geometry for Extrinsic Shape Analysis
We propose using the Dirichlet-to-Neumann operator as an extrinsic
alternative to the Laplacian for spectral geometry processing and shape
analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator,
cannot capture the spatial embedding of a shape up to rigid motion, and many
previous extrinsic methods lack theoretical justification. Instead, we consider
the Steklov eigenvalue problem, computing the spectrum of the
Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable
property of this operator is that it completely encodes volumetric geometry. We
use the boundary element method (BEM) to discretize the operator, accelerated
by hierarchical numerical schemes and preconditioning; this pipeline allows us
to solve eigenvalue and linear problems on large-scale meshes despite the
density of the Dirichlet-to-Neumann discretization. We further demonstrate that
our operators naturally fit into existing frameworks for geometry processing,
making a shift from intrinsic to extrinsic geometry as simple as substituting
the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator.Comment: Additional experiments adde
Compressive Wave Computation
This paper considers large-scale simulations of wave propagation phenomena.
We argue that it is possible to accurately compute a wavefield by decomposing
it onto a largely incomplete set of eigenfunctions of the Helmholtz operator,
chosen at random, and that this provides a natural way of parallelizing wave
simulations for memory-intensive applications.
This paper shows that L1-Helmholtz recovery makes sense for wave computation,
and identifies a regime in which it is provably effective: the one-dimensional
wave equation with coefficients of small bounded variation. Under suitable
assumptions we show that the number of eigenfunctions needed to evolve a sparse
wavefield defined on N points, accurately with very high probability, is
bounded by C log(N) log(log(N)), where C is related to the desired accuracy and
can be made to grow at a much slower rate than N when the solution is sparse.
The PDE estimates that underlie this result are new to the authors' knowledge
and may be of independent mathematical interest; they include an L1 estimate
for the wave equation, an estimate of extension of eigenfunctions, and a bound
for eigenvalue gaps in Sturm-Liouville problems.
Numerical examples are presented in one spatial dimension and show that as
few as 10 percents of all eigenfunctions can suffice for accurate results.
Finally, we argue that the compressive viewpoint suggests a competitive
parallel algorithm for an adjoint-state inversion method in reflection
seismology.Comment: 45 pages, 4 figure
Semiclassical bounds for spectra of biharmonic operators
We provide complementary semiclassical bounds for the Riesz means of
the eigenvalues of various biharmonic operators, with a second term in the
expected power of . The method we discuss makes use of the averaged
variational principle (AVP), and yields two-sided bounds for individual
eigenvalues, which are semiclassically sharp. The AVP also yields comparisons
with Riesz means of different operators, in particular Laplacians
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