19,287 research outputs found
Bootstrap Multigrid for the Laplace-Beltrami Eigenvalue Problem
This paper introduces bootstrap two-grid and multigrid finite element
approximations to the Laplace-Beltrami (surface Laplacian) eigen-problem on a
closed surface. The proposed multigrid method is suitable for recovering
eigenvalues having large multiplicity, computing interior eigenvalues, and
approximating the shifted indefinite eigen-problem. Convergence analysis is
carried out for a simplified two-grid algorithm and numerical experiments are
presented to illustrate the basic components and ideas behind the overall
bootstrap multigrid approach
A Random Matrix Model of Adiabatic Quantum Computing
We present an analysis of the quantum adiabatic algorithm for solving hard
instances of 3-SAT (an NP-complete problem) in terms of Random Matrix Theory
(RMT). We determine the global regularity of the spectral fluctuations of the
instantaneous Hamiltonians encountered during the interpolation between the
starting Hamiltonians and the ones whose ground states encode the solutions to
the computational problems of interest. At each interpolation point, we
quantify the degree of regularity of the average spectral distribution via its
Brody parameter, a measure that distinguishes regular (i.e., Poissonian) from
chaotic (i.e., Wigner-type) distributions of normalized nearest-neighbor
spacings. We find that for hard problem instances, i.e., those having a
critical ratio of clauses to variables, the spectral fluctuations typically
become irregular across a contiguous region of the interpolation parameter,
while the spectrum is regular for easy instances. Within the hard region, RMT
may be applied to obtain a mathematical model of the probability of avoided
level crossings and concomitant failure rate of the adiabatic algorithm due to
non-adiabatic Landau-Zener type transitions. Our model predicts that if the
interpolation is performed at a uniform rate, the average failure rate of the
quantum adiabatic algorithm, when averaged over hard problem instances, scales
exponentially with increasing problem size.Comment: 9 pages, 7 figure
Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and Arnoldi-Chebyshev algorithm
An efficient way of solving 2D stability problems in fluid mechanics is to
use, after discretization of the equations that cast the problem in the form of
a generalized eigenvalue problem, the incomplete Arnoldi-Chebyshev method. This
method preserves the banded structure sparsity of matrices of the algebraic
eigenvalue problem and thus decreases memory use and CPU-time consumption.
The errors that affect computed eigenvalues and eigenvectors are due to the
truncation in the discretization and to finite precision in the computation of
the discretized problem. In this paper we analyze those two errors and the
interplay between them. We use as a test case the two-dimensional eigenvalue
problem yielded by the computation of inertial modes in a spherical shell. This
problem contains many difficulties that make it a very good test case. It turns
out that that single modes (especially most-damped modes i.e. with high spatial
frequency) can be very sensitive to round-off errors, even when apparently good
spectral convergence is achieved. The influence of round-off errors is analyzed
using the spectral portrait technique and by comparison of double precision and
extended precision computations. Through the analysis we give practical recipes
to control the truncation and round-off errors on eigenvalues and eigenvectors.Comment: 15 pages, 9 figure
Solving eigenvalue problems on curved surfaces using the Closest Point Method
Eigenvalue problems are fundamental to mathematics and science. We present a
simple algorithm for determining eigenvalues and eigenfunctions of the
Laplace--Beltrami operator on rather general curved surfaces. Our algorithm,
which is based on the Closest Point Method, relies on an embedding of the
surface in a higher-dimensional space, where standard Cartesian finite
difference and interpolation schemes can be easily applied. We show that there
is a one-to-one correspondence between a problem defined in the embedding space
and the original surface problem. For open surfaces, we present a simple way to
impose Dirichlet and Neumann boundary conditions while maintaining second-order
accuracy. Convergence studies and a series of examples demonstrate the
effectiveness and generality of our approach
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