14,728 research outputs found
New Iterative Method Based on Laplace Decomposition Algorithm
We introduce a new form of Laplace decomposition algorithm (LDA). By this form a new iterative method was achieved in which there is no need to calculate Adomian polynomials, which require so much computational time for higher-order approximations. We have implemented this method for the solutions of different types of nonlinear pantograph equations to support the proposed analysis
Boundary Element Methods for the Laplace Hypersingular Integral Equation on Multiscreens: a two-level Substructuring Preconditioner
We present a preconditioning method for the linear systems arising from the
boundary element discretization of the Laplace hypersingular equation on a
-dimensional triangulated surface in . We allow
to belong to a large class of geometries that we call polygonal
multiscreens, which can be non-manifold. After introducing a new, simple
conforming Galerkin discretization, we analyze a substructuring
domain-decomposition preconditioner based on ideas originally developed for the
Finite Element Method. The surface is subdivided into non-overlapping
regions, and the application of the preconditioner is obtained via the solution
of the hypersingular equation on each patch, plus a coarse subspace correction.
We prove that the condition number of the preconditioned linear system grows
poly-logarithmically with , the ratio of the coarse mesh and fine mesh
size, and our numerical results indicate that this bound is sharp. This
domain-decomposition algorithm therefore guarantees significant speedups for
iterative solvers, even when a large number of subdomains is used
A Time-Dependent Dirichlet-Neumann Method for the Heat Equation
We present a waveform relaxation version of the Dirichlet-Neumann method for
parabolic problem. Like the Dirichlet-Neumann method for steady problems, the
method is based on a non-overlapping spatial domain decomposition, and the
iteration involves subdomain solves with Dirichlet boundary conditions followed
by subdomain solves with Neumann boundary conditions. However, each subdomain
problem is now in space and time, and the interface conditions are also
time-dependent. Using a Laplace transform argument, we show for the heat
equation that when we consider finite time intervals, the Dirichlet-Neumann
method converges, similar to the case of Schwarz waveform relaxation
algorithms. The convergence rate depends on the length of the subdomains as
well as the size of the time window. In this discussion, we only stick to the
linear bound. We illustrate our results with numerical experiments.Comment: 9 pages, 5 figures, Lecture Notes in Computational Science and
Engineering, Vol. 98, Springer-Verlag 201
2-D Prony-Huang Transform: A New Tool for 2-D Spectral Analysis
This work proposes an extension of the 1-D Hilbert Huang transform for the
analysis of images. The proposed method consists in (i) adaptively decomposing
an image into oscillating parts called intrinsic mode functions (IMFs) using a
mode decomposition procedure, and (ii) providing a local spectral analysis of
the obtained IMFs in order to get the local amplitudes, frequencies, and
orientations. For the decomposition step, we propose two robust 2-D mode
decompositions based on non-smooth convex optimization: a "Genuine 2-D"
approach, that constrains the local extrema of the IMFs, and a "Pseudo 2-D"
approach, which constrains separately the extrema of lines, columns, and
diagonals. The spectral analysis step is based on Prony annihilation property
that is applied on small square patches of the IMFs. The resulting 2-D
Prony-Huang transform is validated on simulated and real data.Comment: 24 pages, 7 figure
A multidomain spectral method for solving elliptic equations
We present a new solver for coupled nonlinear elliptic partial differential
equations (PDEs). The solver is based on pseudo-spectral collocation with
domain decomposition and can handle one- to three-dimensional problems. It has
three distinct features. First, the combined problem of solving the PDE,
satisfying the boundary conditions, and matching between different subdomains
is cast into one set of equations readily accessible to standard linear and
nonlinear solvers. Second, touching as well as overlapping subdomains are
supported; both rectangular blocks with Chebyshev basis functions as well as
spherical shells with an expansion in spherical harmonics are implemented.
Third, the code is very flexible: The domain decomposition as well as the
distribution of collocation points in each domain can be chosen at run time,
and the solver is easily adaptable to new PDEs. The code has been used to solve
the equations of the initial value problem of general relativity and should be
useful in many other problems. We compare the new method to finite difference
codes and find it superior in both runtime and accuracy, at least for the
smooth problems considered here.Comment: 31 pages, 8 figure
- …