95 research outputs found
Hardware acceleration of reaction-diffusion systems:a guide to optimisation of pattern formation algorithms using OpenACC
Reaction Diffusion Systems (RDS) have widespread applications in computational ecology, biology, computer graphics and the visual arts. For the former applications a major barrier to the development of effective simulation models is their computational complexity - it takes a great deal of processing power to simulate enough replicates such that reliable conclusions can be drawn. Optimizing the computation is thus highly desirable in order to obtain more results with less resources. Existing optimizations of RDS tend to be low-level and GPGPU based. Here we apply the higher-level OpenACC framework to two case studies: a simple RDS to learn the ‘workings’ of OpenACC and a more realistic and complex example. Our results show that simple parallelization directives and minimal data transfer can produce a useful performance improvement. The relative simplicity of porting OpenACC code between heterogeneous hardware is a key benefit to the scientific computing community in terms of speed-up and portability
A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces
The closest point method (Ruuth and Merriman, J. Comput. Phys.
227(3):1943-1961, [2008]) is an embedding method developed to solve a variety
of partial differential equations (PDEs) on smooth surfaces, using a closest
point representation of the surface and standard Cartesian grid methods in the
embedding space. Recently, a closest point method with explicit time-stepping
was proposed that uses finite differences derived from radial basis functions
(RBF-FD). Here, we propose a least-squares implicit formulation of the closest
point method to impose the constant-along-normal extension of the solution on
the surface into the embedding space. Our proposed method is particularly
flexible with respect to the choice of the computational grid in the embedding
space. In particular, we may compute over a computational tube that contains
problematic nodes. This fact enables us to combine the proposed method with the
grid based particle method (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024,
[2009]) to obtain a numerical method for approximating PDEs on moving surfaces.
We present a number of examples to illustrate the numerical convergence
properties of our proposed method. Experiments for advection-diffusion
equations and Cahn-Hilliard equations that are strongly coupled to the velocity
of the surface are also presented
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GRADE: Gibbs Reaction and Diffusion Equations
Recently there have been increasing interests in using nonlinear PDEs for applications in computer vision and image processing. In this paper, we propose a general statistical framework for designing a new class of PDEs. For a given application, a Markov random field model is learned according to the minimax entropy principle so that should characterize the ensemble of images in our application. is a Gibbs distribution whose energy terms can be divided into two categories. Subsequently the partial differential equations given by gradient descent on the Gibbs potential are essentially reaction-diffusion equations, where the energy terms in one category produce anisotropic diffusion while the inverted energy terms in the second category produce reaction associated with pattern formation. We call this new class of PDEs the Gibbs Reaction And Diffusion Equations-GRADE and we demonstrate experiments where GRADE are used for texture pattern formation, denoising, image enhancement, and clutter removal.Mathematic
A narrow-band unfitted finite element method for elliptic PDEs posed on surfaces
The paper studies a method for solving elliptic partial differential
equations posed on hypersurfaces in , . The method allows
a surface to be given implicitly as a zero level of a level set function. A
surface equation is extended to a narrow-band neighborhood of the surface. The
resulting extended equation is a non-degenerate PDE and it is solved on a bulk
mesh that is unaligned to the surface. An unfitted finite element method is
used to discretize extended equations. Error estimates are proved for finite
element solutions in the bulk domain and restricted to the surface. The
analysis admits finite elements of a higher order and gives sufficient
conditions for archiving the optimal convergence order in the energy norm.
Several numerical examples illustrate the properties of the method.Comment: arXiv admin note: text overlap with arXiv:1301.470
A Meshfree Generalized Finite Difference Method for Surface PDEs
In this paper, we propose a novel meshfree Generalized Finite Difference
Method (GFDM) approach to discretize PDEs defined on manifolds. Derivative
approximations for the same are done directly on the tangent space, in a manner
that mimics the procedure followed in volume-based meshfree GFDMs. As a result,
the proposed method not only does not require a mesh, it also does not require
an explicit reconstruction of the manifold. In contrast to existing methods, it
avoids the complexities of dealing with a manifold metric, while also avoiding
the need to solve a PDE in the embedding space. A major advantage of this
method is that all developments in usual volume-based numerical methods can be
directly ported over to surfaces using this framework. We propose
discretizations of the surface gradient operator, the surface Laplacian and
surface Diffusion operators. Possibilities to deal with anisotropic and
discontinous surface properties (with large jumps) are also introduced, and a
few practical applications are presented
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