1,417 research outputs found
Shenfun -- automating the spectral Galerkin method
With the shenfun Python module (github.com/spectralDNS/shenfun) an effort is
made towards automating the implementation of the spectral Galerkin method for
simple tensor product domains, consisting of (currently) one non-periodic and
any number of periodic directions. The user interface to shenfun is
intentionally made very similar to FEniCS (fenicsproject.org). Partial
Differential Equations are represented through weak variational forms and
solved using efficient direct solvers where available. MPI decomposition is
achieved through the {mpi4py-fft} module (bitbucket.org/mpi4py/mpi4py-fft), and
all developed solver may, with no additional effort, be run on supercomputers
using thousands of processors. Complete solvers are shown for the linear
Poisson and biharmonic problems, as well as the nonlinear and time-dependent
Ginzburg-Landau equation.Comment: Presented at MekIT'17, the 9th National Conference on Computational
Mechanic
A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity
The richness of natural images makes the quest for optimal representations in
image processing and computer vision challenging. The latter observation has
not prevented the design of image representations, which trade off between
efficiency and complexity, while achieving accurate rendering of smooth regions
as well as reproducing faithful contours and textures. The most recent ones,
proposed in the past decade, share an hybrid heritage highlighting the
multiscale and oriented nature of edges and patterns in images. This paper
presents a panorama of the aforementioned literature on decompositions in
multiscale, multi-orientation bases or dictionaries. They typically exhibit
redundancy to improve sparsity in the transformed domain and sometimes its
invariance with respect to simple geometric deformations (translation,
rotation). Oriented multiscale dictionaries extend traditional wavelet
processing and may offer rotation invariance. Highly redundant dictionaries
require specific algorithms to simplify the search for an efficient (sparse)
representation. We also discuss the extension of multiscale geometric
decompositions to non-Euclidean domains such as the sphere or arbitrary meshed
surfaces. The etymology of panorama suggests an overview, based on a choice of
partially overlapping "pictures". We hope that this paper will contribute to
the appreciation and apprehension of a stream of current research directions in
image understanding.Comment: 65 pages, 33 figures, 303 reference
Sparse Modelling and Multi-exponential Analysis
The research fields of harmonic analysis, approximation theory and computer algebra are seemingly different domains and are studied by seemingly separated research communities. However, all of these are connected to each other in many ways. The connection between harmonic analysis and approximation theory is not accidental: several constructions among which wavelets and Fourier series, provide major insights into central problems in approximation theory. And the intimate connection between approximation theory and computer algebra exists even longer: polynomial interpolation is a long-studied and important problem in both symbolic and numeric computing, in the former to counter expression swell and in the latter to construct a simple data model. A common underlying problem statement in many applications is that of determining the number of components, and for each component the value of the frequency, damping factor, amplitude and phase in a multi-exponential model. It occurs, for instance, in magnetic resonance and infrared spectroscopy, vibration analysis, seismic data analysis, electronic odour recognition, keystroke recognition, nuclear science, music signal processing, transient detection, motor fault diagnosis, electrophysiology, drug clearance monitoring and glucose tolerance testing, to name just a few. The general technique of multi-exponential modeling is closely related to what is commonly known as the Padé-Laplace method in approximation theory, and the technique of sparse interpolation in the field of computer algebra. The problem statement is also solved using a stochastic perturbation method in harmonic analysis. The problem of multi-exponential modeling is an inverse problem and therefore may be severely ill-posed, depending on the relative location of the frequencies and phases. Besides the reliability of the estimated parameters, the sparsity of the multi-exponential representation has become important. A representation is called sparse if it is a combination of only a few elements instead of all available generating elements. In sparse interpolation, the aim is to determine all the parameters from only a small amount of data samples, and with a complexity proportional to the number of terms in the representation. Despite the close connections between these fields, there is a clear lack of communication in the scientific literature. The aim of this seminar is to bring researchers together from the three mentioned fields, with scientists from the varied application domains.Output Type: Meeting Repor
Efficient algorithms for the fast computation of space charge effects caused by charged particles in particle accelerators
In this dissertation, a Poisson solver is improved with three parts: the efficient integrated Green's function; the discrete cosine transform of the efficient integrated Green's function values; the implicitly zero-padded fast Fourier transform for charge density. In addition, the high performance computing technology is utilized for the further improvement of efficiency, such as: OpenMP API, OpenMP+CUDA, MPI, and MPI+OpenMP parallelizations. The examples and simulation results are matched with the results of the commonly used Poisson solver to demonstrate the accuracy performance
Computing the Newton Potential in the Boundary Integral Equation for the Dirichlet Problem of the Poisson Equation
Evaluating the Newton potential is crucial for efficiently solving the boundary integral equation of the Dirichlet boundary value problem of the Poisson equation. In the context of the Fourier-Garlerkin method for solving the boundary integral equation, we propose a fast algorithm for evaluating Fourier coefficients of the Newton potential by using a sparse grid approximation. When the forcing function of the Poisson equation expressed in the polar coordinates has mth-order bounded mixed derivatives, the proposed algorithm achieves an accuracy of order (n-m log3 n), with requiring (n log2 n) number of arithmetics for the computation, where n is the number of quadrature points used in one coordinate direction. With the help of this algorithm, the boundary integral equation derived from the Poisson equation can be efficiently solved by a fast fully discrete Fourier-Garlerkin method
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