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Adaptive Solution Of Partial Differential Equations In Multiwavelet Bases
 J. Comput. Phys
, 1999
"... . Representations of derivative and exponential operators, with linear boundary conditions, are constructed in multiwavelet bases, leading to a simple adaptive scheme for the solution of nonlinear timedependent partial differential equations. The emphasis on hierarchical representations of function ..."
Abstract

Cited by 36 (5 self)
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. Representations of derivative and exponential operators, with linear boundary conditions, are constructed in multiwavelet bases, leading to a simple adaptive scheme for the solution of nonlinear timedependent partial differential equations. The emphasis on hierarchical representations of functions on intervals helps to address issues both of highorder approximation and of efficient application of integral operators, and the lack of regularity of multiwavelets does not preclude their use in representing differential operators. Comparisons with finite difference, finite element, and spectral element methods are presented, as are numerical examples with the heat equation and Burgers' equation. Key words. adaptive techniques, Burgers' equation, exact linear part, highorder approximation, integrodifferential operators, Legendre polynomials, Runge phenomenon AMS subject classifications. 65D15, 65M60, 65M70, 65N30, 65N35 1. Introduction. In this paper we construct representations of oper...
MULTIRESOLUTION REPRESENTATION OF OPERATORS WITH BOUNDARY CONDITIONS ON SIMPLE DOMAINS
"... Abstract. We develop a multiresolution representation of a class of integral operators satisfying boundary conditions on simple domains in order to construct fast algorithms for their application. We also elucidate some delicate theoretical issues related to the construction of periodic Green’s func ..."
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Cited by 2 (1 self)
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Abstract. We develop a multiresolution representation of a class of integral operators satisfying boundary conditions on simple domains in order to construct fast algorithms for their application. We also elucidate some delicate theoretical issues related to the construction of periodic Green’s functions for Poisson’s equation. By applying the method of images to the nonstandard form of the free space operator, we obtain lattice sums that converge absolutely on all scales, except possibly on the coarsest scale. On the coarsest scale the lattice sums may be only conditionally convergent and, thus, allow for some freedom in their definition. We use the limit of square partial sums as a definition of the limit and obtain a systematic, simple approach to the construction (in any dimension) of periodized operators with sparse nonstandard forms. We illustrate the results on several examples in dimensions one and three: the Hilbert transform, the projector on divergence free functions, the nonoscillatory Helmholtz Green’s function and the Poisson operator. Remarkably, the limit of square partial sums yields a periodic Poisson Green’s function which is not a convolution. Using a short sum of decaying Gaussians to approximate periodic Green’s functions, we arrive at fast algorithms for their application. We further show that the results obtained for operators with periodic boundary conditions extend to operators with Dirichlet, Neumann, or mixed boundary conditions. 1.
Fast electrostatic halftoning
, 2011
"... Electrostatic halftoning is a high quality method for stippling, dithering, and sampling, but it suffers from a high runtime. This made it difficult to use this technique for most realworld applications. A recently proposed minimisation scheme based on the nonequispaced fast Fourier transform (NFF ..."
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Electrostatic halftoning is a high quality method for stippling, dithering, and sampling, but it suffers from a high runtime. This made it difficult to use this technique for most realworld applications. A recently proposed minimisation scheme based on the nonequispaced fast Fourier transform (NFFT) lowers the complexity in the particle number M from O(M 2) to O(M log M). However, the NFFT is hard to parallelise, and the runtime on modern CPUs lies still in the orders of an hour for about 50’000 particles, to a day for 1 million particles. Our contributions to remedy this problem are threefold: We design the first GPUbased NFFT algorithm without special structural assumptions on the positions of nodes, we introduce a novel nearestneighbour identification scheme for continuous point distributions, and we optimise the whole algorithm for nbody problems such as electrostatic halftoning. For 1 million particles, this new algorithm runs 50 times
unknown title
"... operators in multiwavelet bases We review some recent results on multiwavelet methods for solving integral and partial differential equations and present an efficient representation of operators using discontinuous multiwavelet bases, including the case for singular integral operators. Numerical cal ..."
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operators in multiwavelet bases We review some recent results on multiwavelet methods for solving integral and partial differential equations and present an efficient representation of operators using discontinuous multiwavelet bases, including the case for singular integral operators. Numerical calculus using these representations produces fast O(N) methods for multiscale solution of integral equations when combined with low separation rank methods. Using this formulation, we compute the Hilbert transform and solve the Poisson and Schrödinger equations. For a fixed order of multiwavelets and for arbitrary but finiteprecision computations, the computational complexity is O(N). The computational structures are similar to fast multipole methods but are more generic in yielding fast O(N) algorithm development.
unknown title
, 2006
"... www.elsevier.com/locate/acha Multiresolution separated representations of singular and weakly singular operators ✩ ..."
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www.elsevier.com/locate/acha Multiresolution separated representations of singular and weakly singular operators ✩