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Fractal Functions of Discontinuous Approximation
"... Abstract: A procedure for the definition of discontinuous real functions is developed, based on a fractal methodology. For this purpose, a binary operation in the space of bounded functions on an interval is established. Two functions give rise to a new one, called in the paper fractal convolution o ..."
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of the originals, whose graph is discontinuous and has a fractal structure in general. The new function approximates one of the chosen pair and, under certain conditions, is continuous. The convolution is used for the definition of discontinuous bases of the space of square integrable functions, whose elements
On the Power of Discontinuous Approximate Computations
, 1992
"... The set of operations S 1 = f+; \Gamma; ; =; ?g is used in algebraic computations to avoid degeneracies (e.g., division by zero), but is also used in numerical computations to avoid huge roundoff errors (e.g., division by a small quantity). On the other hand, the classes of algorithms using operatio ..."
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, and S 3 for computing with approximations. We prove that S 2 does very efficiently simulate S 1 , while S 3 does not; this
Interior penalty discontinuous approximations of elliptic problems
, 2000
"... Abstract. This paper studies an interior penalty discontinuous approximation of elliptic problems on non–matching grids. Error analysis, interface domain decomposition type preconditioners, as well as numerical results illustrating both, discretization errors and condition number estimates of the pr ..."
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Cited by 5 (2 self)
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Abstract. This paper studies an interior penalty discontinuous approximation of elliptic problems on non–matching grids. Error analysis, interface domain decomposition type preconditioners, as well as numerical results illustrating both, discretization errors and condition number estimates
The RungeKutta discontinuous Galerkin method for conservation laws V: multidimensional systems
, 1997
"... This is the fifth paper in a series in which we construct and study the socalled RungeKutta Discontinuous Galerkin method for numerically solving hyperbolic conservation laws. In this paper, we extend the method to multidimensional nonlinear systems of conservation laws. The algorithms are describ ..."
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Cited by 508 (43 self)
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This is the fifth paper in a series in which we construct and study the socalled RungeKutta Discontinuous Galerkin method for numerically solving hyperbolic conservation laws. In this paper, we extend the method to multidimensional nonlinear systems of conservation laws. The algorithms
Linear and discontinuous approximations for optimal control problems
 Numer. Funct. Anal. Optim
"... Abstract. An optimal control problem for 2d and 3d elliptic equations is investigated with pointwise control constraints. This paper is concerned with discretization of the control by piecewise linear, but discontinuous functions. The state and the adjoint state are discretized by linear finite e ..."
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Cited by 2 (1 self)
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Abstract. An optimal control problem for 2d and 3d elliptic equations is investigated with pointwise control constraints. This paper is concerned with discretization of the control by piecewise linear, but discontinuous functions. The state and the adjoint state are discretized by linear finite
The Contourlet Transform: An Efficient Directional Multiresolution Image Representation
 IEEE TRANSACTIONS ON IMAGE PROCESSING
"... The limitations of commonly used separable extensions of onedimensional transforms, such as the Fourier and wavelet transforms, in capturing the geometry of image edges are well known. In this paper, we pursue a “true” twodimensional transform that can capture the intrinsic geometrical structure t ..."
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Cited by 519 (20 self)
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link between the developed filter bank and the associated continuousdomain contourlet expansion via a directional multiresolution analysis framework. We show that with parabolic scaling and sufficient directional vanishing moments, contourlets achieve the optimal approximation rate for piecewise smooth
Almost Optimal Interior Penalty Discontinuous Approximations Of Symmetric Elliptic Problems On NonMatching Grids
 Numer. Math
, 2001
"... . We consider an interior penalty discontinuous approximation for symmetric elliptic problems of second order on nonmatching grids in this paer. The main result is an almost optimal error estimate for the interior penalty approximation of the original problem based on the partition of the domain i ..."
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Cited by 3 (0 self)
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. We consider an interior penalty discontinuous approximation for symmetric elliptic problems of second order on nonmatching grids in this paer. The main result is an almost optimal error estimate for the interior penalty approximation of the original problem based on the partition of the domain
New tight frames of curvelets and optimal representations of objects with piecewise C² singularities
 COMM. ON PURE AND APPL. MATH
, 2002
"... This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along C2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needleshap ..."
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Cited by 429 (21 self)
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for discontinuities along C 2 curves and is essentially optimal. In comparison, the squared error of nterm wavelet approximations only converges as n −1 as n → ∞, which is considerably worst than the optimal behavior.
Meshless methods: An overview and recent developments
 Computer Methods in Applied Mechanics and Engineering
, 1996
"... Meshless approximations based on moving least squares, kernels, and partitions of unity are examined. It is shown that the three methods are in most cases identical except for the important fact that partitions of unity enablepadaptivity to be achieved. Methods for constructing discontinuous approx ..."
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Cited by 334 (18 self)
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Meshless approximations based on moving least squares, kernels, and partitions of unity are examined. It is shown that the three methods are in most cases identical except for the important fact that partitions of unity enablepadaptivity to be achieved. Methods for constructing discontinuous
Results 1  10
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347,509