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347
Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
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Cited by 341 (27 self)
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This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (antialiasing) prefilters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
A Massively Parallel Adaptive Finite Element Method with Dynamic Load Balancing
 Appl. Numer. Math
, 1993
"... We construct massively parallel adaptive finite element methods for the solution of hyperbolic conservation laws. Spatial discretization is performed by a discontinuous Galerkin finite element method using a basis of piecewise Legendre polynomials. Temporal discretization utilizes a RungeKutta meth ..."
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Cited by 110 (14 self)
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We construct massively parallel adaptive finite element methods for the solution of hyperbolic conservation laws. Spatial discretization is performed by a discontinuous Galerkin finite element method using a basis of piecewise Legendre polynomials. Temporal discretization utilizes a RungeKutta method. Dissipative fluxes and projection limiting prevent oscillations near solution discontinuities. The resulting method is of high order and may be parallelized efficiently on MIMD computers. We demonstrate parallel efficiency through computations on a 1024processor nCUBE/2 hypercube. We present results using adaptiverefinement to reduce the computational cost of the method, and tiling, a dynamic, elementbased data migration system that maintains global load balance of the adaptive method by overlapping neighborhoods of processors that each perform local balancing. 1. Introduction We are studying massively parallel adaptive finite element methods for solving systems ofdimensional hyper...
Modeling Holes and Inclusions by Level Sets in the Extended Finite Element Method
, 2000
"... A methodology to model arbitrary holes and material interfaces (inclusions) without meshing the internal boundaries is proposed. The numerical method couples the level set method (Osher and Sethian, 1988) to the eXtended Finite Element Method (XFEM) (Moes et al., 1999). In the XFEM, the finite ..."
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Cited by 73 (10 self)
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A methodology to model arbitrary holes and material interfaces (inclusions) without meshing the internal boundaries is proposed. The numerical method couples the level set method (Osher and Sethian, 1988) to the eXtended Finite Element Method (XFEM) (Moes et al., 1999). In the XFEM, the finite element approximation is enriched by additional functions through the notion of partition of unity. The level set method is used for representing the location of holes and material interfaces, and in addition, the level set function is used to develop the local enrichment for material interfaces. Numerical examples in 2dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique.
The Natural Element Method In Solid Mechanics
, 1998
"... The application of the Natural Element Method (NEM) (Traversoni, 1994; Braun and Sambridge, 1995) to boundary value problems in twodimensional small displacement elastostatics is presented. The discrete model of the domain \Omega consists of a set of distinct nodes N , and a polygonal descripti ..."
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Cited by 61 (14 self)
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The application of the Natural Element Method (NEM) (Traversoni, 1994; Braun and Sambridge, 1995) to boundary value problems in twodimensional small displacement elastostatics is presented. The discrete model of the domain \Omega consists of a set of distinct nodes N , and a polygonal description of the boundary @ In the Natural Element Method, the trial and test functions are constructed using natural neighbor interpolants. These interpolants are based on the Voronoi tessellation of the set of nodes N . The interpolants are smooth (C NEM is identical to linear finite elements. The NEM interpolant is strictly linear between adjacent nodes on the boundary of the convex hull, which facilitates imposition of essential boundary conditions. A methodology to model material discontinuities and nonconvex bodies (cracks) using NEM is also described.
Adaptive Multilevel Methods in Three Space Dimensions
 Int. J. Numer. Methods Eng
, 1993
"... this paper to collect wellknown results on 3D mesh refinement ..."
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Cited by 56 (13 self)
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this paper to collect wellknown results on 3D mesh refinement
A Posteriori Error Estimates for Elliptic Problems in Two and Three Space Dimensions
 SIAM J. Numer. Anal
, 1993
"... . Let u 2 H be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some ~ u 2 S, S being a suitable finite element space. Efficient and reliable a posteriori estimates of the error jj u \Gamma ~ u jj, measuring the (local) quality of ~ u, play a cruc ..."
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Cited by 52 (9 self)
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. Let u 2 H be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some ~ u 2 S, S being a suitable finite element space. Efficient and reliable a posteriori estimates of the error jj u \Gamma ~ u jj, measuring the (local) quality of ~ u, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A wellknown class of error estimates can be derived systematically by localizing the discretized defect problem using domain decomposition techniques. In the present paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations. Key words: adaptive finite element methods, aposteriori error estimates AMS (MOS) subject classifications: 65N30, 65N50, 65N55, 35J25 1 submi...
Adaptive Multilevel  Methods for Obstacle Problems
, 1992
"... We consider the discretization of obstacle problems for second order elliptic differential operators by piecewise linear finite elements. Assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies, the linear problems are solved iteratively by ..."
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Cited by 44 (8 self)
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We consider the discretization of obstacle problems for second order elliptic differential operators by piecewise linear finite elements. Assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies, the linear problems are solved iteratively by preconditioned cgiterations. The proposed preconditioners are treated theoretically as abstract additive Schwarz methods and are implemented as truncated hierarchical basis preconditioners. To allow for local mesh refinement we derive semilocal and local a posteriori error estimates, providing lower and upper estimates for the discretization error. The theoretical results are illustrated by numerical computations.
A posteriori error estimation techniques in practical finite element analysis
, 2005
"... In this paper we review the basic concepts to obtain a posteriori error estimates for the finite element solution of an elliptic linear model problem. We give the basic ideas to establish global error estimates for the energy norm as well as goaloriented error estimates. While we show how these er ..."
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Cited by 41 (5 self)
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In this paper we review the basic concepts to obtain a posteriori error estimates for the finite element solution of an elliptic linear model problem. We give the basic ideas to establish global error estimates for the energy norm as well as goaloriented error estimates. While we show how these error estimation techniques are employed for our simple model problem, the emphasis of the paper is on assessing whether these procedures are ready for use in practical linear finite element analysis. We conclude that the actually practical error estimation techniques do not provide mathematically proven bounds on the error and need to be used with care. The more accurate estimation procedures also do not provide proven bounds that, in general, can be computed efficiently. We also briefly comment upon the state of error estimations
A comparison of a posteriori error estimators for mixed finite element discretizations by raviartthomas elements
 MATH. COMP
, 1999
"... We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. In particular, we present and analyze four different kinds of error estimators: a resid ..."
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Cited by 36 (5 self)
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We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. In particular, we present and analyze four different kinds of error estimators: a residual based estimator, a hierarchical one, error estimators relying on the solution of local subproblems and on a superconvergence result, respectively. Finally, we examine the relationship between the presented error estimators and compare their local components.
A WAVELETOPTIMIZED, VERY HIGH ORDER ADAPTIVE GRID and ORDER NUMERICAL METHOD
, 1996
"... Diggerencing operators of arbitrarily high order can be constructed by interpolating a polynomial through a set of data followed by diggerentiation of this polynomial and nally evaluation of the polynomial at the point where a derivative approximation is desired. Furthermore, the interpolating polyn ..."
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Cited by 31 (2 self)
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Diggerencing operators of arbitrarily high order can be constructed by interpolating a polynomial through a set of data followed by diggerentiation of this polynomial and nally evaluation of the polynomial at the point where a derivative approximation is desired. Furthermore, the interpolating polynomial can be constructed from algebraic, trigonometric, or, perhaps exponential polynomials. This paper begins with a comparison of such diggerencing operator construction. Next, the issue of proper grids for high order polynomials is addressed. Finally, an adaptive numerical method is introduced which adapts the numerical grid and the order of the diggerencing operator depending on the data. The numerical grid adaptation is performed on a Chebyshev grid. That is, at each level of reginement the grid is a Chebyshev