A general framework of compactly supported splines and wavelets

Chui, Charles K.; Jian-zhong, Wang

doi:10.1016/0021-9045(92)90120-d

Cited by 114 publications

(48 citation statements)

References 7 publications

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“…Any multilevel basis for the one-particle space with a sufficient decay property may be used as basic ingredient for our sparse grid approach with similar results. Candidates are other hierarchical global polynomial systems [8,10,12,51,73] or function families with localization properties like wavelets [18], prewavelets [15,42], interpolets [19,22], and related wavelet-like constructs, see [12,16] for a survey. But also multiscale finite element systems and frames [37,40,41,67] or multiscale Gaussians [63] may be used 1,2 .…”

Section: Introductionmentioning

confidence: 99%

Sparse grids for the Schrödinger equation

Griebel¹,

Hamaekers²

2007

ESAIM: M2AN

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Abstract. We present a sparse grid/hyperbolic cross discretization for many-particle problems. It involves the tensor product of a one-particle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization. Here, depending on the norms involved, different variants of sparse grid techniques for many-particle spaces can be derived that, in the best case, result in complexities and error estimates which are independent of the number of particles. Furthermore we introduce an additional constraint which gives antisymmetric sparse grids which are suited to fermionic systems. We apply the antisymmetric sparse grid discretization to the electronic Schrödinger equation and compare costs, accuracy, convergence rates and scalability with respect to the number of electrons present in the system. Mathematics Subject Classification. 35J10, 65N25, 65N30, 65T40, 65Z05.

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Section: Introductionmentioning

confidence: 99%

Sparse grids for the Schrödinger equation

Griebel¹,

Hamaekers²

2007

ESAIM: M2AN

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“…fcez If cj) further satisfies the refinement equation (1), then it is desirable to characterize the orthogonality of the integer translates of (j) in terms of the mask b . To this end, we take Fourier-Laplace transforms of both sides of (1) and obtain (2) 0(C) = 4>(Z/2)b(e-«'2)/2 for all C € C, where b(z) is the symbol of b :…”

Section: Jrmentioning

confidence: 99%

Stability and Linear Independence Associated with Wavelet Decompositions

Jia¹,

Wang²

1993

Proceedings of the American Mathematical Society

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“…In the early 1990s, the two-band LPME arose in the study of the construction of compactly-supported wavelets [3][4][5][6][7]. In the construction of multi-wavelets, the LPME problems arise [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Euclidean Algorithm for Extension of Symmetric Laurent Polynomial Matrix and Its Application in Construction of Multiband Symmetric Perfect Reconstruction Filter Bank

Jian-zhong

2017

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Abstract:For a given pair of s-dimensional real Laurent polynomials ( a(z), b(z)), which has a certain type of symmetry and satisfies the dual condition b(z) T a(z) = 1, an s × s Laurent polynomial matrix A(z) (together with its inverse A −1 (z)) is called a symmetric Laurent polynomial matrix extension of the dual pair ( a(z), b(z)) if A(z) has similar symmetry, the inverse A −1 (Z) also is a Laurent polynomial matrix, the first column of A(z) is a(z) and the first row of A −1 (z) is ( b(z)) T . In this paper, we introduce the Euclidean symmetric division and the symmetric elementary matrices in the Laurent polynomial ring and reveal their relation. Based on the Euclidean symmetric division algorithm in the Laurent polynomial ring, we develop a novel and effective algorithm for symmetric Laurent polynomial matrix extension. We also apply the algorithm in the construction of multi-band symmetric perfect reconstruction filter banks.

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A general framework of compactly supported splines and wavelets

Cited by 114 publications

References 7 publications

Sparse grids for the Schrödinger equation

Sparse grids for the Schrödinger equation

Stability and Linear Independence Associated with Wavelet Decompositions

Euclidean Algorithm for Extension of Symmetric Laurent Polynomial Matrix and Its Application in Construction of Multiband Symmetric Perfect Reconstruction Filter Bank

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