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## A multiresolution approach to regularization of singular operators and fast summation (2002)

Venue: | SIAM J. Sci. Comp |

Citations: | 6 - 2 self |

### Citations

2927 |
Classical Electrodynamics
- Jackson
- 1962
(Show Context)
Citation Context ...rnels also includes freespace Green’s functions for the Poisson problem in two and three dimensions, namely, K(x, y) = log(x2 + y2)−1/2 and K(x, y, z) = (x2 + y2 + z2)−1/2. The expression (see, e.g., =-=[12]-=-) Φ(r, φ, θ) = 1 4π ∫ Φ(r′, φ′, θ′) r′(r2 − r′2) (r2 + r′ 2 − 2rr′ cos γ)3/2 dΩ ′ , which involves a homogeneous kernel, gives the electrostatic or gravitational potential outside a sphere of radius r... |

2193 | Orthonormal bases of compactly supported wavelets
- Daubechies
- 1988
(Show Context)
Citation Context ...al compactly supported scaling functions, and we will consider applications using them elsewhere. We may also choose to use a biorthogonal system, which involves a pair of dual scaling functions (see =-=[8]-=-). If both scaling functions are compactly supported, then there is no difference from the case of a single scaling function described above. If one of the scaling functions is not compactly supported... |

1146 | A fast algorithm for particle simulations
- Greengard, Rokhlin
- 1987
(Show Context)
Citation Context ...t the particle locations {xn}, and the function g(x) is well defined provided that x = xn for n = 1, . . . , N . Such sums MULTIRESOLUTION REGULARIZATION 83 can be computed using FMM-type algorithms =-=[11]-=-, [6]. Alternatively, the sum can be interpreted as an integral with a hypersingular kernel, regularized using our approach, then evaluated using a fast algorithm. This application is developed in sec... |

1036 |
An Introduction to Wavelets
- Chui
- 1992
(Show Context)
Citation Context ...tion, and we indicate below how to handle this case using the B-splines as an example. The B-splines do not form an orthogonal basis but may be considered as part of a biorthogonal system (see, e.g., =-=[7]-=-, [8]). Let us choose a dual scaling function that forms a basis for the same subspace as that spanned by the B-splines but is not compactly supported. This leads to an infinite matrix in the regulari... |

709 |
Théorie des distributions
- Schwartz
- 1966
(Show Context)
Citation Context ...eralized function, operating on a class of test functions. The origin of the mathematical treatment of generalized functions (distributions) goes back to the theory introduced by Schwartz (see, e.g., =-=[16]-=-). Such a functional, appropriately constructed, provides the definition for the classical regularization. We consider a natural regularization (see [10] or section 1). We note that divergent integral... |

541 |
Fast wavelet transforms and numerical algorithms
- Beylkin, Coifman, et al.
- 1991
(Show Context)
Citation Context ...nt bases we would of course obtain different coefficients). 3.5. Nonstandard form. We note that if the operators Tj on all scales j ∈ Z are available, then the nonstandard form is also available (see =-=[5]-=-). The nonstandard form consists of triplets {Aj , Bj ,Γj}, on all scales j ∈ Z, where Aj :Wj →Wj , Bj : Vj →Wj , Γj :Wj → Vj , and where Wj = Vj−1 − Vj are the wavelet spaces associated with the chos... |

192 |
On the representation of operators in bases of compactly supported wavelets
- Beylkin
- 1992
(Show Context)
Citation Context ... δ(k)(x) ,(26) where c1, c2, and C are constants and x+ and x− are defined in Appendix A. If the kernel K(x) is δ(k)(x), then the operator T is simply kth derivative. This case has been considered in =-=[4]-=-, where it was shown that if the wavelet basis has a sufficient number of vanishing moments, then the two-scale difference equation (25) reduces to 2−kt = At plus an additional normalization condition... |

144 | A fast adaptive multipole algorithm in three dimensions - Cheng, Greengard, et al. - 1999 |

107 |
Waveletlike bases for the fast solution of second-kind integral equations,”
- Alpert, Beylkin, et al.
- 1993
(Show Context)
Citation Context ...using compactly supported, orthonormal bases involving a single scaling function. However, for practical application, we may prefer to use other bases—for example, bases of multiwavelets (see [1] and =-=[2]-=-). Such bases involve several compactly supported scaling functions, and we will consider applications using them elsewhere. We may also choose to use a biorthogonal system, which involves a pair of d... |

87 |
and sufficient conditions for constructing orthonormal wavelet bases,J
- Lawton, Necessary
- 1991
(Show Context)
Citation Context ...see, e.g., [9]). Let the matrix A be defined as in (25), and let M be the number of vanishing moments in the MRA. Then, for k = 0, 1, . . . , 2M − 1, 2−k is an eigenvalue of A. It has been shown (see =-=[14]-=-) that eigenvalue 1, corresponding to k = 0, must be simple. Eigenvalues 2−k for k = 1, 2, . . . may have higher multiplicities depending on the choice of the basis. In what follows, we assume that th... |

61 |
A class of bases in L2 for the sparse representation of integral operators
- Alpert
- 1993
(Show Context)
Citation Context ...ization using compactly supported, orthonormal bases involving a single scaling function. However, for practical application, we may prefer to use other bases—for example, bases of multiwavelets (see =-=[1]-=- and [2]). Such bases involve several compactly supported scaling functions, and we will consider applications using them elsewhere. We may also choose to use a biorthogonal system, which involves a p... |

49 |
A method of local corrections for computing the velocity field due to a distribution of vortex blobs
- Anderson
- 1986
(Show Context)
Citation Context ...). The FMM (see, e.g., [11]) has been highly successful in constructing fast algorithms for a variety of summation problems and incorporates several ideas which are common to such algorithms. The MLC =-=[3]-=- was introduced as a vortex method for problems in fluid mechanics, though the main ideas are applicable in a wider context. After discussing these two algorithms, we present an algorithm based on mul... |

21 |
Numerical quadratures for singular and hypersingular integrals
- Kolm, Rokhlin
- 2001
(Show Context)
Citation Context ... in the numerical evaluation of such operators on functions. Since formal integrals defining the action of these operators are divergent, the construction of quadratures is quite delicate (see, e.g., =-=[13]-=-). Our goal in this paper is to provide a multiresolution definition and, hence, regularization of such operators. Using only the degree of homogeneity of the kernel K(x) and its asymptotic behavior f... |