B. Alpert, G. Beylkin, R. R. Coifman, and V. Rokhlin. Wavelet-like bases for the fast solution of second-kind integral equations. SIAM J. Sci. Comput. , 14:159-174, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....polynomial used. We prove that the Frobenius norm kA Gamma Bk F 6 ffl if k is of O(log ffl ) for smooth kernels (including log jx Gamma tj) and of O(log log n log ffl ) for weakly singular kernels such as jx Gamma tj . Comparison with the waveletlike method by Alpert et al. [2] shows that our method requires less memory and is more accurate. Keywords: Fredholm integral equation, polynomial interpolation, conjugate gradients 1. INTRODUCTION In this paper, we consider the fast solutions of Fredholm integral equation of the second kind: a(x; t)f(t)dt = g(x) x 2 [0; ....

....the form Ay, see [8] Therefore even for well conditioned problems, such as the second kind integral equations, the methods require O(n ) operations, which for large scale problems is often prohibitive. In recent years, a number of fast algorithms for (1. 2) have been developed, see for instance [9, 13, 3, 2]. The fast multipole method proposed in [9] combines the use of low order polynomial interpolation of the kernel functions with a divideand conquer strategy. For kernel functions that are Coulombic or gravitational in nature, it results in an order O(n) algorithm for the matrix vector ....

[Article contains additional citation context not shown here]

B. Alpert, G. Beylkin, R. Coifman, and V. Rokhlin, Wavelet-like bases for the fast solution of second-kind integral equations. SIAM J. Sci. Comput., (1993) 14, 159--184.

....vectors for each square s on finest level (lev= maxlev) Get G using (4.26) and Ls8 combine solves technique 4. 4 Fine to coarse sweep In this part of the algorithm, the goal is to use the row basis representation just obtained to obtain a representation which is wavelet like in structure [22, 42, 41], obtaining G (Gws( In this section, whenever we use the notation Gab, it means the approximate Gab obtained from (4.16) This is a simpler representation to work with and has the advantage that further sparsity can be obtained by thresholding out small entries in Gws to form an even sparser ....

B. Alpert, G. Beylkin, R. Coifman, and V. Rokhlin, "Wavelet-like bases for the fast solution of second-kind integral equations," SIAM J. Sci. Cornput., vol. 14, no. 1, pp. 159 184, 1993.

....Pan and Schreiber [107] show how to accelerate the convergence by at least a factor 2 via scaling parameters and how to use the iteration to carry out rank and projection calculations. For an interesting application of the Schulz iteration to a sparse matrix possessing a sparse inverse, see [1]. 4 The Least Squares Problem In this section we consider the least squares (LS) problem min x kAx Gamma bk 2 , where with m n. The most thorough and up to date treatment of the LS problem is given by Bjorck s book [19] 4.1 The Seminormal Equations Given a QR factorization A = QR, where Q ....

B. Alpert, G. Beylkin, R. Coifman, and V. Rokhlin. Wavelet-like bases for the fast solution of second-kind integral equations. SIAM J. Sci. Comput., 14(1):159--184, 1993.

....the model. Therefore, the use of wavelets results in a sparse moment matrix which can be solved efficiently by sparse solvers. Mathematicians originally applied wavelets to solve integral operator equations with essentially smooth, nonoscillatory kernels (similar to those in electrostatics) 9] [10]. They have demonstrated that using wavelets can obtain a solution in operations, where is the number of unknowns. This is in contrast with Manuscript received March 14, 1996; revised February 7, 1997. The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological ....

....solution. More recently, Wagner and Chew [8] analyzed the effect of wavelet methods in terms of the radiation receiving characteristics of the wavelet basis functions. The orthonormal wavelet matrix transforms were used for solutions of EM integral equations, following the method in [9] and [10]. However, few studies have been done on the two challenging problems: how to efficiently use wavelet transforms and what kind of wavelet transforms is more effective for solutions of EM problems. In this paper, both orthonormal and nonorthonormal wavelets [11] 14] are studied and discussed. An ....

[Article contains additional citation context not shown here]

B. Alpert, G. Beylkin, R. Coifman, and V. Rokhlin, "Wavelet-like bases for the fast solution of second-kind integral equations," SIAM J. Sci. Comput., vol. 14, no. 1, pp. 159--184, Jan. 1993.

....to generate an orthogonal discrete time basis having an octave band characteristic, for a fixed number of zero moments it does not yield a discretetime basis that is optimal with respect to time localization. This paper examines a special case of a class of bases originally described by Alpert in [4, 5, 6] in a multiwavelet context, the construction of which relies on Gram Schmidt orthogonalization. We describe the basis from a filter bank viewpoint, give explicit solutions for the filter coefficients, and describe an efficient algorithm for the transform. The DWT described in this paper is based ....

....improved results. For example, the shift invariant transform of [10, 20] mentioned above, and the hidden Markov model based approach of [12] 2. 11 Underlying Continuous time Wavelets As noted in the Introduction, the slantlet basis is a special case of the multiwavelet bases, described by Alpert [4, 5, 6], comprised of r scaling functions and r wavelet functions with r vanishing moments. The continuoustime multiwavelet basis of [6] with r = 2 is piecewise linear and discontinuous 2 . However, it is important to note that for these bases, the relationship between continuous time and discrete time ....

B. Alpert, G. Beylkin, R. Coifman, and V. Rokhlin. Wavelet-like bases for the fast solution of secondkind integral equations. SIAM J. on Sci. Comp., 14(1):159--184, January 1993. 20

....of these two multiscale techniques is the basic idea behind the novel multiscale approach for assimilation of data proposed here. TIMELY COMMUNICATION 951 We have to emphasize that standard multiscale techniques, such as the multigrid methods [11] 12] 13] 14] and the wavelet methods [15] [16], cannot be applied to the problem of data assimilation in a straightforward way. As we have already mentioned, a global data assimilation system has to provide a reasonable mechanism for information transport between data rich and data sparse areas. This strong inhomogeneity in the observation ....

....value of # that would give an essential reduction of all spectral error components. The e#ect described above is well studied for the case when (6) is obtained as a grid approximation of the continuous integral equation. A few multiscale techniques based on multigrid [8] 9] and wavelet [16] approaches were developed in the 1990s and successfully applied to a range of problems. Unfortunately, these techniques cannot be applied to the considered problem in a straightforward way. As we mentioned above, the global data assimilation system has to carefully transport information between ....

B. ALPERT,G.BEYLKIN,R.COIFMAN, AND V. ROKHLIN, Wavelet-like bases for the fast solution of second-kind integral equations, SIAM J. Sci. Comput., 14 (1993), pp. 159--184.

....2 s 1 2 ; 9.2.7) that is, both spaces agree as sets and the norms are equivalent. However, there is, of course, the restriction s 1=2, which will be seen later to be an unfortunate obstruction. 9. 3 Multi Wavelets The above geometric setting suggests the following natural concept; see [2, 3, 153, 154]. Let Pi d be the set of polynomials of total degree less than d on IR n and let P : fP : jj = 1 : n dg be an orthonormal basis of Pi d on 2, which can be generated by the Gram Schmidt process from the monomial basis. For simplicity, let us now choose 2 = 0; 1) n . A similar ....

A. Alpert, G. Beylkin, R. Coifman, V. Rokhlin, Waveletlike bases for the fast solution of second-kind integral equations, SIAM J. Sci. Statist. Comp. 14 (1993), 159--184.

....Daubechies FIR filter coefficients in this finite dimensional context. This paper is an attempt to fill this gap, which is making its presence felt as researchers use discrete time periodic wavelets in practical applications ranging from image compression [17, Section 13.10] to numerical analysis [1] [3] 8] Because discrete time periodic signals constitute a finitedimensional vector space, our presentation is based primarily on concepts from linear algebra and discrete Fourier transforms, though some details in Appendix A require results about the factorization of polynomials. Hence, we ....

B. Alpert, G. Beylkin, R. Coifman and V. Rokhlin, "Waveletlike bases for the fast solution of second-kind integral equations ", SIAM J. Sci. Comput., Vol. 14, No. 1, January 1993, pp. 159--184.

....to generate an orthogonal discrete time basis havinganoctave band characteristic, for a fixed number of zero moments it does not yield a discrete time basis that is optimal with respect to time localization. This paper examines a special case of a class of bases originally described by Alpert in [3, 4] in a multiwavelet context, the construction of which relies on Gram Schmidt orthogonalization. We describe the basis from a filter bank viewpoint and give explicit solutions for the filter coefficients. The DWT described in this paper is based on a filter bank structure, where different filters ....

B. Alpert, G. Beylkin, R. Coifman, and V. Rokhlin. Waveletlike bases for the fast solution of second-kind integral equations. SIAM J. on Sci. Comp., 14:159--184, 1993.

....value of which would give an essential reduction of all spectral error components. The effect described above is well studied for the case when (4) is obtained as a grid approximation of the continuous integral equation. A few multiscale techniques based on multigrid ( 6] 7] and wavelet ([8]) approaches were developed in 1990 s. Unfortunately, these techniques cannot be applied to the considered problem straightforwardly because of the strong inhomogeneity of the observation network. The central idea of the approach developed below is to filter sequentially spectral components of r ....

Alpert, B., G. Beylkin, R. Coifman, and V. Rokhlin, 1993: Wavelet-like bases for the fast solution of second-kind integral equations. SIAM J. Sci. Comput., 14, 159-184.

.... mentioned above, e.g. using far field expansions [16] 15] 13] hierarchical solvers in many body simulations [1] 3] 10] 12] and FFT based schemes [17] for the solution of the corresponding integral equations) In the past decade wavelet techniques have become popular, e.g. see [5] [2], where the complexity reduction is obtained by representation on a suitable set of increasingly coarse base functions. Most of these techniques have restrictions, e.g. a limited accuracy, limitation to potential type kernels, or they require a significant amount of matrix manipulations to arrive ....

B. Alpert, G. Beylkin, R. Coifman, and V. Rokhlin, Wavelet-like bases for the fast solution of second-kind integral equations, SIAM J. Sci. Comput., 14,1 (1993), pp. 159--184.

....1, the G i are scalars: G 0 = 1= p 2; G 1 = Gamma1= p 2. In the case m = 2, the G i obey G 0 = j ffi 0 # G 1 = Gammaj ffi 0 Gamma # : The functions h j;k are piecewise polynomials supported in I j;k with knots at the endpoints and midpoint of I j;k . For pictures see [3] or [2]. We mention that each basis functions from Alpert s construction is either symmetric or antisymmetric. A key fact about the two scale matrices we have just defined is that the 2m Theta 2m matrix U = H 0 H 1 G 0 G 1 # is orthogonal. This is equivalent to saying we have two different ....

B. Alpert, G. Beylkin, R. Coifman, and V. Rokhlin. Wavelet-like bases for the fast solution of second-kind integral equations. SIAM Journal of Mathematical Analysis, 14(1):159--184, 1993.

....and multiplication, with compressed matrices, rather than with the discretized versions of T , then we may significant speed up of the numerical treatment. This program of using the wavelet representations for the efficient numerical treatment of operators was initiated in [10] We also refer to [1, 2] for related material and many more details. In a different direction, because of the close similarities between the scaling function and finite elements, it seems natural to try wavelets where traditionally finite element methods are used, e.g. for solving boundary value problems [72] There are ....

B. Alpert, G. Beylkin, R. Coifman, and V. Rokhlin. Wavelet-like bases for the fast solution of second-kind integral equations. SIAM J. Scient. Comp., 14, 1.

....RADIOSITY IN COMPUTER GRAPHICS The final topic in this paper is that of illumination and radiosity. There has been a number of contributions to this field over the past few years, and most recently it has been pointed out that wavelets (and in particular the work of the wavelet group at Yale [3] [1]) can play a significant role in reducing the computational time in the simulation of such images. Some models of reflection from surfaces use electromagnetic reflection models in the spirit of our discussion of SAR images in the preceding section [7] 17] 34] and these have their deficiencies ....

B. Alpert, G. Beylkin, R. Coifman, and V. Rokhlin. Wavelet-like bases for the fast solution of second-kind integral equations. SIAM Journal on Scientific Computing, 14(1), Jan 1993.

....which allows to design fast numerical algorithms for these operators. There exists a series of papers on the application of wavelet methods to the computation of integral operators and the solution of integral equations, where different types of scaling functions and wavelets are used (see [1], 2] 5] and [13] the references therein) Since the scaling function OE has to satisfy the so called refinement equation OE(x) X m2Z n am OE(2x Gamma m) 1.4) as a rule these functions are piecewise polynomials satisfying some smoothness and vanishing moment conditions. Many interesting ....

Alpert, B., Beylkin, G., Coifman, R., Rokhlin, V., Waveletlike bases for the fast solution of secondkind integral equations, SIAM J. Sci. Statist. Comp. 14 (1993), 159--184.

....In this paper we construct representations of operators in bases of multiwavelets, with the goal of developing adaptive solvers for both linear and nonlinear partial differential equations, and we demonstrate success with a simple solver. We use multiwavelet bases constructed in [2] following [3, 5]. These bases were also considered in [14] although not for numerical purposes. Multiwavelet bases retain the properties of wavelet bases, such as vanishing moments, orthogonality, and compact support. The basis functions do not overlap on a given scale and are organized in small groups of ....

....basis functions do not overlap on a given scale and are organized in small groups of several functions (thus, multiwavelets) sharing the same support. On the other hand, the basis functions are discontinuous, similar to the Haar basis and in contrast to wavelets with regularity. As was shown in [3] (discrete version of multiwavelets) and [2] multiwavelet bases can be successfully used for representing integral operators. A wide class of integrodifferential operators has effectively sparse representations in these bases, due to vanishing moments of the basis functions. An effectively sparse ....

[Article contains additional citation context not shown here]

B. Alpert, G. Beylkin, R. R. Coifman, and V. Rokhlin. Wavelet-like bases for the fast solution of second-kind integral equations. SIAM J. Sci. Comput., 14(1):159--174, 1993.

....in integral equation formulations of problems in potential theory, wave propagation, and other application areas. Fast algorithms, including those by Rokhlin [1] 2] Greengard and Rokhlin [3] Hackbusch and Nowak [4] Beylkin, Coifman, and Rokhlin [5] Alpert, Beylkin, Coifman, and Rokhlin [6], and Kelley [7] have generally reduced the computational complexity to O(n)orO(n log n) operations, with n unknowns, for the application of an integral operator or its inverse. The appearance of these fast algorithms has increased the urgency of developing accurate quadratures for the ....

....is determined by combining (97) 101) The computation of all m 2 coe#cients requires m(m 2j 2a 1) evaluations of the kernel K and therefore order O(m 2 ) operations. This cost can often be substantially reduced using techniques that exploit kernel smoothness (see, for example, [6], 5] A slightly di#erent application of the quadratures is the computation of Fourier transforms of functions that fail to satisfy the assumptions usually made when using the discrete Fourier transform. In particular, if a function decays slowly for large argument or is compactly supported and ....

B. Alpert, G. Beylkin, R. Coifman, and V. Rokhlin, Wavelet-like bases for the fast solution of second-kind integral equations, SIAM J. Sci. Comput., 14 (1993), pp. 159--184.

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B. Alpert, G. Beylkin, R. R. Coifman, and V. Rokhlin. Wavelet-like bases for the fast solution of second-kind integral equations. SIAM J. Sci. Comput. , 14:159-174, 1993.

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ALPERT, B., BEYLKIN, G., COIFMAN, R., AND ROKHLIN, V. 1993. Wavelet-like bases for the fast solution of second-kind integral equations. SIAM Journal on Scientific Computing 14, 1 (Jan.), 159--184.

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A. Alpert, G. Beylkin, R. Coifman, V. Rokhlin, Waveletlike bases for the fast solution of second-kind integral equations, SIAM J. Sci. Statist. Comp., 14, 1993, pp. 159--184.

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B. Alpert, G. Beylkin, R. Coifman, and V. Rokhlin. Waveletlike bases for the fast solution of second-kind integral equations. SIAM Journal on Scientific Computing, 14(1):159--184, January 1993. 5

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B. Alpert, G. Beylkin, R. Coifman, and V. Rokhlin, \Wavelet-like bases for the fast solution of second-kind integral equations," SIAM J. Sci. Comput. , 1993.

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B. Alpert, G. Beylkin, R. Coifman and V. Rokhlin, Waveletlike bases for the fast solution of second-kind integral equations SIAM J. Sci. Statist. Comp. , 14 (1993), 159--184.

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B. Alpert, G. Beylkin, R. Coifman and V. Rokhlin, "Wavelet-like Bases for the Fast Solution of Second-kind Integral Equations," SIAM Journal of Science Computations, Vol. 14, no. 1, pp. 159-184, January 1993.

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A. Alpert, G. Beylkin, R. Coifman, V. Rokhlin, Waveletlike bases for the fast solution of second-kind integral equations, SIAM J. Sci. Statist. Comp., 14 (1993), 159--184.

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