Baker, C. (1977). The Numerical Treatment of Integral Equations. Clarendon Press, Oxford, UK.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....equations is reduced to O(N ) 11] and in particular cases to O(N log N) 8] thus justifying the extra effort in programming. Now we define our numerical problem and we explain how the multi grid scheme works for solving it. In discretising the TBA equations (7) we use the trapezoidal rule [10] on a grid with mesh size h so that our system yields ffl a (fi) rM a cosh fi Gammaffl b (fi ) 11) a = 1; 2; n, fi h , where Omega h is the set of grid points with grid spacing h. The weights are w(fi) 1 unless on the boundary where w(fi) 1=2. Now let ....

C.T.H. Baker,The numerical treatment of integral equations (Oxford University Press, London, 1977) p.110.

....should scale with the number of segments rather than the number of pixels in the image. 3.4 Related Work on Approximation E. J. Nystrom published his method in the late 1920 s [11] Its use in approximating solutions to integral equations is well known for its simplicity and accuracy [2, 6, 13]. The Nystrom method has also been recently applied in the kernel learning community [24] for fast approximate Gaussian process classification and regression. As noted in [24] this approximation method directly corresponds to the kernel PCA feature space projection technique of [17] The authors ....

C. T. H. Baker. The numerical treatment of integral equations. Oxford: Clarendon Press, 1977.

....should scale with the number of segments rather than the number of pixels in the image. 3.4 Related Work on Approximation E. J. Nystrom published his method in the late 1920 s [10] Its use in approximating solutions to integral equations is well known for its simplicity and accuracy [2] [6] 12] The Nystrom method has also been recently applied in the kernel learning community [23] for fast approximate Gaussian process classification and regression. As noted in [23] this approximation method directly corresponds to the kernel PCA feature space projection technique of [16] The ....

C. T. H. Baker. The numerical treatment of integral equations. Oxford: Clarendon Press, 1977.

....this computational problem through the use of a subset of corresponding points. In doing so, we highlight connections to related approaches in the area of Gaussian RBF networks that are relevant to the TPS mapping problem. Finally, we discuss a novel application of the Nystr om approximation [1] to the TPS mapping problem. Our experimental results suggest that the present work should be particularly useful in applications such as shape matching and correspondence recovery (e.g. 2, 6, 4] as well as in graphics applications such as morphing. 2 Review of Thin Plate Splines Let v i ....

....the full set of target values. The approach is based on a technique known as the Nystr om method. The Nystr om method provides a means of approximating the eigenvectors of K without using C. It was originally developed in the late 1920s for the numerical solution of eigenfunction problems [1] and was recently used in [10] for fast approximate Gaussian process regression and in [7] implicitly) to speed up several machine learning techniques using Gaussian kernels. Implicit to the Nystr om method is the assumption that C can be approximated by B T A 1 B, i.e. K = A B B T B ....

C. T. H. Baker. The numerical treatment of integral equations. Oxford: Clarendon Press, 1977.

....space Fn by a projection method (# PnK)un = Pn f, 1.3) where un # Fn and Pn : C # Fn is a projection operator, or we use the Nystrom quadrature method (#I Kn )un = f, 1. 4) where Kn approximates K and is obtained by discretization of K by an n point quadrature rule; see [4, 8, 13, 28]. Such discretizations of integral equations give rise to dense linear systems of equations. As is known, these systems can be prohibitively expensive to solve as n, the order of the linear system of algebraic equations, increases. Iterative methods are the natural options for e#cient solutions. ....

C. T. H. Baker, The Numerical Treatment of Integral Equations, Clarendon Press, Oxford, 1977.

....equations is reduced to O(N 2 ) 11] and in particular cases to O(N log N) 8] thus justifying the extra effort in programming. Now we define our numerical problem and we explain how the multi grid scheme works for solving it. In discretising the TBA equations (7) we use the trapezoidal rule [10] on a grid with mesh size h so that our system yields ffl a (fi) rM a cosh fi h 2 n X b=1 X fi 0 2 Omega h w(fi 0 ) ab (fi Gamma fi 0 ) log(1 e Gammaffl b (fi 0 ) 11) a = 1; 2; n, fi 2 Omega h , where Omega h is the set of grid points with grid ....

C.T.H. Baker,The numerical treatment of integral equations (Oxford University Press, London, 1977) p.110.

.... methods for (9) would be to use an implicit method of the form ae (n 1) ae (n) 1T Z 1 0 k(u; w)f(ae (n 1) w) dw = ae (n 1) x : Once again, such methods for nonlinear Fredholm integro differential equations have not received much attention in the literature, but are mentioned in Baker (1978). The main work involved in such schemes invariably concerns the nonlinear equations that must be solved at each time step. Acknowledgement The authors of this report would like to thank everybody who contributed to discussion of this problem. ....

Baker, C.T.H. (1978) The numerical treatment of integral equations, Oxford University Press.

....of their limiting values from the right and left. We use these kernels only as tools in the analysis and do not construct them in the implementation. Km is a degenerate kernel operator. Such operators have been used as approximations for the purposes of solution and error estimation for many years [4], 10] where the kernels are typically constructed using orthonormal bases or operator product formulae for k. In this paper, the operator Km has been constructed as a preconditioner, which is a quite different purpose. We will require the following lemma, which is a direct consequence of ....

C. Baker, The Numerical Treatment of Integral Equations, Oxford University Press, Oxford, 1977.

.... ERROR ESTIMATES Since, for fixed 0, 3) is a Fredholm integral equation with smooth kernel, a wide variety of approximation methods applies to the numerical solution of equation (3) e.g. projection methods (such as Galerkin or collocation methods) and quadrature (Nystrom) methods (see e.g. [1, 3, 4, 13, 16, 23]) Assume that such an approximation method is given by the sequence of equations A (n) n = f n (n 2 N) 39) where f n 2 X n is known and ;n 2 X n is the approximate solution of equation (3) with X n being a closed subspace of L N p ( Gamma; Then ;n can be viewed as an ....

.... (A Gamma f) P n Gamma ;n = A (n) Gamma1 (A (n) P n Gamma f n ) and the triangle inequality jj Gamma ;n jj jj Gamma jj jj Gamma ;n jj : Since for the aforementioned approximation methods estimates of the last three terms are known (see e.g. [1, 3, 4, 13, 16, 23]) the problem of estimating jj Gamma ;n jj is reduced to estimating the term jjA Gamma A jj. The following lemma gives a corresponding estimate in the particular case of a closed curve Gamma. Notice that in this case the solution of (2) has the same regularity as f provided a; b and ....

Baker, C.T.H., The numerical treatment of integral equations, Clarendon Press, Oxford, 1977.

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Baker, C. (1977). The Numerical Treatment of Integral Equations. Clarendon Press, Oxford, UK.

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C. Baker. The Numerical Treatment of Integral Equations. Oxford, UK: Clarendon Press, 1977.

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C. T. H. Baker. The numerical treatment of integral equations. Clarendon Press, Oxford, 1977. Monographs on Numerical Analysis.

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C.T.H. Baker, The Numerical Treatment of Integral Equations. Oxford: Clarendon Press, 1977.

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C. Baker. The numerical treatment of integral equations. Clarendon Press, Oxford, 1977.

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C. T. H. Baker. The Numerical Treatment of Integral Equations. Clarendon Press, Oxford, 1977.

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C. Baker, The numerical treatment of integral equations. Oxford Clarendon Press, 1977.

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C. Baker, The numerical treatment of integral equations, Oxford Clarendon Press, 1977.

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C. Baker. The numerical treatment of integral equations. Clarendon Press, Oxford, 1977.

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Baker, C. T. H. (1977). The Num erical Treatm ent of Integral Equations. Clarendon Press, Oxford.

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