L.M. Delves and J.L. Mohamed. Computational Methods for Integral Equations. Cambridge University Press, Cambridge, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....subject classifications. 45L10, 65R20. Key Words. Fredholm integral equation, polynomial interpolation. 1 Introduction Solution of integral equations of the second kind is a much studied subject and various direct and iterative methods have been proposed for their numerical solutions, see [6] for instance. However, one overriding drawback of these methods is the high cost of working with the associated dense matrices. For problems discretized with n quadrature points, classical direct methods such as Gaussian elimination method requires O(n ) operations to obtain the numerical ....

....Thus a good method for solving these well conditioned equations is the conjugate gradient method or its variants, see for instance [7, 3] They converge to the solution in a linear rate, cf. 9] and Table 2 in x5. To find the solution numerically, we discretize (2) by Nystrom s method (see [6]) at equally spaced points (i Gamma 1) n Gamma 1) i = 1; n, on [0; 1] This results in a matrix equation (I Gamma A)f = g; 3) where I is the identity matrix, g is a given vector and f is the unknown vector. As in [1] we define the entries of the discretization matrix A to be [A] ....

L. Delves and J. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, 1985.

....(I Gamma DA)f = g; 1.2) where I is the identity matrix, D is a diagonal matrix and A is a dense matrix corresponding to the quadrature rule and quadrature points used in discretization of the integral in (1.1) Various direct and iterative methods have been proposed for solving (1. 2) see [7] for instance. However, one overriding drawback of these methods is the high cost of working with the associated dense matrices A. For problems discretized with n quadrature points, direct methods such as Gaussian elimination method require O(n ) operations to obtain the numerical solutions. ....

L. Delves and J. Mohamed, Computational Methods for Integral Equations. Cambridge University Press, Cambridge, 1985.

.... flow, K 0 (w; x) fA (y Gamma x)dy x Gamma w PfA B Gamma xg while for a fluid flow, K 0 (w; x) 1 (C Gamma r 0 ) r 1 Gamma C) fS y Gamma x dy B Gamma w B Gamma x The above integral equation can be solved numerically with the standard method [2]. The steady state performance measures then can be computed as follows. For a customer flow, P1 flossg = v(0) PfD1 DjA1 W1 Bg = v(0) P fCD A Bg PfA B Gamma wg 0 (w)dw] For a fluid flow, PflossjR(1) r 1 g = v(0) PfS (0)g E[ jR(1) r 1 ] v(0) u 0 ....

L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge, U. K., Cambridge Press, 1988.

....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 ....

L. Delves and J. Mohamed. Computational Methods for Integral Equations. Cambridge University Press, 1985.

....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 ....

L. Delves and J. Mohamed. Computational Methods for Integral Equations. Cambridge University Press, 1985.

....in 3D) In general the differences are limited to more indices to manipulate, or in the case of a program, more array dimensions to iterate over. 1. 2 Numerical Solution A canonical solution technique for integral equations such as the radiosity equation (1) is the weighted residual method [8], often referred to as finite elements. The historical development of radiosity algorithms did not proceed from the theory of integral equations and finite element methods. Instead the first approaches were motivated by power balance arguments [10, 22] Only recently [16] was the traditional ....

....2 ) the number of elements necessary for a given overall accuracy falls faster for a net gain. To see why this is, we use the fact that a Galerkin method using a piecewise polynomial basis of order M 1 will have an accuracy of O(h M ) Where h gives the sidelength of the elements in the mesh [8, 19]. To make this concrete, suppose we are willing to allow an error proportional to 1 256 . Using piecewise constant basis functions, h would have to be on the order of 1 256 to meet our goal. Now consider piecewise linear functions. In this case h would only need to be on the order of q 1 ....

Delves, L. M., and Mohamed, J. L. Computational Methods for Integral Equations. Cambridge University Press, 1985.

....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. ....

L. M. Delves and J. L. Mohammed, Computational Methods for Integral Equations, Cambridge University Press, 1985.

....space was also studied [5] and the resulting discrete problem was then solved effectively by either of the methods previously developed for the matrix case. This method can also be used to estimate the solution at arbitrary points of the phase space by making use of a variant of Nystrom s method [6]. The combination proves to be quite effective, especially because a theoretical error bound is available that is minimized when the discrepancy is least. This suggests that low discrepancy point sets should be used to define the discrete transport problem, especially when the phase space is ....

Delves, L. M., Mohamed, J. L.: Computational Methods for Integral Equations. Cambridge University Press New York 1985

....transport equation obtained by restricting the kernel to a nite set of points in phase space. The resulting matrix equation may then be solved adaptively to obtain an approximation to the solution of the original problem. The technique may be viewed as a variant of Nystom s quadrature method [1], which is normally applied by using a regular grid as the set of nodal points. We prove a theorem that establishes the error resulting from such approximations. This theorem suggests that low discrepancy nodal sets will produce smaller errors than cartesian product and other grid choices, ....

.... (a seventh dimension is needed to describe time dependent transport) This discretization produces a linear matrix problem: M j ) 1 N N X k=1 K(M j ; M k ) M k ) S(M j ) 2) and a natural interpolation formula (P) 1 N N X k=1 K(P;M k ) M k ) S(P) 3) Nystrom s method [1] is based on recognizing the utility of equation (3) and its extensions using more general quadrature formulae) in obtaining approximate solutions of equation (1) However, when the phase space has dimension s, in order to achieve a predetermined accuracy when a conventional product quadrature ....

L.M. Delves and J.L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, New York (1985).

....is a system of n equations in n unknowns. Eqs. 12) may be solved numerically to yield an approximate solution to Eq. 9) given by the expression fR (x) n X i=1 f i b i (x) How large is the error e R = f 0 fR of the approximate solution We follow the derivation by Delves and Mohamed in [7]. Defining g R by the formula g R (x) n X i=1 g i b i (x) we rewrite Eqs. 9) and (12) in terms of operators K and R to obtain (I 0K)f = g (I 0R)fR = g R : Combining the latter equations yields (I 0K)eR = K0R)fR (g 0 g R ) Provided that (I 0K) 01 exists, we obtain the error ....

L. M. Delves and J. L. Mohamed. Computational Methods for Integral Equations. Cambridge University Press, 1985.

.... = 1 (C Gamma r 0 ) r 1 Gamma C) Z B maxfw;xg f T y Gamma w C Gamma r 0 fS y Gamma x r 1 Gamma C dy 1 C Gamma r 0 f T B Gamma w C Gamma r 0 P ae S B Gamma x r 1 Gamma C oe : The above integral equation can be solved numerically with the standard method [2]. The steady state performance measures then can be computed as follows. For a customer flow, P1 flossg = v(0) Z B 0 PfA B Gamma wg 0 (w)dw and PfD1 DjA1 W1 Bg = v(0) P fCD A Bg Z B 0 Pfmaxf0; CD Gamma wg A B Gamma wg PfA B Gamma wg 0 (w)dw] For a fluid flow, ....

L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge, U. K., Cambridge Press, 1988.

....a vector between A i (x) and A j (x # ) # i,j the angles made by this vector with the local surface normals; and V (x, x # ) takes on the values 1, 0 depending whether A i (x) can see A j (x # ) 1 . This formulation is an instance of a Galerkin method for the solution of integral equations [5]. More precisely we wish to solve the radiosity integral equation B(x) E(x) #(x) Z k(x, x # )B(x # ) dx # (Again we will assume that # is constant for each element in the scene. In the Galerkin method all involved functions are expanded with respect to some basis set of functions. In the ....

....angles. In this case a rule derived with our model geometry would be exact. Another approach to dealing with the singularity, which we have not touched upon at all, concerns symbolic methods which subtract out the singularity. Many such techniques are known (for an overview of some of these see [5]) and should be explored for radiosity. It may also be possible to reduce the dimensionality of the integration below four. Many special expressions exist for form factors which reduce the four dimensional integrals to two dimensions (or even less) Similar expressions may be possible for the ....

Delves, L. M., and Mohamed, J. L. Computational Methods for Integral Equations. Cambridge University Press, 1985.

....[ I2 I we assume that there exists a constant C d such that max I2 fa I g C d . In what follows, we form the discrete interpolation formula by brutal discretization of the continuous integral representation, namely, Eqs. 14) 15) and the moment equation (5) by Nystrom quadrature method [11]. For given window function, OE 0, around particle x I , the polynomial basis takes the value P I (x) fP 1I ; Delta Delta Delta ; P iI ; Delta Delta Delta ; P jI ; Delta Delta Delta ; P I g with P iI = i x I Gamma x ae j ff and P jI = i x I Gamma x ae j fi , the discrete ....

L. Delves and J. Mohamed. Computational Methods for Integral Equations. Cambridge University Press, New York, 1985.

....subject classifications. 45L10, 65R20. Key Words. Fredholm integral equation, polynomial interpolation. 1 Introduction Solution of integral equations of the second kind is a much studied subject and various direct and iterative methods have been proposed for their numerical solutions, see [4] for instance. However, one overriding drawback of these methods is the high cost of working with the associated dense matrices. For problems discretized with n quadrature points, classical direct methods such as Gaussian elimination method requires O(n 3 ) operations to obtain the numerical ....

....Thus a good method for solving these well conditioned equations is the conjugate gradient method or its variants, see for instance [5, 3] They converge to the solution in a linear rate, cf. 7] and Table 2 in x5. To find the solution numerically, we discretize (2) by Nystrom s method (see [4]) at equally spaced points (i Gamma 1) n Gamma 1) i = 1; n, on [0; 1] This results in a matrix equation (I Gamma A)f = g; 3) where I is the identity matrix, g is a given vector and f is the unknown vector. As in [1] we define the entries of the discretization matrix A to be [A] i;j ....

L. Delves and J. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, 1985.

.... (w) def = 0(w) v(0) 11) and divide both sides of (10) by v(0) so 0 (w) K0(w; 0) Z B 0 K0(w;x) 0 (x)dx: 12) The above equation is a non homogeneous Fredholm integral equation of the second kind in 0 (w) with kernel K0(w;x) and can be solved numerically with the standard method [4]. With 0 (w) obtained by solving (12) now we can determine v(0) as follows. Since by definition 0 (w) 0 (w) v(0) w) Gamma v(w)ffi(w) v(0) integrating both sides of the above equation from 0 to B, we have Z B 0 0 (w)dw = R B 0 (w)dw Gamma R B 0 v(w)ffi(w)dw v(0) 1 ....

L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge, U. K., Cambridge Press, 1988.

No context found.

L.M. Delves and J.L. Mohamed. Computational Methods for Integral Equations. Cambridge University Press, Cambridge, 1985.

No context found.

L.M. Delves and J.L. Mohamed. Computational Methods for Integral Equations. Cambridge University Press, Cambridge, 1985.

No context found.

L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Great Britain, 1985.

No context found.

L.M. Delves and J.L. Mohamed, Computational methods for integral equations, Cambridge University Press, Great Britain, 1985.

No context found.

L. Delves and J. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, 1985.

No context found.

L. M. Delves and J. L. Mohamed. Computational Methods for Integral Equations. Cambridge University Press, New York, 1985.

No context found.

L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge Univ. Press, Cambridge, 1985.

No context found.

L. Delves and J. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, 1985.

No context found.

L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, Cambridge, England, 1985.

No context found.

L.M.Delves and J.L.Mohamed, Computational methods for integral equations (Cambridge U.P. ,1985).

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC