R. H. Chan and M. K. Ng, "Conjugate gradient methods for Toeplitz systems," SIAM Rev., vol. 38, no. 3, pp. 427--482, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....but convergence rates may be undesirably slow. Preconditioning methods can accelerate the convergence of gradient based iterative methods for tomographic image reconstruction and image restoration. Circulant or Fourier preconditioners have been used extensively for shift invariant problems [6 8], for which This work was supported in part by NIH grants CA 60711 and CA 54362 and by the Whitaker Foundation. the Hessian of the objective function is approximately block Toeplitz or block circulant. However, in applications with nonuniform noise variance (such as arises from Poisson ....

R. H. Chan and M. K. Ng, "Conjugate gradient methods for Toeplitz systems," SIAM Review, vol. 38, no. 3, pp. 427--82, September 1996.

....and the last entry of every row is the first entry of its succeeding row. The use of circulant matrices as preconditioners for solving Toeplitz systems have been studied extensively since 1986. It has been shown that they are good preconditioners for a large class of Toeplitz systems, see [6] and the references therein. The first preconditioner proposed in [2] for the matrix M in (3) is the well known T. Chan circulant preconditioner, see [6] The second one is a new preconditioner that he called the P circulant preconditioner. His numerical results showed that the Krylov subspace ....

....systems have been studied extensively since 1986. It has been shown that they are good preconditioners for a large class of Toeplitz systems, see [6] and the references therein. The first preconditioner proposed in [2] for the matrix M in (3) is the well known T. Chan circulant preconditioner, see [6]. The second one is a new preconditioner that he called the P circulant preconditioner. His numerical results showed that the Krylov subspace methods, when applied to solving both circulant preconditioned systems, converge very quickly. He showed moreover that P is invertible for some special ....

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Chan, R., and Ng, M., 1996. Conjugate Gradient Methods for Toeplitz Systems. SIAM Review 38, 427--482.

....of the Hessian of the objective function, is approximately block Toeplitz, where G is a system matrix described in (1) below. For these problems the diagonal preconditioner is ineffective, but appropriate circulant preconditioners can provide very remarkable improvements in convergence rate. See [15] for a recent thorough review of this subject. Circulant preconditioners are particularly appealing since one can use the fast Fourier transform (FFT) for efficient implementation. Several optimal circulant preconditioners are available for Toeplitz problems [15 19] Such circulant ....

....in convergence rate. See [15] for a recent thorough review of this subject. Circulant preconditioners are particularly appealing since one can use the fast Fourier transform (FFT) for efficient implementation. Several optimal circulant preconditioners are available for Toeplitz problems [15 19]. Such circulant preconditioners, also called Fourier preconditioners, have been applied to both tomographic image reconstruction [20] and image restoration problems [21, 22] Figure 3, described in Section V below, illustrates the well known effectiveness of circulant preconditioners for ....

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R. H. Chan and M. K. Ng, "Conjugate gradient methods for Toeplitz systems," SIAM Review, vol. 38, no. 3, pp. 427--82, Sept. 1996.

....the iterative method to the solution of the system Ax = M g: We remark that, although A may be indefinite, the regularizing preconditioner matrix is always positive definite. A survey on preconditioners for Toeplitz and block Toeplitz matrices has recently been presented by Chan and Ng [12]. A method that explicitly explores the displacement rank is presented in [13] 6 Towards a black box It is desirable that numerical methods be developed which make it possible to solve large linear discrete ill posed problems without user intervention. This is possible when the Morozov ....

R. Chan and M. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Review, 38 (1996), pp. 427--482.

....by (AN ) the spectrum of AN , we suppose that there is a probability measure such that, for every function f 2 C(R) lim N 1 1 N X 2 (AN ) f( Z f( d ( 1. 2) This assumption is natural in a number of applications such as the solution of (block) Toeplitz systems (see, e.g. [ChNg96, BoSi99] and [BeKu99, x4] or the discretization of elliptic PDEs via nite di erence techniques (see, e.g. Ser00, SeTi00, Til98] and the discussions in [BeKu99, x5] or [Be00c] Under some additional weak regularity assumptions stated in x2 below, the following estimate for the error in Conjugate ....

R.H. Chan and M.K. Ng, Conjugate Gradient Methods for Toeplitz systems, SIAM Rev. 38 (1996), pp. 427-482.

....to obtain a quality direction vector in as few CG iterations per subproblem as possible. We have investigated a number of preconditioners, including FFT based preconditioners that model the approximately Toeplitz block Toeplitz nature of CC T with a circulant block circulant approximation [2, 3], high pass lter approximations to the FFT based preconditioner [4] the EM preconditioner XQ 1 [34] the exact diagonal of M , and diagonal Hessian approximations [46] Of the above preconditioners, by far the best performing was the exact diagonal of M , which can be computed at reasonable ....

R.H. Chan and M.K. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Review, 38 (1996), pp. 427-482.

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R. Chan and M. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Review, 38 (1996), pp. 427-482.

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R. Chan and M. Ng. Conjugate gradient method for Toeplitz system, SIAM Review, 38 (1996), pp. 427--482.

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R. Chan, M. Ng, Conjugate Gradient Methods for Toeplitz Systems, SIAM Review 38, No. 3 (1996) 427--482.

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R. H. Chan and M. K. Ng. Conjugate gradient methods of Toeplitz systems. SIAM Review, 38:427 -- 482, 1996.

....a matrix C as follows C = T X X T where X is chosen to make C circulant. Now C can be decomposed as C = F F where F is the N Gammapoint Fourier matrix and is a diagonal matrix containing the eigenvalues of C. For more details on fast methods for Hankel and Toeplitz matrices see, e.g. [CN]. 5. Numerical examples. In examples (i) ii) and (iii) below, we choose OE(i) i and (i) 0: Then u(j; j Gamma : Note that, for the conformal map f(z) from the disk, OE(z) f(z) and the boundary values at the mesh points are given by G(z) 4(Ref(z) Gamma ....

R. H. Chan and M. K. Ng, Conjugate gradient methods for Toeplitz systems, to appear in SIAM Review.

....Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong. E mail: mng maths.hku.hk. Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong. Research supported in part by HKU 7147 99P and HKU CRCG Grant Nos. 10201939 and 10202720. [7]. In this paper, we consider the solution of Hermitian positive definite Toeplitz systems. There are a number of specialized fast direct methods for solving such systems in O(n operations, see for instance [22] Faster methods requiring O(n log n) operations have also been developed, see ....

....for at most 2p 1 outliers, and that their smallest eigenvalues are bounded uniformly away from zero. It follows that the conjugate gradient method, when applied to solving these circulant preconditioned systems, will converge linearly. Since the cost per iteration is O(n log n) operations, see [7], the total complexity of solving these ill conditioned Toeplitz systems is of O(n log n) operations. In the case when f is positive, we show that the spectra of the preconditioned matrices are clustered around 1 and thus the method converges superlinearly. The case where f has multiple zeros is ....

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R. Chan and M. Ng, Conjugate Gradient Methods for Toeplitz Systems, SIAM Review, 38 (1996), pp. 427--482.

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R. Chan and M. Ng, Conjugate Gradient Methods for Toeplitz Systems, SIAM Review, V38, No. 3 (1996), pp. 427--482.

....is a cyclic shift of its preceding column. One important property of circulant matrices is that they can be diagonalized by Fast Fourier Transforms [27] Hence their inverses can be found easily. Circulant matrices have shown to be good preconditioners for Toeplitz systems in many applications [13] and in particular in queueing networks, see for instance [12, 14, 15, 16, 20] The main observation in queueing network applications is that most queueing networks have generator matrices that are close to Toeplitz matrices. These include sophisticated networks such as the Markov modulated ....

....Fast Fourier Transforms [27] Hence their inverses can be found easily. Circulant preconditioners have been used in many applications where the Toeplitz matrices come into play, such as in image processing, partial differential equations, integral equations and in particular queueing networks, see [13] and the references therein. In this section, we consider the construction of circulant preconditioners for SANs. The success of our preconditioners depends on the observation that in many network applications, the matrices D ij and E ij in (9) are low rank perturbations of Toeplitz matrices (cf. ....

R. Chan and K. Ng, Conjugate Gradient Methods for Toeplitz Systems. SIAM Review, 38 (1996), pp. 427--482. 17

....operations by using fast Fourier transforms. Preconditioning techniques for Toeplitz systems have been well studied in the past 10 years. However, most of the papers in this area are concerned with the case where the generating function f is either positive or nonnegative; see, for instance, [5, 3, 20, 7, 17, 10] and the references therein. In this paper, we consider f that has sign changes. The method we propose here will also work for generating functions that are positive or nonnegative. Up to now iterative methods for Toeplitz systems with generating functions having di#erent signs were only ....

R. H. Chan and M. K. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Rev., 38 (1996), pp. 427--482.

.... matrices of circulant integral operators using the rectangular quadrature rule are circulant matrices, see for instance Chan, Jin and Ng [4] Circulant matrices have been proposed and used as preconditioners for Toeplitz matrices in the past ten years, see the survey paper by Chan and Ng [7] and the references therein. It is established theoretically that the circulant preconditioned systems converge superlinearly when the given Toeplitz system is well conditioned, see for instance Chan and Strang [8] However, the performance of circulant preconditioners for ill conditioned Toeplitz ....

R. Chan and M. Ng, Conjugate Gradient Methods for Toeplitz Systems, SIAM Review, 38 (1996), 427-482.

.... ; ffi = 0; n Gamma 1; 1) see Tyrtyshnikov [17] Using the circulant structure of c(B) the inverse [c(B) of c(B) and the matrix vector multiplication [c(B) x for any vector x can be obtained in O(n log n) operations by using fast Fourier transforms, see for instance Chan and Ng [8]. Moreover, c(B) is positive definite whenever B is, see Tyrtyshnikov [17] This makes c(B) a very attractive choice of preconditioner in the preconditioned conjugate gradient method for solving the system By = b. For then, if B is positive definite, the preconditioned matrix is positive definite. ....

.... in (11) can be obtained by multiplying the augmented square circulant matrix to the augmented vector (q 1 ; q 2 ; q m ; 0; 0) This matrix vector product can be obtained efficiently by using three fast Fourier transforms of length 2m Gamma 1, see for instance Chan and Ng [8]. Thus the cost of obtaining the diagonal sums is O(m log m) operations and the storage required is O(m) Corollary 1 Let P and Q be two given k by m matrices. Then the diagonal sums of the product Q can be obtained in O(km log m) operations and the storage required is O(m) Proof: We have ....

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R. Chan and M. Ng, Conjugate Gradient Methods for Toeplitz Systems, SIAM Review, 38 (1996), 427--482.

....vector. The matrix system in (2.4) becomes T f = g. 2.8) We see from (2.6) that the coe#cient matrix T is a Toeplitz matrix. There are many iterative or direct Toeplitz solvers that can solve the Toeplitz system (2. 8) with costs ranging from O(n log n) to O(n ) operations; see, for instance, [19, 16, 1, 7]. In the two dimensional case, the resulting matrices will be block Toeplitz Toeplitz block matrices. Inversion of these matrices is known to be very expensive, e.g. the fastest direct Toeplitz solver is of O(n ) operations for an by n block Toeplitz Toeplitz block matrix; see [17] ....

....For the zero boundary condition, we have to solve a block Toeplitz Toeplitz block system for each . The fastest direct Toeplitz solver requires O(n ) operations; see [17] In our tests, the systems are solved by the preconditioned conjugate gradient method with circulant preconditioners [7]. Table 6.1 shows the numbers of iterations required for the two blurring functions for di#erent . The stopping tolerance is 10 6 . We note that the cost per iteration is about four two dimensional FFTs. Thus the cost is extremely expensive, especially when is small. In conclusion, we see that ....

R. Chan and M. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Rev., 38 (1996), pp. 427--482.

....is also immediately satis ed. Moreover for the case when the generating function a(z) is a function from the Wiener class, positive on the unit circle, the 3rd property for the G.Strang and T.Chan preconditioners was established in [C89] CS89] and in [CY92] resp. A recent survey [CN96] gives a fairly comprehensive review of these and related results, and describes many other preconditioners, including those of R.Chan, E.Tyrtyshnikov, T.Ku and C.Kuo, T.Huckle, and others. A thorough theoretical and numerical comparison of all di erent preconditioners is one of the directions of ....

.... comparison of all di erent preconditioners is one of the directions of current research, indicating that the question of which preconditioner is better may have di erent answers depending upon the particular classes of Toeplitz systems, and their generating functions, see, e.g. TS96] T95] [CN96]. Along with many favorable properties of circulant preconditioners, they unfortunately require complex arithmetic (for computing FFT s) even for real symmetric Toeplitz matrices. To overcome this disadvantage, D.Bini and F.Di Benedetto [BB90] proposed non circulant analogs of the G.Strang and ....

R.Chan and K.Ng, Conjugate gradient method for Toeplitz systems, SIAM Review, 38(1996), 427-482.

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R. H. Chan and M. K. Ng, "Conjugate gradient methods for Toeplitz systems," SIAM Rev., vol. 38, no. 3, pp. 427--482, 1996.

No context found.

R. H. Chan and M. K. Ng, Conjugate gradient methods for Toeplitz systems,SIAM Rev., 38 (1996), pp. 427--482.

No context found.

R. H. Chan and M. K. Ng. Conjugate Gradient Methods for Toeplitz Systems. SIAM Review, Vol. 38(3), pp. 427-482, September 1996.

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R. H. Chan and K. P. Ng, "Conjugate gradient method for Toeplitz systems, " SIAM Rev., vol. 38, no. 3, pp. 427--482, 1996.

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R.H. CHAN AND M.K. NG, Conjugate Gradient Methods for Toeplitz systems, SIAM Rev. 38 (1996), pp. 427--482.

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R. H. Chan and M. K. Ng, "Conjugate gradient method for toeplitz systems," SIAM Rev. 38 (1996) 427--482.

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