Preconditioning Strategies for Hermitian Toeplitz Systems with Nondefinite Generating Functions

A novel sine transform based preconditioned conjugate gradient (PCG) method for signal restoration is presented. If the bandwidth 2? +1 in the model matrix is a constant, then the asymptotic complexity for the new algorithm is O(N) per iterative step, where Nis the order of the model. This complexity is reduced by an order of magnitude in comparison with the known fast Fourier transform (FFT) technique proposed recently in the literature. Moreover, if ? has a size similar to N, then the new PCG method is accomplished by the fast sine transform (FST) and the fast Hartley transform (FHT). The computational and storage cost per preconditioning step for the new PCG method is reduced by 50% as compared to the FFT approach. To stabilize the computation, a special Tikhonov regularization is introduced. Numerical experimentations show that the new PCG method is of a speedy convergence rate. The new PCG methods maintain less computational complexity per iterative step and have a better convergence rate than the other known PCG methods.

In this study, we analyze the real-time solution of a series of augmented least squares problems, which are generated by adding information to an original least squares model repetitively. Instead of solving the least squares problems directly, we transform them into a batch of saddle point linear systems and subsequently solve the linear systems using restrictively preconditioned conjugate gradient (RPCG) methods. Approximation of the new Schur complement is generated effectively based on a previously approximated Schur complement. Owing to the variations of the preconditioned conjugate gradient method, the proposed methods generate convergence results similar to the conjugate gradient method and achieve a very fast convergent iterative sequence when the coefficient matrix is well preconditioned. Numerical tests show that the new methods are more effective than some standard Krylov subspace methods. Updated RPCG methods meet the requirement of real-time computing successfully for multifactor models.

In order to solve Toeplitz linear systems An(f)x=b generated by a nonnegative integrable function f, through use of the preconditioned conjugate gradient (PCG) method, several authors have proposed An(g) as preconditioner in the case where g is a trigonometric polynomial [10, 14, 27, 12, 28]. In preceding works, we studied the distribution and the extremal properties of the spectrum of the preconditioned matrix G=An−1(g) An(f). In this paper we prove that the union of the spectra of all the Gn is dense on the essential range of f/g, i.e.,ER(f/g) and we obtain asymptotic information about the rate of convergence of the smallest eigenvalue λln of Gn to r (and of λnn to R). As a consequence of this second order result, it is possible to handle the case where f has zeros of any order θ, through the PCG methods proposed in [10, 14]. This is a noteworthy extension since the techniques developed in [10, 14, 27, 12, 28] are shown to be effective only when f has zeros of even orders. The cost of this procedure is O(n1+c(θ) log n) arithmetic operations (ops) where the quantity c(θ) belongs to interval [0,2−1] and takes the maximum value 2−1 when f has a zero of odd order. Finally, for the special case of zeros of odd orders, we propose a further algorithm which makes use of the PCG techniques proposed in [10, 14, 27, 12, 28] for theeven order case, reducing the cost to O(n long n) ops.

Convergence analysis of superoptimal PCG algorithm for Toeplitz systems with a Fisher–Hartwig singularity

Gauss-Seidel preconditioned conjugate gradient (GS-PCG) method based on element-by-element (EBE) is derived and the calculation process of finite element method (FEM) is presented in this paper. The corresponding FEM program are developed and implemented on the magnetic field calculations which produced by straight conductor carrying current and 3-phases induction motor respectively. The validity and accuracy of the method have been verified by comparing the result with the analytical solution. Compared with Jacobi preconditioned conjugate gradient (J-PCG) method based on EBE-FEM, less computational time and cost are needed for the proposed method with same accuracy. The proposed method can be applied to electromagnetic field analysis of transformer and motor, which may refer large scale numerical computation.

Various Toeplitz preconditioners $P_N $ have recently been proposed so that an $N \times N$ symmetric positive definite Toeplitz system $T_N {\bf x} = {\bf b}$ can be solved effectively by the preconditioned conjugate gradient (PCG) method. It has been proven that if $T_N $ is generated by a positive function in the Wiener class, the eigenvalues of the preconditioned matrices $P_N^{ - 1} T_N $ are clustered between $( 1 - \epsilon ,1 + \epsilon )$ except for a fixed number independent of N. In this research, the spectra of $P_N^{ - 1} T_N $ are characterized more precisely for rational Toeplitz matrices $T_N $ with preconditioners proposed by Strang [Stud. Appl. Math., 74 (1986), pp. 171–176] and Ku and Kuo [IEEE Trans. Signal Process., 40 (1992), pp. 129–141]. The eigenvalues of $P_N^{ - 1} T_N $ are classified into two classes, i.e., the outliers and the clustered eigenvalues, depending on whether they converge to 1 asymptotically. It is proved that the number of outliers depends on the order of the rat...

High-resolution global ocean general circulation models (OGCMs) play a key role in accurate ocean forecasting. However, the models of the operational forecasting systems are still not in high resolution due to the subsequent high demand for large computation, as well as the low parallel efficiency barrier. Good scalability is an important index of parallel efficiency and is still a challenge for OGCMs. We found that the communication cost in a barotropic solver, namely, the preconditioned conjugate gradient (PCG) method, is the key bottleneck for scalability due to the high frequency of the global reductions. In this work, we developed a new algorithm—a communication-avoiding Krylov subspace method with a PCG (CA-PCG)—to improve scalability and then applied it to the Nucleus for European Modelling of the Ocean (NEMO) as an example. For PCG, inner product operations with global communication were needed in every iteration, while for CA-PCG, inner product operations were only needed every eight iterations. Therefore, the global communication cost decreased from more than 94.5% of the total execution time with PCG to less than 63.4% with CA-PCG. As a result, the execution time of the barotropic modes decreased from more than 17,000 s with PCG to less than 6000 s with CA-PCG, and the total execution time decreased from more than 18,000 s with PCG to less than 6200 s with CA-PCG. Besides, the ratio of the speedup can also be increased from 3.7 to 4.6. In summary, the high process count scalability when using CA-PCG was effectively improved from that using the PCG method, providing a highly effective solution for accurate ocean simulation.

A preconditioned Chebyshev basis communication-avoiding conjugate gradient method (P-CBCG) is applied to the pressure Poisson equation in a multiphase thermal-hydraulic CFD code JUPITER, and its computational performance and convergence properties are compared against a preconditioned conjugate gradient (P-CG) method and a preconditioned communication-avoiding conjugate gradient (P-CACG) method on the Oakforest-PACS, which consists of 8,208 KNLs. The P-CBCG method reduces the number of collective communications with keeping the robustness of convergence properties. Compared with the P-CACG method, an order of magnitude larger communication-avoiding steps are enabled by the improved robustness. It is shown that the P-CBCG method is 1.38 $$\times $$ and 1.17 $$\times $$ faster than the P-CG and P-CACG methods at 2,000 processors, respectively.

A fast fourth-order ADI scheme is proposed to solve 2D Riesz space-fractional diffusion equations (RSFDEs). This scheme involves a sequence of symmetric positive definite (SPD) Toeplitz linear systems, thus allowing the fast sine transform to be employed for computational efficiency. Furthermore, those discretized SPD systems are solved using the preconditioned conjugate gradient (PCG) method. Our theoretical analysis demonstrates that the preconditioned matrix can be expressed as the sum of an identity matrix, a small-norm matrix, and a low-rank matrix. Numerical results verify the method’s efficacy.

Iterative 3D PET reconstruction represents a very computational challenge due to the large number of lines of response (LOR) collected for each data set. This iterative 3D reconstruction also needs a lot of iterations to achieve an acceptable PET reconstructed image. A preconditioned conjugate gradient (PCG) method was previously shown to have faster convergence rate than expectation maximization (EM) type algorithms. For the microPET, images suffer from crystal penetration blurring due to small scanner radius. An exact 2D blur model is needed to achieve high resolution image. A Preconditioned Conjugate Gradient (PCG) method is described for reconstruction of high-resolution 3D images from the microPET Inveon small-animal scanner from Siemens. The projector pair is used as part of a factored system matrix that takes into account detector-pair response by using shift-variant sinogram blur kernels, attenuation correction, and detector efficiency corrections. The system matrix for geometric projection is based on depth dependent solid angle calculation in combination with a spatially variant detector response model. The mircoPET PCG is combined with OSEM to accelerate convergence. This reconstruction model achieves a high resolution animal image; however, it took an hour to reconstruct a frame. Therefore, we describe an ultrafast forward and back projector pair based on Symmetry and SIMD projector (SSP). The proposed method produces similar quality images when compared to those obtained with the software package from Siemens and requires order of magnitude less computation.

This study presents an efficacy comparison of iterative solution methods for solving the compressible-fluid Reynolds equation in modeling air- or gas-lubricated bearings. A direct fixed-point iterative (DFI) method and Newton’s method are employed to transform the Reynolds equation in a form that can be solved iteratively. The iterative solution methods examined are the Gauss–Seidel method, the successive over-relaxation (SOR) method, the preconditioned conjugate gradient (PCG) method, and the multigrid method. The overall solution time is affected by both the transformation method and the iterative method applied. In this study, Newton’s method shows its effectiveness over the straightforward DFI method when the same iterative method is used. It is demonstrated that the use of an optimal relaxation factor is of vital importance for the efficiency of the SOR method. The multigrid method is an order faster than the PCG and optimal SOR methods. Also, the multigrid and PCG methods involve an extended coding work and are less flexible in dealing with gridwork and boundary conditions. Consequently, a compromise has to be made in terms of ease of use as well as programming effort for the solution of the compressible-fluid Reynolds equation.

A new preconditioning strategy for symmetric positive definite banded circulant and Toeplitz systems is described. The optimal tridiagonal preconditioner for tridiagonal circulant systems is modified and applied to both circulant and Toeplitz banded systems. The strategy is extended to block tridiagonal systems. The parallelisation aspects of the PCG algorithm are discussed.

Classical iterative methods when applied to the partial solution of the generalized eigenvalue problem Ax =λBx, may yield very poor convergence rates particularly when ill‐conditioned problems are considered. In this paper the preconditioned conjugate gradient (CG) method via the minimization of the Rayleigh quotient and the reverse power method is employed for the partial eigenproblem. The triangular splitting preconditioners employed are obtained from an incomplete Choleski factorization and a partial Evans preconditioner. This approach can dramatically improve the convergence rate of the basic CG method and is applicable to any symmetric eigenproblem in which one of the matrices A, B is positive definite. Because of the renewed interest in CG techniques for FE work on microprocessors and parallel computers, it is believed that this improved approach to the generalized eigenvalue problem is likely to be very promising.

A parallel implementation of the specialized interior-point algorithm for multicommodity network flows introduced in [6] is presented. In this algorithm, the positive definite systems of each iteration are solved through a scheme that combines direct factorizations and a preconditioned conjugate gradient (PCG) method. Although this numerical procedure works well in practice, it requires the solution of at least k systems of equations at each iteration of the PCG, k being the number of commodities to be routed through the network.

Efficient solver for mixed finite element method and control-volume mixed finite element method in 3-D on Hexahedral grids