Fast Direct Algorithm Research Articles

A Fourth Order Hermitian Box-Scheme with Fast Solver for the Poisson Problem in a Square

A new fourth order box-scheme for the Poisson problem in a square with Dirichlet boundary conditions is introduced, extending the approach in Croisille (Computing 78:329---353, 2006). The design is based on a box approach, combining the approximation of the gradient by the fourth order hermitian derivative, with a conservative discrete formulation on boxes of length 2h. The goal is twofold: first to show that fourth order accuracy is obtained both for the unknown and the gradient; second, to describe a fast direct algorithm, based on the Sherman-Morrison formula and the Fast Sine Transform. Several numerical results in a square are given, indicating an asymptotic O(N 2log?2(N)) computing complexity.

Fast directional multilevel algorithm combined with Calderon multiplicative preconditioner for stable electromagnetic scattering analysis

AbstractIn this article, a new method called fast directional multilevel algorithm (FDMA) is proposed for three‐dimensional electromagnetic problems. Combined with the Caldron identities of Calderon multiplicative preconditioner (CMP), the new algorithm has a fast convergence rate of iterative solvers and is stable at low frequency for the electrical field integral equation solutions. Like the fast multipole method, octree structure is used in our proposed algorithm. However, the new algorithm does not require the implementation of multipole expansions of the Green's function but based only on kernel evaluations. It does not suffer from low‐frequency breakdown when the discretization density is high. The numerical results demonstrate that the CMP‐preconditioned FDMA leads to significant reduction of both the iteration number and the computer processing use (CPU) time for radar cross section (RCS) calculation. © 2010 Wiley Periodicals, Inc. Microwave Opt Technol Lett 52: 1963–1969, 2010; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.25421

Simplified 2-D Cubic Spline Interpolation Scheme Using Direct Computation Algorithm

It has been shown that the 2-D cubic spline interpolation (CSI) proposed by Truong et al. is one of the best algorithms for image resampling or compression. Such a CSI algorithm together with the image coding standard, e.g., JPEG, can be used to obtain a modified image codec while still maintaining a good quality of the reconstructed image for higher compression ratios. In this paper, a fast direct computation algorithm is developed to improve the computational efficiency of the original FFT-based 2-D CSI methods. In fact, this algorithm computes the 2-D CSI directly without explicitly calculating the complex division usually needed in the FFT or Winograd discrete Fourier transform (WDFT) algorithm. In addition, this paper describes a novel way to derivate the 2-D CSI from the 1-D CSI by using the row-column method. This new fast 2-D CSI provides a regular and simple structure based upon linear correlations. Therefore, it can be implemented by the use of a modification of Kung’s pipeline structure and is naturally suitable for VLSI implementations. Experimental results show that the proposed new fast 2-D CSI algorithm can achieve almost the same CSI performance with much fewer arithmetic operations in comparison with existing efficient algorithms.

Fast Directional Continuous Spherical Wavelet Transform Algorithms

We describe the construction of a spherical wavelet analysis through the inverse stereographic projection of the Euclidean planar wavelet framework, introduced originally by Antoine and Vandergheynst and developed further by Wiaux et al. Fast algorithms for performing the directional continuous wavelet analysis on the unit sphere are presented. The fast directional algorithm, based on the fast spherical convolution algorithm developed by Wandelt and Gorski, provides a saving of O(sqrt(Npix)) over a direct quadrature implementation for Npix pixels on the sphere, and allows one to perform a directional spherical wavelet analysis of a 10^6 pixel map on a personal computer.

A fast direct Fourier-based algorithm for subpixel registration of images

This paper presents a new direct Fourier-based algorithm for performing image-to-image registration to subpixel accuracy, where the image differences are restricted to translations and uniform changes of illumination. The algorithm detects the Fourier components that have become unreliable estimators of shift due to aliasing, and removes them from the shift-estimate computation. In the presence of aliasing, the average precision of the registration is a few hundredths of a pixel. Experimental data presented here show that the new algorithm yields superior registration precision in the presence of aliasing when compared to several earlier methods and has comparable precision to the iterative method of P. Thevenaz et al. (1998).

A Fast Direct Algorithm for the Solution of the Laplace Equation on Regions with Fractal Boundaries

An algorithm is presented for the rapid direct solution of the Laplace equation on regions with fractal boundaries. In a typical application, the numerical simulation has to be on a very large scale involving at least tens of thousands of equations with as many unknowns, in order to obtain any meaningful results. Attempts to use conventional techniques have encountered insurmountable difficulties, due to excessive CPU time requirements of the computations involved. Indeed, conventional direct algorithms for the solution of linear systems require order O(N3) operations for the solution of an N × N-problem, while classical iterative methods require order O(N2) operations, with the constant strongly dependent on the problem in question. In either case, the computational expense is prohibitive for large-scale problems. The direct algorithm of the present paper requires O(N) operations with a constant dependent only on the geometry of the boundary, making it considerably more practical for large-scale problems encountered in the computation of harmonic measure of fractals, complex iteration theory, potential theory, and growth phenomena such as crystallization, electrodeposition, viscous fingering, and diffusion-limited aggregation.

Preconditioned Richardson and Minimal Residual Iterative Methods for Piecewise Hermite Bicubic Orthogonal Spline Collocation Equations

The preconditioned Richardson and preconditioned minimal residual iterative methods are presented for the solution of linear equations arising when orthogonal spline collocation with piecewise Hermite bicubics is applied to a selfadjoint elliptic Dirichlet boundary value problem on a rectangle. For both methods, the orthogonal spline collocation discretization of Laplace’s operator is used as a preconditioner. In the preconditioned Richardson method, an approximation of the optimal iteration parameter is computed from knowledge of spectral equivalence constants. Such a priori information is not required for the preconditioned minimal residual method. In each iteration of both methods, orthogonal spline collocation Poisson’s problems are solved by a fast direct algorithm which employs fast Fourier transforms. An application of the preconditioned minimal residual method is also discussed for the solution of linear equations arising from the orthogonal spline collocation discretization of nonselfadjoint elliptic Dirichlet boundary value problems.

A variational modification algorithm for three‐dimensional mass flux non‐divergence

AbstractThe diagnosed three‐dimensional (3D) wind field should strictly satisfy the continuity equation; however, because of analysis, interpolation, and numerical errors this requirement is not met in practice. The resulting noise divergence is of the order of 10−6s−1. Its impact upon energy budgets is fatal. In order to remove the noise divergence it is suggested that a 3D minimum mean‐square modifying field is added to the diagnosed wind under the constraint that the modified wind strictly satisfies mass continuity. It is shown that the solution of the resulting variational problem, namely the modifying wind field, has the property of conserving the 3D vorticity of the diagnosed wind field.In this paper the theory proposed for modification of the diagnosed wind field is rigorously developed for the first time for both the continuous and the discrete situation. The discrete problem—its solution being referred to as variational modification algorithm (VMA)—is approached by converting the diagnosed wind field into mass fluxes across the faces of finite boxes; the modifying mass flux components are obtained as differences of a potential function being the solution of a 3D Poisson equation which—owing to consideration of mass fluxes—is free of coordinate increments. A fast direct algorithm to solve this equation through decoupling coordinate directions is presented and compared with a fast Fourier algorithm.The VMA is applied to the wind field over Europe, as routinely analysed by the European Centre for Medium‐range Weather Forecasts (ECMWF), for the specific case of the strong rain episode over central Europe on 1 August 1991. The wind is converted into mass flux components across the faces of boxes (100 km)2 × 100 hPa in size. The modification of the mass flux enforced by the VMA is about 7 per cent of the analysed mass flux, and 6 per cent of the initialized mass flux; this similarity suggests that the original 3D noise divergence is not in the ECMWF data (analysed or initialized), but is generated by the interpolation procedure from the ECMWF model coordinates to pressure levels. Investigation of analysed and initialized mass fluxes before and after application of the VMA reveals that the modification of the mass flux field through the initialization procedure at the ECMWF is considerably larger than the modification required to enforce non‐divergence through application of the VMA.The impact of the modification upon the imbalance of the heat budgets in the atmosphere over Europe is investigated by a diagnostic model. It is found that the VMA reduces the root‐mean‐square values of the sensible‐heat imbalance from 518 (337) down to 66 (35) W m−2 for analysed (initialized) input data. The reduction is less dramatic for the latent‐heat imbalance (about 30 per cent).The effectiveness of the VMA as well as its generalization to arbitrary coordinate systems and variable resolution are briefly discussed.

Fast FFT-based algorithm for phase estimation in speckle imaging

This paper describes a fast direct algorithm for obtaining least-squares phase estimates from arrays of noisy phase differences. The algorithm uses the fast Fourier transform to diagonalize and decouple the system of equations which results from the application of the least-squares criterion. It is accurate and stable, and is perhaps an order of magnitude faster than the best iterative method. The effectiveness of the algorithm has been demonstrated by using it in connection with the Knox-Thompson speckle-imaging procedure to restore an optical object perturbed by simulated atmospheric turbulence. Representative results are discussed in the paper.

A fast directional algorithm for high frequency acoustic scattering in two dimensions

This paper is concerned with fast solution of high frequency acoustic scattering problems in two dimensions. We introduce a directional multiscale algorithm for the $N$-body problem of the two dimensional Helmholtz kernel. The algorithm follows the approach developed in Engquist and Ying, SIAM J. Sci. Comput., 29 (4), 2007, where the three dimensional case was studied. The main observation is that, for two regions that follow a directional parabolic geometric conguration, the interaction between these two regions through the 2D Helmholtz kernel is approximately low rank. We propose an improved randomized procedure for generating the low rank separated representation for the interaction between these regions. Based on this representation, the computation of the far field interaction is organized in a multidirectional and multiscale way to achieve maximum efficiency. The proposed algorithm is accurate and has the optimal $O(NlogN)$ complexity for problems from two dimensional scattering applications. Finally, we combine this fast directional algorithm with standard boundary integral formulations to solve acoustic scattering problems that are of thousands of wavelengths in size.