Fast Collocation Method Research Articles

A fast collocation method for solving the weakly singular fractional integro-differential equation

A fast collocation method for solving the weakly singular fractional integro-differential equation

Fast collocation method for a two-dimensional variable-coefficient linear nonlocal diffusion model

In this paper, a fast collocation method is developed for a two-dimensional variable-coefficient linear nonlocal diffusion model. By carefully dealing with the variable coefficient in the integral operator and then analyzing the structure of the coefficient matrix, we can reduce the computational operations in each Krylov subspace iteration from O(N^{2}) to O(Nlog N) and the memory requirement for the coefficient matrix from O(N^{2}) to O(N). Numerical experiments are carried out to show the utility of the fast collocation method.

A preconditioned fast collocation method for a linear bond-based peridynamic model

We develop a fast collocation method for a static bond-based peridynamic model. Based on the analysis of the structure of the stiffness matrix, a fast matrix-vector multiplication technique was found, which can be used in the Krylov subspace iteration method. In this paper, we also present an effective preconditioner to accelerate the convergence of the Krylov subspace iteration method. Using the block-Toeplitz–Toeplitz-block (BTTB)-type structure of the stiffness matrix, we give a block-circulant-circulant-block (BCCB)-type preconditioner. The numerical experiments show the utility of the preconditioned fast collocation method.

A Preconditioned Fast Collocation Method for a Linear Nonlocal Diffusion Model in Convex Domains

Recently, there are many papers dedicated to develop fast numerical methods for nonlocal diffusion and peridynamic models. However, these methods require the physical domain where we solve the governing equations is rectangular. To relax this restriction, in this article, we develop a fast collocation method for a static linear nonlocal diffusion model in convex domains via a volume-penalization approach. Based on the analysis of the structure of the coefficient matrix, we also present a fast solution technique of the linear system arising from the collocation discretization by accelerating the matrix-vector multiplications in the usual Krylov subspace iteration method. This technique helps to reduce the computational work in each Krylov subspace iteration from $O(N^{2})$ to $O(N\log N)$ and the storage requirement of the coefficient matrix from $O(N^{2})$ to $O(N)$ without any lossy compression, where $N$ is the number of unknowns. Moreover, an efficient preconditioner was also provided in this work to accelerate the convergence of the Krylov subspace iteration method. Numerical experiments show the applicability of this method.

A fast collocation method for a variable-coefficient nonlocal diffusion model

We develop a fast collocation scheme for a variable-coefficient nonlocal diffusion model, for which a numerical discretization would yield a dense stiffness matrix. The development of the fast method is achieved by carefully handling the variable coefficients appearing inside the singular integral operator and exploiting the structure of the dense stiffness matrix. The resulting fast method reduces the computational work from O(N3) required by a commonly used direct solver to O(Nlog⁡N) per iteration and the memory requirement from O(N2) to O(N). Furthermore, the fast method reduces the computational work of assembling the stiffness matrix from O(N2) to O(N). Numerical results are presented to show the utility of the fast method.

A fast collocation method for high-frequency acoustic scattering

% Based on the boundary integral equation reduced from the sound-soft high frequency scattering problem in two dimensions, we firstly convert the density to slow oscillatory from high oscillatory by using the asymptotic behaviour as $k \\to +\\infty$ of density in a strictly convex bounded region. Furthermore, the equivalence of second Fredholm integral equation is obtained. Using the composite mesh with a polynomial grading on the illuminated zone and a geometric grading on the shadow zone, we obtain that the convergence is $O(k^{-\\frac {1} {24}}) $ $ (k\\gg 1)$. Moreover, examples are given for demonstrating that our method has good accuracy.

A fast collocation method for a static bond-based linear peridynamic model

We develop a fast collocation method for a steady-state bond-based linear peridynamic model in two space dimensions. The method reduces the computational work from O(N2) per Krylov subspace iteration in a traditional collocation method to O(NlogN) and the memory requirement from O(N2) in a collocation method to O(N), where N is the number of unknowns in the discrete system. Furthermore, the method reduces the computational work of evaluating and assembling the stiffness matrix, which often constitutes a major portion of the overall computational work, from O(N2) in a collocation method to O(N). The significant reduction of the CPU times and storage in the fast collocation method is achieved by carefully exploring the structure of the stiffness matrix of the collocation method without any lossy compression involved. In other words, the fast method is evaluated in an equivalent but more efficient manner. Therefore, the fast method generates identical numerical solutions as the traditional collocation method does and naturally inherits the stability and convergence properties that were already proved for a traditional collocation method. Numerical results are presented to show the utility of the fast method.

Fast wavelet collocation method for three-dimensional Hammerstein equation

ABSTRACTIn this paper, we extend the wavelet collocation method to a class of three-dimensional Hammerstein equation with weakly singular kernel. Three-dimensional wavelets and collocation polynomials on the unit cube are constructed. A fast wavelets collocation method is obtained by the use of a practical block truncation strategy. Numerical results are also included.

Fast and Accurate Collocation of the Visible Infrared Imaging Radiometer Suite Measurements with Cross-Track Infrared Sounder

Given the fact that Cross-track Infrared Sounder (CrIS) and the Visible Infrared Imaging Radiometer Suite (VIIRS) are currently onboard the Suomi National Polar-orbiting Partnership (Suomi NPP) satellite and will continue to be carried on the same platform as future Joint Polar Satellite System (JPSS) satellites for the next decade, it is desirable to develop a fast and accurate collocation scheme to collocate VIIRS products and measurements with CrIS for applications that rely on combining measurements from two sensors such as inter-calibration, geolocation assessment, and cloud detection. In this study, an accurate and fast collocation method to collocate VIIRS measurements within CrIS instantaneous field of view (IFOV) directly based on line-of-sight (LOS) pointing vectors is developed and discussed in detail. We demonstrate that this method is not only accurate and precise from a mathematical perspective, but also easy to implement computationally. More importantly, with optimization, this method is very fast and efficient and thus can meet operational requirements. Finally, this collocation method can be extended to a wide variety of sensors on different satellite platforms.

A fast Fourier-collocation method for second boundary integral equations

In this paper we develop a fast collocation method for second boundary integral equations by the trigonometric polynomials. We propose a convenient way to compress the dense matrix representation of a compact integral operator with a smooth kernel under the Fourier basis and the corresponding collocation functionals. The compression leads to a sparse matrix with only O ( n log 2 n ) number of nonzero entries, where 2 n + 1 denotes the order of the matrix. Thus we develop a fast Fourier-collocation method. We prove that the fast Fourier-collocation method gives the optimal convergence order up to a logarithmic factor. Moreover, we design a fast scheme for solving the corresponding truncated linear system. We establish that this algorithm preserves the quasi-optimal convergence of the approximate solution with requiring a number of O ( n log 3 n ) multiplications.

A Fast Collocation Method for Eigen-Problems of Weakly Singular Integral Operators

A multiscale collocation method is developed for solving the eigen-problem of weakly singular integral operators. We employ a matrix truncation strategy of Chen, Micchelli and Xu to compress the collocation matrix, which the compressed matrix has only $\mathcal{O}(N\log N)$ nonzero entries, where N denotes the order of the matrix. This truncation leads to a fast collocation method for solving the eigen-problem. We prove that the fast collocation method has the optimal convergence order for approximation of the eigenvalues and eigenvectors. The power iteration method is used for solving the corresponding discrete eigen-problem. We present a numerical example to demonstrate how the methods can be used to compute a nonzero eigenvalue rapidly and efficiently.

A Fast Collocation Method for Solving Stochastic Integral Equations

Based on sparse grid multiscale piecewise polynomial bases, we develop a fast collocation method for solving Fredholm integral equations of the second kind with stochastic loading terms. It is proved that the proposed method preserves the optimal rate of convergence and has linear (up to a logarithmic factor) computational complexity. The reduction in computational costs come from three aspects: the adoption of the collocation principle, the use of the sparse grid for the solution space, and the employment of a truncation strategy for the coefficient matrix of the resulting linear system. Numerical experiments confirm the theoretical results on approximation accuracy and computational efficiency of the proposed method.

A fast collocation method for acoustic scattering in shallow oceans

In this paper, we investigate the numerical solution of the integral equation of the second kind reduced by acoustic scattering in shallow oceans with Dirichlet condition. Based on analyzing the singularity of the truncating kernel with a sum of infinite series, using our trigonometric interpolatory wavelets and collocation method, we obtain the numerical solution which possesses a fast convergence rate like o ( 2 − j ) . Moreover, the entries of the stiffness matrix can be obtained by FFT, which lead the computational complexity to decrease obviously.

A fast collocation method for the radiosity equation, based on the hierarchical algorithm of Hanrahan and Salzman: the 1d case

In this paper we study a collocation method with piecewise constant trial functions for the solution of the planar radiosity equation. The matrix of the collocation method is approximated by a method which was developed by Hanrahan et al. [9,10]. We prove that the modified collocation method results in a reduction of work while the order of convergence stays the same. Numerical examples demonstrate the theoretical results.

A fast collocation method for an inverse boundary value problem

AbstractIn this paper, we present an implementation of a fast multiscale collocation method for boundary integral equations of the second kind, and its application to solving an inverse boundary value problem of recovering a coefficient function from a boundary measurement. We illustrate by numerical examples the insensitive nature of the map from the coefficient to measurement, and design and test a Gauss–Newton iteration algorithm for obtaining the best estimate of the unknown coefficient from the given measurement based on a least‐squares formulation. Copyright © 2004 John Wiley & Sons, Ltd.

A Gravimetric Geoid Computation and Comparison with GPS Results in Northern Andalusia (Spain)

Two new GPS surveys have been carried out to check the accuracy of an existing gravimetric geoid in a test area located in northern Andalusia (Spain). The fast collocation method and the remove-restore procedure have been used for the computation of the quasigeoid model. The Spanish height system is based on orthometric heights, so the gravimetrically determined quasigeoid has been transformed to a geoid model and then compared to geoid undulations provided by GPS and levelling at benchmarks belonging to the Spanish first-order levelling network. The discrepancies between the gravimetric solution and GPS/levelling undulations amount to ±2 cm for one survey and ±5 cm for another after fitting a plane to the geoid model.