• Geophysics and Geophysical Exploration
• /
• v.9 no.1
• /
• pp.121-128
• /
• 2006

Magnetic survey flights by helicopters are usually parallel to the topographic surface, with a nominal clearance, but especially in high-resolution surveys the altitudes at which observations are made may be too variable to be regarded as a smooth surface. We have developed a reduction procedure for such data using the method of equivalent sources, where surrounding sources are included to control edge effects, and data from points distributed randomly in three dimensions are directly modelled. Although the problem is generally underdetermined, the method of conjugate gradients can be used to find a minimum-norm solution. There is freedom to select the harmonic function that relates the magnetic anomaly with the source. When the upward continuation function operator is selected, the equivalent source is the magnetic anomaly itself. If we select as source a distribution of magnetic dipoles in the direction of the ambient magnetic field, we can easily derive reduction-to-pole anomalies by rotating the direction of the magnetic dipoles to vertical.

• Journal of applied mathematics & informatics
• /
• v.25 no.1_2
• /
• pp.435-443
• /
• 2007

Computing the interior spectrum of large sparse generalized eigenvalue problems $Ax\;=\;{\lambda}Bx$, where A and b are large sparse and SPD(Symmetric Positive Definite), is often required in areas such as structural mechanics and quantum chemistry, to name a few. Recently, CG-type methods have been found useful and hence, very amenable to parallel computation for very large problems. Also, as in the case of linear systems proper choice of preconditioning is known to accelerate the rate of convergence. After the smallest eigenpair is found we use the orthogonal deflation technique to find the next m-1 eigenvalues, which is also suitable for parallelization. This offers advantages over Jacobi-Davidson methods with partial shifts, which requires re-computation of preconditioner matrx with new shifts. We consider as preconditioners Incomplete LU(ILU)(0) in two variants, ever-relaxation(SOR), and Point-symmetric SOR(SSOR). We set m to be 5. We conducted our experiments on matrices from discretizations of partial differential equations by finite difference method. The generated matrices has dimensions up to 4 million and total number of processors are 32. MPI(Message Passing Interface) library was used for interprocessor communications. Our results show that in general the Multi-Color ILU(0) gives the best performance.

• Nuclear Engineering and Technology
• /
• v.52 no.3
• /
• pp.485-498
• /
• 2020

A diffusion synthetic acceleration (DSA) technique for the S_N transport equation discretized with the linear discontinuous expansion method with subcell balance (LDEM-SCB) on unstructured tetrahedral meshes is presented. The LDEM-SCB scheme solves the transport equation with the discrete ordinates method by using the subcell balances and linear discontinuous expansion of the flux. Discretized DSA equations are derived by consistently discretizing the continuous diffusion equation with the LDEM-SCB method, however, the discretized diffusion equations are not fully consistent with the discretized transport equations. In addition, a fine mesh rebalance (FMR) method is devised to accelerate the discretized diffusion equation coupled with the preconditioned conjugate gradient (CG) method. The DSA method is applied to various test problems to show its effectiveness in speeding up the iterative convergence of the transport equation. The results show that the DSA method gives small spectral radii for the tetrahedral meshes having various minimum aspect ratios even in highly scattering dominant mediums for the homogeneous test problems. The numerical tests for the homogeneous and heterogeneous problems show that DSA with FMR (with preconditioned CG) gives significantly higher speedups and robustness than the one with the Gauss-Seidel-like iteration.

• International Union of Geodesy and Geophysics Korean Journal of Geophysical Research
• /
• v.25 no.1
• /
• pp.23-34
• /
• 1997

Seismic tomography using the graph theoretical method of ray tracing is performed in two synthetic data sets with laterally varying velocity structures. The straight-ray tomography shows so poor results in imaging the laterally varying velocity structure that the ray-traced tomographic techniques should be used. Conventional ray tracing methods have serious drawbacks, i.e. problems of convergence and local minima, when they are applied to seismic tomography. The graph theretical method finds good approximated raypaths in rapidly varying media even in shadow zones, where shooting methods meet with convergence problems. The graph theoretical method ensures the globally minimal traveltime raypath while bending methods often cause local minima problems. Especially, the graph theoretical method is efficient in case that many sources and receivers exist, since it can find the traveltimes and corresponding raypaths to all receivers from a specific source at one time. Moreover, the algorithm of graph theoretical method is easily applicable to the ray tracing in anisotropic media, and even to the three dimensional case. Among the row-active inversion techniques, the conjugate gradient (CG) method is used because of fast convergence and high efficiency. The iterative sequence of the ray tracing by the graph theoretical method and the inversion by the CG method is an efficient and robust algorithm for seismic tomography in laterally varying velocity structures.

• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.22 no.1
• /
• pp.35-44
• /
• 2009

This paper describes an application of domain decomposition method for parallel finite element analysis which is required to large scale 3D structural analysis. A parallel finite element method system which adopts a domain decomposition method is developed. Node is generated if its distance from existing node points is similar to the node spacing function at the point. The node spacing function is well controlled by the fuzzy knowledge processing. The Delaunay triangulation method is introduced as a basic tool for element generation. Domain decomposition method using automatic mesh generation system holds great benefits for 3D analyses. Aa parallel numerical algorithm for the finite element analyses, domain decomposition method was combined with an iterative solver, i.e. the conjugate gradient(CG) method where a whole analysis domain is fictitiously divided into a number of subdomains without overlapping. Practical performance of the present system are demonstrated through several examples.

• Journal of the Korea Academia-Industrial cooperation Society
• /
• v.12 no.8
• /
• pp.3384-3390
• /
• 2011

This paper describes a domain decomposition method of parallel finite element analysis for elasto-plastic structural problems. As a parallel numeral algorithm for the finite element analysis, the authors have utilized the domain decomposition method combined with an iterative solver such as the conjugate gradient method. Here the domain decomposition method algorithm was applied directly to elasto-plastic problem. The present system was successfully applied to three-dimensional elasto-plastic structural problems.

• Journal of applied mathematics & informatics
• /
• v.36 no.3_4
• /
• pp.317-330
• /
• 2018

We first provide how to apply the global preconditioned conjugate gradient ($G{\ell}-PCG$) method with Kronecker product preconditioners to image deblurring problems with nearly separable point spread functions. We next provide a coarse-grained parallel image deblurring algorithm using the $G{\ell}-PCG$. Lastly, we provide numerical experiments for image deblurring problems to evaluate the effectiveness of the $G{\ell}-PCG$ with Kronecker product preconditioner by comparing its performance with those of the $G{\ell}-CG$, CGLS and preconditioned CGLS (PCGLS) methods.

• Steel and Composite Structures
• /
• v.20 no.5
• /
• pp.1119-1131
• /
• 2016

As a first attempt, an inverse hybrid numerical method for small scale parameter estimation of functionally graded (FG) nanobeams using measured frequencies is presented. The governing equations are obtained with the Eringen's nonlocal elasticity assumptions and the first-order shear deformation theory (FSDT). The equations are discretized by using the differential quadrature method (DQM). The discretized equations are transferred from temporal domain to frequency domain and frequencies of the nanobeam are obtained. By applying random error to these frequencies, measured frequencies are generated. The measured frequencies are considered as input data and inversely, the small scale parameter of the beam is obtained by minimizing a defined functional. The functional is defined as root mean square error between the measured frequencies and calculated frequencies by the DQM. Then, the conjugate gradient (CG) optimization method is employed to minimize the functional and the small scale parameter is obtained. Efficiency, convergence and accuracy of the presented hybrid method for small scale parameter estimation of the beams for different applied random error, boundary conditions, length-to-thickness ratio and volume fraction coefficients are demonstrated.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.3 no.2
• /
• pp.1-4
• /
• 1999

The Helmholtz equation is very important in physics and engineering. However, solution of the Helmholtz equation is in general known as a very difficult phenomenon. For if the ${\omega}$ is negative, the FDM discretized linear system becomes indefinite, whose solution by iterative method requires a very clever preconditioner. In this paper we assume that ${\omega}$ is nonnegative, and determine the optimal ${\rho}$ parameter for the three dimensional ADI iteration for the Helmholtz equation. The ADI(Alternating Direction Implicit) method is also getting new attentions due to the fact that it is very suitable to the vector/parallel computers, for example, as a preconditioner to the Krylov subspace methods. However, classical ADI was developed for two dimensions, and for three dimensions it is known that its convergence behaviour is quite different from that in two dimensions. So far, in three dimensions the so-called Douglas-Rachford form of ADI was developed. It is known to converge for a relatively wide range of ${\rho}$ values but its convergence is very slow. In this paper we determine the necessary conditions of the ${\rho}$ parameter for the convergence and optimal ${\rho}$ for the three dimensional ADI iteration of the Peaceman-Rachford form for the real Helmholtz equation with nonnegative ${\omega}$. Also, we conducted some experiments which is in close agreement with our theory. This straightforward extension of Peaceman-rachford ADI into three dimensions will be useful as an iterative solver itself or as a preconditioner to the the Krylov subspace methods, such as CG(Conjugate Gradient) method or GMRES(m).

• Investigative Magnetic Resonance Imaging
• /
• v.25 no.3
• /
• pp.141-155
• /
• 2021

Purpose: To develop a 3D magnetic resonance fingerprinting (MRF) method for application in high resolution knee cartilage PD, T₁, T₂ mapping. Materials and Methods: A novel 3D acquisition trajectory with golden-angle rotating radial in k_xy direction and interleaved echo planar imaging (EPI) acquisition in the k_z direction was implemented in the MRF framework. A centric order was applied to the interleaved EPI acquisition to reduce Nyquist ghosting artifact due to field inhomogeneity. For the reconstruction, singular value decomposition (SVD) compression method was used to accelerate reconstruction time and conjugate gradient sensitivity-encoding (CG-SENSE) was performed to overcome low SNR of the high resolution data. Phantom experiments were performed to verify the proposed method. In vivo experiments were performed on 6 healthy volunteers and 2 early osteoarthritis (OA) patients. Results: In the phantom experiments, the T₁ and T₂ values of the proposed method were in good agreement with the spin-echo references. The results from the in vivo scans showed high quality proton density (PD), T₁, T₂ map with EPI echo train length (N_ETL = 4), acceleration factor in through plane (R_z = 5), and number of radial spokes (N_spk = 4). In patients, high T₂ values (50-60 ms) were seen in all transverse, sagittal, and coronal views and the damaged cartilage regions were in agreement with the hyper-intensity regions shown on conventional turbo spin-echo (TSE) images. Conclusion: The proposed 3D MRF method can acquire high resolution (0.5 mm³) quantitative maps in practical scan time (~ 7 min and 10 sec) with full coverage of the knee (FOV: 160 × 160 × 120 mm³).