Orthogonal Conjugate Gradient Research Articles

階層型領域分割法による電磁界解析における四倍精度演算の検討

This paper deals with a finite element analysis (FEA) of an electromagnetic problem with a quadruple precision floating point number. A Conjugate Gradient (CG) method for real symmetric matrices and a Conjugate Orthogonal Conjugate Gradient (COCG) method for complex symmetric matrices are implemented in the quadruple precision. Then, an FEA solver with the quadruple precision arithmetic for the electromagnetic problem is implemented by the C language. As a result, the convergence of the iterative method is improved, and the minimum residual norm is reduced by approximately three orders of magnitude.

A Discrete Dipole Approximation Solver Based on the COCG-FFT Algorithm and Its Application to Microwave Breast Imaging.

We introduce the discrete dipole approximation (DDA) for efficiently calculating the two-dimensional electric field distribution for our microwave tomographic breast imaging system. For iterative inverse problems such as microwave tomography, the forward field computation is the time limiting step. In this paper, the two-dimensional algorithm is derived and formulated such that the iterative conjugate orthogonal conjugate gradient (COCG) method can be used for efficiently solving the forward problem. We have also optimized the matrix-vector multiplication step by formulating the problem such that the nondiagonal portion of the matrix used to compute the dipole moments is block-Toeplitz. The computation costs for multiplying the block matrices times a vector can be dramatically accelerated by expanding each Toeplitz matrix to a circulant matrix for which the convolution theorem is applied for fast computation utilizing the fast Fourier transform (FFT). The results demonstrate that this formulation is accurate and efficient. In this work, the computation times for the direct solvers, the iterative solver (COCG), and the iterative solver using the fast Fourier transform (COCG-FFT) are compared with the best performance achieved using the iterative solver (COCG-FFT) in C++. Utilizing this formulation provides a computationally efficient building block for developing a low cost and fast breast imaging system to serve under-resourced populations.

Converting BiCR method for linear equations with complex symmetric matrices

The Bi-Conjugate Gradient (BiCG) method for Symmetric Complex matrices (SCBiCG), which can be derived from BiCG, has been proposed for solving linear equations with complex symmetric matrices. However, an alternative method derived from the Bi-Conjugate Residual (BiCR) method for complex symmetric matrices has not previously been proposed. We therefore design BiCR for Symmetric Complex matrices (SCBiCR) by using the same analogy as that discussed in SCBiCG. Coefficients ci with real number defined in SCBiCG need to be set by users before starting the iteration, and we have had the numerical results, with several combinations when the coefficients ci are real, that the residual norms of SCBiCG do not converge. We therefore design an alternative implementation such that the coefficients ci can be complex and are appropriately determined at each step of the algorithm. We give the preconditioned algorithms. Moreover, the factor in the loss of convergence speed is analyzed to clarify the difference of convergence between SCBiCG and our proposed SCBiCR. Numerical experiments demonstrate that the residual norms of our proposed variant with the complex coefficients ci converge fairly faster than those of the Conjugate Orthogonal Conjugate Gradient (COCG) method and several implementations of SCBiCG.

Coordinate-space solver for superfluid many-fermion systems with the shifted conjugate-orthogonal conjugate-gradient method

Self-consistent approaches to superfluid many-fermion systems in 3-dimensions (and subsequent time-dependent approaches) require a large number of diagonalizations of very large dimension hermitian matrices, which results in enormous computational costs. We present an approach based on the shifted conjugate-orthogonal conjugate-gradient (COCG) method for the evaluation of the Green's function, from which we subsequently extract various densities (particle number, spin, current, kinetic energy, etc.) of a nuclear system needed in self-consistent approaches. The approach eschews the construction of the quasiparticle wavefunctions and their corresponding quasiparticle energies, which are never explicitly needed in any density functional approaches. As benchmarks we present calculations for nuclei with axial symmetry, including the ground state of spherical (magic or semi-magic) and axially deformed nuclei, the saddle-point in the $^{240}$Pu constrained fission path, and a vortex in the neutron star crust.

Multi-GPU Accelerated Admittance Method for High-Resolution Human Exposure Evaluation.

A multi-graphics processing unit (GPU) accelerated admittance method solver is presented for solving the induced electric field in high-resolution anatomical models of human body when exposed to external low-frequency magnetic fields. In the solver, the anatomical model is discretized as a three-dimensional network of admittances. The conjugate orthogonal conjugate gradient (COCG) iterative algorithm is employed to take advantage of the symmetric property of the complex-valued linear system of equations. Compared against the widely used biconjugate gradient stabilized method, the COCG algorithm can reduce the solving time by 3.5 times and reduce the storage requirement by about 40%. The iterative algorithm is then accelerated further by using multiple NVIDIA GPUs. The computations and data transfers between GPUs are overlapped in time by using asynchronous concurrent execution design. The communication overhead is well hidden so that the acceleration is nearly linear with the number of GPU cards. Numerical examples show that our GPU implementation running on four NVIDIA Tesla K20c cards can reach 90 times faster than the CPU implementation running on eight CPU cores (two Intel Xeon E5-2603 processors). The implemented solver is able to solve large dimensional problems efficiently. A whole adult body discretized in 1-mm resolution can be solved in just several minutes. The high efficiency achieved makes it practical to investigate human exposure involving a large number of cases with a high resolution that meets the requirements of international dosimetry guidelines.

The SCBiCG class of algorithms for complex symmetric linear systems with applications in several electromagnetic model problems

Frequency domain formulations of computational electromagnetic problems often require the solutions of complex-valued non-Hermitian systems of equations, which are still symmetric. For this kind of problems a whole class of sub-variant solver methods derived from the complex-valued Bi-Conjugate Gradient method is available. This class of methods contains established iterative methods as the Conjugate Orthogonal Conjugate Gradient (COCG) method, Bi-Conjugate Gradient Conjugate Residual (BiCGCR) method and Conjugate A-Orthogonal Conjugate Residual (COCR) method. The mathematical equivalence of the BiCGCR method and COCR method is shown and preconditioned variants of the various solvers are derived. An efficient kind of two-step preconditioning technique is also proposed. Numerical experiments involving e.g. electro-quasistatic frequency domain simulation are employed to show the difference in the convergence behaviors of these iterative methods and effectiveness of the two-step preconditioning techniques.

Preconditioning Complex Symmetric Linear Systems

A new preconditioner for symmetric complex linear systems based on Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear systems is herein presented. It applies to conjugate orthogonal conjugate gradient (COCG) or conjugate orthogonal conjugate residual (COCR) iterative solvers and does not require any estimation of the spectrum of the coefficient matrix. An upper bound of the condition number of the preconditioned linear system is provided. To reduce the computational cost the preconditioner is approximated with an inexact variant based on incomplete Cholesky decomposition or on orthogonal polynomials. Numerical results show that the present preconditioner and its inexact variant are efficient and robust solvers for this class of linear systems. A stability analysis of the inexact polynomial version completes the description of the preconditioner.

Quasi-Minimal Residual Variants of the COCG and COCR Methods for Complex Symmetric Linear Systems in Electromagnetic Simulations

The conjugate orthogonal conjugate gradient (COCG) method has been considered an attractive part of the Lanczos-type Krylov subspace method for solving complex symmetric linear systems. However, it is often faced with apparently irregular convergence behaviors in practical electromagnetic simulations. To avoid such a problem, the symmetric quasi-minimal residual (QMR) method has been developed. On the other hand, the conjugate A-orthogonal conjugate residual (COCR) method, which can be regarded as an extension of the conjugate residual method, also had been established. It shows that the COCR often gives smoother convergence behavior than the COCG method. The purpose of this paper is to apply the QMR approaches to the COCG and COCR to derive two new methods (including their preconditioned versions), and to report the benefits of the modified methods by some practical examples arising in electromagnetic simulations.

An Efficient MAINV Preconditioned COCG Method for FEM Analysis of Millimeter Wave Filters

The modified AINV (MAINV) sparse approximate inverse preconditioner is applied to the conjugate orthogonal conjugate gradient (COCG) iterative method for solving a large systems of linear equations resulting from the use of edge finite element method (FEM). The proposed preconditioner is derived from basic AINV process by adding pivots compensation strategy to avoid the potential breakdowns. Numerical experiments on several typical millimeter wave structrues demonstrate the effectiveness of the MAINV-COCG method, in comparison with other conventional methods.

Novel algorithm of large-scale simultaneous linear equations

We review our recently developed methods of solving large-scale simultaneous linearequations and applications to electronic structure calculations both in one-electron theoryand many-electron theory. This is the shifted COCG (conjugate orthogonal conjugategradient) method based on the Krylov subspace, and the most important issue forapplications is the shift equation and the seed switching method, which greatly reduce thecomputational cost. The applications to nano-scale Si crystals and the double orbitalextended Hubbard model are presented.

Folded Preconditioner: A New Class of Preconditioners for Krylov Subspace Methods to Solve Redundancy-Reduced Linear Systems of Equations

The A-phi formulation, which is widely used in electromagnetic analysis, leads to a redundant linear system of equations that includes a substantial number of redundant degrees of freedom (DOF). We can derive a redundancy-reduced linear system of equations by eliminating the redundant DOF, thereby decreasing the computation costs per iteration for iterative solvers, such as the incomplete Cholesky conjugate gradient (ICCG) solver. This does not, however, result in a reduction in total computation time, due to significant convergence deterioration. In this paper, we present a solution to this problem in the form of folded preconditioners. First, the theorem presented reveals that, for any preconditioned Krylov subspace method for the original redundant linear systems, we can derive the equivalent Krylov subspace method for the redundancy-reduced linear systems by using the corresponding folded preconditioner. As an uncomplicated example, the standard ICCG solver for the original redundant systems has exactly the same convergence property as the CG solver for the redundancy-reduced systems using the folded variant of the IC preconditioner (the folded IC preconditioner). Furthermore, we discuss efficient computational procedures for the folded preconditioners and the design of Krylov subspace algorithms using the preconditioners. A sample full-wave analysis demonstrates the good performance of a newly developed solver, the conjugate orthogonal conjugate gradient (COCG) method with the folded IC preconditioner. The new solver not only lowers the computation costs per iteration by reducing the number of DOF, but also completely avoids the convergence deterioration.

Shifted Conjugate-Orthogonal–Conjugate-Gradient Method and Its Application to Double Orbital Extended Hubbard Model

We discuss the shifted conjugate-orthogonal–conjugate-gradient (COCG) method, which can be used to solve a series of linear equations generated by a number of scalar shifts, without performing time-consuming matrix–vector operations, except at a single reference energy. This is a type of CG method and is robust and has an expression of the accuracy of the calculated result that requires no additional calculation cost. The shifted COCG is useful for calculating the Green's function of a many-electron Hamiltonian, which has a very large dimension. We applied the shifted COCG method to the double orbital extended Hubbard model with twelve electrons on a periodic \(\sqrt{8} \times \sqrt{8}\) site system, with the dimension of the Hamiltonian equal to 64,128,064, and we found that the ground state is an insulator. We also discuss the main points of reducing the amount of memory required to apply COCG algorithm.

Theory of large-scale matrix computation and applications in electronic structurecalculation

We review the methods that we recently developed for making large-scale electronicstructure calculations, both using one-electron theory and using many-electron theory. Themethods are based on the density matrix representation, together with the Wannier staterepresentation and the Krylov subspace method, using one-electron theory ofsystems on a scale of a few tens of nanometers. The hybrid method of quantummechanical molecular dynamical simulation is explained. The Krylov subspacemethod, the CG (conjugate gradient) method and the shifted COCG (conjugateorthogonal conjugate gradient) method can be applied in the investigation ofthe ground state and the excitation spectra using many-electron theory. Themathematical foundation of the Krylov subspace method for large-scale matrixcomputation is focused on, and the key techniques of the shifted COCG method,i.e. using the collinear residual and ‘seed-switching’, are explained. A wide variety ofapplications of the extended novel algorithm are also explained. These include studies offracture formation and propagation, liquid carbon, and formation processes ofgold nanowires, together with the application to the extended Hubbard model.

A Preconditioned COCG Method for Solving Complex Symmetric Linear Systems Arising from Scattering Problems

In this paper, we study a class of modified incomplete LDLT preconditioners followed by the conjugate orthogonal conjugate gradient (COCG) method for solving the large complex symmetric system with linear equations Ax = b, arising from the scattering problems. The preconditioner is derived from the incomplete Cholesky factorization with diagonal compensation. With an ordering scheme, our factorized preconditioners can reduce the memory requirement as well as the COCG iteration numbers. Numerical tests on harmonic analysis for plane wave scattering from a metallic plate and a metallic sphere coated by a lossy dielectric layer show the efficiency of our method.

NUMERICAL ANALYSIS OF LARGE-SCALE SOUND FIELDS USING ITERATIVE METHODS PART II: APPLICATION OF KRYLOV SUBSPACE METHODS TO FINITE ELEMENT ANALYSIS

Krylov subspace iterative solvers are applied to large-scale finite element sound-field analyses of architectural rooms. First, convergence behaviors are compared among four iterative solvers. Results show that the Conjugate Orthogonal Conjugate Gradient (COCG) method offers the best characteristics for finite-element (FE) analysis from the viewpoint of robustness of convergence and computation time. Two investigations to reduce the computation time of the COCG method were carried out. Results show the following. (1) The mean residual of sound pressure levels between COCG method and direct method is less than 0.1 dB if the convergence criterion is set to 10-4 and the maximum residual of those between COCG method and direct method is less than 0.2 dB if the convergence criterion is set to 10-6. (2) The computation time of the COCG method with diagonal preconditioning is about 30% shorter than that of COCG method without preconditioning. Finally, sound pressure level distributions obtained using the authors' FEM are compared to those obtained using fast multipole BEM (FMBEM) and measurements.

A COCR method for solving complex symmetric linear systems

The Conjugate Orthogonal Conjugate Gradient (COCG) method has been recognized as an attractive Lanczos-type Krylov subspace method for solving complex symmetric linear systems; however, it sometimes shows irregular convergence behavior in practical applications. In the present paper, we propose a Conjugate A -Orthogonal Conjugate Residual (COCR) method, which can be regarded as an extension of the Conjugate Residual (CR) method. Numerical examples show that COCR often gives smoother convergence behavior than COCG.

有限要素音場解析へのKrylov部分空間法の適用 : 反復解法を利用した大規模音場数値解析 その2

Krylov subspace iterative solvers are applied to large-scale finite element sound field analyses of architectural rooms. First, convergence behaviors are compared among four iterative solvers. Results show that Conjugate Orthogonal Conjugate Gradient (COCG) method offers the best characteristics for finite-element (FE) analysis from the viewpoint of robustness of convergence and computation time. Then, two investigations to reduce computation time of the COCG method were carried out. Results show the following. (1) The averaged residual of sound pressure levels between COCG method and direct method are less than 0.1dB if the convergence criterion is set to 10^<-4> and the maximum residual of those between COCG method and direct method is less than 0.2dB if the convergence criterion is set to 10^<-6>. (2) The computation time of the COCG method with diagonal preconditioning is about 30% shorter than that of COCG method without preconditioning. Finally, sound pressure level distributions obtained using the authors' FEM are compared with those obtained by fast multipole BEM (FMBEM) and measurements.