Randomized Kaczmarz with averaging

Finite-Difference Frequency-Domain methods (FDFD) require solution of large linear systems of equations. These large systems are represented by matrix equations including highly sparse coefficient matrices, and they can often only be solved by using iterative methods. This paper presents an algorithm in which the matrix-equation solution approach in an iterative method is replaced by a multi-step solution process. Instead of using a coefficient matrix, the coefficients in the FDFD formulations are kept as three-dimensional arrays, and they are treated as operators. The algorithm is used together with the Bi-Conjugate Gradients Stabilized (BICGSTAB) method. This is applied to a three-dimensional FDFD method to solve for scattering from dielectric objects. It is also applied to two other FDFD methods (a single-grid and a double-grid FDFD) to solve for scattering from chiral objects. It has been shown that the presented algorithm effectively reduces the solution time and memory requirements.

This paper describes and tests a parallel implementation of a factorized approximate inverse preconditioner (FSAI) to accelerate iterative linear system solvers. Such a preconditioner reveals an efficient accelerator of both Conjugate gradient and BiCGstab iterative methods in the parallel solution of large linear systems arising from the discretization of the advection-diffusion equation. The resulting message passing code allows the solution of large problems leading to a very cost-effective algorithm for the solution of large and sparse linear systems.

Two variants of modified incomplete block-matrix factorization with additive correction are proposed for the iterative solution of large linear systems of equations. Both rigorous theoretical support and numerical evidence are given displaying their efficiency when applied to discrete second order partial differential equations (PDEs), even in the case of quasi-singular problems.

Efficient vectorizable PDE solvers

Electron-impact excitation and ionization of helium is studied in the $S$-wave model. The problem is treated in full dimensionality using a time-dependent formulation of the exterior complex scaling method that does not involve the solution of large linear systems of equations. We discuss the steps that must be taken to compute stable ionization amplitudes. We present total excitation, total ionization, and single differential cross sections from the ground and $n=2$ excited states and compare our results with those obtained by others using a frozen-core model.

A new randomized vector algorithm for iterative solution of large linear systems

Accelerated generalized successive overrelaxation method for least squares problems

On the application of orthogonal polynomials to the iterative solution of linear systems of equations with indefinite or non-Hermitian matrices

Projected equation methods for approximate solution of large linear systems

Computer Solution of Large Linear Systems

Iterative Tikhonov regularization of tensor equations based on the Arnoldi process and some of its generalizations

Large scale parallel FEM computations of far/near stress field changes in rocks

A new stride of 4 reduction algorithm is proposed for the solution of large linear systems of tridiagonal equations. The new strategy incorporates a larger granularity in the solution process thus increasing the efficiency of the parallel computation.

Multi-scale methods such as wavelet and multigrid schemes are efficient algorithms for the numerical solution of large classes of partial differential equations (PDEs). In this chapter we illustrate the basic ideas behind iterative methods for the solution of large linear systems which arise from the discretisation of PDEs. Besides describing the concepts of wavelet preconditioning, we summarise the main ingredients of multigrid methods. Examples from computational geosciences will show the importance of multi-scale methods for the simulation of large and complex systems.

Numerical simulation of scattering in the time domain offers potential implementation advantages in comparison with frequency-domain methods. By using explicit numerical time integration schemes, elastic stresses and velocities can typically be advanced using only spatially local information which makes parallelization by domain decomposition straightforward to implement in comparison with the solution of a large linear system of equations. On the other hand, incorporation of frequency-dependent sound speed and attenuation appears more challenging in the time domain with the general need to compute convolutions under non-FFT friendly conditions. This talk presents efficiency gains for the elastodynamic finite integration technique (EFIT) which is similar to the finite-difference time-domain method (FDTD). We present results on using explicit decomposition of scattered and incident wave fields and far-field projection in the time domain with the discrete Helmholtz–Kirchhoff integral. We also present incorporation of frequency-dependent material parameters by efficient recursive convolution techniques that have been used successfully to create time-domain perfectly matched layers [J. A. Roden and S. D. Gedney, Microwave Opt. Technol. Lett. 27, 334–338, (2000)]. As an application example, we present results on computing bistatic target strength variance for a two-dimensional object located near a statistically rough seafloor. [Work sponsored by the Office of Naval Research.]