Iterative Matrix Solver Research Articles

Optimized Analytical Computation of Partial Elements Using a Retarded Taylor Series Expansion

The aim of this article is to efficiently and accurately calculate the integrals of the full-wave (FW) partial element equivalent circuit (PEEC) method. The accuracy of the analytical formulas calculated by the standard precision can be compromised when using nonuniform mesh to properly model the high-frequency effects. The numerical errors can be avoided by using a high-precision arithmetic, i.e., higher number of digits, however, at the expense of significantly higher computation time. This article presents an analytical approach for calculating the FW-PEEC interaction integrals of two elementary volumes/surfaces based on the Taylor expansion, which allows a high computational speed preserving the accuracy with a relative error of less than 0.1%. The proposed solution is verified compared to the high-precision arithmetic and the standard Gaussian integration for two examples of strip lines. Moreover, it is shown that the accuracy of FW-PEEC integrals can affect the convergence of an iterative PEEC matrix solver.

Jacobian weighted element differential method for solid mechanics

In this paper, a novel collocation method is proposed to solve the problems in solid mechanics. In the proposed method, the gradients across the element boundaries are evaluated by a weight-averaging algorithm to achieve higher order continuity. Then, the collocation method is employed to discretize the governing equations for all domain nodes, including the nodes at the interfaces of elements for which the earlier collocation methods have to use the traction-equilibrium equations. As a result of this difference, linear and serendipity elements can be used in the proposed method to reduce the total number of freedoms and easily used for very complex geometries. Moreover, the proposed method significantly improves the accuracy compared with the earlier collocation methods. The provided numerical test cases show that the method can also avoid the degeneration of accuracy when odd-order elements are employed, which further demonstrates the advantages of the proposed method. The proposed method are also leads to better convergency in nonlinear cases and iterative matrix solver. By analyzing the coefficient matrix, we found that the proposed method is better conditioned. The proposed method also yields promising advantages in transient problems for explicit time marching because of the advantage of the collocation methods.

Simple immersed boundary matrix‐cut‐method for cartesian grids using FV and FD discretizations

AbstractThis work presents a means of implementing a simple implicit Cartesian grid matrix‐cut method to model the heat and mass transfer in irregular geometries using any implicit Cartesian grid solver in finite‐volume and finite‐difference formulations. This method allows the precise imposing of Dirichlet, Neumann, and Robin boundary conditions in any internal node of the grid. The main idea of this method comes from a finite element method where the coefficients of the implicit matrix of a discretized equation are modified directly to ensure the consistency of the presence of the boundary conditions at immersed boundaries represented by grid nodes. One of the advantages of this method is its capability to solve a transport equation implicitly in steady and unsteady modes, applied to complex geometry immersed into the Cartesian grid. This method can be easily implemented in any direct matrix solver, e.g., Gauss elimination, or any iterative matrix solver. To validate this method, different cases are considered.

Application-based fault tolerance techniques for sparse matrix solvers

High-performance computing systems continue to increase in size in the quest for ever higher performance. The resulting increased electronic component count, coupled with the decrease in feature sizes of the silicon manufacturing processes used to build these components, may result in future exascale systems being more susceptible to soft errors caused by cosmic radiation than in current high-performance computing systems. Through the use of techniques such as hardware-based error-correcting codes and checkpoint-restart, many of these faults can be mitigated at the cost of increased hardware overhead, run-time, and energy consumption that can be as much as 10–20%. Some predictions expect these overheads to continue to grow over time. For extreme scale systems, these overheads will represent megawatts of power consumption and millions of dollars of additional hardware costs, which could potentially be avoided with more sophisticated fault-tolerance techniques. In this paper we present new software-based fault tolerance techniques that can be applied to one of the most important classes of software in high-performance computing: iterative sparse matrix solvers. Our new techniques enables us to exploit knowledge of the structure of sparse matrices in such a way as to improve the performance, energy efficiency, and fault tolerance of the overall solution.

Iterative Speedup by Utilizing Symmetric Data in Pricing Options with Two Risky Assets

The Crank–Nicolson method can be used to solve the Black–Scholes partial differential equation in one-dimension when both accuracy and stability is of concern. In multi-dimensions, however, discretizing the computational grid with a Crank–Nicolson scheme requires significantly large storage compared to the widely adopted Operator Splitting Method (OSM). We found that symmetrizing the system of equations resulting from the Crank–Nicolson discretization help us to use the standard pre-conditioner for the iterative matrix solver and reduces the number of iterations to get an accurate option values. In addition, the number of iterations that is required to solve the preconditioned system, resulting from the proposed iterative Crank–Nicolson scheme, does not grow with the size of the system. Thus, we can effectively reduce the order of complexity in multidimensional option pricing. The numerical results are compared to the one with implicit Operator Splitting Method (OSM) to show the effectiveness.

Laplace-domain waveform modeling and inversion for the 3D acoustic–elastic coupled media

Laplace-domain waveform inversion reconstructs long-wavelength subsurface models by using the zero-frequency component of damped seismic signals. Despite the computational advantages of Laplace-domain waveform inversion over conventional frequency-domain waveform inversion, an acoustic assumption and an iterative matrix solver have been used to invert 3D marine datasets to mitigate the intensive computing cost. In this study, we develop a Laplace-domain waveform modeling and inversion algorithm for 3D acoustic–elastic coupled media by using a parallel sparse direct solver library (MUltifrontal Massively Parallel Solver, MUMPS). We precisely simulate a real marine environment by coupling the 3D acoustic and elastic wave equations with the proper boundary condition at the fluid-solid interface. In addition, we can extract the elastic properties of the Earth below the sea bottom from the recorded acoustic pressure datasets. As a matrix solver, the parallel sparse direct solver is used to factorize the non-symmetric impedance matrix in a distributed memory architecture and rapidly solve the wave field for a number of shots by using the lower and upper matrix factors. Using both synthetic datasets and real datasets obtained by a 3D wide azimuth survey, the long-wavelength component of the P-wave and S-wave velocity models is reconstructed and the proposed modeling and inversion algorithm are verified. A cluster of 80CPU cores is used for this study.

Three-Dimensional Induced Polarization Parallel Inversion Using Nonlinear Conjugate Gradients Method

Four kinds of array of induced polarization (IP) methods (surface, borehole-surface, surface-borehole, and borehole-borehole) are widely used in resource exploration. However, due to the presence of large amounts of the sources, it will take much time to complete the inversion. In the paper, a new parallel algorithm is described which uses message passing interface (MPI) and graphics processing unit (GPU) to accelerate 3D inversion of these four methods. The forward finite differential equation is solved by ILU0 preconditioner and the conjugate gradient (CG) solver. The inverse problem is solved by nonlinear conjugate gradients (NLCG) iteration which is used to calculate one forward and two “pseudo-forward” modelings and update the direction, space, and model in turn. Because each source is independent in forward and “pseudo-forward” modelings, multiprocess modes are opened by calling MPI library. The iterative matrix solver within CULA is called in each process. Some tables and synthetic data examples illustrate that this parallel inversion algorithm is effective. Furthermore, we demonstrate that the joint inversion of surface and borehole data produces resistivity and chargeability results are superior to those obtained from inversions of individual surface data.

Evaluate the performance of an accelerated initial stiffness method in three dimensional finite element analysis

Newton’s method is a commonly used algorithm for elasto-plastic finite element analysis and has three common variations: the full Newton–Raphson method, the modified Newton–Raphson method and the initial stiffness method. The Newton–Raphson methods can converge to the solution in a small number of iterations when the system is stable; however, the methods can be quite computationally expensive in some types of problems, for example where the tangent stiffness matrix is unsymmetric or the plasticity is highly localized. The initial stiffness method is robust in those cases but requires a larger number of iterations. This prompted the formulation of many acceleration techniques in literature. In this paper, those techniques will be briefly discussed. This will be followed by the development of a modified acceleration technique for the initial stiffness method. The performance of the modified accelerated initial stiffness method will be examined in elasto-plastic analyses, using both direct and iterative matrix solvers. The results will be compared – in terms of the required number of iterations and the computation time – with an existing accelerated initial stiffness method, the non-accelerated initial stiffness method and the Newton–Raphson tangent stiffness method.

3D Laplace-domain full waveform inversion using a single GPU card

The Laplace-domain full waveform inversion is an efficient long-wavelength velocity estimation method for seismic datasets lacking low-frequency components. However, to invert a 3D velocity model, a large cluster of CPU cores have commonly been required to overcome the extremely long computing time caused by a large impedance matrix and a number of source positions. In this study, a workstation with a single GPU card (NVIDIA GTX 580) is successfully used for the 3D Laplace-domain full waveform inversion rather than a large cluster of CPU cores. To exploit a GPU for our inversion algorithm, the routine for the iterative matrix solver is ported to the CUDA programming language for forward and backward modeling parts with minimized modification of the remaining parts, which were originally written in Fortran 90. Using a uniformly structured grid set, nonzero values in the sparse impedance matrix can be arranged according to certain rules, which efficiently parallelize the preconditioned conjugate gradient method for a number of threads contained in the GPU card. We perform a numerical experiment to verify the accuracy of a floating point operation performed by a GPU to calculate the Laplace-domain wavefield. We also measure the efficiencies of the original CPU and modified GPU programs using a cluster of CPU cores and a workstation with a GPU card, respectively. Through the analysis, the parallelized inversion code for a GPU achieves the speedup of 14.7–24.6x compared to a CPU-based serial code depending on the degrees of freedom of the impedance matrix. Finally, the practicality of the proposed algorithm is examined by inverting a 3D long-wavelength velocity model using wide azimuth real datasets in 3.7 days.

Using <i>ScalIT</i> for Performing Accurate Rovibrational Spectroscopy Calculations for Triatomic Molecules: A Practical Guide

This paper presents a practical guide for use of the ScalIT software package to perform highly accurate bound rovibrational spectroscopy calculations for triatomic molecules. At its core, ScalIT serves as a massively scalable iterative sparse matrix solver, while assisting modules serve to create rovibrational Hamiltonian matrices, and analyze computed energy levels (eigenvalues) and wavefunctions (eigenvectors). Some of the methods incorporated into the package include: phase space optimized discrete variable representation, preconditioned inexact spectral transform, and optimal separable basis preconditioning. ScalIT has previously been implemented successfully for a wide range of chemical applications, allowing even the most state-of-the-art calculations to be computed with relative ease, across a large number of computational cores, in a short amount of time.

An algorithm for 3D acoustic time-Laplace-Fourier-domain hybrid full waveform inversion

When full waveform inversion (FWI) is performed in the Laplace-Fourier domain, Laplace-Fourier-domain wave modeling, using finite-difference or finite-element methods, is performed to compute the gradient direction. In this case, a complex impedance matrix is composed and factored by direct matrix solvers. Alternatively, iterative matrix solvers may be used. However, solving 3D problems with such methods requires excessive computer memory or computing time, which causes problems in the application of the Laplace-Fourier-domain FWI. To avoid computational overburden in 3D problems, we propose a time-Laplace-Fourier hybrid FWI, in which forward and backward modeling are performed in the time domain and other procedures are conducted in the Laplace-Fourier domain. Our hybrid FWI is applied to two groups of frequencies. Inversions for the first and second groups of frequencies correspond to the Laplace-domain FWI and the Laplace-Fourier-domain FWI, respectively. The graphic processing unit is used to speed up the hybrid inversion algorithm. To verify the feasibility of this technique, the 3D hybrid FWI is applied to the data recorded along the A1 line of the synthetic SEG/EAGE 3D salt model and 3D wide-azimuth real exploration data. Numerical examples show that the hybrid FWI yields reasonable subsurface velocity structures that contribute to the enhancement of reverse-time migration images.

A partly matrix-free solver for the gyrokinetic field equation in three-dimensional geometry

In the case of adiabatic electrons the gyrokinetic field equation for the electrostatic potential includes an averaging operator acting on flux surfaces. For realistic three-dimensional configurations, as e.g. in stellarator devices, the discretisation of this integro-differential equation leads to very large nearly dense matrices (full matrix approach) which typically cannot be stored in computer memory explicitly. A low memory consuming partly matrix-free approach, based on a preconditioned iterative matrix solver, has been developed where the Helmholtz part of the field equation is used in a matrix formulation while the averaging term is treated matrix-free. For matrices which could still be stored in memory explicitly, it is demonstrated that this approach is also much faster than the full matrix approach.

Fast and efficient sensitivity calculation using adjoint method for three-phase field-scale history matching

Adjoint method-based sensitivity for field-scale history matching with large numbers of parameters suffers from several limitations. First, the CPU time depends on the data points which are large for any brown fields of long history; second, it requires large memory to save the gridblock pressure and saturation per each time step used in the forward model. Third, it is computationally expensive as it requires solving the Adjoint system of equations backward in time per each forward time step which is usually of high magnitude in case of field scale applications of long history. Lastly, the solver used for solving the Adjoint system of equations needs to be efficient for large-scale applications. We propose an efficient and fast approach for sensitivity calculation based on the Adjoint method to overcome much of the current limitations. First, we use a commercial finite difference simulator, ECLIPSE, as a forward model, which is general and can account for complex physical behavior that dominates most field applications. Second, the production data misfit is represented by a single generalized travel time misfit per well, thus effectively reducing the number of data points into one per well. Third, we solve the Adjoint system of equations backward in time in a larger time step that is equivalent to the time of severe changes in pressure and saturation due to changing well conditions or introducing new infill wells rather than using the forward model time steps. This approach reduces the computational effort and memory allocated for the sensitivity calculation. Fourth, we use an iterative sparse matrix solver, LSQR, for solving the Adjoint system of equations which shows high stability for field-scale applications. We demonstrate the power and utility of our approach using synthetic and pseudo field examples. The synthetic examples show the robustness and efficiency of our sensitivity calculation approach compared to the perturbation. The pseudo-field example had 10 years of production history with an original gas cap and a strong aquifer support. Using well log data, core data, water cut and gas–oil ratio history from producing wells; we characterize the permeability at each cell, thus demonstrating the feasibility of our approach for field applications.

LAPACKrc: Fast linear algebra kernels/solvers for FPGA accelerators

We present LAPACKrc, a family of FPGA-based linear algebra solvers able to achieve more than 100x speedup per commodity processor on certain problems. LAPACKrc subsumes some of the LAPACK and ScaLAPACK functionalities, and it also incorporates sparse direct and iterative matrix solvers. Current LAPACKrc prototypes demonstrate between 40x-150x speedup compared against top-of-the-line hardware/software systems. A technology roadmap is in place to validate current performance of LAPACKrc in HPC applications, and to increase the computational throughput by factors of hundreds within the next few years.

Wide-Angle Alternating-Direction Implicit Finite-Difference Beam Propagation Method

An alternating direction implicit (ADI), scheme for the wide-angle finite-difference beam propagation method (FD-BPM) based on the wide angled Pade multistep method is presented. The scheme incorporates an iterative technique for correction of the operator splitting error. The resulting equations are efficiently solved by the Thomas algorithm for tri-diagonal band matrices. The dispersion characteristics, accuracy and stability of the scheme is verified analytically and numerically for the cases of a plane wave propagating at fixed angle to the assumed propagation direction of the algorithm and the three dimensional angled propagation of a Gaussian beam. The computational requirements of the method are assessed against a standard wide-angle Pade multistep method that uses direct and iterative sparse matrix solvers.

Full-Wave Analysis of Large-Scale Interconnects Using the Multilevel UV Method With the Sparse Matrix Iterative Approach (SMIA)

Moving towards the goal of analyzing whole printed circuit boards (PCBs) and packages using full-wave electromagnetic (EM) methods, the multilevel UV method is applied to the method-of-moments (MoM) solution of the current on large-scale interconnects. The MoM solution uses the layered media Green's functions computed using the numerical modified steepest-descent path (NMSP) method, and is applied to the exterior layers of the interconnect structure. The sparse matrix iterative approach (SMIA) is used to speed up the solution of the iterative matrix solver. The iterative solver is also accelerated by using larger blocks in the block diagonal inverse preconditioner. With the multilevel UV method, a fast solution is presented for solving the current on large-scale interconnects on thin layered structures at high frequencies. We show an example of an interconnect structure that has horizontal dimensions of 12.675 lambda t 12.876 lambda with 24\thinspace 848 current unknowns and an interconnect fractional area of approximately 31%. This problem takes a total of 21 min 20 s to solve for the current on the traces on a Pentium 3.2-GHz CPU with 4 GB of RAM.

A SIMPLE BUT EFFICIENT PRECONDITIONING FOR CONJUGATE GRADIENT POISSON SOLVER USING HAAR WAVELET

This paper describes a wavelet-based preconditioning technique for conjugate gradient method for linear systems derived from the Poisson equation. The linear systems solved with a conventional iterative matrix solver resulted in a marked increase in computing time with respect to an increase in grid points. Use of our wavelet-based technique leads to a matrix with a bounded condition number so that computing time is reduced significantly. In this study, one of the simplest wavelets, the Haar wavelet, is used for the purpose of developing a simple but efficient preconditioning algorithm. Simple wavelets having low data communication property such as the Haar wavelet are expected to be suitable for the purpose of improving computing performance. In this study, we also pay attention to the basic characteristics of the Haar-wavelet-based preconditioning method for a Poisson equation solver.

Survey and Performance Assessment of Solution Methods for Elastic Rough Contact Problems

Recent developments in the solution of models of rough contact have demonstrated the advantages of fast matrix multiplication methods in increasing the size of problem that can usefully be solved. However, to date, no attempt has been made to compare these methods with existing approaches. In this paper a survey of solution methods reveals that three main classes of solution method have been developed: active set methods with direct matrix solution; active set methods with iterative matrix solvers; iterative methods based on fast matrix multiplications. The best features of each class are assembled from the literature and combined to formulate a “best-possible” algorithm in each class. These algorithms are implemented and compared, with emphasis on their solution speed, accuracy and memory requirements. For smaller problems with fewer than 2000 grid points in contact, the direct matrix solution method proves to be most efficient, and is always the most accurate. However, the method is limited by memory requirements as the contact region increases. For large problems with more than 2000 grid points in contact, the multilevel multisummation approach requires less memory, and performs faster, but gives errors in predicted pressure and contact region of approximately 0.5%. The active set method with iterative matrix solver requires less memory than direct matrix solution, is more accurate than the multilevel multisummation approach, but is always slower than one of the other methods.

Application of the Multilevel UV Method to Analyze Large Microstrip Patch Structures

The analysis of a plane wave incident on a large microstrip patch is studied using the method of moments (MoM) with the layered medium Green's functions and the rooftop basis functions. A fast numerical approach, the multilevel UV method, is employed to speed up the method of moment solution. A conjugate gradient method solver with preconditioning is applied as an iterative matrix solver to accelerate the matrix-vector multiplication. The multilevel UV method has O(N log N) complexity for CPU per iteration and memory.

A Single-Level Dual Rank IE-QR Algorithm to Model Large Microstrip Antenna Arrays

A single-level dual rank IE-QR algorithm is introduced so that the resulting dense method of moments (MOM) matrix is efficiently compressed. For a system of N equations, an amount of work of the order O(N/sup 2/) has traditionally been required for both matrix assembly and matrix-vector multiplication in an iterative matrix solver. The algorithm of the present paper reduces the memory requirement and CPU time for both matrix assembly and matrix-vector multiplication to O(N/sup 3/2/) making it practical to solve for large antenna arrays with full wave approach. In conjunction with a "geometric-neighboring" preconditioner for matrix solution using GMRES, the current approach solves problems involving large antenna arrays using only a fraction of what are needed by conventional MOM both in term of memory and total CPU time.