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Recent advances in the field of iterative methods for solving large linear systems are reviewed. The main focus is on developments in the area of conjugate gradient-type algorithms and Krylov subspace methods for nonHermitian matrices.
A model of a general class of asynchronous, iterative solution methods for linear systems is developed. In the model, the system is solved by creating several cooperating tasks that each compute a portion of the solution vector. A data transfer model predicting both the probability that data must be transferred between two tasks and the amount of data to be transferred is presented. This model is used to derive an execution time model for predicting parallel execution time and an optimal number of tasks given the dimension and sparsity of the coefficient matrix and the costs of computation, synchronization, and communication. The suitability of different parallel architectures for solving randomly sparse linear systems is discussed. Based on the complexity of task scheduling, one parallel architecture, based on a broadcast bus, is presented and analyzed.
Solving large, sparse, linear systems of equations is a fundamental problem in large scale scientific and engineering computation. A model of a general class of asynchronous, iterative solution methods for linear systems is developed. In the model, the system is solved by creating several cooperating tasks that each compute a portion of the solution vector. A data transfer model predicting both the probability that data must be transferred between two tasks and the amount of data to be transferred is presented. This model is used to derive an execution time model for predicting parallel execution time and an optimal number of tasks given the dimension and sparsity of the coefficient matrix and the costs of computation, synchronization, and communication. The suitability of different parallel architectures for solving randomly sparse linear systems is discussed. Based on the complexity of task scheduling, one parallel architecture, based on a broadcast bus, is presented and analyzed.
The applicability of static data flow architectures to the iterative solution of sparse linear systems of equations is investigated. An analytic performance model of a static data flow computation is developed. This model includes both spatial parallelism, concurrent execution in multiple PE's, and pipelining, the streaming of data from array memories through the PE's. The performance model is used to analyze a row partitioned iterative algorithm for solving sparse linear systems of algebraic equations. Based on this analysis, design parameters for the static data flow architecture as a function of matrix sparsity and dimension are proposed.
Parallel algorithms are developed for a class of scientific computational problems by partitioning the problems into smaller problems which may be solved concurrently. The effectiveness of the resulting parallel solutions is determined by the amount and frequency of communication and synchronization and the extent to which communication can be overlapped with computation. Three different parallel algorithms for solving the same class of problems are presented, and their effectiveness is analyzed from this point of view. The algorithms are programmed using a new programming environment. Run-time statistics and experience obtained from the execution of these programs assist in measuring the effectiveness of these algorithms.
A family of secant methods based on general rank-1 updates was revisited in view of the construction of iterative solvers for large non-Hermitian linear systems. As it turns out, both Broyden's good and bad update techniques play a special role, but should be associated with two different line search principles. For Broyden's bad update technique, a minimum residual principle is natural, thus making it theoretically comparable with a series of well known algorithms like GMRES. Broyden's good update technique, however, is shown to be naturally linked with a minimum next correction principle, which asymptotically mimics a minimum error principle. The two minimization principles differ significantly for sufficiently large system dimension. Numerical experiments on discretized partial differential equations of convection diffusion type in 2-D with integral layers give a first impression of the possible power of the derived good Broyden variant.
Fast dynamics simulation of large-scale power systems is a computational challenge because of the need to solve a large set of stiff, nonlinear differential-algebraic equations at every time step. The main bottleneck in dynamic simulations is the solution of a linear system during each nonlinear iteration of Newton’s method. In this paper, we present a parallel Krylov- Schwarz linear solution scheme that uses the Krylov subspacebased iterative linear solver GMRES with an overlapping restricted additive Schwarz preconditioner. As a result, performance tests of the proposed Krylov-Schwarz scheme for several large test cases ranging from 2,000 to 20,000 buses, including a real utility network, show good scalability on different computing architectures.
A new method for the solution of non-linear partial differential equations by an ADI procedure is described. Although the method is second order accurate in time, it does not require either iterations or predictor corrector methods to overcome the nonlinearity of the equations. Thus the computational effort required for the solution of the non-linear problem becomes similar to that required for the linear case. The method is applied to a two-dimensional 'extended Burgers equation'. Linear stability is studied, and some numerical solutions obtained. The improved accuracy obtained by the 2nd order truncation error is clearly manifested.
MVL-15 is the most recent version of the Modified Vortex-Lattice (MVL) code developed within the Aerodynamics Systems Analysis Branch (ASAB) at NASA LaRC. The term "modified" refers to the primary modification of the core vortex-lattice methodology: inclusion of viscous aerodynamics tables that are linked to the linear solution via iterative processes. The inclusion of the viscous aerodynamics inherently converts the MVL-15 from a purely analytic linearized method to a semi-empirical blend which retains the rapid execution speed of the linearized method while empirically characterizing the section aerodynamics at all spanwise lattice points. The modification provides a means to assess non-linear effects on lift that occur at angles of attack near stall, and provides a means to determine the drag associated with the application of design strategies for lift augmentation such as the use of flaps or blowing. The MVL-15 code is applicable to the analyses of aircraft aerodynamics during cruise, but it is most advantageously applied to the analysis of aircraft operating in various high-lift configurations. The MVL methodology has been previously conceived and implemented; the initial concept version was delivered to the ASAB in 2001 (van Dam, C.), subsequently revised (Gelhausen, P. and Ozoroski, T. 2002 / AVID Inc., Gelhausen, P., and Roberts, M. 2004), and then overhauled (Ozoroski, T., Hahn, A. 2008). The latest version, MVL-15 has been refined to provide analysis transparency and enhanced to meet the analysis requirements of the Environmentally Responsible Aviation (ERA) Project. Each revision has been implemented with reasonable success. Separate applications of the methodology are in use, including a similar in-house capability, developed by Olson, E. that is tailored for structural and acoustics analyses. A central premise of the methodology is that viscous aerodynamic data can be associated with analytic inviscid aerodynamic results at each spanwise wing section, thereby providing a pathway to map viscous data to the inviscid results. However, a number of factors can sidetrack the analysis consistency during various stages of this process. For example, it should be expected that the final airplane lift curve and drag polar results depend strongly on the geometry and aerodynamics of the airfoil section; however, flap deflections and flap chord extensions change the local reference geometry of the input airfoil, the airplane wing, the tabulated non-dimensional viscous aerodynamics, and the spanwise links between the linear and the viscous aerodynamics. These changes also affect the bound circulation and therefore, calculation and integration of the induced angle of attack and induced drag. MVL-15 is configured to ensure these types of challenges are properly addressed. This report is a comprehensive manual describing the theory, use, and validation of the MVL-15 analysis tool. Section 3 summarizes theoretical, procedural, and characteristic features of MVL-15, and includes a list of the files required to setup, execute, and summarize an analysis. Section 4, Section 5, Section 6, and Section 7 combine to comprise the User's Guide portions of this report. The MVL-15 input and output files are described in Section 4 and Section 5, respectively; the descriptions are supplemented with example files and information about the file formats, parameter definitions, and typical parameter values. Section 6 describes the Wing Geometry Setup Utility and the 2d-Variants Utility files that simplify and assist setting up a consistent set of MVL-15 geometry and aerodynamics input parameters and input files. Section 7 describes the use of the 3d-Results Presentation Utility file that can be used to automatically create summary tables and charts from the MVL-15 output files. Section 8 documents the Validation Results of an extensive and varied validation test matrix, including results of an airplane analysis representative of the ERA Program. A start-to-finish example of the airplane analysis procedure is described in Section 7.
Second-order solutions of supersonic-flow problems are sought by iteration, using the linearized solution as the first step. For plane and axially symmetric flows, particular solutions of the iteration equation are discovered which reduce the second-order problem to an equivalent linearized problem. Comparison of second-order solutions with exact and numerical results shows great improvement over linearized theory. For full three-dimensional flow, only a partial particular solution is found. The inclined cone is solved, and the possibility of treating more general problems is considered.
We present a computational methodology (a novel Nystrom approach based on use of a non-overlapping patch technique and Chebyshev discretizations) for efficient solution of problems of acoustic and electromagnetic scattering by open surfaces. Our integral equation formulations (1) Incorporate, as ansatz, the singular nature of open-surface integral-equation solutions, and (2) For the Electric Field Integral Equation (EFIE), use analytical regularizes that effectively reduce the number of iterations required by iterative linear-algebra solution based on Krylov-subspace iterative solvers.
We describe the algorithmic details and a performance evaluation of a Langevin approach to a strongly interacting electron-phonon system, and show it has a near linear scaling with lattice size N s . Many of the limitations of previous attempts to employ such methods to condensed matter lattice Hamiltonians are absent. In particular, the iterative linear algebra solution remains well behaved at strong coupling and low temperatures. The use of Fourier Acceleration is crucial for efficiency, and its use makes the method competitive with the widely-used local update methods, which scale as ${N}_{s}^{3}$ for on-site interactions and ${N}_{s}^{4}$ for long range electron-phonon coupling, even on rather small lattice sizes.
In this brief paper we examine a Riccati equation decomposition due to Reid and Lainiotis and apply the result to the continuous time-invariant linear filtering problem. Exploitation of the time-invariant structure leads to integration-free covariance recursions which are of use in covariance analyses and in filter implementations. A super-linearly convergent iterative solution to the algebraic Riccati equation (ARE) is developed. The resulting algorithm, arranged in a square-root form, is thought to be numerically stable and competitive with other ARE solution methods. Certain covariance relations that are relevant to the fixed-point and fixed-lag smoothing problems are also discussed.
Some recent developments in acceleration of convergence methods for vector sequences are reviewed. The methods considered are the minimal polynomial extrapolation, the reduced rank extrapolation, and the modified minimal polynomial extrapolation. The vector sequences to be accelerated are those that are obtained from the iterative solution of linear or nonlinear systems of equations. The convergence and stability properties of these methods as well as different ways of numerical implementation are discussed in detail. Based on the convergence and stability results, strategies that are useful in practical applications are suggested. Two applications to computational fluid mechanics involving the three dimensional Euler equations for ducted and external flows are considered. The numerical results demonstrate the usefulness of the methods in accelerating the convergence of the time marching techniques in the solution of steady state problems.
The minimal polynomial extrapolation (MPE) and reduced rank extrapolation (RRE) are two effective techniques that have been used in accelerating the convergence of vector sequences, such as those that are obtained from iterative solution of linear and nonlinear systems of equation. Their definitions involve some linear least squares problems, and this causes difficulties in their numerical implementation. Timewise efficient and numerically stable implementations for MPE and RRE are developed. A computer program written in FORTRAN 77 is also appended and applied to some model problems.
Some recent developments in acceleration of convergence methods for vector sequences are reviewed. The methods considered are the minimal polynomial extrapolation, the reduced rank extrapolation, and the modified minimal polynomial extrapolation. The vector sequences to be accelerated are those that are obtained from the iterative solution of linear or nonlinear systems of equations. The convergence and stability properties of these methods as well as different ways of numerical implementation are discussed in detail. Based on the convergence and stability results, strategies that are useful in practical applications are suggested. Two applications to computational fluid mechanics involving the three dimensional Euler equations for ducted and external flows are considered. The numerical results demonstrate the usefulness of the methods in accelerating the convergence of the time marching techniques in the solution of steady state problems.
The goal of this work is to present a fast and viable approach for the numerical solution of the high-contrast state problems arising in topology optimization. The optimization process is iterative, and the gradients are obtained by an adjoint analysis, which requires the numerical solution of large high-contrast linear elastic problems with features spanning several length scales. The size of the discretized problems forces the utilization of iterative linear solvers with solution time dependent on the quality of the preconditioner. The lack of clear separation between the scales, as well as the high-contrast, imposes severe challenges on the standard preconditioning techniques. Thus, here we propose new methods for the high-contrast elasticity equation with performance independent of the high-contrast and the multi-scale structure of the elasticity problem. The solvers are based on two-levels domain decomposition techniques with a carefully constructed coarse level to deal with the high-contrast and multi-scale nature of the problem. The construction utilizes spectral equivalence between scalar diffusion and each displacement block of the elasticity problems and, in contrast to previous solutions proposed in the literature, is able to select the appropriate dimension of the coarse space automatically. The new methods inherit the advantages of domain decomposition techniques, such as easy parallelization and scalability. Finally, the presented numerical experiments demonstrate the excellent performance of the proposed methods.
The numerical solution of stiff initial value problems, which lead to the problem of solving large systems of mildly nonlinear equations are considered. For many problems derived from engineering and science, a solution is possible only with methods derived from iterative linear equation solvers. A common approach to solving the nonlinear equations is to employ an approximate solution obtained from an explicit method. The error is examined to determine how it is distributed among the stiff and non-stiff components, which bears on the choice of an iterative method. The conclusion is that error is (roughly) uniformly distributed, a fact that suggests the Chebyshev method (and the accompanying Manteuffel adaptive parameter algorithm). This method is described, also commenting on Richardson's method and its advantages for large problems. Richardson's method and the Chebyshev method with the Mantueffel algorithm are applied to the solution of the nonlinear equations by Newton's method.