VTechWorks Repository :: Browsing by Author "Beattie, Christopher A."

Browsing by Author "Beattie, Christopher A."

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Active and Semi-Active Control of Civil Structures under Seismic Excitation
Matheu, Enrique E. (Virginia Tech, 1997-05-06)
The main focus of this study is on the active and semi-active control of civil engineering structures subjected to seismic excitations. Among different candidate control strategies, the sliding mode control approach emerges as a convenient alternative, because of its superb robustness under parametric and input uncertainties. The analytical developments and numerical results presented in this dissertation are directed to investigate the feasibility of application of the sliding mode control approach to civil structures. In the first part of this study, a unified treatment of active and semi-active sliding mode controllers for civil structures is presented. A systematic procedure, based on a special state transformation, is also presented to obtain the regular form of the state equations which facilitates the design of the control system. The conditions under which this can be achieved in the general case of control redundancy are also defined. The importance of the regular form resides in the fact that it allows to separate the design process in two basic steps: (a) selection of a target sliding surface and (b) determination of the corresponding control actions. Several controllers are proposed and extensive numerical results are presented to investigate the performance of both active and semi-active schemes, examining in particular the feasibility of application to real size civil structures. These numerical studies show that the selection of the sliding surface constitutes a crucial step in the implementation of an efficient control design. To improve this design process, a generalized sliding surface definition is used which is based on the incorporation of two auxiliary dynamical systems. Numerical simulations show that this definition renders a controller design which is more flexible, facilitating its tuning to meet different performance specifications. This study also considers the situation in which not all the state information is available for control purposes. In practical situations, only a subset of the physical variables, such as displacements and velocities, can be directly measured. A general approach is formulated to eliminate the explicit effect of the unmeasured states on the design of the sliding surface and the associated controller. This approach, based on a modified regular form transformation, permits the utilization of arbitrary combinations of measured and unmeasured states. The resulting sliding surface design problem is discussed within the framework of the classical optimal output feedback theory, and an efficient algorithm is proposed to solve the corresponding matrix nonlinear equations. A continuous active controller is proposed based only on bounding values of the unmeasured states and the input ground motion. Both active and semi-active schemes are evaluated by numerical simulations, which show the applicability and performance of the proposed approach.
Active Antenna Bandwidth Control Using Reconfigurable Antenna Elements
Cummings, Nathan Patrick (Virginia Tech, 2003-12-08)
Reconfigurable antennas represent a recent innovation in antenna design that changes from classical fixed-form, fixed-function antennas to modifiable structures that can be adapted to fit the requirements of a time varying system. Advances in microwave semiconductor processing technologies have enabled the use of compact, ultra-high quality RF and microwave switches in novel aspects of antenna design. This dissertation introduces the concept of reconfigurable antenna bandwidth control and how advances in switch technology have made these designs realizable. Specifically, it details the development of three new antennas capable of reconfigurable bandwidth control. The newly developed antennas include the reconfigurable ring patch, the reconfigurable planar inverted-F and the reconfigurable parasitic folded dipole. The relevant background work to these designs is described and then design details along with computer simulations and measured experimental results are given.
An Algebraic Approach to Reverse Engineering with an Application to Biochemical Networks
Stigler, Brandilyn Suzanne (Virginia Tech, 2005-08-04)
One goal of systems biology is to predict and modify the behavior of biological networks by accurately monitoring and modeling their responses to certain types of perturbations. The construction of mathematical models based on observation of these responses, referred to as reverse engineering, is an important step in elucidating the structure and dynamics of such networks. Continuous models, described by systems of differential equations, have been used to reverse engineer biochemical networks. Of increasing interest is the use of discrete models, which may provide a conceptual description of the network. In this dissertation we introduce a discrete modeling approach, rooted in computational algebra, to reverse-engineer networks from experimental time series data. The algebraic method uses algorithmic tools, including Groebner-basis techniques, to build the set of all discrete models that fit time series data and to select minimal models from this set. The models used in this work are discrete-time finite dynamical systems, which, when defined over a finite field, are described by systems of polynomial functions. We present novel reverse-engineering algorithms for discrete models, where each algorithm is suitable for different amounts and types of data. We demonstrate the effectiveness of the algorithms on simulated networks and conclude with a description of an ongoing project to reverse-engineer a real gene regulatory network in yeast.
An Alternative to Full Configuration Interaction Based on a Tensor Product Decomposition
Senese, Frederick A.; Beattie, Christopher A.; Schug, John C.; Viers, Jimmy W.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1989)
A new direct full variational approach exploits a tensor (Kronecker) product decompositions of the Hamiltonian. Explicit assembly and storage of the Hamiltonian matrix is avoided by using the Kronecker product structure to form matrix-vector products directly from the molecular integrals. Computation-intensive integral transformations and formula tapes are unnecessary. The wave function is expanded in terms of spin-free primitive sets rather than Slater determinants or configuration state functions and is equivalent to a full configuration interaction expansion. The approach suggests compact storage schemes and algorithms which are naturally suited to parallel and pipelined machines.
Antenna Array Systems: Propagation and Performance
Ertel, Richard Brian (Virginia Tech, 1999-07-28)
Due to the enormous performance gains associated with the use of antenna arrays in wireless networks, it is inevitable that these technologies will become an integral part of future systems. This report focuses on signal propagation modeling for antenna array systems and on its relationship to the performance of these systems. Accurate simulation and analytical models are prerequisite to the characterization of antenna array system performance. Finally, an understanding of the performance of these systems in various environments is needed for effective overall network design. This report begins with an overview of the fundamentals of antenna array systems. A survey of vector channel models is presented. Angle of arrival and time of arrival statistics for the circular and elliptical (Liberti's Model) models are derived. A generalized optimum output SINR analysis is derived for space-time processing structures in frequency selective fading channels. The hardware and software of the MPRG Antenna Array Testbed (MAAT) is described. A literature review of previous antenna array propagation measurements is given. Antenna array measurement results obtained with the MAAT are used to compare the properties of the received signal vector in the various environmental conditions. The influence of channel parameters on the ability of antenna arrays to separate the signals of two users on the reverse link is studied using simulation. Finally, forward link beamforming techniques are reviewed.
Approximation of integro-partial differential equations of hyperbolic type
Fabiano, Richard H. (Virginia Polytechnic Institute and State University, 1986)
A state space model is developed for a class of integro-partial differential equations of hyperbolic type which arise in viscoelasticity. An approximation scheme is developed based on a spline approximation in the spatial variable and an averaging approximation in the de1ay variable. Techniques from linear semigroup theory are used to discuss the well-posedness of the state space model and the convergence properties of the approximation scheme. We give numerical results for a sample problem to illustrate some properties of the approximation scheme.
Approximation of Parametric Dynamical Systems
Carracedo Rodriguez, Andrea (Virginia Tech, 2020-09-02)
Dynamical systems are widely used to model physical phenomena and, in many cases, these physical phenomena are parameter dependent. In this thesis we investigate three prominent problems related to the simulation of parametric dynamical systems and develop the analysis and computational framework to solve each of them. In many cases we have access to data resulting from simulations of a parametric dynamical system for which an explicit description may not be available. We introduce the parametric AAA (p-AAA) algorithm that builds a rational approximation of the underlying parametric dynamical system from its input/output measurements, in the form of transfer function evaluations. Our algorithm generalizes the AAA algorithm, a popular method for the rational approximation of nonparametric systems, to the parametric case. We develop p-AAA for both scalar and matrix-valued data and study the impact of parameter scaling. Even though we present p-AAA with parametric dynamical systems in mind, the ideas can be applied to parametric stationary problems as well, and we include such examples. The solution of a dynamical system can often be expressed in terms of an eigenvalue problem (EVP). In many cases, the resulting EVP is nonlinear and depends on a parameter. A common approach to solving (nonparametric) nonlinear EVPs is to approximate them with a rational EVP and then to linearize this approximation. An existing algorithm can then be applied to find the eigenvalues of this linearization. The AAA algorithm has been successfully applied to this scheme for the nonparametric case. We generalize this approach by using our p-AAA algorithm to find a rational approximation of parametric nonlinear EVPs. We define a corresponding linearization that fits the format of the compact rational Krylov (CORK) algorithm for the approximation of eigenvalues. The simulation of dynamical systems may be costly, since the need for accuracy may yield a system of very large dimension. This cost is magnified in the case of parametric dynamical systems, since one may be interested in simulations for many parameter values. Interpolatory model order reduction (MOR) tackles this problem by creating a surrogate model that interpolates the original, is of much smaller dimension, and captures the dynamics of the quantities of interest well. We generalize interpolatory projection MOR methods from parametric linear to parametric bilinear systems. We provide necessary subspace conditions to guarantee interpolation of the subsystems and their first and second derivatives, including the parameter gradients and Hessians. Throughout the dissertation, the analysis is illustrated via various benchmark numerical examples.
Bilinear Immersed Finite Elements For Interface Problems
He, Xiaoming (Virginia Tech, 2009-04-20)
In this dissertation we discuss bilinear immersed finite elements (IFE) for solving interface problems. The related research works can be categorized into three aspects: (1) the construction of the bilinear immersed finite element spaces; (2) numerical methods based on these IFE spaces for solving interface problems; and (3) the corresponding error analysis. All of these together form a solid foundation for the bilinear IFEs. The research on immersed finite elements is motivated by many real world applications, in which a simulation domain is often formed by several materials separated from each other by curves or surfaces while a mesh independent of interface instead of a body-fitting mesh is preferred. The bilinear IFE spaces are nonconforming finite element spaces and the mesh can be independent of interface. The error estimates for the interpolation of a Sobolev function in a bilinear IFE space indicate that this space has the usual approximation capability expected from bilinear polynomials, which is O(h²) in L² norm and O(h) in H¹ norm. Then the immersed spaces are applied in Galerkin, finite volume element (FVE) and discontinuous Galerkin (DG) methods for solving interface problems. Numerical examples show that these methods based on the bilinear IFE spaces have the same optimal convergence rates as those based on the standard bilinear finite element for solutions with certain smoothness. For the symmetric selective immersed discontinuous Galerkin method based on bilinear IFE, we have established its optimal convergence rate. For the Galerkin method based on bilinear IFE, we have also established its convergence. One of the important advantages of the discontinuous Galerkin method is its flexibility for both p and h mesh refinement. Because IFEs can use a mesh independent of interface, such as a structured mesh, the combination of a DG method and IFEs allows a flexible adaptive mesh independent of interface to be used for solving interface problems. That is, a mesh independent of interface can be refined wherever needed, such as around the interface and the singular source. We also develop an efficient selective immersed discontinuous Galerkin method. It uses the sophisticated discontinuous Galerkin formulation only around the locations needed, but uses the simpler Galerkin formulation everywhere else. This selective formulation leads to an algebraic system with far less unknowns than the immersed DG method without scarifying the accuracy; hence it is far more efficient than the conventional discontinuous Galerkin formulations.
Block-Circulant Approximation of the Precision Matrix for Assimilating SWOT Altimetry Data
Yaremchuk, Max; Beattie, Christopher A.; Panteleev, Gleb; D’Addezio, Joseph (MDPI, 2024-05-29)
The recently deployed Surface Water and Ocean Topography (SWOT) mission for the first time has observed the ocean surface at a spatial resolution of 1 km, thus giving an opportunity to directly monitor submesoscale sea surface height (SSH) variations that have a typical magnitude of a few centimeters. This progress comes at the expense of the necessity to take into account numerous uncertainties in calibration of the quality-controlled altimeter data. Of particular importance is the proper filtering of spatially correlated errors caused by the uncertainties in geometry and orientation of the on-board interferometer. These “systematic” errors dominate the SWOT error budget and are likely to have a notable signature in the SSH products available to the oceanographic community. In this study, we explore the utility of the block-circulant (BC) approximation of the SWOT precision matrix developed by the Jet Propulsion Laboratory for assessment of a mission’s accuracy, including the possible impact of the systematic errors on the assimilation of the wide-swath altimeter data into numerical models. It is found that BC approximation of the precision matrix has sufficient (90–99%) accuracy for a wide range of significant wave heights of the ocean surface, and, therefore, could potentially serve as an efficient preconditioner for data assimilation problems involving altimetry observations by space-borne interferometers. An extensive set of variational data assimilation (DA) experiments demonstrates that BC approximation provides more accurate SSH retrievals compared to approximations, assuming a spatially uncorrelated observation error field as is currently adopted in operational DA systems.
A Class of Immersed Finite Element Spaces and Their Application to Forward and Inverse Interface Problems
Camp, Brian David (Virginia Tech, 2003-11-19)
A class of immersed finite element (IFE) spaces is developed for solving elliptic boundary value problems that have interfaces. IFE spaces are finite element approximation spaces which are based upon meshes that can be independent of interfaces in the domain. Three different quadratic IFE spaces and their related biquadratic IFE spaces are introduced here for the purposes of solving both forward and inverse elliptic interface problems in 1D and 2D. These different spaces are constructed by (i) using a hierarchical approach, (ii) imposing extra continuity requirements or (iii) using a local refinement technique. The interpolation properties of each space are tested against appropriate testing functions in 1D and 2D. The IFE spaces are also used to approximate the solution of a forward elliptic interface problem using the Galerkin finite element method and the mixed least squares finite element method. Finally, one appropriate space is selected to solve an inverse interface problem using either an output least squares approach or the least squares with mixed equation error method.
Coherence and Phase Synchrony Analysis of Electroencephalogram
Tcheslavski, Gleb V. (Virginia Tech, 2005-12-12)
Phase Synchrony (PS) and coherence analyses of stochastic time series - tools to discover brain tissue pathways traveled by electrical signals - are considered for the specific purpose of processing of the electroencephalogram (EEG). We propose the Phase Synchrony Processor (PSP), as a tool for implementing phase synchrony analysis, and examine its properties on the basis of known signals. Long observation times and wide filter bandwidths can decrease bias in PS estimates. The value of PS is affected by the difference in frequency of the sequences being analyzed and can be related to that frequency difference by the periodic sinc function. PS analysis of the EEG shows that the average PS is higher - for a number of electrode pairs - for non-ADHD than for ADHD participants. The difference is more pronounced in the δ rhythm (0-3 Hz) and in the γ rhythm (30-50 Hz) PS. The Euclidean classifier with electrode masking yields 66 % correct classification on average for ADHD and non-ADHD subjects using the δ and γ1 rhythms. We observed that the average γ1 rhythm PS is higher for the eyes closed condition than for the eyes open condition. The latter may potentially be used for vigilance monitoring. The Euclidean discriminator with electrode masking shows an average percentage of correct classification of 78 % between the eyes open and eyes closed subject conditions. We develop a model for a pair of EEG electrodes and a model-based MS coherence estimator aimed at processing short (i.e. 20 samples) EEG frames. We verify that EEG sequences can be modeled as AR(3) processes degraded by additive white noise with an average SNR of approximately 11-12 dB. Application of the MS coherence estimator to the EEG suggests that MS coherence is generally higher for non-ADHD individuals than for ADHD participants when evaluated for the Î¸ rhythm of EEG. Also, MS coherence is consistently higher for ADHD subjects than for the majority of non-ADHD individuals when computed for the low end of the δ rhythm (i.e. below 1 Hz). ADHD produces more measurable effects in the frontal lobe EEG and for participants performing attention intensive tasks.
Coherent Mitigation of Radio Frequency Interference in 10-100 MHz
Lee, Kyehun (Virginia Tech, 2008-08-28)
This dissertation describes methods of mitigating radio frequency interference (RFI) in the frequency range 10-100 MHz, developing and evaluating coherent methods with which RFI is subtracted from the afflicted data, nominally resulting in no distortion of the underlying signals. This approach is of interest in weak signal applications such as radio astronomy, where the signal of interest may have interference-to-noise ratio much less than one, and so can be easily distorted by other methods. Environmental noise in this band is strong and non-white, so a realistic noise model is developed, with which we characterize the performance of signal parameter estimation, a key component of the proposed algorithms. Two classes of methods are considered: "generic" parameter estimation/subtraction (PE/S) and a modulation-specific form known as demodulation-remodulation ("demod--remod") PE/S. It is demonstrated for RFI in the form of narrowband FM and Broadcast FM that generic PE/S has the problem of severely distorting underlying signals of interest and demod-remod PE/S is less prone to this problem. Demod-remod PE/S is also applied and evaluated for RFI in the form of Digital TV signals. In both cases, we compare the performance of the demod-remod PE/S with that of a traditional adaptive canceling method employing a reference antenna, and propose a hybrid method to further improve performance. A new metric for "toxicity" is defined and employed to determine the degree to which RFI mitigation damages the underlying signal of interest.
Computation of a Damping Matrix for Finite Element Model Updating
Pilkey, Deborah F. (Virginia Tech, 1998-04-21)
The characterization of damping is important in making accurate predictions of both the true response and the frequency response of any device or structure dominated by energy dissipation. The process of modeling damping matrices and experimental verification of those is challenging because damping can not be determined via static tests as can mass and stiffness. Furthermore, damping is more difficult to determine from dynamic measurements than natural frequency. However, damping is extremely important in formulating predictive models of structures. In addition, damping matrix identification may be useful in diagnostics or health monitoring of structures. The objective of this work is to find a robust, practical procedure to identify damping matrices. All aspects of the damping identification procedure are investigated. The procedures for damping identification presented herein are based on prior knowledge of the finite element or analytical mass matrices and measured eigendata. Alternately, a procedure is based on knowledge of the mass and stiffness matrices and the eigendata. With this in mind, an exploration into model reduction and updating is needed to make the problem more complete for practical applications. Additionally, high performance computing is used as a tool to deal with large problems. High Performance Fortran is exploited for this purpose. Finally, several examples, including one experimental example are used to illustrate the use of these new damping matrix identification algorithms and to explore their robustness.
Contragredient Transformations Applied to the Optimal Projection Equations
Zigic, Dragan; Watson, Layne T.; Beattie, Christopher A. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1992)
The optimal projection approach to solving the H2 reduced order model problem produces two coupled, highly nonlinear matrix equations with rank conditions as constraints. It is not obvious from their original form how they can be differentiated and how some algorithm for solving nonlinear equations can be applied to them. A contragredient transformation, a transformation which simultaneously diagonalizes two symmetric positive semi-definite matrices, is used to transform the equations into forms suitable for algorithms for solving nonlinear problems. Three different forms of the equations obtained using contragredient transformations are given. An SVD-based algorithm for the contragredient transformation and a homotopy algorithm for the transformed equations are given, together with a numerical example.
Convergence theorems for intermediate problems. II
Beattie, Christopher A.; Greenlee, W. M. (Cambridge University Press, 2002)
Convergence theorems for the practical eigenvector free methods of Gay and Goerisch are obtained under a variety of hypotheses, so that our theorems apply to both traditional boundary-value problems and atomic problems. In addition, we prove convergence of the T*T method of Bazley and Fox without an alignment of projections hypothesis required in previous literature.
Damping optimization of parameter dependent mechanical systems by rational interpolation
Tomljanović, Z.; Beattie, Christopher A.; Gugercin, Serkan (2017-07-06)
We consider an optimization problem related to semi-active damping of vibrating systems. The main problem is to determine the best damping matrix able to minimize influence of the input on the output of the system. We use a minimization criteria based on the $\mathcal{H}_2$ system norm. The objective function is non-convex and the associated optimization problem typically requires a large number of objective function evaluations. We propose an optimization approach that calculates `interpolatory' reduced order models, allowing for significant acceleration of the optimization process. In our approach, we use parametric model reduction (PMOR) based on the Iterative Rational Krylov Algorithm, which ensures good approximations relative to the $\mathcal{H}_2$ system norm, aligning well with the underlying damping design objectives. For the parameter sampling that occurs within each PMOR cycle, we consider approaches with predetermined sampling and approaches using adaptive sampling, and each of these approaches may be combined with three possible strategies for internal reduction. In order to preserve important system properties, we maintain second-order structure, which through the use of modal coordinates, allows for very efficient implementation. The methodology proposed here provides a significant acceleration of the optimization process; the gain in efficiency is illustrated in numerical experiments.
Data-driven structured realization
Schulze, P.; Unger, Benjamin; Beattie, Christopher A.; Gugercin, Serkan (Elsevier, 2018-01-15)
We present a framework for constructing structured realizations of linear dynamical systems having transfer functions of the form C̃(∑_k=1^K h_k(s)Ã_k)^-1B̃ where h₁, h₂, ..., h_k are prescribed functions that specify the surmised structure of the model. Our construction is data-driven in the sense that an interpolant is derived entirely from measurements of a transfer function. Our approach extends the Loewner realization framework to more general system structure that includes second-order (and higher) systems as well as systems with internal delays. Numerical examples demonstrate the advantages of this approach.
Design and Evaluation of a Data-distributed Massively Parallel Implementation of a Global Optimization Algorithm---DIRECT
He, Jian (Virginia Tech, 2007-11-15)
The present work aims at an efficient, portable, and robust design of a data-distributed massively parallel DIRECT, the deterministic global optimization algorithm widely used in multidisciplinary engineering design, biological science, and physical science applications. The original algorithm is modified to adapt to different problem scales and optimization (exploration vs.\ exploitation) goals. Enhanced with a memory reduction technique, dynamic data structures are used to organize local data, handle unpredictable memory requirements, reduce the memory usage, and share the data across multiple processors. The parallel scheme employs a multilevel functional and data parallelism to boost concurrency and mitigate the data dependency, thus improving the load balancing and scalability. In addition, checkpointing features are integrated to provide fault tolerance and hot restarts. Important algorithm modifications and design considerations are discussed regarding data structures, parallel schemes, error handling, and portability. Using several benchmark functions and real-world applications, the present work is evaluated in terms of optimization effectiveness, data structure efficiency, memory usage, parallel performance, and checkpointing overhead. Modeling and analysis techniques are used to investigate the design effectiveness and performance sensitivity under various problem structures, parallel schemes, and system settings. Theoretical and experimental results are compared for two parallel clusters with different system scale and network connectivity. An analytical bounding model is constructed to measure the load balancing performance under different schemes. Additionally, linear regression models are used to characterize two major overhead sources---interprocessor communication and processor idleness, and also applied to the isoefficiency functions in scalability analysis. For a variety of high-dimensional problems and large scale systems, the data-distributed massively parallel design has achieved reasonable performance. The results of the performance study provide guidance for efficient problem and scheme configuration. More importantly, the generalized design considerations and analysis techniques are beneficial for transforming many global search algorithms to become effective large scale parallel optimization tools.
Domain decomposition and high order discretization of elliptic partial differential equations
Pitts, George G. (Virginia Tech, 1994)
Numerical solutions of partial differential equations (PDEs) resulting from problems in both the engineering and natural sciences result in solving large sparse linear systems Au = b. The construction of such linear systems and their solutions using either direct or iterative methods are topics of continuing research. The recent advent of parallel computer architectures has resulted in a search for good parallel algorithms to solve such systems, which in turn has led to a recent burgeoning of research into domain decomposition algorithms. Domain decomposition is a procedure which employs subdivision of the solution domain into smaller regions of convenient size or shape and, although such partitionings have proven to be quite effective on serial computers, they have proven to be even more effective on parallel computers. Recent work in domain decomposition algorithms has largely been based on second order accurate discretization techniques. This dissertation describes an algorithm for the numerical solution of general two-dimensional linear elliptic partial differential equations with variable coefficients which employs both a high order accurate discretization and a Krylov subspace iterative solver in which a preconditioner is developed using domain decomposition. Most current research into such algorithms has been based on symmetric systems; however, variable PDE coefficients generally result in a nonsymmetric A, and less is known about the use of preconditioned Krylov subspace iterative methods for the solution of nonsymmetric systems. The use of the high order accurate discretization together with a domain decomposition based preconditioner results in an iterative technique with both high accuracy and rapid convergence. Supporting theory for both the discretization and the preconditioned iterative solver is presented. Numerical results are given on a set of test problems of varying complexity demonstrating the robustness of the algorithm. It is shown that, if only second order accuracy is required, the algorithm becomes an extremely fast direct solver. Parallel performance of the algorithm is illustrated with results from a shared memory multiprocessor.
Domain Decomposition Preconditioners for Hermite Collocation Problems
Mateescu, Gabriel (Virginia Tech, 1998-12-14)
Accelerating the convergence rate of Krylov subspace methods with parallelizable preconditioners is essential for obtaining effective iterative solvers for very large linear systems of equations. Substructuring provides a framework for constructing robust and parallel preconditioners for linear systems arising from the discretization of boundary value problems. Although collocation is a very general and effective discretization technique for many PDE problems, there has been relatively little work on preconditioners for collocation problems. This thesis proposes two preconditioning methods for solving linear systems of equations arising from Hermite bicubic collocation discretization of elliptic partial differential equations on square domains with mixed boundary conditions. The first method, called edge preconditioning, is based on a decomposition of the domain in parallel strips, and the second, called edge-vertex preconditioning, is based on a two-dimensional decomposition. The preconditioners are derived in terms of two special rectangular grids -- a coarse grid with diameter H and a hybrid coarse/fine grid -- which together with the fine grid of diameter h provide the framework for approximating the interface problem induced by substructuring. We show that the proposed methods are effective for nonsymmetric indefinite problems, both from the point of view of the cost per iteration and of the number of iterations. For an appropriate choice of H, the edge preconditioner requires O(N) arithmetic operations per iteration, while the edge-vertex preconditioner requires O(N^4/3) operations, where N is the number of unknowns. For the edge-vertex preconditioner, the number of iterations is almost constant when h and H decrease such that H/h is held constant and it increases very slowly with H when h is held constant. For both the edge- and edge-vertex preconditioners the number of iterations depends only weakly on h when H is constant. The edge-vertex preconditioner outperforms the edge-preconditioner for small enough H. Numerical experiments illustrate the parallel efficiency of the preconditioners which is similar or even better than that provided by the well-known PETSc parallel software library for scientific computing.

Browsing by Author "Beattie, Christopher A."

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