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Residual Vector (residual + vector)
Selected AbstractsA stopping criterion for the conjugate gradient algorithm in the framework of anisotropic adaptive finite elementsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 4 2009M. Picasso Abstract We propose a simple stopping criterion for the conjugate gradient (CG) algorithm in the framework of anisotropic, adaptive finite elements for elliptic problems. The goal of the adaptive algorithm is to find a triangulation such that the estimated relative error is close to a given tolerance TOL. We propose to stop the CG algorithm whenever the residual vector has Euclidian norm less than a small fraction of the estimated error. This stopping criterion is based on a posteriori error estimates between the true solution u and the computed solution u (the superscript n stands for the CG iteration number, the subscript h for the typical mesh size) and on heuristics to relate the error between uh and u to the residual vector. Numerical experiments with anisotropic adaptive meshes show that the total number of CG iterations can be divided by 10 without significant discrepancy in the computed results. Copyright © 2008 John Wiley & Sons, Ltd. [source] Coupled Navier,Stokes,Molecular dynamics simulations using a multi-physics flow simulation frameworkINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2010R. Steijl Abstract Simulation of nano-scale channel flows using a coupled Navier,Stokes/Molecular Dynamics (MD) method is presented. The flow cases serve as examples of the application of a multi-physics computational framework put forward in this work. The framework employs a set of (partially) overlapping sub-domains in which different levels of physical modelling are used to describe the flow. This way, numerical simulations based on the Navier,Stokes equations can be extended to flows in which the continuum and/or Newtonian flow assumptions break down in regions of the domain, by locally increasing the level of detail in the model. Then, the use of multiple levels of physical modelling can reduce the overall computational cost for a given level of fidelity. The present work describes the structure of a parallel computational framework for such simulations, including details of a Navier,Stokes/MD coupling, the convergence behaviour of coupled simulations as well as the parallel implementation. For the cases considered here, micro-scale MD problems are constructed to provide viscous stresses for the Navier,Stokes equations. The first problem is the planar Poiseuille flow, for which the viscous fluxes on each cell face in the finite-volume discretization are evaluated using MD. The second example deals with fully developed three-dimensional channel flow, with molecular level modelling of the shear stresses in a group of cells in the domain corners. An important aspect in using shear stresses evaluated with MD in Navier,Stokes simulations is the scatter in the data due to the sampling of a finite ensemble over a limited interval. In the coupled simulations, this prevents the convergence of the system in terms of the reduction of the norm of the residual vector of the finite-volume discretization of the macro-domain. Solutions to this problem are discussed in the present work, along with an analysis of the effect of number of realizations and sample duration. The averaging of the apparent viscosity for each cell face, i.e. the ratio of the shear stress predicted from MD and the imposed velocity gradient, over a number of macro-scale time steps is shown to be a simple but effective method to reach a good level of convergence of the coupled system. Finally, the parallel efficiency of the developed method is demonstrated. Copyright © 2009 John Wiley & Sons, Ltd. [source] Fault detection and isolation in robotic manipulators via neural networks: A comparison among three architectures for residual analysisJOURNAL OF FIELD ROBOTICS (FORMERLY JOURNAL OF ROBOTIC SYSTEMS), Issue 7 2001Marco Henrique Terra In this article we discuss artificial neural networks-based fault detection and isolation (FDI) applications for robotic manipulators. The artificial neural networks (ANNs) are used for both residual generation and residual analysis. A multilayer perceptron (MLP) is employed to reproduce the dynamics of the robotic manipulator. Its outputs are compared with actual position and velocity measurements, generating the so-called residual vector. The residuals, when properly analyzed, provides an indication of the status of the robot (normal or faulty operation). Three ANNs architectures are employed in the residual analysis. The first is a radial basis function network (RBFN) which uses the residuals of position and velocity to perform fault identification. The second is again an RBFN, except that it uses only the velocity residuals. The third is an MLP which also performs fault identification utilizing only the velocity residuals. The MLP is trained with the classical back-propagation algorithm and the RBFN is trained with a Kohonen self-organizing map (KSOM). We validate the concepts discussed in a thorough simulation study of a Puma 560 and with experimental results with a 3-joint planar manipulator. © 2001 John Wiley & Sons, Inc. [source] Convergence conditions for a restarted GMRES method augmented with eigenspacesNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 4 2005Jan ZķtkoArticle first published online: 9 AUG 200 Abstract We consider the GMRES(m,k) method for the solution of linear systems Ax=b, i.e. the restarted GMRES with restart m where to the standard Krylov subspace of dimension m the other subspace of dimension k is added, resulting in an augmented Krylov subspace. This additional subspace approximates usually an A -invariant subspace. The eigenspaces associated with the eigenvalues closest to zero are commonly used, as those are thought to hinder convergence the most. The behaviour of residual bounds is described for various situations which can arise during the GMRES(m,k) process. The obtained estimates for the norm of the residual vector suggest sufficient conditions for convergence of GMRES(m,k) and illustrate that these augmentation techniques can remove stagnation of GMRES(m) in many cases. All estimates are independent of the choice of an initial approximation. Conclusions and remarks assessing numerically the quality of proposed bounds conclude the paper. Copyright © 2004 John Wiley & Sons, Ltd. [source] Computing projections via Householder transformations and Gram,Schmidt orthogonalizationsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 7 2004Achiya Dax Abstract Let x* denote the solution of a linear least-squares problem of the form where A is a full rank m × n matrix, m > n. Let r*=b - Ax* denote the corresponding residual vector. In most problems one is satisfied with accurate computation of x*. Yet in some applications, such as affine scaling methods, one is also interested in accurate computation of the unit residual vector r*/,r*,2. The difficulties arise when ,r*,2 is much smaller than ,b,2. Let x, and r, denote the computed values of x* and r*, respectively. Let ,denote the machine precision in our computations, and assume that r, is computed from the equality r, =b - Ax,. Then, no matter how accurate x, is, the unit residual vector ū =r,/,r,,2 contains an error vector whose size is likely to exceed ,,b,2/,r*,2. That is, the smaller ,r*,2 the larger the error. Thus although the computed unit residual should satisfy ATū=0, in practice the size of ,ATū,2 is about ,,A,2,b,2/,r*,2. The methods discussed in this paper compute a residual vector, r,, for which ,ATr,,2 is not much larger than ,,A,2,r,,2. Numerical experiments illustrate the difficulties in computing small residuals and the usefulness of the proposed safeguards. Copyright © 2004 John Wiley & Sons, Ltd. [source] Block s-step Krylov iterative methodsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2010Anthony T. Chronopoulos Abstract Block (including s-step) iterative methods for (non)symmetric linear systems have been studied and implemented in the past. In this article we present a (combined) block s-step Krylov iterative method for nonsymmetric linear systems. We then consider the problem of applying any block iterative method to solve a linear system with one right-hand side using many linearly independent initial residual vectors. We present a new algorithm which combines the many solutions obtained (by any block iterative method) into a single solution to the linear system. This approach of using block methods in order to increase the parallelism of Krylov methods is very useful in parallel systems. We implemented the new method on a parallel computer and we ran tests to validate the accuracy and the performance of the proposed methods. It is expected that the block s-step methods performance will scale well on other parallel systems because of their efficient use of memory hierarchies and their reduction of the number of global communication operations over the standard methods. Copyright © 2009 John Wiley & Sons, Ltd. 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