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Event(s) on October 2007
- Tuesday, 2nd October, 2007
Title: Numerical Computation for the Hamiltonian Schur form Speaker: Prof. Delin Chu, Department of Mathematics, National University of Singapore, Singapore Time/Place: 11:30 - 12:30
Abstract: We introduce a new numerical method for computing the Hamiltonian Schur form of a 2n-by-2n Hamiltonian matrix M that has no purely imaginary eigenvalues. We demonstrate the properties of the new method by showing its performance for the benchmark collection of continuous-time algebraic Riccati equations. Despite the fact that no complete error analysis for the method is yet available, the numerical results indicate that if no eigenvalues of M are close to the imaginary axis then the method computes the exact Hamiltonian Schur form of a nearby Hamiltonian matrix and thus is numerically strongly backward stable. The new method is of complexity O(n^3).
- Tuesday, 9th October, 2007
Title: A Posteriori Error Estimation and Its Applications Speaker: Prof. Jun Cao, Department of Mechanical & Industrial Engineering, Ryerson University, Canada Time/Place: 11:30 - 12:30
Abstract: Despite significant progress made in the Computational Fluid Dynamics (CFD) study over past decades, challenges remain in efficiently achieving a high-quality numerical solution of large-scale fluid mechanics problems with geometries of practical interests. In order to resolve the conflict between mesh size and solution accuracy, adaptive grid generation methods become more and more attractive to the CFD researchers. One of the main issues arising in connection with adaptive algorithms is how to determine where adaptive re-meshing is needed. In addressing this issue, a mathematical criterion based on a posteriori error estimates has been studied - the errors within each element on a mesh are estimated by solving a local Neumann problem corresponding to the mathematical model of interests; then, the a posteriori error estimates are used to form both local and global energy norms involving all the physical variables in the model chosen; this energy norm serves as error index to be reduced through local mesh refinements and, ultimately, to be approximately equi-distributed over the mesh through simultaneous mesh refinement and coarsening, along with node moving. For applications ranging from environmental engineering to aerospace engineering, a variety of numerical experiments covering the potential flow, Stokes flow, and Navier-Stokes flow models have been performed using finite element mesh adaptive algorithms oriented by a posteriori error estimates. Through this series of test cases, the generality and reliability of the a posteriori error estimation approach have been demonstrated for guiding mesh adaptation to obtain a more accurate numerical solution to engineering problems without requiring a significant increase of computational cost.
- Tuesday, 23rd October, 2007
Title: Numerical Stability of GMRES Speaker: Prof. Miro Rozloznik, Department of Computational Methods, Czech Academy of Sciences, Czech Republic Time/Place: 11:30 - 12:30
Abstract: In this contribution we consider the GMRES method, the most widely known and used representative of the class of nonsymmetric Krylov subspace method. This method consists of constructing the basis of associated Krylov subspace via the Arnoldi method and then solving the transformed Hessenberg least squares problem at each iteration step. In exact arithmetic the Arnoldi vectors are orthogonal. However, in finite precision computation the orthogonality is lost, which may potentially affect both the convergence rate and the ultimate attainable accuracy of the computed approximate solution. One may therefore want to keep the computed orthogonality as close to the machine precision as possible using proper orthogonal transformations, e.g. Householder orthogonalizations. The price is, unfortunately, too high for most of the applications. The Gram-Schmidt process is a cheaper alternative and its modified version represents the most frequently used compromise. Although, the (classical or modified) Gram-Schmidt orthogonalization may end up with the basis which lost its orthogonality completely, in the GMRES context, however, there is a very important relation between the loss of orthogonality among the Arnoldi vectors and the decrease of the residual of the computed approximation close to its final value. It was proved that, for the modified Gram-Schmidt GMRES, the Arnoldi vectors loose their orthogonality completely only after the residual of the computed approximation is reduced close to its final level of accuracy, which is proportional to the machine precision multiplied by the condition number of the system matrix. For the classical Gram-Schmidt the corresponding level of limiting accuracy, of course, is significantly different. The modified Gram-Schmidt GMRES however performs almost exactly as well as the Householder implementation and both implementations are backward stable. This suggests that unless the system matrix is extremely ill-conditioned, the use of the Householder or modified Gram-Schmidt GMRES is theoretically well justified. Presented results lead to important conclusions about the parallel implementation and application of the GMRES method. The theoretical analysis has not been finished yet. In the end of our talk we mention some questions related to the rate of convergence of the implementation in the finite precision arithmetic which are open and still need some effort.
- Friday, 26th October, 2007
Title: Discontinuous Galerkin Finite Element Method for Convection-Diffusion Problems: Numerical Analysis Speaker: Prof. Vit Dolejsi, Department of Numerical Mathematics, Charles University, Czech Republic Time/Place: 11:30 - 12:30
Abstract: Our aim is to develop a sufficiently robust, accurate and efficient numerical scheme for the solution of the system of the compressible Navier-Stokes equations which describe a motion of viscous compressible flows. For simplicity, we start with a model problem represented by a scalar nonlinear non-stationary convection-diffusion equation. We discretize this equation with the aid of the discontinuous Galerkin finite element method (DGFEM) which is based on a piecewise polynomial but discontinuous approximation. In order to replace the inter-element continuity, some stabilization and penalty terms have to be added to the scheme. We present several variants of DGFEM and derive a priori error estimates in the L^2-norm and the H^1-seminorm. Moreover, we deal with the time discretization using the so-called backward difference formulae (BDF) scheme. The resulting BDF-DGFEM represents an efficient tool for the solution of convection-diffusion problems. Several numerical examples verifying the theoretical results are presented.
- Friday, 26th October, 2007
Title: Numerics Stability of Gram-Schmidt Orthogonalization Speaker: Prof. Miro Rozloznik, Department of Computational Methods, Czech Academy of Sciences, Czech Republic Time/Place: 14:30 - 15:30
Abstract: In this contribution we compare the numerical properties of the classical (CGS) and modified (MGS) Gram-Schmidt orthogonalization. The theory for the MGS algorithm is well-known thanks to results of Bjorck and Paige, who have shown that for a numerically nonsingular matrix A the loss of orthogonality in MGS occurs in a predictable way and it can be bounded by a term proportional to the condition number cond(A) and to the roundoff unit u. Generally accepted view for the CGS process is that due to rounding errors the orthogonality of computed vectors can be lost very quickly or may be even lost totally. The textbooks usually contain examples with progressive deterioration in the orthogonality withoutany bound relating it to the condition number of initial vectors. We will show, however, that one can derive a bound for the loss of orthogonality in the CGS algorithm. This bound depends only quadratically on the condition number of the system. This result sounds very well with all examples used for demonstrating the unreliability of the CGS process, what will be also illustrated on examples. If we want to keep the computed orthogonality as close to the machine precision as possible, we need to use the Gram-Schmidt orthogonalization scheme with reorthogonalization. We will show that either for the CGS or MGS process with (one) reorthogonalzation the loss of orthogonality is preserved on the machine precision level. On the other hand, the price for it is rather high, it is actually doubled in comparison to the standard CGS or MGS algorithm. From a practical point of view, the CGS algorithm is a better candidate for parallel implementation than the MGS variant of the same algorithm and this aspect could not be overlooked in certain computing environments. Moreover, a new trend is emerging nowadays, several experiments are reporting that even if performing twice as much operations as MGS, the CGS algorithm with one (complete) reorthogonalization may be faster. This indicates that such results to certain extent may lead to reinstating of the CGS algorithm as a suitable alternative for parallel implementation of the Gram-Schmidt orthogonalization process.
- Friday, 26th October, 2007
Title: Discontinuous Galerkin Finite Element Method for Convection-Diffusion Problems: Application to Compressible Flows Simulations Speaker: Prof. Vit Dolejsi, Department of Numerical Mathematics, Charles University, Czech Republic Time/Place: 15:45 - 16:45
Abstract: We apply the BDF-DGFEM to the system of the compressible Navier-Stokes equations. A special attention is paid to the definition of stabilization terms since an extension from the scalar case is not too much straightforward. Moreover, in order to avoid a solution of a nonlinear algebraic problem at each time step, we propose a linearization of inviscid as well as viscous fluxes and then linear terms are treated implicitly and the nonlinear ones explicitly. The resulting scheme is practically unconditionally stable, has a high degree of accuracy with respect to space and time coordinates and requires a solution of one linear algebraic problem at each time step. Some implementation aspects are mentioned and a set of numerical experiments is presented.
- Monday, 29th October, 2007
Title: Saddle-Point Problems in Contact Mechanics Speaker: Prof. Michel Fortin, GIREF, Université Laval, Québec, Canada Time/Place: 15:30 - 16:30
Abstract: Contact problems in solid mechanics lead to saddle-point problems which present some interesting challenges. We shall introduce the problem and discuss its finite element discretisation. The condition number of the dual problem is mesh dependent. Nevertheless, solution methods are easily available for slipping contact. Projected conjugate gradient methods are indeed quite efficient. Frictional contact, however, leads to non linear constraints on the multiplier. We shall consider a few ways of addressing this issue: a standard Uzawa's method, SQP, multipliers for the constraints … Many open questions remain. We shall end with a few numerical results.