A GCV based Arnoldi-Tikhonov regularization method

For the solution of linear discrete ill-posed problems, in this paper we consider the Arnoldi-Tikhonov method coupled with the Generalized Cross Validation for the computation of the regularization parameter at each iteration. We study the convergence behavior of the Arnoldi method and its properties for the approximation of the (generalized) singular values, under the hypothesis that Picard condition is satisfied. Numerical experiments on classical test problems and on image restoration are presented

An Arnoldi-frontal approach for the stability analysis of flows in a collapsible channel

In this paper, we present a new approach based on a combination of the Arnoldi and frontal methods for solving large sparse asymmetric and generalized complex eigenvalue problems. The new eigensolver seeks the most unstable eigensolution in the Krylov subspace and makes use of the efficiency of the frontal solver developed for the finite element methods. The approach is used for a stability analysis of flows in a collapsible channel and is found to significantly improve the computational efficiency compared to the traditionally used QZ solver or a standard Arnoldi method. With the new approach, we are able to validate the previous results obtained either on a much coarser mesh or estimated from unsteady simulations. New neutral stability solutions of the system have been obtained which are beyond the limits of previously used methods

Computing eigenvalues of ordinary differential equations

Discretisations of differential eigenvalue problems have a sensitivity to perturbations which is asymptotically least as h ? 0 when the differential equation is in first order system form. Both second and fourth order accurate discretisations of the first order system are straightforward to derive and lead to generalised eigenvalue problems of the form ?? where both A and B are narrow-banded, block bidiagonal (hence unsymmetric) matrices, and typically B is singular. Solutions of the differential equation associated with eigenvalues of small magnitude are best determined by the discretisations. Thus Krylov subspace methods (for example) require A to be invertible and seek large solutions of ?? This already requires rational methods in principle. It follows that rapidly convergent methods based on inverse iteration applied to the original formulation as a nonstandard generalised eigenvalue problem prove attractive for the narrow-banded systems considered here. Also they have the advantage that they are applicable under the weaker condition A ? ?B ? =? . We have had extensive experience with a method combining aspects of Newton's method and inverse iteration and having a convergence rate of 3.56 . Our implementation combines this basic algorithm with a limiting form of Weilandt deflation to find a sequence of eigenvalues. It has proved extremely satisfactory in a range of applications. This formulation has the further advantage that it is easy to insert the eigenvalue calculation inside an outer loop to satisfy a constraint on an auxiliary parameter. Examples to illustrate both the robustness of the deflation and the flexibility of the approach are provided

Hard hexagon partition function for complex fugacity

We study the analyticity of the partition function of the hard hexagon model in the complex fugacity plane by computing zeros and transfer matrix eigenvalues for large finite size systems. We find that the partition function per site computed by Baxter in the thermodynamic limit for positive real values of the fugacity is not sufficient to describe the analyticity in the full complex fugacity plane. We also obtain a new algebraic equation for the low density partition function per site.Comment: 49 pages, IoP styles files, lots of figures (png mostly) so using PDFLaTeX. Some minor changes added to version 2 in response to referee report

Towards Gradient-Based Design Optimization of Flexible Transport Aircraft with Flutter Constraints

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/140443/1/6.2014-2726.pd

BiGlobal stability analysis in curvilinear coordinates of massively separated lifting bodies

A methodology based on spectral collocation numerical methods for global flow stability analysis of incompressible external flows is presented. A potential shortcoming of spectral methods, namely the handling of the complex geometries encountered in global stability analysis, has been dealt with successfully in past works by the development of spectral-element methods on unstructured meshes. The present contribution shows that a certain degree of regularity of the geometry may be exploited in order to build a global stability analysis approach based on a regular spectral rectangular grid in curvilinear coordinates and conformal mappings. The derivation of the stability linear operator in curvilinear coordinates is presented along with the discretisation method. Unlike common practice to the solution of the same problem, the matrix discretising the eigenvalue problem is formed and stored. Subspace iteration and massive parallelisation are used in order to recover a wide window of its leading Ritz system. The method is applied to two external flows, both of which are lifting bodies with separation occurring just downstream of the leading edge. Specifically the flow configurations are a NACA 0015 airfoil, and an ellipse of aspect ratio 8 chosen to closely approximate the geometry of the airfoil. Both flow configurations are at an angle of attack of 18, with a Reynolds number based on the chord length of 200. The results of the stability analysis for both geometries are presented and illustrate analogous features