Uncertain Flow Visualization using LIC

In this paper we look at the Line Integral Convolution method for flow visualization and ways in which this can be applied to the visualization of two dimensional, steady flow fields in the presence of uncertainty. To achieve this, we start by studying the method and reviewing the history of modifications other authors have made to it in order to improve its efficiency or capabilities, and using these as a base for the visualization of uncertain flow fields. Finally, we apply our methodology to a case study from the field of oceanography

Gaining understanding of multivariate and multidimensional data through visualization

High dimensionality is a major challenge for data visualization. Parameter optimization problems require an understanding of the behaviour of the objective function in the n-dimensional space around the optimum—this is multidimensional visualization and is the traditional domain of scientific visualization. Large data tables require us to understand the relationship between attributes in the table—this is multivariate visualization and is an important aspect of information visualization. Common to both types of ‘high-dimensional’ visualization is a need to reduce the dimensionality for display. In this paper we present a uniform approach to the filtering of both multidimensional and multivariate data, to allow extraction of data subject to constraints on their position or value within an n-dimensional window, and on choice of dimensions for display. A simple example of understanding the trajectory of solutions from an optimization algorithm is given—this involves a combination of multidimensional and multivariate data

Corrections to matrices expressed in non-symmetric product form

It is shown that all non-symmetric rank-one and rank-two corrections to non-singular matrices can be expressed in a non-symmetric product form. This extends the earlier work of Brodlie, Gourlay & Greenstadt (1973) on symmetric correction formulae

An assessment of two approaches to variable metric methods

Two recent suggestions in the field of variable metric methods for function minimization are reviewed: the self-scaling method, first introduced by Oren and Luenberger, and the method of Biggs. The two proposals are considered both from a theoretical and computational aspect. They are compared with methods which use correction formulae from the Broyden one-parameter family, in particular the BFGS formula and the Fletcher switching strategy

A new direction set method for unconstrained minimization without evaluating derivatives

A new direction set method for unconstrained minimization without evaluating derivatives is presented. The algorithm can be regarded as an application to function minimization of Jacobi's method for determining the eigenvalues and eigenvectors of a real symmetric matrix. Numerical results are presented, illustrating the performance of the new algorithm on well-known test problems; a comparison with other methods is also given

Computational steering in visualization dataflow environments

This paper traces the evolution of computational steering within visualization dataflow environments. In computational steering we integrate simulation and visualization into a single environment, in which the scientist can control the simulation on the basis of the visualization of the current results. This paper is essentially a review paper, in which we describe a number of projects with which we have been involved over the last twenty years. We begin with a historical look-back to the early development of dataflow visualization systems. These are also known as modular visualization environments. They typically consist of a library of elementary visualization components that can be wired together in a visual editor in order to compose a pipeline forming the visualization application. From an early stage these have been used for computational steering, because it is possible to include user code, for example a simulation, as a component in the pipeline. Moreover modules are able to run on a distributed set of resources, allowing simulation code to run on a remote resource. Indeed, if the simulation is producing large volumes of data, then the visualization modules can be co-located with the simulation thus reducing the amount of data returned to the desktop for visualization. This very useful facility has been surprisingly little used. Our own involvement in computational steering began in 1993 with the GRASPARC project, and interest was re-kindled in 2000 with the emergence of Grid computing and the UK e-science programme. Our first task was to re-work the distributed computing model of IRIS Explorer (the dataflow system we use at Leeds) so as to provide the security that is expected in modern computing environments. This was followed by a re-working of the architecture for computational steering, in the gViz project, where we separated the simulation code from the visualization dataflow, and allowed the simulation to run autonomously. This gives the advantage of disconnecting simulation lifetime from visualization system execution time. In doing this it allows simulations to run over greater time scales than the time period a user may wish to actively interact with them in any one session. Previously, shutting down the visualization would have shut down the simulation component. This was achieved using the gViz computational steering library. Our more recent work in the e-viz project has extended the architecture further, basing the work around an abstract description of the visualization pipeline from which a user interface for steering can be automatically generated. Likewise the pipeline description can be interpreted in terms of different visualization systems, providing an extra level of abstraction. As interest in service-oriented architectures develops, we are seeing a reworking of visualization systems as a pipeline of services, rather than modules. We can expect this trend to carry over to computational steering in future research

On the Convergence of Cyclic Jacobi Methods

In a cyclic Jacobi method for calculating the eigenvalues and eigenvectors of a symmetric matrix, the pivots are chosen in any fixed cyclic order. It is not known in theory whether convergence to the solution is always obtained, although convergence has been proved subject to a restriction on the angle of rotation about each pivot (Henrici, 1958). Now we report an actual computer calculation where a cyclic Jacobi method failed, due to computer rounding errors, so in practice the angle restriction may be needed. A new bound for the angle restriction is given that is less severe than the one proposed originally

Visualization notations, models and taxonomies

Visualization taxonomies are an important means of imposing some structure on a rather diverse field. We review some earlier work in this area, particularly work based on the use of a notation to label classes of visualization techniques that are appropriate to particular entities. We propose a new notation introducing it in the context of a new visualization reference model, one we hope will lead eventually to a means of describing visualizations in a clear and unambiguous way