Chaire d'Analyse Appliquée

Overview of Parameter Continuation Research

General Research Description

The group's work in this area is focussed on the development and use of visualization techniques to interactively steer multi-parameter continuation computations on various parallel architectures (both work-station clusters and remote massively parallel machines). There is an existing one-parameter continuation code called VBM (which was the subject of a talk at the 1998 International Congress of Mathematicians) which is written in C++/Python/TK/OpenGL to run on a variety of UNIX operating systems using sockets, PVM or MPI for communications, and driving a variety of computation engines (the Fortan code AUTO being one of the most heavily used in the lab). Current research involves extending this code to multi-parameter continuation, with the primary questions of interest being

how to visualize higher dimension continuation computations,

how to make efficient use of many processors for a multi-level parallelism involving a coarse grain parallelization of computations at different parameter values, and a fine grain parallelization of the computation at a given set of parameters, and

various data reduction strategies.

Some more details regarding these points are described in a conference presentation 3rd SOS Workshop

The main target application is currently models of equilibrium configurations of DNA (which involves the solution of elliptic boundary value problems for differential equations and integro-differential equations). Our models of self-contact are computationally quite intensive and require parallel computation.

The group's work in the development of analytical, numerical and visualization techniques for steered parameter continuation computations is organized around the development of several software packages.

DataViewer is a general OpenGL based 3D visualization software package (analogous to the GeomView program, various OpenInventor tools, or the GeometryViewer module inside AVS) but tailored specifically to our needs of full source code, the ability to build flexible, problem dependent, interfaces, and allowing large data throughput. All of our current software packages exploit DataViewer for their 3D graphics capabilities.
VBM (or Visualization of Bifurcation Manifolds) is the main software that provides interactive steering and post-processing capabilities in parameter continuation computations.
MC³ (or Multi-parameter Computation and Continuation Code) is an API between VBM and various continuation algorithms.
MC² (for Multiplier and Constraint Continuation) is an older code implemented within AVS that was our first step toward visualization and steered parameter continuation computations.

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