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Analysis and Visualization of Stochastic Simulation Solutions

Acknowledgement:

This material is based upon collaborative work supported by the National Science Foundation under Grant No.IIS-0914564 (Kirby) and IIS-0914447 (Xiu).

Disclaimer:

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Award title:

AF: Small: Collaborative Research: Analysis and Visualization of Stochastic Simulation Solutions

Duration: September 15, 2009 through August 31, 2012

Research Challenges

In this age of scientific computing, the simulation science pipeline of mathematical modeling, simulation and evaluation is a commonly employed rendition of the scientific method. In addition to the traditional components of the pipeline, there has been a recent surge of interest in uncertainty quantification (UQ). Visualization is the window through which scientists examine their data for deriving new science, and hence visualization methods which depict underlying uncertainty information are crucial. This research addresses the questions of how does one accurately and efficiently post-process stochastic simulation fields and how does one effectively and succinctly convey the results. This is accomplished by developing strategies and techniques for augmenting current visualization techniques used for visualizing spatio-temporal fields with UQ information in a seamless way.

Research Team

Dr. Robert M. Kirby: Dr. Kirby is currently an Associate Professor in the School of Computing and a faculty member of the Scientific Computing and Imaging Institute at University of Utah. His research focus is on large-scale scientific computing and visualization, with an emphasis on the scientific cycle of mathematical modeling, computation, visualization, evaluation, and understanding.

Dr. Dongbin Xiu: Dr. Xiu is currently an Associate Professor in Department of Mathematics at Purdue University. His research focus is on Stochastic computations and uncertainty quantification, Design and optimization under uncertainty, Data assimilation, High-order numerical methods, Modeling and simulation of complex systems.

Chao Yang: Chao Yang is currently an PhD student in School of Computing at University of Utah. His research interest is Uncertainty Visualization and Computer Graphics.

Project Goals

Our current research plan is to develop methods for visualizing the uncertainty in isosurface position when generated/informed by stochastic finite element methods. We are also concurrently examining the visualization of probability density functions (PDF) obtained from stochastic finite element simulations.

Research Challenges

This project has at its core a seemingly simple question to a challenging problem: given the tremendous number of recent simulation results consisting of a vast amount of uncertainty data produced through some collection of stochastic computational techniques, how does one analyze and visualize these results in a way that scientists and engineers can get meaningful answers to queries that they might have about the impact of variability on their results. The goal of this research is to address the issue of how does one accurately and efficiently post-process stochastic simulation fields and how does one effectively and succinctly convey the results.

Current/Final Results

In our most recent work, we present a numerical technique to visualize covariance and cross-covariance fields of a stochastic simulation. The method is local in the sense that it demonstrates the covariance structure of the solution at a point with its neighboring locations. When coupled with an efficient stochastic simulation solver, our framework allows one to effectively concurrently visualize both the mean and (cross-)covariance information for two-dimensional (spatial) simulation results. Most importantly, the visualization provides the scientist a means to identify interesting correlation structure of the solution field. The mathematical setup is discussed, along with several examples to demonstrate the efficacy of this approach.

Publications

P.K. Jimack and R.M. Kirby, "Towards the development of an h-p refinement strategy based upon error estimate sensitivity", Computers and Fluids, In Press, 2011.

C. Yang, D. Xiu and R.M. Kirby, "Visualization of Covariance and Cross-covariance Fields", International Journal for Uncertainty Quantification, Accepted, 2011.

T. Patz, R.M. Kirby and T. Preusser, "Ambrosio-Tortorelli Segmentation of Stochastic Images", International Journal for Computer Vision, Under Review, 2011.

H. Tiesler, R.M. Kirby, D. Xiu and T. Preusser, "Stochastic Collocation for Optimal Control Problems with Stochastic PDE Constraints", SIAM Journal on Optimization, Under Review, 2011.

J. Li and D. Xiu, "Evaluation of Failure Probability via Surrogate Models", Journal of Computational Physics, Vol. 229, 8966-8980, 2010.

Broader Impacts

The broader impacts of this work are that (1) proper techniques for UQ will have large impact on many scientific disciplines from medical/bioengineering to aeronautics, and (2) developed visualization techniques might be put to use when higher dimensional data is available for each point in space. The educational objectives are focused on training a new generation of scientists who are proficient not in both visualization techniques and in UQ. The project will produce a series of methods and algorithms for stochastic visualization. These pioneering results will be disseminated in archival publications as well as via the project website (http://www.cs.utah.edu/~kirby/StochasticVis.html). Workshops on stochastic methods and tutorial sessions in SIAM and IEEE conferences are also planned to raise the visibility and impact of the project.

References

W. Aigner, S. Miksch, B. Thurnher, and S. Biff. PlanningLines: novel glyphs for representing temporal uncertainties and their evaluation. In Proc. of the 9th International Conf. on Information Visualisation 2005, pages 457--463, 2005.

Ralf P. Botchen, Daniel Weiskopf, and Thomas Ertl. Texture-based visualization of uncertainty in flow fields. In Visualization, 2005. VIS 05. IEEE, pages 647--654, 2005.

Alex T. Pang, Craig M. Wittenbrink, and Suresh K. Lodha. Approaches to uncertainty visualization. The Visual Computer, 13(8):370--390, 1997.
ISSN 0178-2789.

D. Xiu, Fast stochastic algorithms for robust optimization and parameter estimation, Air Force Office of Scientific Research Computational Mathematics Program Review, Arlington, Virginia, August 13-15, 2008.