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Analysis and Visualization of Stochastic Simulation Solutions


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


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, 2013


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

mike kirby dongbin xiu chao yang nazmus saquib

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 a Professor in the Department of Mathematics at the University of Utah, and is a member of the Scientific Computing and Imaging Institute. 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 a PhD student in School of Computing at University of Utah. His research interest is Uncertainty Visualization and Computer Graphics. Chao was an active participant during the first two years of the project.

Nazmus Saquib

Nazmus Saquib is currently an MS student in the Computational Engineering and Science (CES) Program at the University of Utah. His research focuses on problems in uncertainty quantification, visualization, and generalized polynomial chaos (gPC) based methods for stochastic PDE simulations in different application areas, such as biophysics and fluid dynamics. Saquib has been an active participant for the second two years of the project.


Project Goals

As a collaborative project, we have attempted to maintain three different goals throughout the life of the project:

  • Investigation of mathematical and computational techniques that would help bridge the simulation-to-visualization gap when one is dealing with stochastic simulation data – in particular data generated through the generalized Polynomial Chaos method;
  • Development of algorithms and implementation strategies for visualizing simulation data produced as part of the uncertainty quantification (UQ), and
  • Participation in discipline-specific scientific case studies that help elucidate the issues that arise when attempting to visualize data produced through the UQ process.

For the Utah Group, our current research focus has been on the development of a mathematical and computational framework for visualizing "intrinsic" changes (such as local isosurface shape) in an isosurface due to parameter perturbations in the input of the simulation responsible for data generation.

For the Purdue Group, our current research focus has been on development of mathematical and computational models that provide surrogates that can be efficiently acted upon when bridging the gap between final UQ simulation output (which might be quite voluminous) and what is needed for doing accurate and interactive visualization.


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 initial work, we presented 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.

In the next phase of the grant, we examine the visualization of the probability density function resulting from stochastic simulations. The probability density function (PDF), and its corresponding cumulative density function (CDF), provide direct statistical insight into the characterization of a random process or field. Typically displayed as a histogram, one can infer probabilities of the occurrence of particular events. When examining a field over some two-dimensional domain in which at each point a PDF of the function values is available, it is challenging to assess the global (stochastic) features present within the field.

In this work, we present a visualization system that allows the user to examine two-dimensional datasets in which PDF (or CDF) information is available at any position within the domain. Our tool provides a contour display showing the normed difference between the PDFs and an ansatz PDF selected by the user, and furthermore allows the user to interactively examine the PDF at any particular position. Canonical examples of the tool are provided to help guide the reader into the mapping of stochastic information to visual cues along with a description of the use of the tool for examining data generated from a uncertainty quantification exercise accomplished within the field of electrophysiology.

In our most recent work, we have turned our focus to understanding uncertainty in isosurfaces. It is common to extract isosurfaces from simulation field data to visualize and gain understanding of the underlying physical phenomenon being simulated. As the input parameters of the simulation change, the resulting isosurface varies, and there has been increased interest in quantifying and visualization these variations as part of the larger interest in uncertainty quantification. In our most recent work, we propose an analysis and visualization pipeline for examining the intrinsic variation in isosurfaces caused by simulation parameter perturbation. Drawing inspiration from the shape modeling community, we incorporate the use of heat-kernel signatures (HKS) with a simple finite-different approach for quantifying the degree to which a region (or even a point) on an isosurface has undergone intrinsic change. Coupled with a clustering technique and the use of color maps, our pipeline allows the user to select the level of fidelity with which they wish to evaluate and visualize the amount of intrinsic change.




T. Patz, R.M. Kirby and T. Preusser, "Ambrosio-Tortorelli Segmentation of Stochastic Images", International Journal for Computer Vision, In Press, 2013.

Ross Whitaker, Mahsa Mirzargar and Robert M. Kirby, "Contour Boxplots: A Method for Characterizing Uncertainty in Feature Sets from Simulation Ensembles", IEEE Transactions on Visualization and Computer Graphics (IEEE Visualization Issue), Accepted for Publication, 2013.

C. Yang, D. Xiu and R.M. Kirby, "Visualization of Covariance and Cross-covariance Fields", International Journal for Uncertainty Quantification, Vol. 3, No. 1, pages 25-38, 2013.


Inga Altrogge, Tobias Preusser, Tim Kroger, Sabrina Haase, Torben Patz and Robert M. Kirby, "Sensitivity Analysis for the Optimization of Radiofrequency Ablation in the Presence of Material Parameter Uncertainty", International Journal for Uncertainty Quantification, Vol. 2, Issue 3, 295-321, 2012.

Kristin Potter, Robert M. Kirby, Dongbin Xiu and Chris R. Johnson, "Interactive Visualization of Probability And Cumulative Density Functions", International Journal for Uncertainty Quantification, Vol. 2, Issue 4, 397-412, 2012.

H. Tiesler, R.M. Kirby, D. Xiu and T. Preusser, "Stochastic Collocation for Optimal Control Problems with Stochastic PDE Constraints", SIAM Journal on Optimization, Vol. 50, Issue 5, pages 2659-2682, 2012.


P.K. Jimack and R.M. Kirby, “Towards the development of an h-p refinement strategy based upon error estimate sensitivity”, Computers and Fluids, Vol. 46, Issue 1, 277-281, 2011.

J. Li, J. Li and D. Xiu, "An Efficient Surrogate-based Method for Computing Rare Failure Probability," Journal of Computational Physics, Vol. 230, 8683-8697, 2011.

A. Narayan and D. Xiu, "Distributional Sensitivity for Uncertainty Quantification," Communications in Computational Physics, 10(1), 140-160, 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 ( Workshops on stochastic methods and tutorial sessions in SIAM and IEEE conferences are also planned to raise the visibility and impact of the project.



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.
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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.