2013

R.T. Whitaker, M. Mirzargar, R.M. Kirby.
**“Contour Boxplots: A Method for Characterizing Uncertainty in Feature Sets from Simulation Ensembles,”** In *IEEE Transactions on Visualization and Computer Graphics*, Vol. 19, No. 12, pp. 2713--2722. December, 2013.

DOI: 10.1109/TVCG.2013.143

PubMed ID: 24051838

Ensembles of numerical simulations are used in a variety of applications, such as meteorology or computational solid mechanics, in order to quantify the uncertainty or possible error in a model or simulation. Deriving robust statistics and visualizing the variability of an ensemble is a challenging task and is usually accomplished through direct visualization of ensemble members or by providing aggregate representations such as an average or pointwise probabilities. In many cases, the interesting quantities in a simulation are not dense fields, but are sets of features that are often represented as thresholds on physical or derived quantities. In this paper, we introduce a generalization of boxplots, called contour boxplots, for visualization and exploration of ensembles of contours or level sets of functions. Conventional boxplots have been widely used as an exploratory or communicative tool for data analysis, and they typically show the median, mean, confidence intervals, and outliers of a population. The proposed contour boxplots are a generalization of functional boxplots, which build on the notion of data depth. Data depth approximates the extent to which a particular sample is centrally located within its density function. This produces a center-outward ordering that gives rise to the statistical quantities that are essential to boxplots. Here we present a generalization of functional data depth to contours and demonstrate methods for displaying the resulting boxplots for two-dimensional simulation data in weather forecasting and computational fluid dynamics.

2012

J. King, H. Mirzaee, J.K. Ryan, R.M. Kirby.
**“Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering for discontinuous Galerkin Solutions: Improved Errors Versus Higher-Order Accuracy,”** In *Journal of Scientific Computing*, Vol. 53, pp. 129--149. 2012.

DOI: 10.1007/s10915-012-9593-8

Smoothness-increasing accuracy-conserving (SIAC) filtering has demonstrated its effectiveness in raising the convergence rate of discontinuous Galerkin solutions from order *k* + 1/2 to order 2*k* + 1 for specific types of translation invariant meshes (Cockburn et al. in Math. Comput. 72:577–606, 2003; Curtis et al. in SIAM J. Sci. Comput. 30(1):272– 289, 2007; Mirzaee et al. in SIAM J. Numer. Anal. 49:1899–1920, 2011). Additionally, it improves the weak continuity in the discontinuous Galerkin method to *k* - 1 continuity. Typically this improvement has a positive impact on the error quantity in the sense that it also reduces the absolute errors. However, not enough emphasis has been placed on the difference between superconvergent accuracy and improved errors. This distinction is particularly important when it comes to understanding the interplay introduced through meshing, between geometry and filtering. The underlying mesh over which the DG solution is built is important because the tool used in SIAC filtering—convolution—is scaled by the geometric mesh size. This heavily contributes to the effectiveness of the post-processor. In this paper, we present a study of this mesh scaling and how it factors into the theoretical errors. To accomplish the large volume of post-processing necessary for this study, commodity streaming multiprocessors were used; we demonstrate for structured meshes up to a 50× speed up in the computational time over traditional CPU implementations of the SIAC filter.

B. Nelson, E. Liu, R.M. Kirby, R. Haimes.
**“ElVis: A System for the Accurate and Interactive Visualization of High-Order Finite Element Solutions,”** In *IEEE Transactions on Visualization and Computer Graphics (TVCG)*, Vol. 18, No. 12, pp. 2325--2334. Dec, 2012.

DOI: 10.1109/TVCG.2012.218

This paper presents the Element Visualizer (ElVis), a new, open-source scientific visualization system for use with high-order finite element solutions to PDEs in three dimensions. This system is designed to minimize visualization errors of these types of fields by querying the underlying finite element basis functions (e.g., high-order polynomials) directly, leading to pixel-exact representations of solutions and geometry. The system interacts with simulation data through runtime plugins, which only require users to implement a handful of operations fundamental to finite element solvers. The data in turn can be visualized through the use of cut surfaces, contours, isosurfaces, and volume rendering. These visualization algorithms are implemented using NVIDIA's OptiX GPU-based ray-tracing engine, which provides accelerated ray traversal of the high-order geometry, and CUDA, which allows for effective parallel evaluation of the visualization algorithms. The direct interface between ElVis and the underlying data differentiates it from existing visualization tools. Current tools assume the underlying data is composed of linear primitives; high-order data must be interpolated with linear functions as a result. In this work, examples drawn from aerodynamic simulations-high-order discontinuous Galerkin finite element solutions of aerodynamic flows in particular-will demonstrate the superiority of ElVis' pixel-exact approach when compared with traditional linear-interpolation methods. Such methods can introduce a number of inaccuracies in the resulting visualization, making it unclear if visual artifacts are genuine to the solution data or if these artifacts are the result of interpolation errors. Linear methods additionally cannot properly visualize curved geometries (elements or boundaries) which can greatly inhibit developers' debugging efforts. As we will show, pixel-exact visualization exhibits none of these issues, removing the visualization scheme as a source of - ncertainty for engineers using ElVis.

K. Potter, R.M. Kirby, D. Xiu, C.R. Johnson.
**“Interactive visualization of probability and cumulative density functions,”** In *International Journal of Uncertainty Quantification*, Vol. 2, No. 4, pp. 397--412. 2012.

DOI: 10.1615/Int.J.UncertaintyQuantification.2012004074

PubMed ID: 23543120

PubMed Central ID: PMC3609671

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 paper, we present a visualization system that allows the user to examine two-dimensional data sets in which PDF (or CDF) information is available at any position within the domain. The 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.

**Keywords:** visualization, probability density function, cumulative density function, generalized polynomial chaos, stochastic Galerkin methods, stochastic collocation methods

H. Tiesler, R.M. Kirby, D. Xiu, T. Preusser.
**“Stochastic Collocation for Optimal Control Problems with Stochastic PDE Constraints,”** In *SIAM Journal on Control and Optimization*, Vol. 50, No. 5, pp. 2659--2682. 2012.

DOI: 10.1137/110835438

We discuss the use of stochastic collocation for the solution of optimal control problems which are constrained by stochastic partial differential equations (SPDE). Thereby the constraining, SPDE depends on data which is not deterministic but random. Assuming a deterministic control, randomness within the states of the input data will propagate to the states of the system. For the solution of SPDEs there has recently been an increasing effort in the development of efficient numerical schemes based upon the mathematical concept of generalized polynomial chaos. Modal-based stochastic Galerkin and nodal-based stochastic collocation versions of this methodology exist, both of which rely on a certain level of smoothness of the solution in the random space to yield accelerated convergence rates. In this paper we apply the stochastic collocation method to develop a gradient descent as well as a sequential quadratic program (SQP) for the minimization of objective functions constrained by an SPDE. The stochastic function involves several higher-order moments of the random states of the system as well as classical regularization of the control. In particular we discuss several objective functions of tracking type. Numerical examples are presented to demonstrate the performance of our new stochastic collocation minimization approach.

**Keywords:** stochastic collocation, optimal control, stochastic partial differential equations

2011

I. Altrogge, T. Preusser, T. Kroeger, S. Haase, T. Paetz, R.M. Kirby.
**“Sensitivity Analysis for the Optimization of Radiofrequency Ablation in the Presence of Material Parameter Uncertainty,”** In *International Journal for Uncertainty Quantification*, 2011.

We present a sensitivity analysis of the optimization of the probe placement in radiofrequency (RF) ablation which takes the uncertainty associated with bio-physical tissue properties (electrical and thermal conductivity) into account. Our forward simulation of RF ablation is based upon a system of partial differential equations (PDEs) that describe the electric potential of the probe and the steady state of the induced heat. The probe placement is optimized by minimizing a temperature-based objective function such that the volume of destroyed tumor tissue is maximized. The resulting optimality system is solved with a multi-level gradient descent approach. By evaluating the corresponding optimality system for certain realizations of tissue parameters (i.e. at certain, well-chosen points in the stochastic space) the sensitivity of the system can be analyzed with respect to variations in the tissue parameters. For the interpolation in the stochastic space we use a stochastic finite element approach with piecewise multilinear ansatz functions on adaptively refined, hierarchical grids. We underscore the significance of the approach by applying the optimization to CT data obtained from a real RF ablation case.

**Keywords:** netl, stochastic sensitivity analysis, stochastic partial dierential equations, stochastic nite element method, adaptive sparse grid, heat transfer, multiscale modeling, representation of uncertainty

C.D. Cantwell, S.J. Sherwin, R.M. Kirby, P.H.J. Kelly.
**“From h to p Efficiently: Strategy Selection for Operator Evaluation on Hexahedral and Tetrahedral Elements,”** In *Computers and Fluids*, Vol. 43, No. 1, pp. 23--28. 2011.

DOI: 10.1016/j.compfluid.2010.08.012

C.D. Cantwell, S.J. Sherwin, R.M. Kirby, P.H.J. Kelly.
**“From h to p Efficiently: Selecting the Optimal Spectral/hp Discretisation in Three Dimensions,”** In *Mathematical Modelling of Natural Phenomena*, Vol. 6, No. 3, pp. 84--96. 2011.

T. Etiene, L.G. Nonato, C. Scheidegger, J. Tierny, T.J. Peters, V. Pascucci, R.M. Kirby, C.T. Silva.
**“Topology Verfication for Isosurface Extraction,”** In *IEEE Transactions on Visualization and Computer Graphics*, pp. (accepted). 2011.

The broad goals of verifiable visualization rely on correct algorithmic implementations. We extend a framework for verification of isosurfacing implementations to check topological properties. Specifically, we use stratified Morse theory and digital topology to design algorithms which verify topological invariants. Our extended framework reveals unexpected behavior and coding mistakes in popular publicly-available isosurface codes.

Z. Fu, W.-K. Jeong, Y. Pan, R.M. Kirby, R.T. Whitaker.
**“A fast iterative method for solving the Eikonal equation on triangulated surfaces,”** In *SIAM Journal of Scientific Computing*, Vol. 33, No. 5, pp. 2468--2488. 2011.

DOI: 10.1137/100788951

PubMed Central ID: PMC3360588

This paper presents an efficient, fine-grained parallel algorithm for solving the Eikonal equation on triangular meshes. The Eikonal equation, and the broader class of Hamilton–Jacobi equations to which it belongs, have a wide range of applications from geometric optics and seismology to biological modeling and analysis of geometry and images. The ability to solve such equations accurately and efficiently provides new capabilities for exploring and visualizing parameter spaces and for solving inverse problems that rely on such equations in the forward model. Efficient solvers on state-of-the-art, parallel architectures require new algorithms that are not, in many cases, optimal, but are better suited to synchronous updates of the solution. In previous work [W. K. Jeong and R. T. Whitaker, SIAM J. Sci. Comput., 30 (2008), pp. 2512–2534], the authors proposed the fast iterative method (FIM) to efficiently solve the Eikonal equation on regular grids. In this paper we extend the fast iterative method to solve Eikonal equations efficiently on triangulated domains on the CPU and on parallel architectures, including graphics processors. We propose a new local update scheme that provides solutions of first-order accuracy for both architectures. We also propose a novel triangle-based update scheme and its corresponding data structure for efficient irregular data mapping to parallel single-instruction multiple-data (SIMD) processors. We provide detailed descriptions of the implementations on a single CPU, a multicore CPU with shared memory, and SIMD architectures with comparative results against state-of-the-art Eikonal solvers.

S.E. Geneser, J.D. Hinkle, R.M. Kirby, Bo Wang, B. Salter, S. Joshi.
**“Quantifying variability in radiation dose due to respiratory-induced tumor motion,”** In *Medical Image Analysis*, Vol. 15, No. 4, pp. 640--649. 2011.

DOI: 10.1016/j.media.2010.07.003

G. Gopalakrishnan, R.M. Kirby, S. Siegel, R. Thakur, W. Gropp, E. Lusk, B.R. de Supinski, M. Schultz, G. Bronevetsky.
**“Formal Analysis of MPI-Based Parallel Programs: Present and Future,”** In *Communications of the ACM*, pp. (accepted). 2011.

S.A. Isaacson, R.M. Kirby.
**“Numerical Solution of Linear Volterra Integral Equations of the Second Kind with Sharp Gradients,”** In *Journal of Computational and Applied Mathematics*, Vol. 235, No. 14, pp. 4283--4301. 2011.

Collocation methods are a well-developed approach for the numerical solution of smooth and weakly singular Volterra integral equations. In this paper, we extend these methods through the use of partitioned quadrature based on the qualocation framework, to allow the efficient numerical solution of linear, scalar Volterra integral equations of the second kind with smooth kernels containing sharp gradients. In this case, the standard collocation methods may lose computational efficiency despite the smoothness of the kernel. We illustrate how the qualocation framework can allow one to focus computational effort where necessary through improved quadrature approximations, while keeping the solution approximation fixed. The computational performance improvement introduced by our new method is examined through several test examples. The final example we consider is the original problem that motivated this work: the problem of calculating the probability density associated with a continuous-time random walk in three dimensions that may be killed at a fixed lattice site. To demonstrate how separating the solution approximation from quadrature approximation may improve computational performance, we also compare our new method to several existing Gregory, Sinc, and global spectral methods, where quadrature approximation and solution approximation are coupled.

P.K. Jimack, R.M. Kirby.
**“Towards the Development on an h-p-Refinement Strategy Based Upon Error Estimate Sensitivity,”** In *Computers and Fluids*, Vol. 46, No. 1, pp. 277--281. 2011.

The use of (*a posteriori*) error estimates is a fundamental tool in the application of adaptive numerical methods across a range of fluid flow problems. Such estimates are incomplete however, in that they do not necessarily indicate where to refine in order to achieve the most impact on the error, nor what type of refinement (for example h-refinement or p-refinement) will be best. This paper extends preliminary work of the authors (Comm Comp Phys, 2010;7:631–8), which uses adjoint-based sensitivity estimates in order to address these questions, to include application with p-refinement to arbitrary order and the use of practical a posteriori estimates. Results are presented which demonstrate that the proposed approach can guide both the h-refinement and the p-refinement processes, to yield improvements in the adaptive strategy compared to the use of more orthodox criteria.

R.M. Kirby, B. Cockburn, S.J. Sherwin.
**“To CG or to HDG: A Comparative Study,”** In *Journal of Scientific Computing*, Note: *published online*, 2011.

DOI: 10.1007/s10915-011-9501-7

Hybridization through the border of the elements (hybrid unknowns) combined with a Schur complement procedure (often called *static condensation* in the context of continuous Galerkin linear elasticity computations) has in various forms been advocated in the mathematical and engineering literature as a means of accomplishing domain decomposition, of obtaining increased accuracy and convergence results, and of algorithm optimization. Recent work on the hybridization of mixed methods, and in particular of the discontinuous Galerkin (DG) method, holds the promise of capitalizing on the three aforementioned properties; in particular, of generating a numerical scheme that is discontinuous in both the primary and flux variables, is locally conservative, and is computationally competitive with traditional continuous Galerkin (CG) approaches. In this paper we present both implementation and optimization strategies for the Hybridizable Discontinuous Galerkin (HDG) method applied to two dimensional elliptic operators. We implement our HDG approach within a spectral*/hp* element framework so that comparisons can be done between HDG and the traditional CG approach.

We demonstrate that the HDG approach generates a global trace space system for the unknown that although larger in rank than the traditional static condensation system in CG, has significantly smaller bandwidth at moderate polynomial orders. We show that if one ignores set-up costs, above approximately fourth-degree polynomial expansions on triangles and quadrilaterals the HDG method can be made to be as efficient as the CG approach, making it competitive for time-dependent problems even before taking into consideration other properties of DG schemes such as their superconvergence properties and their ability to handle *hp*-adaptivity.

G. Li, R. Palmer, M. DeLisi, G. Gopalakrishnan, R.M. Kirby.
**“Formal Specification of MPI 2.0: Case Study in Specifying a Practical Concurrent Programming API,”** In *Science of Computer Programming*, Vol. 76, pp. 65--81. 2011.

DOI: 10.1016/j.scico.2010.03.007

We describe the first formal specification of a non-trivial subset of MPI, the dominant communication API in high performance computing. Engineering a formal specification for a non-trivial concurrency API requires the right combination of rigor, executability, and traceability, while also serving as a smooth elaboration of a pre-existing informal specification. It also requires the modularization of reusable specification components to keep the length of the specification in check. Long-lived APIs such as MPI are not usually 'textbook minimalistic' because they support a diverse array of applications, a diverse community of users, and have efficient implementations over decades of computing hardware. We choose the TLA+ notation to write our specifications, and describe how we organized the specification of around 200 of the 300 MPI 2.0 functions. We detail a handful of these functions in this paper, and assess our specification with respect to the aforementioned requirements. We close with a description of possible approaches that may help render the act of writing, understanding, and validating the specifications of concurrency APIs much more productive.

T. Martin, E. Cohen, R.M. Kirby.
**“Direct Isosurface Visualization of Hex-Based High-Order Geometry and Attribute Representations,”** In *IEEE Transactions on Visualization and Computer Graphics (TVCG)*, Vol. PP, No. 99, pp. 1--14. 2011.

ISSN: 1077-2626

DOI: 10.1109/TVCG.2011.103

In this paper, we present a novel isosurface visualization technique that guarantees the accuarate visualization of isosurfaces with complex attribute data defined on (un-)structured (curvi-)linear hexahedral grids. Isosurfaces of high-order hexahedralbased finite element solutions on both uniform grids (including MRI and CT scans) and more complex geometry represent a domain of interest that can be rendered using our algorithm. Additionally, our technique can be used to directly visualize solutions and attributes in isogeometric analysis, an area based on trivariate high-order NURBS (Non-Uniform Rational B-splines) geometry and attribute representations for the analysis. Furthermore, our technique can be used to visualize isosurfaces of algebraic functions. Our approach combines subdivision and numerical root-finding to form a robust and efficient isosurface visualization algorithm that does not miss surface features, while finding all intersections between a view frustum and desired isosurfaces. This allows the use of view-independent transparency in the rendering process. We demonstrate our technique through a straightforward CPU implementation on both complexstructured and complex-unstructured geometry with high-order simulation solutions, isosurfaces of medical data sets, and isosurfaces of algebraic functions.

H. Mirzaee, Liangyue, J.K. Ryan, R.M. Kirby.
**“Smoothness-Increasing Accuracy-Conserving (SIAC) Postprocessing for Discontinuous Galerkin Solutions Over Structured Triangular Meshes,”** In *SIAM Journal of Numerical Analysis*, Vol. 49, No. 5, pp. 1899--1920. 2011.

Theoretically and computationally, it is possible to demonstrate that the order of accuracy of a discontinuous Galerkin (DG) solution for linear hyperbolic equations can be improved from order

H. Mirzaee, J.K. Ryan, R.M. Kirby.
**“Efficient Implementation of Smoothness-Increasing Accuracy-Conserving (SIAC) Filters for Discontinuous Galerkin Solutions,”** In *Journal of Scientific Computing*, pp. (in press). 2011.

DOI: 10.1007/s10915-011-9535-x

The discontinuous Galerkin (DG) methods provide a high-order extension of the finite volume method in much the same way as high-order or spectral*/hp* elements extend standard finite elements. However, lack of inter-element continuity is often contrary to the smoothness assumptions upon which many post-processing algorithms such as those used in visualization are based. Smoothness-increasing accuracy-conserving (SIAC) filters were proposed as a means of ameliorating the challenges introduced by the lack of regularity at element interfaces by eliminating the discontinuity between elements in a way that is consistent with the DG methodology; in particular, high-order accuracy is preserved and in many cases increased. The goal of this paper is to explicitly define the steps to efficient computation of this filtering technique as applied to both structured triangular and quadrilateral meshes. Furthermore, as the SIAC filter is a good candidate for parallelization, we provide, for the first time, results that confirm anticipated performance scaling when parallelized on a shared-memory multi-processor machine.

B. Nelson, R. Haimes, R.M. Kirby.
**“GPU-Based Interactive Cut-Surface Extraction From High-0rder Finite Element Fields,”** In *IEEE Transactions on Visualization and Computer Graphics (IEEE Visualization Issue)*, Vol. 17, No. 12, pp. 1803--1811. 2011.

We present a GPU-based ray-tracing system for the accurate and interactive visualization of cut-surfaces through 3D simulations of physical processes created from spectral/hp high-order finite element methods. When used by the numerical analyst to debug the solver, the ability for the imagery to precisely reflect the data is critical. In practice, the investigator interactively selects from a palette of visualization tools to construct a scene that can answer a query of the data. This is effective as long as the implicit contract of image quality between the individual and the visualization system is upheld. OpenGL rendering of scientific visualizations has worked remarkably well for exploratory visualization for most solver results. This is due to the consistency between the use of first-order representations in the simulation and the linear assumptions inherent in OpenGL (planar fragments and color-space interpolation). Unfortunately, the contract is broken when the solver discretization is of higher-order. There have been attempts to mitigate this through the use of spatial adaptation and/or texture mapping. These methods do a better job of approximating what the imagery should be but are not exact and tend to be view-dependent. This paper introduces new rendering mechanisms that specifically deal with the kinds of native data generated by high-order finite element solvers. The exploratory visualization tools are reassessed and cast in this system with the focus on image accuracy. This is accomplished in a GPU setting to ensure interactivity.