T. Etiene, D. Jonsson, T. Ropinski, C. Scheidegger, J. Comba, L. Gustavo Nonato, R.M. Kirby, A. Ynnerman, C.T. Silva. Verifying Volume Rendering Using Discretization Error Analysis, SCI Technical Report, No. UUSCI-2013-001, SCI Institute, University of Utah, 2013.
Keywords: discretization errors, volume rendering, verifiable visualization
Z. Fu, R.M. Kirby, R.T. Whitaker. A Fast Iterative Method for Solving the Eikonal Equation on Tetrahedral Domains, In SIAM Journal on Scientific Computing, Vol. 35, No. 5, pp. C473--C494. 2013.
L.K. Ha, J. King, Z. Fu, R.M. Kirby. A High-Performance Multi-Element Processing Framework on GPUs, SCI Technical Report, No. UUSCI-2013-005, SCI Institute, University of Utah, 2013.
Many computational engineering problems ranging from finite element methods to image processing involve the batch processing on a large number of data items. While multielement processing has the potential to harness computational power of parallel systems, current techniques often concentrate on maximizing elemental performance. Frameworks that take this greedy optimization approach often fail to extract the maximum processing power of the system for multi-element processing problems. By ultilizing the knowledge that the same operation will be accomplished on a large number of items, we can organize the computation to maximize the computational throughput available in parallel streaming hardware. In this paper, we analyzed weaknesses of existing methods and we proposed efficient parallel programming patterns implemented in a high performance multi-element processing framework to harness the processing power of GPUs. Our approach is capable of levering out the performance curve even on the range of small element size.
M. Hall, R.M. Kirby, F. Li, M.D. Meyer, V. Pascucci, J.M. Phillips, R. Ricci, J. Van der Merwe, S. Venkatasubramanian. Rethinking Abstractions for Big Data: Why, Where, How, and What, In Cornell University Library, 2013.
Big data refers to large and complex data sets that, under existing approaches, exceed the capacity and capability of current compute platforms, systems software, analytical tools and human understanding . Numerous lessons on the scalability of big data can already be found in asymptotic analysis of algorithms and from the high-performance computing (HPC) and applications communities. However, scale is only one aspect of current big data trends; fundamentally, current and emerging problems in big data are a result of unprecedented complexity |in the structure of the data and how to analyze it, in dealing with unreliability and redundancy, in addressing the human factors of comprehending complex data sets, in formulating meaningful analyses, and in managing the dense, power-hungry data centers that house big data.
The computer science solution to complexity is finding the right abstractions, those that hide as much triviality as possible while revealing the essence of the problem that is being addressed. The "big data challenge" has disrupted computer science by stressing to the very limits the familiar abstractions which define the relevant subfields in data analysis, data management and the underlying parallel systems. Efficient processing of big data has shifted systems towards increasingly heterogeneous and specialized units, with resilience and energy becoming important considerations. The design and analysis of algorithms must now incorporate emerging costs in communicating data driven by IO costs, distributed data, and the growing energy cost of these operations. Data analysis representations as structural patterns and visualizations surpass human visual bandwidth, structures studied at small scale are rare at large scale, and large-scale high-dimensional phenomena cannot be reproduced at small scale.
As a result, not enough of these challenges are revealed by isolating abstractions in a traditional soft-ware stack or standard algorithmic and analytical techniques, and attempts to address complexity either oversimplify or require low-level management of details. The authors believe that the abstractions for big data need to be rethought, and this reorganization needs to evolve and be sustained through continued cross-disciplinary collaboration.
In what follows, we first consider the question of why big data and why now. We then describe the where (big data systems), the how (big data algorithms), and the what (big data analytics) challenges that we believe are central and must be addressed as the research community develops these new abstractions. We equate the biggest challenges that span these areas of big data with big mythological creatures, namely cyclops, that should be conquered.
R.M. Kirby, M.D. Meyer. Visualization Collaborations: What Works and Why, In IEEE Computer Graphics and Applications: Visualization Viewpoints, Vol. 33, No. 6, pp. 82--88. 2013.
D. Wang, R.M. Kirby, R.S. MacLeod, C.R. Johnson.
Inverse Electrocardiographic Source Localization of Ischemia: An Optimization Framework and Finite Element Solution, In Journal of Computational Physics, Vol. 250, Academic Press, pp. 403--424. 2013.
Keywords: cvrti, 2P41 GM103545-14
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 2k + 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.
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.
PubMed ID: 23543120
PubMed Central ID: PMC3609671
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.
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
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.
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: 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.
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.
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.