V. Zala, V. Shankar, S.P. Sastry, R.M. Kirby. Curvilinear Mesh Adaptation Using Radial Basis Function Interpolation and Smoothing, In Journal of Scientific Computing, Springer Nature, pp. 1--22. April, 2018.
We present a new iterative technique based on radial basis function (RBF) interpolation and smoothing for the generation and smoothing of curvilinear meshes from straight-sided or other curvilinear meshes. Our technique approximates the coordinate deformation maps in both the interior and boundary of the curvilinear output mesh by using only scattered nodes on the boundary of the input mesh as data sites in an interpolation problem. Our technique produces high-quality meshes in the deformed domain even when the deformation maps are singular due to a new iterative algorithm based on modification of the RBF shape parameter. Due to the use of RBF interpolation, our technique is applicable to both 2D and 3D curvilinear mesh generation without significant modification.
A. Bhaduri, Y. He, M.D. Shields, L. Graham-Brady, R.M. Kirby. Stochastic collocation approach with adaptive mesh refinement for parametric uncertainty analysis, In CoRR, 2017.
Presence of a high-dimensional stochastic parameter space with discontinuities poses major computational challenges in analyzing and quantifying the effects of the uncertainties in a physical system. In this paper, we propose a stochastic collocation method with adaptive mesh refinement (SCAMR) to deal with high dimensional stochastic systems with discontinuities. Specifically, the proposed approach uses generalized polynomial chaos (gPC) expansion with Legendre polynomial basis and solves for the gPC coefficients using the least squares method. It also implements an adaptive mesh (element) refinement strategy which checks for abrupt variations in the output based on the second order gPC approximation error to track discontinuities or non-smoothness. In addition, the proposed method involves a criterion for checking possible dimensionality reduction and consequently, the decomposition of the full-dimensional problem to a number of lower-dimensional subproblems. Specifically, this criterion checks all the existing interactions between input dimensions of a specific problem based on the high-dimensional model representation (HDMR) method, and therefore automatically provides the subproblems which only involve interacting dimensions. The efficiency of the approach is demonstrated using both smooth and non-smooth function examples with input dimensions up to 300, and the approach is compared against other existing algorithms.
J. Docampo-Sánchez, J.K. Ryan, M. Mirzargar, R.M. Kirby. Multi-Dimensional Filtering: Reducing the Dimension Through Rotation Read More: https://epubs.siam.org/doi/abs/10.1137/16M1097845, In SIAM Journal on Scientific Computing, Vol. 39, No. 5, SIAM, pp. A2179--A2200. Jan, 2017.
Over the past few decades there has been a strong effort toward the development of Smoothness-Increasing Accuracy-Conserving (SIAC) filters for discontinuous Galerkin (DG) methods, designed to increase the smoothness and improve the convergence rate of the DG solution through this postprocessor. These advantages can be exploited during flow visualization, for example, by applying the SIAC filter to DG data before streamline computations [M. Steffen, S. Curtis, R. M. Kirby, and J. K. Ryan, IEEE Trans. Vis. Comput. Graphics, 14 (2008), pp. 680--692]. However, introducing these filters in engineering applications can be challenging since a tensor product filter grows in support size as the field dimension increases, becoming computationally expensive. As an alternative, [D. Walfisch, J. K. Ryan, R. M. Kirby, and R. Haimes, J. Sci. Comput., 38 (2009), pp. 164--184] proposed a univariate filter implemented along the streamline curves. Until now, this technique remained a numerical experiment. In this paper we introduce the line SIAC filter and explore how the orientation, structure, and filter size affect the order of accuracy and global errors. We present theoretical error estimates showing how line filtering preserves the properties of traditional tensor product filtering, including smoothness and improvement in the convergence rate. Furthermore, numerical experiments are included, exhibiting how these filters achieve the same accuracy at significantly lower computational costs, becoming an attractive tool for the scientific visualization community.
C. Gritton, J. Guilkey, J. Hooper, D. Bedrov, R. M. Kirby, M. Berzins. Using the material point method to model chemical/mechanical coupling in the deformation of a silicon anode, In Modelling and Simulation in Materials Science and Engineering, Vol. 25, No. 4, pp. 045005. 2017.
The lithiation and delithiation of a silicon battery anode is modeled using the material point method (MPM). The main challenges in modeling this process using the MPM is to simulate stress dependent diffusion coupled with concentration dependent stress within a material that undergoes large deformations. MPM is chosen as the numerical method of choice because of its ability to handle large deformations. A method for modeling diffusion within MPM is described. A stress dependent model for diffusivity and three different constitutive models that fully couple the equations for stress with the equations for diffusion are considered. Verifications tests for the accuracy of the numerical implementations of the models and validation tests with experimental results show the accuracy of the approach. The application of the fully coupled stress diffusion model implemented in MPM is applied to modeling the lithiation and delithiation of silicon nanopillars.
M. Mirzargar, A. Jallepalli, J.K. Ryan, R.M. Kirby. Hexagonal Smoothness-Increasing Accuracy-Conserving Filtering, In Journal of Scientific Computing, Vol. 73, No. 2-3, Springer Nature, pp. 1072--1093. Aug, 2017.
Discontinuous Galerkin (DG) methods are a popular class of numerical techniques to solve partial differential equations due to their higher order of accuracy. However, the inter-element discontinuity of a DG solution hinders its utility in various applications, including visualization and feature extraction. This shortcoming can be alleviated by postprocessing of DG solutions to increase the inter-element smoothness. A class of postprocessing techniques proposed to increase the inter-element smoothness is SIAC filtering. In addition to increasing the inter-element continuity, SIAC filtering also raises the convergence rate from order k+1 to order 2k+1. Since the introduction of SIAC filtering for univariate hyperbolic equations by Cockburn et al. (Math Comput 72(242):577–606, 2003), many generalizations of SIAC filtering have been proposed. Recently, the idea of dimensionality reduction through rotation has been the focus of studies in which a univariate SIAC kernel has been used to postprocess a two-dimensional DG solution (Docampo-Sánchez et al. in Multi-dimensional filtering: reducing the dimension through rotation, 2016. arXiv preprint arXiv:1610.02317). However, the scope of theoretical development of multidimensional SIAC filters has never gone beyond the usage of tensor product multidimensional B-splines or the reduction of the filter dimension. In this paper, we define a new SIAC filter called hexagonal SIAC (HSIAC) that uses a nonseparable class of two-dimensional spline functions called hex splines. In addition to relaxing the separability assumption, the proposed HSIAC filter provides more symmetry to its tensor-product counterpart. We prove that the superconvergence property holds for a specific class of structured triangular meshes using HSIAC filtering and provide numerical results to demonstrate and validate our theoretical results.
M. Mirzargar, R.T. Whitaker, R.M. Kirby. Exploration of Heterogeneous Data Using Robust Similarity, In CoRR, 2017.
Heterogeneous data pose serious challenges to data analysis tasks, including exploration and visualization. Current techniques often utilize dimensionality reductions, aggregation, or conversion to numerical values to analyze heterogeneous data. However, the effectiveness of such techniques to find subtle structures such as the presence of multiple modes or detection of outliers is hindered by the challenge to find the proper subspaces or prior knowledge to reveal the structures. In this paper, we propose a generic similarity-based exploration technique that is applicable to a wide variety of datatypes and their combinations, including heterogeneous ensembles. The proposed concept of similarity has a close connection to statistical analysis and can be deployed for summarization, revealing fine structures such as the presence of multiple modes, and detection of anomalies or outliers. We then propose a visual encoding framework that enables the exploration of a heterogeneous dataset in different levels of detail and provides insightful information about both global and local structures. We demonstrate the utility of the proposed technique using various real datasets, including ensemble data.
T.A.J. Ouermi, A. Knoll, R.M. Kirby, M. Berzins. OpenMP 4 Fortran Modernization of WSM6 for KNL, In Proceedings of the Practice and Experience in Advanced Research Computing 2017 on Sustainability, Success and Impact, PEARC17, No. 12, ACM, pp. 12:1--12:8. 2017.
Parallel code portability in the petascale era requires modifying existing codes to support new architectures with large core counts and SIMD vector units. OpenMP is a well established and increasingly supported vehicle for portable parallelization. As architectures mature and compiler OpenMP implementations evolve, best practices for code modernization change as well. In this paper, we examine the impact of newer OpenMP features (in particular OMP SIMD) on the Intel Xeon Phi Knights Landing (KNL) architecture, applied in optimizing loops in the single moment 6-class microphysics module (WSM6) in the US Navy's NEPTUNE code. We find that with functioning OMP SIMD constructs, low thread invocation overhead on KNL and reduced penalty for unaligned access compared to previous architectures, one can leverage OpenMP 4 to achieve reasonable scalability with relatively minor reorganization of a production physics code.
T.A.J. Ouermi, A. Knoll, R.M. Kirby, M. Berzins. Optimization Strategies for WRF Single-Moment 6-Class Microphysics Scheme (WSM6) on Intel Microarchitectures, In Proceedings of the fifth international symposium on computing and networking (CANDAR 17). Awarded Best Paper , IEEE, 2017.
Optimizations in the petascale era require modifications of existing codes to take advantage of new architectures with large core counts and SIMD vector units. This paper examines high-level and low-level optimization strategies for numerical weather prediction (NWP) codes. These strategies employ thread-local structures of arrays (SOA) and an OpenMP directive such as OMP SIMD. These optimization approaches are applied to the Weather Research Forecasting single-moment 6-class microphysics schemes (WSM6) in the US Navy NEPTUNE system. The results of this study indicate that the high-level approach with SOA and low-level OMP SIMD improves thread and vector parallelism by increasing data and temporal locality. The modified version of WSM6 runs 70x faster than the original serial code. This improvement is about 23.3x faster than the performance achieved by Ouermi et al., and 14.9x faster than the performance achieved by Michalakes et al.
M. Rautenhaus, M. Böttinger, S. Siemen, R. Hoffman, R.M. Kirby, M. Mirzargar, N. Rober, R. Westermann. Visualization in Meteorology---A Survey of Techniques and Tools for Data Analysis Tasks, In IEEE Transactions on Visualization and Computer Graphics, IEEE, pp. 1--1. 2017.
This article surveys the history and current state of the art of visualization in meteorology, focusing on visualization techniques and tools used for meteorological data analysis. We examine characteristics of meteorological data and analysis tasks, describe the development of computer graphics methods for visualization in meteorology from the 1960s to today, and visit the state of the art of visualization techniques and tools in operational weather forecasting and atmospheric research. We approach the topic from both the visualization and the meteorological side, showing visualization techniques commonly used in meteorological practice, and surveying recent studies in visualization research aimed at meteorological applications. Our overview covers visualization techniques from the fields of display design, 3D visualization, flow dynamics, feature-based visualization, comparative visualization and data fusion, uncertainty and ensemble visualization, interactive visual analysis, efficient rendering, and scalability and reproducibility. We discuss demands and challenges for visualization research targeting meteorological data analysis, highlighting aspects in demonstration of benefit, interactive visual analysis, seamless visualization, ensemble visualization, 3D visualization, and technical issues.
T. Etiene, R.M. Kirby, C. Silva. An Introduction to Verification of Visualization Techniques, Morgan & Claypool Publishers, 2015.
C. Gritton, M. Berzins, R. M. Kirby. Improving Accuracy In Particle Methods Using Null Spaces and Filters, In Proceedings of the IV International Conference on Particle-Based Methods - Fundamentals and Applications, Barcelona, Spain, Edited by E. Onate and M. Bischoff and D.R.J. Owen and P. Wriggers and T. Zohdi, CIMNE, pp. 202-213. September, 2015.
While particle-in-cell type methods, such as MPM, have been very successful in providing solutions to many challenging problems there are some important issues that remain to be resolved with regard to their analysis. One such challenge relates to the difference in dimensionality between the particles and the grid points to which they are mapped. There exists a non-trivial null space of the linear operator that maps particles values onto nodal values. In other words, there are non-zero particle values values that when mapped to the nodes are zero there. Given positive mapping weights such null space values are oscillatory in nature. The null space may be viewed as a more general form of the ringing instability identified by Brackbill for PIC methods. It will be shown that it is possible to remove these null-space values from the solution and so to improve the accuracy of PIC methods, using a matrix SVD approach. The expense of doing this is prohibitive for real problems and so a local method is developed for doing this.
R.M. Kirby, M. Berzins, J.S. Hesthaven (Editors). Spectral and High Order Methods for Partial Differential Equations, Subtitled Selected Papers from the ICOSAHOM'14 Conference, June 23-27, 2014, Salt Lake City, UT, USA., In Lecture Notes in Computational Science and Engineering, Springer, 2015.
H. Bhatia, V. Pascucci, R.M. Kirby, P.-T. Bremer. Extracting Features from Time-Dependent Vector Fields Using Internal Reference Frames, In Computer Graphics Forum, Vol. 33, No. 3, pp. 21--30. June, 2014.
T. Etiene, D. Jonsson, T. Ropinski, C. Scheidegger, J.L.D. Comba, L. G. Nonato, R. M. Kirby, A. Ynnerman,, C. T. Silva. Verifying Volume Rendering Using Discretization Error Analysis, In IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, Vol. 20, No. 1, IEEE, pp. 140-154. January, 2014.
We propose an approach for verification of volume rendering correctness based on an analysis of the volume rendering integral, the basis of most DVR algorithms. With respect to the most common discretization of this continuous model (Riemann summation), we make assumptions about the impact of parameter changes on the rendered results and derive convergence curves describing the expected behavior. Specifically, we progressively refine the number of samples along the ray, the grid size, and the pixel size, and evaluate how the errors observed during refinement compare against the expected approximation errors. We derive the theoretical foundations of our verification approach, explain how to realize it in practice, and discuss its limitations. We also report the errors identified by our approach when applied to two publicly available volume rendering packages.
M. Mirzargar, R. Whitaker, R. M. Kirby. Curve Boxplot: Generalization of Boxplot for Ensembles of Curves, In IEEE Transactions on Visualization and Computer Graphics, Vol. 20, No. 12, IEEE, pp. 2654-63. December, 2014.
In simulation science, computational scientists often study the behavior of their simulations by repeated solutions with variations in parameters and/or boundary values or initial conditions. Through such simulation ensembles, one can try to understand or quantify the variability or uncertainty in a solution as a function of the various inputs or model assumptions. In response to a growing interest in simulation ensembles, the visualization community has developed a suite of methods for allowing users to observe and understand the properties of these ensembles in an efficient and effective manner. An important aspect of visualizing simulations is the analysis of derived features, often represented as points, surfaces, or curves. In this paper, we present a novel, nonparametric method for summarizing ensembles of 2D and 3D curves. We propose an extension of a method from descriptive statistics, data depth, to curves. We also demonstrate a set of rendering and visualization strategies for showing rank statistics of an ensemble of curves, which is a generalization of traditional whisker plots or boxplots to multidimensional curves. Results are presented for applications in neuroimaging, hurricane forecasting and fluid dynamics
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.