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
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.
PubMed ID: 24051838
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.
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.
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.
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: Strategy Selection for Operator Evaluation on Hexahedral and Tetrahedral Elements, In Computers and Fluids, Vol. 43, No. 1, pp. 23--28. 2011.
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.
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.
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.
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.
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.
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.
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.
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.
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.