Designed especially for neurobiologists, FluoRender is an interactive tool for multi-channel fluorescence microscopy data visualization and analysis.
Large scale visualization on the Powerwall.
BrainStimulator is a set of networks that are used in SCIRun to perform simulations of brain stimulation such as transcranial direct current stimulation (tDCS) and magnetic transcranial stimulation (TMS).
Developing software tools for science has always been a central vision of the SCI Institute.

Scientific Computing

Numerical simulation of real-world phenomena provides fertile ground for building interdisciplinary relationships. The SCI Institute has a long tradition of building these relationships in a win-win fashion – a win for the theoretical and algorithmic development of numerical modeling and simulation techniques and a win for the discipline-specific science of interest. High-order and adaptive methods, uncertainty quantification, complexity analysis, and parallelization are just some of the topics being investigated by SCI faculty. These areas of computing are being applied to a wide variety of engineering applications ranging from fluid mechanics and solid mechanics to bioelectricity.


martin

Martin Berzins

Parallel Computing
GPUs
mike

Mike Kirby

Finite Element Methods
Uncertainty Quantification
GPUs
pascucci

Valerio Pascucci

Scientific Data Management
chris

Chris Johnson

Problem Solving Environments
ross

Ross Whitaker

GPUs
chuck

Chuck Hansen

GPUs
       

Scientific Computing Project Sites:


Publications in Scientific Computing:


Demonstrating GPU Code Portability and Scalability for Radiative Heat Transfer Computations
B. Peterson, A. Humphrey, J. Holmen T. Harman, M. Berzins, D. Sunderland, H.C. Edwards. In Journal of Computational Science, Elsevier BV, June, 2018.
ISSN: 1877-7503
DOI: 10.1016/j.jocs.2018.06.005

High performance computing frameworks utilizing CPUs, Nvidia GPUs, and/or Intel Xeon Phis necessitate portable and scalable solutions for application developers. Nvidia GPUs in particular present numerous portability challenges with a different programming model, additional memory hierarchies, and partitioned execution units among streaming multiprocessors. This work presents modifications to the Uintah asynchronous many-task runtime and the Kokkos portability library to enable one single codebase for complex multiphysics applications to run across different architectures. Scalability and performance results are shown on multiple architectures for a globally coupled radiation heat transfer simulation, ranging from a single node to 16384 Titan compute nodes.



Numerical integration in multiple dimensions with designed quadrature
V. Keshavarzzadeh, R.M. Kirby, A. Narayan. In CoRR, 2018.

We present a systematic computational framework for generating positive quadrature rules in multiple dimensions on general geometries. A direct moment-matching formulation that enforces exact integration on polynomial subspaces yields nonlinear conditions and geometric constraints on nodes and weights. We use penalty methods to address the geometric constraints, and subsequently solve a quadratic minimization problem via the Gauss-Newton method. Our analysis provides guidance on requisite sizes of quadrature rules for a given polynomial subspace, and furnishes useful user-end stability bounds on error in the quadrature rule in the case when the polynomial moment conditions are violated by a small amount due to, e.g., finite precision limitations or stagnation of the optimization procedure. We present several numerical examples investigating optimal low-degree quadrature rules, Lebesgue constants, and 100-dimensional quadrature. Our capstone examples compare our quadrature approach to popular alternatives, such as sparse grids and quasi-Monte Carlo methods, for problems in linear elasticity and topology optimization.



On the treatment of field quantities and elemental continuity in fem solutions
A. Jallepalli, J. Docampo-Sánchez, J.K. Ryan, R. Haimes, R.M. Kirby. In IEEE Transactions on Visualization and Computer Graphics, Vol. 24, No. 1, IEEE, pp. 903--912. Jan, 2018.
DOI: 10.1109/tvcg.2017.2744058

As the finite element method (FEM) and the finite volume method (FVM), both traditional and high-order variants, continue their proliferation into various applied engineering disciplines, it is important that the visualization techniques and corresponding data analysis tools that act on the results produced by these methods faithfully represent the underlying data. To state this in another way: the interpretation of data generated by simulation needs to be consistent with the numerical schemes that underpin the specific solver technology. As the verifiable visualization literature has demonstrated: visual artifacts produced by the introduction of either explicit or implicit data transformations, such as data resampling, can sometimes distort or even obfuscate key scientific features in the data. In this paper, we focus on the handling of elemental continuity, which is often only C0 continuous or piecewise discontinuous, when visualizing primary or derived fields from FEM or FVM simulations. We demonstrate that traditional data handling and visualization of these fields introduce visual errors. In addition, we show how the use of the recently proposed line-SIAC filter provides a way of handling elemental continuity issues in an accuracy-conserving manner with the added benefit of casting the data in a smooth context even if the representation is element discontinuous.



Weighted approximate fekete points: sampling for least-squares polynomial approximation
L. Guo, A. Narayan, L. Yan, T. Zhou. In SIAM Journal on Scientific Computing, Vol. 40, No. 1, SIAM, pp. A366--A387. Jan, 2018.
DOI: 10.1137/17m1140960

We propose and analyze a weighted greedy scheme for computing deterministic sample configurations in multidimensional space for performing least-squares polynomial approximations on $L^2$ spaces weighted by a probability density function. Our procedure is a particular weighted version of the approximate Fekete points method, with the weight function chosen as the (inverse) Christoffel function. Our procedure has theoretical advantages: when linear systems with optimal condition number exist, the procedure finds them. In the one-dimensional setting with any density function, our greedy procedure almost always generates optimally conditioned linear systems. Our method also has practical advantages: our procedure is impartial to the compactness of the domain of approximation and uses only pivoted linear algebraic routines. We show through numerous examples that our sampling design outperforms competing randomized and deterministic designs when the domain is both low and high dimensional.



Spectral Element and hp Methods,
Y. Yu, R.M. Kirby, G.E. Karniadakis. In Encyclopedia of Computational Mechanics Second Edition, John Wiley & Sons, Ltd, pp. 1--43. 2018.

Spectral/hp element methods provide high‐order discretization, which is essential in the longtime integration of advection–diffusion systems and for capturing dynamic instabilities in solids. In this chapter, we review the main formulations for simulations of incompressible and compressible viscous flows as well as for solid mechanics and present several examples with some emphasis on fluid–structure interactions and interfaces. The first generation of (nodal) spectral elements was limited to relatively simple geometries and smooth solutions. However, the new generation of spectral/hp elements, consisting of both nodal and modal forms, can handle very complex geometries using unstructured grids and can capture strong shocks by employing discontinuous Galerkin methods. New flexible formulations allow simulations of multiphysics problems including extremely complex geometries and multiphase flows. Several implementation strategies have also been developed on the basis of multilevel parallel algorithms that allow dynamic p ‐refinement at constant wall clock time. After three decades of intense developments, spectral element and hp methods are mature and efficient to be used effectively in applications of industrial complexity. They provide the capabilities that standard finite element and finite volume methods do, but, in addition, they exhibit high‐order accuracy and error control.



Curvilinear Mesh Adaptation Using Radial Basis Function Interpolation and Smoothing
V. Zala, V. Shankar, S.P. Sastry, R.M. Kirby. In Journal of Scientific Computing, Springer Nature, pp. 1--22. April, 2018.
DOI: 10.1007/s10915-018-0711-0

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.



Flexible Live‐Wire: Image Segmentation with Floating Anchors
B. Summa, N. Faraj, C. Licorish, V. Pascucci. In Computer Graphics Forum, Vol. 37, No. 2, Wiley, pp. 321-328. May, 2018.
DOI: 10.1111/cgf.13364

We introduce Flexible Live‐Wire, a generalization of the Live‐Wire interactive segmentation technique with floating anchors. In our approach, the user input for Live‐Wire is no longer limited to the setting of pixel‐level anchor nodes, but can use more general anchor sets. These sets can be of any dimension, size, or connectedness. The generality of the approach allows the design of a number of user interactions while providing the same functionality as the traditional Live‐Wire. In particular, we experiment with this new flexibility by designing four novel Live‐Wire interactions based on specific primitives: paint, pinch, probable, and pick anchors. These interactions are only a subset of the possibilities enabled by our generalization. Moreover, we discuss the computational aspects of this approach and provide practical solutions to alleviate any additional overhead. Finally, we illustrate our approach and new interactions through several example segmentations.



Uncertainty quantification guided robust design for nanoparticles' morphology
Y. He, M. Razi, C. Forestiere, L. Dal Negro, R.M. Kirby. In Computer Methods in Applied Mechanics and Engineering, Elsevier BV, pp. 578--593. July, 2018.
DOI: 10.1016/j.cma.2018.03.027

The automatic inverse design of three-dimensional plasmonic nanoparticles enables scientists and engineers to explore a wide design space and to maximize a device's performance. However, due to the large uncertainty in the nanofabrication process, we may not be able to obtain a deterministic value of the objective, and the objective may vary dramatically with respect to a small variation in uncertain parameters. Therefore, we take into account the uncertainty in simulations and adopt a classical robust design model for a robust design. In addition, we propose an efficient numerical procedure for the robust design to reduce the computational cost of the process caused by the consideration of the uncertainty. Specifically, we use a global sensitivity analysis method to identify the important random variables and consider the non-important ones as deterministic, and consequently reduce the dimension of the stochastic space. In addition, we apply the generalized polynomial chaos expansion method for constructing computationally cheaper surrogate models to approximate and replace the full simulations. This efficient robust design procedure is performed by varying the particles' material among the most commonly used plasmonic materials such as gold, silver, and aluminum, to obtain different robust optimal shapes for the best enhancement of electric fields.



A gradient enhanced ℓ1-minimization for sparse approximation of polynomial chaos expansions
L. Guo, A. Narayan, T. Zhou. In Journal of Computational Physics, Vol. 367, Elsevier BV, pp. 49--64. Aug, 2018.

We investigate a gradient enhanced ℓ 1-minimization for constructing sparse polynomial chaos expansions. In addition to function evaluations, measurements of the function gradient is also included to accelerate the identification of expansion coefficients. By designing appropriate preconditioners to the measurement matrix, we show gradient enhanced ℓ 1 minimization leads to stable and accurate coefficient recovery. The framework for designing preconditioners is quite general and it applies to recover of functions whose domain is bounded or unbounded. Comparisons between the gradient enhanced approach and the standard ℓ 1-minimization are also presented and numerical examples suggest that the inclusion of derivative information can guarantee sparse recovery at a reduced computational cost.



Practical error bounds for a non-intrusive bi-fidelity approach to parametric/stochastic model reduction
J. Hampton, HR. Fairbanks, A. Narayan, A. Doostan. In Journal of Computational Physics, Vol. 368, Elsevier BV, pp. 315--332. September, 2018.
DOI: 10.1016/j.jcp.2018.04.015

For practical model-based demands, such as design space exploration and uncertainty quantification (UQ), a high-fidelity model that produces accurate outputs often has high computational cost, while a low-fidelity model with less accurate outputs has low computational cost. It is often possible to construct a bi-fidelity model having accuracy comparable with the high-fidelity model and computational cost comparable with the low-fidelity model. This work presents the construction and analysis of a non-intrusive (i.e., sample-based) bi-fidelity model that relies on the low-rank structure of the map between model parameters/uncertain inputs and the solution of interest, if exists. Specifically, we derive a novel, pragmatic estimate for the error committed by this bi-fidelity model. We show that this error bound can be used to determine if a given pair of low- and high-fidelity models will lead to an accurate bi-fidelity approximation. The cost of this error bound is relatively small and depends on the solution rank. The value of this error estimate is demonstrated using two example problems in the context of UQ, involving linear and non-linear partial differential equations.



Fast predictive models based on multi-fidelity sampling of properties in molecular dynamics simulations
M. Razi, A. Narayan, RM. Kirby, D. Bedrov. In Computational Materials Science, Vol. 152, Elsevier BV, pp. 125--133. September, 2018.
DOI: 10.1016/j.commatsci.2018.05.029

In this paper we introduce a novel approach for enhancing the sampling convergence for properties predicted by molecular dynamics. The proposed approach is based upon the construction of a multi-fidelity surrogate model using computational models with different levels of accuracy. While low fidelity models produce result with a lower level of accuracy and computational cost, in this framework they can provide the basis for identification of the optimal sparse sampling pattern for high fidelity models to construct an accurate surrogate model. Such an approach can provide a significant computational saving for the estimation of the quantities of interest for the underlying physical/engineering systems. In the present work, this methodology is demonstrated for molecular dynamics simulations of a Lennard-Jones fluid. Levels of multi-fidelity are defined based upon the integration time step employed in the simulation. The proposed approach is applied to two different canonical problems including (i) single component fluid and (ii) binary glass-forming mixture. The results show about 70% computational saving for the estimation of averaged properties of the systems such as total energy, self diffusion coefficient, radial distribution function and mean squared displacements with a reasonable accuracy.



A Preliminary Port and Evaluation of the Uintah AMT Runtime on Sunway TaihuLight
Z. Yang, D. Sahasrabudhe, A. Humphrey, M. Berzins. In 9th IEEE International Workshop on Parallel and Distributed Scientific and Engineering Computing (PDSEC 2018), IEEE, May, 2018.

The Sunway TaihuLight is the world's fastest supercomputer at the present time with a low power consumption per flop and a unique set of architectural features. Applications performance depends heavily on being able to adapt codes to make best use of these features. Porting large codes to novel architectures such as Sunway is both time-consuming and expensive, as modifications throughout the code may be needed. One alternative to conventional porting is to consider an approach based upon Asynchronous Many Task (AMT) Runtimes such as the Uintah framework considered here. Uintah structures the problem as a series of tasks that are executed by the runtime via a task scheduler. The central challenge in porting a large AMT runtime like Uintah is thus to consider how to devise an appropriate scheduler and how to write tasks to take advantage of a particular architecture. It will be shown how an asynchronous Sunway-specific scheduler, based upon MPI and athreads, may be written and how individual taskcode for a typical but model structured-grid fluid-flow problem needs to be refactored. Preliminary experiments show that it is possible to obtain a strong-scaling efficiency ranging from 31.7% to 96.1% for different problem sizes with full optimizations. The asynchronous scheduler developed here improves the overall performance over a synchronous alternative by at most 22.8%, and the fluid-flow simulation reaches 1.17% the theoretical peak of the running nodes. Conclusions are drawn for the porting of full-scale Uintah applications.



Scalable Data Management of the Uintah Simulation Framework for Next-Generation Engineering Problems with Radiation
S. Kumar, A. Humphrey, W. Usher, S. Petruzza, B. Peterson, J. A. Schmidt, D. Harris, B. Isaac, J. Thornock, T. Harman, V. Pascucci,, M. Berzins. In Supercomputing Frontiers, Springer International Publishing, pp. 219--240. 2018.
ISBN: 978-3-319-69953-0
DOI: 10.1007/978-3-319-69953-0_13

The need to scale next-generation industrial engineering problems to the largest computational platforms presents unique challenges. This paper focuses on data management related problems faced by the Uintah simulation framework at a production scale of 260K processes. Uintah provides a highly scalable asynchronous many-task runtime system, which in this work is used for the modeling of a 1000 megawatt electric (MWe) ultra-supercritical (USC) coal boiler. At 260K processes, we faced both parallel I/O and visualization related challenges, e.g., the default file-per-process I/O approach of Uintah did not scale on Mira. In this paper we present a simple to implement, restructuring based parallel I/O technique. We impose a restructuring step that alters the distribution of data among processes. The goal is to distribute the dataset such that each process holds a larger chunk of data, which is then written to a file independently. This approach finds a middle ground between two of the most common parallel I/O schemes--file per process I/O and shared file I/O--in terms of both the total number of generated files, and the extent of communication involved during the data aggregation phase. To address scalability issues when visualizing the simulation data, we developed a lightweight renderer using OSPRay, which allows scientists to visualize the data interactively at high quality and make production movies. Finally, this work presents a highly efficient and scalable radiation model based on the sweeping method, which significantly outperforms previous approaches in Uintah, like discrete ordinates. The integrated approach allowed the USC boiler problem to run on 260K CPU cores on Mira.



Nonlinear stability and time step selection for the MPM method
M. Berzins. In Computational Particle Mechanics, Jan, 2018.
ISSN: 2196-4386
DOI: 10.1007/s40571-018-0182-y

The Material Point Method (MPM) has been developed from the Particle in Cell (PIC) method over the last 25 years and has proved its worth in solving many challenging problems involving large deformations. Nevertheless there are many open questions regarding the theoretical properties of MPM. For example in while Fourier methods, as applied to PIC may provide useful insight, the non-linear nature of MPM makes it necessary to use a full non-linear stability analysis to determine a stable time step for MPM. In order to begin to address this the stability analysis of Spigler and Vianello is adapted to MPM and used to derive a stable time step bound for a model problem. This bound is contrasted against traditional Speed of sound and CFL bounds and shown to be a realistic stability bound for a model problem.



Research and Education in Computational Science and Engineering
Subtitled “Report from a workshop sponsored by the Society for Industrial and Applied Mathematics (SIAM) and the European Exascale Software Initiative (EESI-2), August 4-6, 2014, Breckenridge, Colorado,” U. Ruede, K. Willcox, L. C. McInnes, H. De Sterck, G. Biros, H. Bungartz, J. Corones, E. Cramer, J. Crowley, O. Ghattas, M. Gunzburger, M. Hanke, R. Harrison, M. Heroux, J. Hesthaven, P. Jimack, C. Johnson, K. E. Jordan, D. E. Keyes, R. Krause, V. Kumar, S. Mayer, J. Meza, K. M. Mrken, J. T. Oden, L. Petzold, P. Raghavan, S. M. Shontz, A. Trefethen, P. Turner, V. Voevodin, B. Wohlmuth,, C. S. Woodward. Vol. abs/1610.02608, 2018.

This report presents challenges, opportunities and directions for computational science and engineering (CSE) research and education for the next decade. Over the past two decades the field of CSE has penetrated both basic and applied research in academia, industry, and laboratories to advance discovery, optimize systems, support decision-makers, and educate the scientific and engineering workforce. Informed by centuries of theory and experiment, CSE performs computational experiments to answer questions that neither theory nor experiment alone is equipped to answer. CSE provides scientists and engineers with algorithmic inventions and software systems that transcend disciplines and scales. CSE brings the power of parallelism to bear on troves of data. Mathematics-based advanced computing has become a prevalent means of discovery and innovation in essentially all areas of science, engineering, technology, and society; and the CSE community is at the core of this transformation. However, a combination of disruptive developments—including the architectural complexity of extreme-scale computing, the data revolution and increased attention to data-driven discovery, and the specialization required to follow the applications to new frontiers—is redefining the scope and reach of the CSE endeavor. With these many current and expanding opportunities for the CSE field, there is a growing demand for CSE graduates and a need to expand CSE educational offerings. This need includes CSE programs at both the undergraduate and graduate levels, as well as continuing education and professional development programs, exploiting the synergy between computational science and data science. Yet, as institutions consider new and evolving educational programs, it is essential to consider the broader research challenges and opportunities that provide the context for CSE education and workforce development.



Stochastic collocation approach with adaptive mesh refinement for parametric uncertainty analysis
A. Bhaduri, Y. He, M.D. Shields, L. Graham-Brady, R.M. Kirby. 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.



Multi-Dimensional Filtering: Reducing the Dimension Through Rotation Read More: https://epubs.siam.org/doi/abs/10.1137/16M1097845
J. Docampo-Sánchez, J.K. Ryan, M. Mirzargar, R.M. Kirby. In SIAM Journal on Scientific Computing, Vol. 39, No. 5, SIAM, pp. A2179--A2200. Jan, 2017.
DOI: 10.1137/16m1097845

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.



Exploration of Heterogeneous Data Using Robust Similarity
M. Mirzargar, R.T. Whitaker, R.M. Kirby. 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.



Hexagonal Smoothness-Increasing Accuracy-Conserving Filtering
M. Mirzargar, A. Jallepalli, J.K. Ryan, R.M. Kirby. In Journal of Scientific Computing, Vol. 73, No. 2-3, Springer Nature, pp. 1072--1093. Aug, 2017.
DOI: 10.1007/s10915-017-0517-5

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.



Offline-Enhanced Reduced Basis Method Through Adaptive Construction of the Surrogate Training Set
J. Jiang, Y. Chen, A. Narayan. In Journal of Scientific Computing, Vol. 73, No. 2-3, Springer Nature, pp. 853--875. September, 2017.
DOI: 10.1007/s10915-017-0551-3

The reduced basis method (RBM) is a popular certified model reduction approach for solving parametrized partial differential equations. One critical stage of the offline portion of the algorithm is a greedy algorithm, requiring maximization of an error estimate over parameter space. In practice this maximization is usually performed by replacing the parameter domain continuum with a discrete "training" set. When the dimension of parameter space is large, it is necessary to significantly increase the size of this training set in order to effectively search parameter space. Large training sets diminish the attractiveness of RBM algorithms since this proportionally increases the cost of the offline phase. In this work we propose novel strategies for offline RBM algorithms that mitigate the computational difficulty of maximizing error estimates over a training set. The main idea is to identify a subset of the training set, a "surrogate training set" (STS), on which to perform greedy algorithms. The STS we construct is much smaller in size than the full training set, yet our examples suggest that it is accurate enough to induce the solution manifold of interest at the current offline RBM iteration. We propose two algorithms to construct the STS: our first algorithm, the successive maximization method, is inspired by inverse transform sampling for non-standard univariate probability distributions. The second constructs an STS by identifying pivots in the Cholesky decomposition of an approximate error correlation matrix. We demonstrate the algorithm through numerical experiments, showing that it is capable of accelerating offline RBM procedures without degrading accuracy, assuming that the solution manifold has rapidly decaying Kolmogorov width.