Philippe Pierre Pebay

Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes

This paper gives an overview of the use of polynomial chaos (PC) expansions to represent stochastic processes in numerical simulations. Several methods are presented for performing arithmetic on, as well as for evaluating polynomial and nonpolynomial functions of variables represented by PC expansions. These methods include {Taylor} series, a newly developed integration method, as well as a sampling-based spectral projection method for nonpolynomial function evaluations. A detailed analysis of the accuracy of the PC representations, and of the different methods for nonpolynomial function evaluations, is performed. It is found that the integration method offers a robust and accurate approach for evaluating nonpolynomial functions, even when very high-order information is present in the PC expansions.

Combining in-situ and in-transit processing to enable extreme-scale scientific analysis

With the onset of extreme-scale computing, I/O constraints make it increasingly difficult for scientists to save a sufficient amount of raw simulation data to persistent storage. One potential solution is to change the data analysis pipeline from a post-process centric to a concurrent approach based on either in-situ or in-transit processing. In this context computations are considered in-situ if they utilize the primary compute resources, while in-transit processing refers to offloading computations to a set of secondary resources using asynchronous data transfers. In this paper we explore the design and implementation of three common analysis techniques typically performed on large-scale scientific simulations: topological analysis, descriptive statistics, and visualization. We summarize algorithmic developments, describe a resource scheduling system to coordinate the execution of various analysis workflows, and discuss our implementation using the DataSpaces and ADIOS frameworks that support efficient data movement between in-situ and in-transit computations. We demonstrate the efficiency of our lightweight, flexible framework by deploying it on the Jaguar XK6 to analyze data generated by S3D, a massively parallel turbulent combustion code. Our framework allows scientists dealing with the data deluge at extreme scale to perform analyses at increased temporal resolutions, mitigate I/O costs, and significantly improve the time to insight.

Resource monitoring and management with OVIS to enable HPC in cloud computing environments

Using the cloud computing paradigm, a host of companies promise to make huge compute resources available to users on a pay-as-you-go basis. These resources can be configured on the fly to provide the hardware and operating system of choice to the customer on a large scale. While the current target market for these resources in the commercial space is web development/hosting, this model has the lure of savings of ownership, operation, and maintenance costs, and thus sounds like an attractive solution for people who currently invest millions to hundreds of millions of dollars annually on High Performance Computing (HPC) platforms in order to support large-scale scientific simulation codes. Given the current interconnect bandwidth and topologies utilized in these commercial offerings, however, the only current viable market in HPC would be small-memory-footprint embarrassingly parallel or loosely coupled applications, which inherently require little to no inter-processor communication. While providing the appropriate resources (bandwidth, latency, memory, etc.) for the HPC community would increase the potential to enable HPC in cloud environments, this would not address the need for scalability and reliability, crucial to HPC applications. Providing for these needs is particularly difficult in commercial cloud offerings where the number of virtual resources can far outstrip the number of physical resources, the resources are shared among many users, and the resources may be heterogeneous. Advanced resource monitoring, analysis, and configuration tools can help address these issues, since they bring the ability to dynamically provide and respond to information about the platform and application state and would enable more appropriate, efficient, and flexible use of the resources key to enabling HPC. Additionally such tools could be of benefit to non-HPC cloud providers, users, and applications by providing more efficient resource utilization in general.

Analysis of large-scale scalar data using hixels

One of the greatest challenges for todays visualization and analysis communities is the massive amounts of data generated from state of the art simulations. Traditionally, the increase in spatial resolution has driven most of the data explosion, but more recently ensembles of simulations with multiple results per data point and stochastic simulations storing individual probability distributions are increasingly common. This paper introduces a new data representation for scalar data, called hixels, that stores a histogram of values for each sample point of a domain. The histograms may be created by spatial down-sampling, binning ensemble values, or polling values from a given distribution. In this manner, hixels form a compact yet information rich approximation of large scale data. In essence, hixels trade off data size and complexity for scalar-value “uncertainty”. Based on this new representation we propose new feature detection algorithms using a combination of topological and statistical methods. In particular, we show how to approximate topological structures from hixel data, extract structures from multi-modal distributions, and render uncertain isosurfaces. In all three cases we demonstrate how using hixels compares to traditional techniques and provide new capabilities to recover prominent features that would otherwise be either infeasible to compute or ambiguous to infer. We use a collection of computer tomography data and large scale combustion simulations to illustrate our techniques.

Numerically stable, single-pass, parallel statistics algorithms

Statistical analysis is widely used for countless scientific applications in order to analyze and infer meaning from data. A key challenge of any statistical analysis package aimed at large-scale, distributed data is to address the orthogonal issues of parallel scalability and numerical stability. In this paper we derive a series of formulas that allow for single-pass, yet numerically robust, pairwise parallel and incremental updates of both arbitrary-order centered statistical moments and co-moments. Using these formulas, we have built an open source parallel statistics framework that performs principal component analysis (PCA) in addition to computing descriptive, correlative, and multi-correlative statistics. The results of a scalability study demonstrate numerically stable, near-optimal scalability on up to 128 processes and results are presented in which the statistical framework is used to process large-scale turbulent combustion simulation data with 1500 processes.

Ovis-2: A robust distributed architecture for scalable RAS

Resource utilization in High Performance Compute clusters can be improved by increased awareness of system state information. Sophisticated run-time characterization of system state in increasingly large clusters requires a scalable fault-tolerant RAS framework. In this paper we describe the architecture of OVIS-2 and how it meets these requirements. We describe some of the sophisticated statistical analysis, 3-D visualization, and use cases for these. Using this framework and associated tools allows the engineer to explore the behaviors and complex interactions of low level system elements while simultaneously giving the system administrator their desired level of detail with respect to ongoing system and component health.

Methods and framework for visualizing higher-order finite elements

The finite element method is an important, widely used numerical technique for solving partial differential equations. This technique utilizes basis functions for approximating the geometry and the variation of the solution field over finite regions, or elements, of the domain. These basis functions are generally formed by combinations of polynomials. In the past, the polynomial order of the basis has been low-typically of linear and quadratic order. However, in recent years so-called p and hp methods have been developed, which may elevate the order of the basis to arbitrary levels with the aim of accelerating the convergence of the numerical solution. The increasing complexity of numerical basis functions poses a significant challenge to visualization systems. In the past, such systems have been loosely coupled to simulation packages, exchanging data via file transfer, and internally reimplementing the basis functions in order to perform interpolation and implement visualization algorithms. However, as the basis functions become more complex and, in some cases, proprietary in nature, it becomes increasingly difficult if not impossible to reimplement them within the visualization system. Further, most visualization systems typically process linear primitives, in part to take advantage of graphics hardware and, in part, due to the inherent simplicity of the resulting algorithms. Thus, visualization of higher-order finite elements requires tessellating the basis to produce data compatible with existing visualization systems. In this paper, we describe adaptive methods that automatically tessellate complex finite element basis functions using a flexible and extensible software framework. These methods employ a recursive, edge-based subdivision algorithm driven by a set of error metrics including geometric error, solution error, and error in image space. Further, we describe advanced pretessellation techniques that guarantees capture of the critical points of the polynomial basis. The framework has been designed using the adaptor design pattern, meaning that the visualization system need not reimplement basis functions, rather it communicates with the simulation package via simple programmatic queries. We demonstrate our method on several examples, and have implemented the framework in the open-source visualization system VTK.

The verdict geometric quality library.

Verdict is a collection of subroutines for evaluating the geometric qualities of triangles, quadrilaterals, tetrahedra, and hexahedra using a variety of metrics. A metric is a real number assigned to one of these shapes depending on its particular vertex coordinates. These metrics are used to evaluate the input to finite element, finite volume, boundary element, and other types of solvers that approximate the solution to partial differential equations defined over regions of space. The geometric qualities of these regions is usually strongly tied to the accuracy these solvers are able to obtain in their approximations. The subroutines are written in C++ and have a simple C interface. Each metric may be evaluated individually or in combination. When multiple metrics are evaluated at once, they share common calculations to lower the cost of the evaluation.

New Applications of the Verdict Library for Standardized Mesh Verification Pre, Post, and End-to-End Processing

verdict is a collection of subroutines for evaluating the geometric qualities of triangles, quadrilaterals, tetrahedra, and hexahedra using a variety of functions. A quality is a real number assigned to one of these shapes depending on its particular vertex coordinates. These functions are used to evaluate the input to finite element, finite volume, boundary element, and other types of solvers that approximate the solution to partial differential equations defined over regions of space. This article describes the most recent version of verdict and provides a summary of the main properties of the quality functions offered by the library. It finally demonstrates the versatility and applicability of verdict by illustrating its use in several scientific applications that pertain to pre, post, and end-to-end processing.

Design and Performance of a Scalable, Parallel Statistics Toolkit

Most statistical software packages implement a broad range of techniques but do so in an ad hoc fashion, leaving users who do not have a broad knowledge of statistics at a disadvantage since they may not understand all the implications of a given analysis or how to test the validity of results. These packages are also largely serial in nature, or target multicore architectures instead of distributed-memory systems, or provide only a small number of statistics in parallel. This paper surveys a collection of parallel implementations of statistics algorithm developed as part of a common framework over the last 3 years. The framework strategically groups modeling techniques with associated verification and validation techniques to make the underlying assumptions of the statistics more clear. Furthermore it employs a design pattern specifically targeted for distributed-memory parallelism, where architectural advances in large-scale high-performance computing have been focused. Moment-based statistics (which include descriptive, correlative, and multicorrelative statistics, principal component analysis (PCA), and k-means statistics) scale nearly linearly with the data set size and number of processes. Entropy-based statistics (which include order and contingency statistics) do not scale well when the data in question is continuous or quasi-diffuse but do scale well when the data is discrete and compact. We confirm and extend our earlier results by now establishing near-optimal scalability with up to 10,000 processes.