Miroslav Stoyanov

NUMERICAL ANALYSIS OF FIXED POINT ALGORITHMS IN THE PRESENCE OF HARDWARE FAULTS

The exponential growth of computational power of the extreme scale machines over the past few decades has led to a corresponding decrease in reliability and a sharp increase of the frequency of hardware faults. Our research focuses on the mathematical challenges presented by the silent hardware faults; i.e., faults that can perturb the result of computations in an inconspicuous way. Using the approach of selective reliability, we present an analytic fault mode that can be used to study the resilience properties of a numerical algorithm. We apply our approach to the classical fixed point iteration and demonstrate that in the presence of hardware faults, the classical method fails to converge in expectation. We preset a modified resilient algorithm that detects and rejects faults resulting in error with large magnitude, while small faults are negated by the natural self-correcting properties of the algorithm. We show that our method is convergent (in first and second statistical moments) even in the presenc...

Application of High Performance Computing for Simulating the Unstable Dynamics of Dilute Spark-Ignited Combustion

In collaboration with a major automotive manufacturer, we are using computational simulations of in-cylinder combustion to understand the multi-scale nonlinear physics of the dilute stability limit. Because some key features of dilute combustion can take thousands of successive cycles to develop, the computation time involved in using complex models to simulate these effects has limited industrys ability to exploit simulations in optimizing advanced engines. We describe a novel approach for utilizing parallel computations to reveal long-timescale features of dilute combustion without the need to simulate many successive engine cycles in series. Our approach relies on carefully guided, concurrent, single-cycle simulations to create metamodels that preserve the long-timescale features of interest. We use a simplified combustion model to develop and demonstrate our strategy for adaptively guiding the concurrent simulations to generate metamodels. We next will implement this strategy with higher-fidelity, multi-scale combustion models on large computing facilities to generate more refined metamodels. The refined metamodels can then be used to accelerate engine development because of their efficiency. Similar approaches might also be used for rapidly exploring the dynamics of other complex multi-scale systems that evolve with serial dependency on time.

Hierarchy-Direction Selective Approach for Locally Adaptive Sparse Grids

We consider the problem of multidimensional adaptive hierarchical interpolation. We use sparse grids points and functions that are induced from a one dimensional hierarchical rule via tensor products. The classical locally adaptive sparse grid algorithm uses an isotropic refinement from the coarser to the denser levels of the hierarchy. However, the multidimensional hierarchy provides a more complex structure that allows for various anisotropic and hierarchy selective refinement techniques. We consider the more advanced refinement techniques and apply them to a number of simple test functions chosen to demonstrate the various advantages and disadvantages of each method. While there is no refinement scheme that is optimal for all functions, the fully adaptive family-direction-selective technique is usually more stable and requires fewer samples.

Power-Law Noises over General Spatial Domains and on Nonstandard Meshes ∗

Power-law noises abound in nature and have been observed extensively in both time series and spatially varying environmental parameters. Although, recent years have seen the extension of traditional stochastic partial differential equations to include systems driven by fractional Brownian motion, spatially distributed scale-invariance has received comparatively little attention, especially for parameters defined over non-standard spatial domains. This paper discusses the generalization of power-law noises to general spatial domains by outlining their theoretical underpinnings as well as addressing their numerical simulation on arbitrary meshes. Three computational algorithms are presented for efficiently generating their sample paths, accompanied by numerous numerical illustrations.

Uncertainty quantification techniques for population density estimates derived from sparse open source data

The Population Density Tables (PDT) project at Oak Ridge National Laboratory (www.ornl.gov) is developing population density estimates for specific human activities under normal patterns of life based largely on information available in open source. Currently, activity-based density estimates are based on simple summary data statistics such as range and mean. Researchers are interested in improving activity estimation and uncertainty quantification by adopting a Bayesian framework that considers both data and sociocultural knowledge. Under a Bayesian approach, knowledge about population density may be encoded through the process of expert elicitation. Due to the scale of the PDT effort which considers over 250 countries, spans 50 human activity categories, and includes numerous contributors, an elicitation tool is required that can be operationalized within an enterprise data collection and reporting system. Such a method would ideally require that the contributor have minimal statistical knowledge, require minimal input by a statistician or facilitator, consider human difficulties in expressing qualitative knowledge in a quantitative setting, and provide methods by which the contributor can appraise whether their understanding and associated uncertainty was well captured. This paper introduces an algorithm that transforms answers to simple, non-statistical questions into a bivariate Gaussian distribution as the prior for the Beta distribution. Based on geometric properties of the Beta distribution parameter feasibility space and the bivariate Gaussian distribution, an automated method for encoding is developed that responds to these challenging enterprise requirements. Though created within the context of population density, this approach may be applicable to a wide array of problem domains requiring informative priors for the Beta distribution.

Adaptive Sparse Grid Construction in a Context of Local Anisotropy and Multiple Hierarchical Parents

We consider general strategy for hierarchical multidimensional interpolation based on sparse grids, where the interpolation nodes and locally supported basis functions are constructed from tensors of a one dimensional hierarchical rule. We consider four different hierarchies that are tailored towards general functions, high or low order polynomial approximation, or functions that satisfy homogeneous boundary conditions. The main advantage of the locally supported basis is the ability to choose a set of functions based on the observed behavior of the target function. We present an alternative to the classical surplus refinement techniques, where we exploit local anisotropy and refine using functions with not strictly decreasing support. The more flexible refinement strategy improves stability and reduces the total number of expensive simulations, resulting in significant computational saving. We demonstrate the advantages of the different hierarchies and refinement techniques by application to series of simple functions as well as a system of ordinary differential equations given by the Kermack-McKendrick SIR model.

Application of High Performance Computing for Studying Cyclic Variability in Dilute Internal Combustion Engines

Combustion instabilities in dilute internal combustion engines are manifest in cyclic variability (CV) in engine performance measures such as integrated heat release or shaft work. Understanding the factors leading to CV is important in model-based control, especially with high dilution where experimental studies have demonstrated that deterministic effects can become more prominent.Observation of enough consecutive engine cycles for significant statistical analysis is standard in experimental studies but is largely wanting in numerical simulations because of the computational time required to compute hundreds or thousands of consecutive cycles. We have proposed and begun implementation of an alternative approach to allow rapid simulation of long series of engine dynamics based on a low-dimensional mapping of ensembles of single-cycle simulations which map input parameters to output engine performance.This paper details the use Titan at the Oak Ridge Leadership Computing Facility to investigate CV in a gasoline direct-injected spark-ignited engine with a moderately high rate of dilution achieved through external exhaust gas recirculation. The CONVERGE™ CFD software was used to perform single-cycle simulations with imposed variations of operating parameters and boundary conditions selected according to a sparse grid sampling of the parameter space. Using an uncertainty quantification technique, the sampling scheme is chosen similar to a design of experiments grid but uses algorithms designed to minimize the number of samples required to achieve a desired degree of accuracy. The simulations map input parameters to output metrics of engine performance for a single cycle, and by mapping over a large parameter space, results can be interpolated from within that space. This interpolation scheme forms the basis for a low-dimensional ‘metamodel’ (or model of a model) which can be used to mimic the dynamical behavior of corresponding high-dimensional simulations.Simulations of high-EGR spark-ignition combustion cycles within a parametric sampling grid were performed and analyzed statistically, and sensitivities of the physical factors leading to high CV are presented. With these results, the prospect of producing low-dimensional metamodels to describe engine dynamics at any point in the parameter space will be discussed. Additionally, modifications to the methodology to account for nondeterministic effects in the numerical solution environment are proposed.Copyright