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Title: Advanced Dynamically Adaptive Algorithms for Stochastic Simulations on Extreme Scales

Abstract

The focus of the project is the development of mathematical methods and high-performance com- putational tools for stochastic simulations, with a particular emphasis on computations on extreme scales. The core of the project revolves around the design of highly e cient and scalable numer- ical algorithms that can adaptively and accurately, in high dimensional spaces, resolve stochastic problems with limited smoothness, even containing discontinuities.

Authors:
 [1]
  1. Purdue Univ., West Lafayette, IN (United States)
Publication Date:
Research Org.:
Purdue Univ., West Lafayette, IN (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1258292
Report Number(s):
DOE-PU-0005173-1
104857
DOE Contract Number:
SC0005173
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING

Citation Formats

Xiu, Dongbin. Advanced Dynamically Adaptive Algorithms for Stochastic Simulations on Extreme Scales. United States: N. p., 2016. Web. doi:10.2172/1258292.
Xiu, Dongbin. Advanced Dynamically Adaptive Algorithms for Stochastic Simulations on Extreme Scales. United States. doi:10.2172/1258292.
Xiu, Dongbin. 2016. "Advanced Dynamically Adaptive Algorithms for Stochastic Simulations on Extreme Scales". United States. doi:10.2172/1258292. https://www.osti.gov/servlets/purl/1258292.
@article{osti_1258292,
title = {Advanced Dynamically Adaptive Algorithms for Stochastic Simulations on Extreme Scales},
author = {Xiu, Dongbin},
abstractNote = {The focus of the project is the development of mathematical methods and high-performance com- putational tools for stochastic simulations, with a particular emphasis on computations on extreme scales. The core of the project revolves around the design of highly e cient and scalable numer- ical algorithms that can adaptively and accurately, in high dimensional spaces, resolve stochastic problems with limited smoothness, even containing discontinuities.},
doi = {10.2172/1258292},
journal = {},
number = ,
volume = ,
place = {United States},
year = 2016,
month = 6
}

Technical Report:

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  • The focus of the project is the development of mathematical methods and high-performance computational tools for stochastic simulations, with a particular emphasis on computations on extreme scales. The core of the project revolves around the design of highly efficient and scalable numerical algorithms that can adaptively and accurately, in high dimensional spaces, resolve stochastic problems with limited smoothness, even containing discontinuities.
  • We discuss new global optimization algorithms that are related to the stochastic methods of Rinnooy Kan and Timmer, and to our previous static, synchronous parallel version of this method. The new algorithms have two main new features. First, they adaptively concentrate the computation in the areas of the domain space that appear most likely to produce the global minimum. Secondly, on parallel computers, they use an asynchronous approach, combined with a central work scheduler, to avoid load balancing problems. We investigate several mechanisms for deciding when and how to make the adaptive adjustments. We also describe both algorithmic and implementationmore » considerations involved in constructing the parallel asynchronous algorithm. Computational tests on sequential and parallel computers show that the adaptive and asynchronous features of our new method can substantially reduce the number of function evaluations, and the execution time, required by previous stochastic methods to solve global optimization problems.« less
  • Highlights: {yields} New approach for stochastic computations based on polynomial chaos. {yields} Development of dynamically adaptive wavelet multiscale solver using space refinement. {yields} Accurate capture of steep gradients and multiscale features in stochastic problems. {yields} All scales of each random mode are captured on independent grids. {yields} Numerical examples demonstrate the need for different space resolutions per mode. - Abstract: In stochastic computations, or uncertainty quantification methods, the spectral approach based on the polynomial chaos expansion in random space leads to a coupled system of deterministic equations for the coefficients of the expansion. The size of this system increases drasticallymore » when the number of independent random variables and/or order of polynomial chaos expansions increases. This is invariably the case for large scale simulations and/or problems involving steep gradients and other multiscale features; such features are variously reflected on each solution component or random/uncertainty mode requiring the development of adaptive methods for their accurate resolution. In this paper we propose a new approach for treating such problems based on a dynamically adaptive wavelet methodology involving space-refinement on physical space that allows all scales of each solution component to be refined independently of the rest. We exemplify this using the convection-diffusion model with random input data and present three numerical examples demonstrating the salient features of the proposed method. Thus we establish a new, elegant and flexible approach for stochastic problems with steep gradients and multiscale features based on polynomial chaos expansions.« less
  • This project addresses both communication-avoiding algorithms, and reproducible floating-point computation. Communication, i.e. moving data, either between levels of memory or processors over a network, is much more expensive per operation than arithmetic (measured in time or energy), so we seek algorithms that greatly reduce communication. We developed many new algorithms for both dense and sparse, and both direct and iterative linear algebra, attaining new communication lower bounds, and getting large speedups in many cases. We also extended this work in several ways: (1) We minimize writes separately from reads, since writes may be much more expensive than reads on emergingmore » memory technologies, like Flash, sometimes doing asymptotically fewer writes than reads. (2) We extend the lower bounds and optimal algorithms to arbitrary algorithms that may be expressed as perfectly nested loops accessing arrays, where the array subscripts may be arbitrary affine functions of the loop indices (eg A(i), B(i,j+k, k+3*m-7, …) etc.). (3) We extend our communication-avoiding approach to some machine learning algorithms, such as support vector machines. This work has won a number of awards. We also address reproducible floating-point computation. We define reproducibility to mean getting bitwise identical results from multiple runs of the same program, perhaps with different hardware resources or other changes that should ideally not change the answer. Many users depend on reproducibility for debugging or correctness. However, dynamic scheduling of parallel computing resources, combined with nonassociativity of floating point addition, makes attaining reproducibility a challenge even for simple operations like summing a vector of numbers, or more complicated operations like the Basic Linear Algebra Subprograms (BLAS). We describe an algorithm that computes a reproducible sum of floating point numbers, independent of the order of summation. The algorithm depends only on a subset of the IEEE Floating Point Standard 754-2008, uses just 6 words to represent a “reproducible accumulator,” and requires just one read-only pass over the data, or one reduction in parallel. New instructions based on this work are being considered for inclusion in the future IEEE 754-2018 floating-point standard, and new reproducible BLAS are being considered for the next version of the BLAS standard.« less