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Litvinenko, Alexander (24)

Tempone, Raul (8)Genton, Marc G. (6)Matthies, Hermann G. (6)Sun, Ying (6)View MoreDepartmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division (18)Applied Mathematics and Computational Science Program (7)Extreme Computing Research Center (4)Center for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ) (3)Statistics Program (3)View MoreSubjecthierarchical matrices (5)Bayesian (4)Matern covariance (3)Uncertainty Quantification (3)HLIB (2)View MoreTypePoster (24)Year (Issue Date)2019 (1)2018 (1)2017 (4)2016 (3)2015 (9)View MoreItem AvailabilityOpen Access (24)

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Computation of Electromagnetic Fields Scattered From Objects With Uncertain Shapes Using Multilevel Monte Carlo Method

Litvinenko, Alexander; Yucel, Abdulkadir; Bagci, Hakan; Oppelstrup, Jesper; Tempone, Raul; Michielssen, Eric (2019-02-14) [Poster]

Computational tools for characterizing electromagnetic scattering from objects with uncertain shapes are needed in various applications ranging from remote sensing at microwave frequencies to Raman spectroscopy at optical frequencies. Often, such computational tools use the Monte Carlo (MC) method to sample a parametric space describing geometric uncertainties. For each sample, which corresponds to a realization of the geometry, a deterministic electromagnetic solver computes the scattered fields. However, for an accurate statistical characterization the number of MC samples has to be large. In this work, to address this challenge, the continuation multilevel Monte Carlo (CMLMC) method is used together with a surface integral equation solver. The CMLMC method optimally balances statistical errors due to sampling of the parametric space, and numerical errors due to the discretization of the geometry using a hierarchy of discretizations, from coarse to fine. The number of realizations of finer discretizations can be kept low, with most samples computed on coarser discretizations to minimize computational cost. Consequently, the total execution time is significantly reduced, in comparison to the standard MC scheme.

Likelihood Approximation With Parallel Hierarchical Matrices For Large Spatial Datasets

Litvinenko, Alexander (2018-04-10) [Poster]

Use H-matrices to approximate large covariance matrices in spatial statistics

Likelihood Approximation With Parallel Hierarchical Matrices For Large Spatial Datasets

Litvinenko, Alexander; Sun, Ying; Genton, Marc G.; Keyes, David E. (2017-11-01) [Poster]

The main goal of this article is to introduce the parallel hierarchical matrix library HLIBpro to the statistical community.
We describe the HLIBCov package, which is an extension of the HLIBpro library for approximating large covariance matrices and maximizing likelihood functions. We show that an approximate Cholesky factorization of a dense matrix of size $2M\times 2M$ can be computed on a modern multi-core desktop in few minutes.
Further, HLIBCov is used for estimating the unknown parameters such as the covariance length, variance and smoothness parameter of a Matérn covariance function by maximizing the joint Gaussian log-likelihood function. The computational bottleneck here is expensive linear algebra arithmetics due to large and dense covariance matrices. Therefore covariance matrices are approximated in the hierarchical ($\H$-) matrix format with computational cost $\mathcal{O}(k^2n \log^2 n/p)$ and storage $\mathcal{O}(kn \log n)$, where the rank $k$ is a small integer (typically $k<25$), $p$ the number of cores and $n$ the number of locations on a fairly general mesh. We demonstrate a synthetic example, where the true values of known parameters are known.
For reproducibility we provide the C++ code, the documentation, and the synthetic data.

Risk assessment of salt contamination of groundwater under uncertain aquifer properties

Litvinenko, Alexander; Keyes, David E.; Logashenko, Dmitry; Tempone, Raul; Wittum, Gabriel (2017-10-01) [Poster]

One of the central topics in hydrogeology and environmental science is the investigation of salinity-driven groundwater flow in heterogeneous porous media. Our goals are to model and to predict pollution of water resources.
We simulate a density driven groundwater flow with uncertain porosity and permeability. This strongly non-linear model describes the unstable transport of salt water with building ‘fingers’-shaped patterns. The computation requires
a very fine unstructured mesh and, therefore, high computational resources.
We run the highly-parallel multigrid solver, based on ug4, on supercomputer Shaheen II. A MPI-based parallelization is done in the geometrical as well as in the stochastic spaces. Every scenario is computed on 32 cores and
requires a mesh with ~8M grid points and 1500 or more time steps. 200 scenarios are computed concurrently. The total number of cores in parallel computation is 200x32=6400. The main goal of this work is to estimate propagation of uncertainties through the model, to investigate sensitivity of the solution to the input uncertain parameters. Additionally, we demonstrate how the multigrid ug4-based solver can be applied as a black-box in the uncertainty quantification framework.

Kriging accelerated by orders of magnitude: combining low-rank with FFT techniques

Litvinenko, Alexander; Nowak, Wolfgang (2014-05-04) [Poster]

Kriging algorithms based on FFT, the separability of certain covariance functions and low-rank representations of covariance functions have been investigated. The current study combines these ideas, and so combines the individual speedup factors of all ideas. The reduced computational complexity is O(dLlogL), where L := max ini, i = 1

Computation of Electromagnetic Fields Scattered From Dielectric Objects of Uncertain Shapes Using MLMC

Litvinenko, Alexander; Haji Ali, Abdul Lateef; Uysal, Ismail Enes; Ulku, Huseyin Arda; Oppelstrup, Jesper; Tempone, Raul; Bagci, Hakan (2016-01-06) [Poster]

Simulators capable of computing scattered fields from objects of uncertain shapes are highly useful in electromagnetics and photonics, where device designs are typically subject to fabrication tolerances. Knowledge of statistical variations in scattered fields is useful in ensuring error-free functioning of devices. Oftentimes such simulators use a Monte Carlo (MC) scheme to sample the random domain, where the variables parameterize the uncertainties in the geometry. At each sample, which corresponds to a realization of the geometry, a deterministic electromagnetic solver is executed to compute the scattered fields. However, to obtain accurate statistics of the scattered fields, the number of MC samples has to be large. This significantly increases the total execution time.
In this work, to address this challenge, the Multilevel MC (MLMC [1]) scheme is used together with a (deterministic) surface integral equation solver. The MLMC achieves a higher efficiency by balancing the statistical errors due to sampling of the random domain and the numerical errors due to discretization of the geometry at each of these samples. Error balancing results in a smaller number of samples requiring coarser discretizations. Consequently, total execution time is significantly shortened.

Solving inverse problem via non-linear update of PCE coefficients

Litvinenko, Alexander; Matthies, Hermann G.; Rosic, Bojana V.; Zander, Elmar (2016-01-06) [Poster]

Hierarchical matrix approximation of large covariance matrices

Litvinenko, Alexander; Genton, Marc G.; Sun, Ying (2016-01-06) [Poster]

Litvinenko, Alexander; Haji Ali, Abdul Lateef; Uysal, Ismail Enes; Ulku, Huseyin Arda; Tempone, Raul; Bagci, Hakan; Oppelstrup, Jesper (2015-01-07) [Poster]

Simulators capable of computing scattered fields from objects of uncertain shapes are highly useful in electromagnetics and photonics, where device designs are typically subject to fabrication tolerances. Knowledge of statistical variations in scattered fields is useful in ensuring error-free functioning of devices. Oftentimes such simulators use a Monte Carlo (MC) scheme to sample the random domain, where the variables parameterize the uncertainties in the geometry. At each sample, which corresponds to a realization of the geometry, a deterministic electromagnetic solver is executed to compute the scattered fields. However, to obtain accurate statistics of the scattered fields, the number of MC samples has to be large.
This significantly increases the total execution time. In this work, to address this challenge, the Multilevel MC (MLMC [1]) scheme is used together with a (deterministic) surface integral equation solver. The MLMC achieves a higher efficiency by “balancing” the statistical errors due to sampling of the random domain and the numerical errors due to discretization of the geometry at each of these samples. Error balancing results in a smaller number of samples requiring coarser discretizations. Consequently, total execution time is significantly shortened.

Scalable Hierarchical Algorithms for stochastic PDEs and UQ

Litvinenko, Alexander; Chavez Chavez, Gustavo Ivan; Keyes, David E.; Ltaief, Hatem; Yokota, Rio (2015-01-07) [Poster]

H-matrices and Fast Multipole (FMM) are powerful methods to approximate linear operators coming from partial differential and integral equations as well as speed up computational cost from quadratic or cubic to log-linear (O(n log n)), where n number of degrees of freedom in the discretization. The storage is reduced to the log-linear as well. This hierarchical structure is a good starting point for parallel algorithms. Parallelization on shared and distributed memory systems was pioneered by Kriemann [1,2]. Since 2005, the area of parallel architectures and software is developing very fast. Progress in GPUs and Many-Core Systems (e.g. XeonPhi with 64 cores) motivated us to extend work started in [1,2,7,8].

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