skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: High-Dimensional Intrinsic Interpolation Using Gaussian Process Regression and Diffusion Maps

Abstract

This article considers the challenging task of estimating geologic properties of interest using a suite of proxy measurements. The current work recast this task as a manifold learning problem. In this process, this article introduces a novel regression procedure for intrinsic variables constrained onto a manifold embedded in an ambient space. The procedure is meant to sharpen high-dimensional interpolation by inferring non-linear correlations from the data being interpolated. The proposed approach augments manifold learning procedures with a Gaussian process regression. It first identifies, using diffusion maps, a low-dimensional manifold embedded in an ambient high-dimensional space associated with the data. It relies on the diffusion distance associated with this construction to define a distance function with which the data model is equipped. This distance metric function is then used to compute the correlation structure of a Gaussian process that describes the statistical dependence of quantities of interest in the high-dimensional ambient space. The proposed method is applicable to arbitrarily high-dimensional data sets. Here, it is applied to subsurface characterization using a suite of well log measurements. The predictions obtained in original, principal component, and diffusion space are compared using both qualitative and quantitative metrics. Considerable improvement in the prediction of themore » geological structural properties is observed with the proposed method.« less

Authors:
 [1];  [2];  [3];  [1]
  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing
  2. Univ. of Southern California, Los Angeles, CA (United States). Sonny Astani Dept. of Civil and Environmental Engineering
  3. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Atmospheric, Earth and Energy Division
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE Laboratory Directed Research and Development (LDRD) Program
OSTI Identifier:
1400095
Report Number(s):
LLNL-JRNL-728760
Journal ID: ISSN 1874-8961; TRN: US1703098
Grant/Contract Number:  
AC52-07NA27344
Resource Type:
Accepted Manuscript
Journal Name:
Mathematical Geosciences
Additional Journal Information:
Journal Volume: 50; Journal Issue: 1; Journal ID: ISSN 1874-8961
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
58 GEOSCIENCES; 97 MATHEMATICS AND COMPUTING; Gaussian process regression; Kriging; Interpolation on manifold; Intrinsic interpolation; Intrinsic metrics; Diffusion distance

Citation Formats

Thimmisetty, Charanraj A., Ghanem, Roger G., White, Joshua A., and Chen, Xiao. High-Dimensional Intrinsic Interpolation Using Gaussian Process Regression and Diffusion Maps. United States: N. p., 2017. Web. doi:10.1007/s11004-017-9705-y.
Thimmisetty, Charanraj A., Ghanem, Roger G., White, Joshua A., & Chen, Xiao. High-Dimensional Intrinsic Interpolation Using Gaussian Process Regression and Diffusion Maps. United States. doi:10.1007/s11004-017-9705-y.
Thimmisetty, Charanraj A., Ghanem, Roger G., White, Joshua A., and Chen, Xiao. Tue . "High-Dimensional Intrinsic Interpolation Using Gaussian Process Regression and Diffusion Maps". United States. doi:10.1007/s11004-017-9705-y. https://www.osti.gov/servlets/purl/1400095.
@article{osti_1400095,
title = {High-Dimensional Intrinsic Interpolation Using Gaussian Process Regression and Diffusion Maps},
author = {Thimmisetty, Charanraj A. and Ghanem, Roger G. and White, Joshua A. and Chen, Xiao},
abstractNote = {This article considers the challenging task of estimating geologic properties of interest using a suite of proxy measurements. The current work recast this task as a manifold learning problem. In this process, this article introduces a novel regression procedure for intrinsic variables constrained onto a manifold embedded in an ambient space. The procedure is meant to sharpen high-dimensional interpolation by inferring non-linear correlations from the data being interpolated. The proposed approach augments manifold learning procedures with a Gaussian process regression. It first identifies, using diffusion maps, a low-dimensional manifold embedded in an ambient high-dimensional space associated with the data. It relies on the diffusion distance associated with this construction to define a distance function with which the data model is equipped. This distance metric function is then used to compute the correlation structure of a Gaussian process that describes the statistical dependence of quantities of interest in the high-dimensional ambient space. The proposed method is applicable to arbitrarily high-dimensional data sets. Here, it is applied to subsurface characterization using a suite of well log measurements. The predictions obtained in original, principal component, and diffusion space are compared using both qualitative and quantitative metrics. Considerable improvement in the prediction of the geological structural properties is observed with the proposed method.},
doi = {10.1007/s11004-017-9705-y},
journal = {Mathematical Geosciences},
number = 1,
volume = 50,
place = {United States},
year = {2017},
month = {10}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Save / Share:

Works referenced in this record:

The origins of kriging
journal, April 1990


Practical application of fuzzy logic and neural networks to fractured reservoir characterization
journal, October 2000


Principles of geostatistics
journal, December 1963


Detecting intrinsic slow variables in stochastic dynamical systems by anisotropic diffusion maps
journal, August 2009

  • Singer, A.; Erban, R.; Kevrekidis, I. G.
  • Proceedings of the National Academy of Sciences, Vol. 106, Issue 38
  • DOI: 10.1073/pnas.0905547106

Data-driven probability concentration and sampling on manifold
journal, September 2016


Automated manifold surgery: constructing geometrically accurate and topologically correct models of the human cerebral cortex
journal, January 2001

  • Fischl, B.; Liu, A.; Dale, A. M.
  • IEEE Transactions on Medical Imaging, Vol. 20, Issue 1
  • DOI: 10.1109/42.906426

Diffusion maps and coarse-graining: a unified framework for dimensionality reduction, graph partitioning, and data set parameterization
journal, September 2006

  • Lafon, S.; Lee, A. B.
  • IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 28, Issue 9, p. 1393-1403
  • DOI: 10.1109/TPAMI.2006.184

COGNITION: The Manifold Ways of Perception
journal, December 2000


Statistical shape analysis: clustering, learning, and testing
journal, April 2005

  • Srivastava, A.; Joshi, S. H.; Mio, W.
  • IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 27, Issue 4
  • DOI: 10.1109/TPAMI.2005.86

Diffusion maps
journal, July 2006

  • Coifman, Ronald R.; Lafon, St√©phane
  • Applied and Computational Harmonic Analysis, Vol. 21, Issue 1
  • DOI: 10.1016/j.acha.2006.04.006

Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps
journal, May 2005

  • Coifman, R. R.; Lafon, S.; Lee, A. B.
  • Proceedings of the National Academy of Sciences, Vol. 102, Issue 21
  • DOI: 10.1073/pnas.0500334102

Bayesian Prediction of Transformed Gaussian Random Fields
journal, December 1997

  • De Oliveira, Victor; Kedem, Benjamin; Short, David A.
  • Journal of the American Statistical Association, Vol. 92, Issue 440
  • DOI: 10.1080/01621459.1997.10473663

Prediction of Hydrocarbon Reservoirs Permeability Using Support Vector Machine
journal, January 2012

  • Gholami, R.; Shahraki, A. R.; Jamali Paghaleh, M.
  • Mathematical Problems in Engineering, Vol. 2012
  • DOI: 10.1155/2012/670723

Screening, Predicting, and Computer Experiments
journal, February 1992

  • Welch, William J.; Buck, Robert. J.; Sacks, Jerome
  • Technometrics, Vol. 34, Issue 1
  • DOI: 10.2307/1269548

Past, present and future intelligent reservoir characterization trends
journal, November 2001