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Principal Component Geostatistical Approach for large-dimensional inverse problems

Journal Article · · Water Resources Research
DOI:https://doi.org/10.1002/2013wr014630· OSTI ID:1623438
 [1];  [2]
  1. Stanford Univ., CA (United States). Civil and Environmental Engineering; DOE/OSTI
  2. Stanford Univ., CA (United States). Civil and Environmental Engineering
+ Show Author Affiliations
The quasi-linear geostatistical approach is for weakly nonlinear underdetermined inverse problems, such as Hydraulic Tomography and Electrical Resistivity Tomography. It provides best estimates as well as measures for uncertainty quantification. However, for its textbook implementation, the approach involves iterations, to reach an optimum, and requires the determination of the Jacobian matrix, i.e., the derivative of the observation function with respect to the unknown. Although there are elegant methods for the determination of the Jacobian, the cost is high when the number of unknowns, m, and the number of observations, n, is high. It is also wasteful to compute the Jacobian for points away from the optimum. Irrespective of the issue of computing derivatives, the computational cost of implementing the method is generally of the order of m2 n, though there are methods to reduce the computational cost. In this work, we present an implementation that utilizes a matrix free in terms of the Jacobian matrix Gauss-Newton method and improves the scalability of the geostatistical inverse problem. For each iteration, it is required to perform K runs of the forward problem, where K is not just much smaller than m but can be smaller that n. The computational and storage cost of implementation of the inverse procedure scales roughly linearly with m instead of m2 as in the textbook approach. For problems of very large m, this implementation constitutes a dramatic reduction in computational cost compared to the textbook approach. Results illustrate the validity of the approach and provide insight in the conditions under which this method perform best.
Research Organization:
Stanford Univ., CA (United States)
Sponsoring Organization:
USDOE Office of Fossil Energy (FE)
Grant/Contract Number:
FE0009260
OSTI ID:
1623438
Journal Information:
Water Resources Research, Journal Name: Water Resources Research Journal Issue: 7 Vol. 50; ISSN 0043-1397
Publisher:
American Geophysical Union (AGU)Copyright Statement
Country of Publication:
United States
Language:
English

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Cited By (7)

A review of surrogate models and their application to groundwater modeling: SURROGATES OF GROUNDWATER MODELS journal August 2015
Scalable subsurface inverse modeling of huge data sets with an application to tracer concentration breakthrough data from magnetic resonance imaging: SCALABLE INVERSE MODELING WITH A HUGE MRI DATA SET journal July 2016
A Reduced‐Order Successive Linear Estimator for Geostatistical Inversion and its Application in Hydraulic Tomography journal March 2018
Riverine Bathymetry Imaging With Indirect Observations journal May 2018
Bayesian Calibration and Sensitivity Analysis for a Karst Aquifer Model Using Active Subspaces journal August 2019
Bayesian Calibration and Sensitivity Analysis for a Karst Aquifer Model Using Active Subspaces text January 2019
Characterization of Hydraulic Heterogeneity of Alluvial Aquifer Using Natural Stimuli: A Field Experience of Northern Italy journal January 2019

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