We propose an alternative method to generate samples of a spatially correlated random field with applications to large-scale problems for forward propagation of uncertainty. A classical approach for generating these samples is the Karhunen--Loève (KL) decomposition. However, the KL expansion requires solving a dense eigenvalue problem and is therefore computationally infeasible for large-scale problems. Sampling methods based on stochastic partial differential equations provide a highly scalable way to sample Gaussian fields, but the resulting parametrization is mesh dependent. We propose a multilevel decomposition of the stochastic field to allow for scalable, hierarchical sampling based on solving a mixed finite element formulation of a stochastic reaction-diffusion equation with a random, white noise source function. Numerical experiments are presented to demonstrate the scalability of the sampling method as well as numerical results of multilevel Monte Carlo simulations for a subsurface porous media flow application using the proposed sampling method.


  1. multilevel methods
  2. PDEs with random input data
  3. mixed finite elements
  4. uncertainty quantification
  5. multilevel Monte Carlo

MSC codes

  1. 65C05
  2. 60H15
  3. 35R60
  4. 65N30
  5. 65M75
  6. 65C30

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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: S543 - S562
ISSN (online): 1095-7197


Submitted: 5 July 2016
Accepted: 21 March 2017
Published online: 26 October 2017


  1. multilevel methods
  2. PDEs with random input data
  3. mixed finite elements
  4. uncertainty quantification
  5. multilevel Monte Carlo

MSC codes

  1. 65C05
  2. 60H15
  3. 35R60
  4. 65N30
  5. 65M75
  6. 65C30



Panayot S. Vassilevski

Funding Information

U.S. Department of Energy https://doi.org/10.13039/100000015 : DE-AC52-07NA27344

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