We show that the expected solution operator of prototypical linear elliptic PDEs with random coefficients is well approximated by a computable sparse matrix. This result is based on a random localized orthogonal multiresolution decomposition of the solution space that allows both the sparse approximate inversion of the random operator represented in this basis as well as its stochastic averaging. The approximate expected solution operator can be interpreted in terms of classical Haar wavelets. When combined with a suitable sampling approach for the expectation, this construction leads to an efficient method for computing a sparse representation of the expected solution operator.


  1. Monte Carlo
  2. random PDEs
  3. expected solution
  4. sparse approximation
  5. uncertainty quantification

MSC codes

  1. 65N30
  2. 65C05

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Information & Authors


Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 3144 - 3164
ISSN (online): 1095-7170


Submitted: 16 March 2020
Accepted: 31 July 2020
Published online: 4 November 2020


  1. Monte Carlo
  2. random PDEs
  3. expected solution
  4. sparse approximation
  5. uncertainty quantification

MSC codes

  1. 65N30
  2. 65C05



Funding Information

Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : PE2143/2-2
Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : CRC 1173

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