Journal
INVERSE PROBLEMS
Volume 32, Issue 1, Pages -Publisher
IOP PUBLISHING LTD
DOI: 10.1088/0266-5611/32/1/015007
Keywords
inverse problems; PDE-constrained optimization; penalty method
Categories
Funding
- Natural Sciences and Engineering Research Council of Canada via the Collaborative Research and Development Grant DNOISEII [375142-08]
- Netherlands Organisation of Scientific Research (NWO) [613.009.032]
- BG Group
- BGP
- CGG
- Chevron
- ConocoPhillips
- DownUnder GeoSolutions
- Hess
- Petrobras
- PGS
- Schlumberger
- Sub Salt Solutions
- Woodside
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Many inverse and parameter estimation problems can be written as PDE-constrained optimization problems. The goal is to infer the parameters, typically coefficients of the PDE, from partial measurements of the solutions of the PDE for several right-hand sides. Such PDE-constrained problems can be solved by finding a stationary point of the Lagrangian, which entails simultaneously updating the parameters and the (adjoint) state variables. For large-scale problems, such an all-at-once approach is not feasible as it requires storing all the state variables. In this case one usually resorts to a reduced approach where the constraints are explicitly eliminated (at each iteration) by solving the PDEs. These two approaches, and variations thereof, are the main workhorses for solving PDE-constrained optimization problems arising from inverse problems. In this paper, we present an alternative method that aims to combine the advantages of both approaches. Our method is based on a quadratic penalty formulation of the constrained optimization problem. By eliminating the state variable, we develop an efficient algorithm that has roughly the same computational complexity as the conventional reduced approach while exploiting a larger search space. Numerical results show that this method indeed reduces some of the nonlinearity of the problem and is less sensitive to the initial iterate.
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